Theoretical Study of Pressure-Induced Phase Transitions and Thermal
Properties for Main-Group Oxides and Nitrides
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classi?ed information.
Bin Xu
Certi?cate of Approval:
Minseo Park
Associate Professor
Physics
Jianjun Dong, Chair
Associate Professor
Physics
Yu Lin
Professor
Physics
Satoshi Hinata
Professor
Physics
George T. Flowers
Dean
Graduate School
Theoretical Study of Pressure-Induced Phase Transitions and Thermal
Properties for Main-Group Oxides and Nitrides
Bin Xu
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful?llment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 10, 2009
Theoretical Study of Pressure-Induced Phase Transitions and Thermal
Properties for Main-Group Oxides and Nitrides
Bin Xu
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Bin Xu, son of Qingshan Xu and Xiufen Cai, was born on April 21, 1980 in Shijiazhuang,
Hebei, P. R. China. He attended University of Science and Technology of China (USTC) in
1998, and graduated with a Bachelor of Science in Physics in 2003. He entered Graduate
School at Auburn University to pursue a Doctoral degree in Physics in September 2003. He
was married to Shi Chen, daughter of Taixuan Chen and Junhua Wang, on April 25, 2007
in China.
iv
Dissertation Abstract
Theoretical Study of Pressure-Induced Phase Transitions and Thermal
Properties for Main-Group Oxides and Nitrides
Bin Xu
Doctor of Philosophy, August 10, 2009
(B.S., University of Science and Technology of China, 2003)
211 Typed Pages
Directed by Jianjun Dong
Main group nitrides and oxides are important solid compounds with applications in
?elds ranging from structural ceramics to catalysts and electronic materials. We have
theoretically investigated the pressure-induced phase transitions and thermal properties for
a series of oxides, nitrides and oxynitrides of IIIB, IVB and IIIB group, including Al2O3,
AlN, Si3N4, Ga2O3, and Ga3O3N.
In this dissertation, thermodynamic potentials at ?nite temperatures were calculated
within the quasi-harmonic approximation (QHA). Structural optimization and total energy
calculation of unit cell models were carried out based on ?rst-principles density functional
theory (DFT) within the local density approximation (LDA). Vibrational spectra were
calculated using the real-space supercell force-constant (SC-FC) method. For pressure-
induced phase transitions, we have studied the equilibrium transition conditions, as well as
the kinetic process, including the investigation of transition pathways, kinetic barriers and
the softening-phonon induced displacive transitions.
In our studies of equilibrium transition conditions, we calculated the T-P phase dia-
grams for Al2O3, AlN, Si3N4, and Ga2O3. Our predicted transition pressures are 84 and 134
GPa at 300 K for corundum?Rh2O3(II)?postperovskite transitions in Al2O3, 9.9 GPa at
v
0 K for wurtzite?rocksalt transition in AlN, 7.5/7.0 GPa at 300 K for ?/??? transitions
in Si3N4 and 0.5 and 39 GPa at 300 K for ????Rh2O3(II) transitions in Ga2O3.
In our studies of the pressure-induced reconstructive phase transition pathways, we ex-
amined the corundum-to-Rh2O3(II) transition in Al2O3 and the wurtzite-to-rocksalt tran-
sition in AlN. We showed that the rhombohedral corundum phase and the orthorhombic
Rh2O3(II) phase are related by intermediate structures with monoclinic symmetry (P2/c).
Using the proposed transition pathway, we calculated the kinetic barriers for the forward
(C-to-R) and backward (R-to-C) transitions and further predict the meta-stabilities of the
two phases. For AlN, di?erent transition pathways were previously proposed. We reinter-
preted the bond-preserving paths with long-range patterns of the ?transition units?, and
our calculated kinetic barriers indicate that the long-range pattern is less important. We
also showed that the bond-breaking paths are not energetically favored. In addition, based
on the pressure dependencies of the barrier heights we explained the discrepancy of tran-
sition pressure between the room temperature observation and the calculated equilibrium
result.
In our studies of displacive transitions in Si3N4, although ? phase is dynamically stable
at low pressure, two competing phonon-softening mechanisms are found under high pressure.
If the ??? transition is bypassed due to kinetic reasons at lower temperatures, the ? phase
is predicted to undergo a ?rst-order ??P3 transition at about 38.5 GPa. This predicted
metastable high-pressure P3 phase is structurally related to ?-Si3N4.
We have also calculated the thermodynamic and elastic properties of these systems,
and selected results are presented. Our predictions are in good agreement with available
experimental and other theoretical data.
Furthermore, we studied the shifted Raman scattering and its correlation with the
growth direction in Ga2O3 nanowires. And collaborated with an experimental study that
synthesized spinel-structured gallium oxynitride from Ga2O3+GaN mixtures at high pres-
sure and high temperature, we showed that the optimal synthesis pressure is predicted to
be close to the ?-to-? transition pressure of Ga2O3.
vi
Acknowledgments
Although only my name appears as the author of this dissertation, its completion is
truly contributed from many great people. I would like to thank everyone and everything
came along into my life, even without mentioning here.
I express my sincere gratitude to my research advisor, Dr. Jianjun Dong, for his
continuous encouragement, support and advice throughout my graduate years. Dr. Dong
taught me not only the knowledge and techniques in this ?eld, but also how to question
thoughts and express ideas. I enjoy the great bene?t from his instruction, especially the
emphasis on details. His tolerance and being understanding is greatly appreciated.
I would like to thank Dr. An-Ban Chen, who introduced me to Auburn University. He
is always there to help with my research and career. And I appreciate the helpful suggestions
from the committee members, Dr. Yu Lin, Dr. Satoshi Hinata, Dr. Minseo Park, and the
outside reader, Dr. Orlando Acevedo. When I was working as a TA, Dr. Satoshi Hinata
gave me great advice on how to teach as a ?rst-time lecturer. And I thank Dr. Robert
Boivin for our discussion about recitations and labs.
I am also thankful to our IT management professional Philip Forrest. He ensured the
computational ability of the computer cluster which is important for my research. And I
would like to acknowledge my graduate friend Xiaoli Tang, for the research discussion and
the in?uence of her diligent attitude, Zengjun Chen for his knowledge on Latex.
Beyond all the above gratitude, the support and care from my family is invaluable. I
especially would like to thank my parents and my wife, Shi Chen, for all their understanding
and care.
Finally, I appreciate the ?nancial support from DOE (DE-FG02-03ER46060) and NSF
(HRD-0317741) that funded part of the research in this dissertation.
vii
Style manual or journal used Bibliography follows American Physical Society (APS)
style
Computer software used The document preparation package TEX (speci?cally LATEX)
together with the departmental style-?le auphd.sty.
viii
Table of Contents
List of Figures xii
List of Tables xviii
1 INTRODUCTION 1
1.1 Pressure-Induced Phase Transitions and Thermodynamic Properties of Main-
Group Oxides and Nitrides . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Al2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 AlN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Si3N4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4 Ga2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Ga3O3N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 First-Principles Theoretical Studies . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 THEORY OF PHASE TRANSITIONS IN SOLIDS 15
2.1 Phases and Crystal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Equilibrium Thermodynamic Theory of Phase Transitions . . . . . . . . . . 19
2.2.1 Thermodynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Phase Diagram and Classi?cation of Phase Transitions . . . . . . . . 21
2.3 Landau Theory for Phase Transitions of Second Kind . . . . . . . . . . . . 25
2.3.1 The Order Parameter and Landau Free Energy . . . . . . . . . . . . 25
2.3.2 Dynamic Lattice Stability and Soft Phonon Modes . . . . . . . . . . 27
2.4 Transition Paths of Reconstructive Phase Transitions . . . . . . . . . . . . . 28
3 FIRST-PRINCIPLES CALCULATIONS OF THERMODYNAMIC PROP-
ERTIES OF CRYSTALS 33
3.1 First-Principles Total Energy Theory . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . 33
3.2 Statistical Theory of Bulk Crystals . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Phonon Theory of Lattice Dynamics . . . . . . . . . . . . . . . . . . 36
3.2.2 Statistical Harmonic Approximation . . . . . . . . . . . . . . . . . . 41
3.2.3 First-Principles Phonon Calculations . . . . . . . . . . . . . . . . . . 42
3.3 Thermodynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 Quasi-Harmonic Approximation . . . . . . . . . . . . . . . . . . . . 46
3.3.2 Equation of State (EOS) Models . . . . . . . . . . . . . . . . . . . . 50
3.3.3 Case Study: MgO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
ix
4 PRESSURE-INDUCED PHASE TRANSITIONS IN ALUMINIUM OXIDE
AND ALUMINIUM NITRIDE 59
4.1 Aluminium Oxide: Al2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 Crystal Structures, Total Energies and Vibrational Properties . . . . 62
4.1.3 T-P Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.4 Transition Pathways in the ?-to-Rh2O3(II) Transition . . . . . . . . 72
4.1.5 Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.6 High Pressure Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Aluminum Nitride: AlN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 Total Energy and Equilibrium Phase Transition . . . . . . . . . . . . 94
4.2.3 Microscopic Mechanism for the Wurtzite-to-Rocksalt Transition . . . 95
4.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 STUDY OF HIGH-PRESSURE PHASE TRANSITIONS IN SILICON NI-
TRIDE 102
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Crystal Structures, Static Binding Energies, and Vibrational Spectra . . . . 106
5.3 Equilibrium Thermodynamic Stability and Phase Transitions . . . . . . . . 114
5.4 Phonon-Softening Induced Structural Instability in ?-Si3N4 at High Pressures 116
5.5 Room Temperature Metastable P3 Phase . . . . . . . . . . . . . . . . . . . 121
5.6 Thermodynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 FIRST-PRINCIPLES STUDY OF GALLIUM OXIDE AND GALLIUM
OXYNITRIDE 132
6.1 Gallium Oxide: Ga2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.2 Total Energy and Vibrational Properties . . . . . . . . . . . . . . . . 136
6.1.3 T-P Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.4 Blue-Shifted Raman Scattering in Gallium Oxide Nanowires . . . . . 141
6.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2 Gallium Oxynitride: Ga3O3N . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.2 Total Energy Calculations of GaN and Ga3O3N . . . . . . . . . . . . 147
6.2.3 Theoretical Study of the Synthesis, Structure, and Stability of
Ga3O3N Spinel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2.4 Electronic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.5 Phonon Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7 CONCLUSIONS AND FUTURE WORK 159
Appendices 180
x
A Total Energy Calculation with VASP 181
B Elasticity Calculation 182
C Beyond Harmonic Approximation: Perturbation Theory of Lattice An-
harmonicity 188
xi
List of Figures
1.1 Photo of recently constructed departmental computer cluster. . . . . . . . . 12
2.1 Phase diagram with two solid phases, one liquid phase and one gas phase.
Two triple points and one critical point are present. . . . . . . . . . . . . . 23
2.2 Variation of the Landau free energy with positive and negative A coe?cients. 26
3.1 LDA (with US-PP) calculated (a) phonon dispersion relation, (b) vibrational
density of states of ?-Al2O3 at zero pressure. High symmetry points are A:parenleftbig
0, 12,0parenrightbig, ?: (0,0,0) Z: parenleftbig12, 12, 12parenrightbig, and D: parenleftbig12,0, 12parenrightbig. Lines denote theoretical
spectrum and discrete squares denote experimental data. . . . . . . . . . . . 47
3.2 Potential energy vs. interatomic distance curve. Dashed line denotes har-
monic potential curve. Minimum potential energy is provided at r = a . . . 48
3.3 LDA calculated thermal free energy at several volume points. (a) At T = 0
K. (b) At T = 2000 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 LDA calculated Fvib?V data at T = 2000 K are ?tted to several EOS models. 55
3.5 Thermal expansion of MgO at zero pressure. Lines represent this work from
LDA calculation. Discrete symbols are reported measurements. . . . . . . . 56
3.6 Thermal expansion of MgO at zero pressure. Solid line denotes our calcula-
tion with Debye model. Discrete symbols are reported measurements. . . . 58
4.1 Crystal structures of four polymorphs of Al2O3: (a) Corundum, (b) Rh2O3-
(II), (c) Pbnm perovskite and (d) post-perovskite. . . . . . . . . . . . . . . 63
4.2 LDA calculated phonon dispersion curves and vibrational density of state of
?-Al2O3 at zero pressure using (a) US-PP and (b) PAW. Discrete squares
denote experimental data172. . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Calculated Raman-active frequencies of ?-Al2O3 as a function of pressure.
(a) US-PP, (b) PAW. Solid and dashed lines represent Eg and A1g modes
from linear ?tting with data sets below 20 GPa. . . . . . . . . . . . . . . . . 67
xii
4.4 Calculated IR-active frequencies of ?-Al2O3 as a function of pressure. (a) TO
modes with US-PP, (b) TO modes with PAW, (c) LO modes with US-PP, (d)
LO modes with PAW. Solid and dashed curves denote Eu and A2u modes
from 2nd order polynomial ?tting. . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Calculated Raman-active frequencies of Rh2O3(II)-Al2O3 as a function of
pressure. (a), (c), (e), (g) US-PP, (b), (d), (f), (h) PAW. Solid curves are
from 2nd order polynomial ?tting. . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Calculated Raman-active frequencies of pPV-Al2O3 as a function of pressure.
(a) US-PP, (b) PAW. Solid, dashed, dotted and dash-dotted curves represent
Ag, B1g, B2g and B3g modes from 2nd order polynomial ?tting. . . . . . . . 72
4.7 Static enthalpies of Al2O3 polymorphs as a function of pressure relative
to that of corundum phase using (a) US-PP and (b) PAW. The static
? ?Rh2O3(II) and Rh2O3(II)?pPV transition pressure are present. The
PV phase is metastable at all pressures considered among the four polymorphs. 73
4.8 Calculated T-P phase diagram of Al2O3. (a) US-PP, (b) PAW. Dashed lines
denote possible phase boundaries. . . . . . . . . . . . . . . . . . . . . . . . . 74
4.9 Comparison of structures of (a) corundum and (b) Rh2O3(II)-Al2O3. The
corundum phase is viewed from an angle in which it appears ?orthorhombic?-
like. The di?erences between two polymorphs are highlighted. . . . . . . . . 75
4.10 Al-O bond lengths as a function of the transition parameter at 84.0 GPa. As
the transition parameter changes from 0 to 1, corundum structure transforms
smoothly to the Rh2O3(II) phase. There are 12 distinct bonds and only one
bond breaks and reforms during the transformation. . . . . . . . . . . . . . 77
4.11 Volume per atom for the corundum?Rh2O3(II) transformation at several
pressures. Same volume ranges are used to illustrate the pressure e?ect. . . 77
4.12 Enthalpies for the corundum?Rh2O3(II) transformation at several pressures. 78
4.13 Enthalpy barrier height for the corundum?Rh2O3(II) and
Rh2O3(II)?corundum transformations as a function of pressure. . . . . . . 79
4.14 Calculated phonon dispersion curves of corundum-Al2O3 at 204 GPa. . . . 79
4.15 Calculated phonon dispersion curves of Rh2O3(II)-Al2O3 at 0 GPa. . . . . . 79
4.16 Comparison of the present theoretical calculation with measured thermal ex-
pansion coe?cients of ?-Al2O3 as a function of temperature at zero pressure.
Solid and dashed lines represent our LDA calculations using US-PP and PAW
method, respectively. Discrete symbols are reported experimental data.16?21 81
xiii
4.17 Theoretical thermal expansion coe?cients of ?-Al2O3 as a function of tem-
perature at several pressures from using (a) US-PP, (b) PAW. . . . . . . . . 82
4.18 Comparison of the present LDA predicted adiabatic bulk modulus as a func-
tion of temperature of ?-Al2O3 with Voigt-Ruess-Hill data by Te?et23, by
Goto et al.25 and the polycrystalline data by sound velocity measurements
by Chung and Simmons24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.19 Comparison of calculated isobaric heat capacity and entropy as a function of
temperature of ?-Al2O3 with experimental data at zero pressure. Discrete
open circles denote measured data from Furukawa22. . . . . . . . . . . . . . 83
4.20 LDA calculated thermal expansion coe?cients of Rh2O3(II)-Al2O3 at several
pressures as a function of temperature up to 3000 K. (a) US-PP, (b) PAW . 84
4.21 LDA calculated thermal expansion coe?cients of pPV-Al2O3 at several pres-
sures as a function of temperature up to 3000 K. (a) US-PP, (b) PAW . . . 84
4.22 Pressure dependence of the elastic constants of corundum. (a) US-PP, (b)
PAW. Discrete symbols represent directly calculated data. Curves are from
the 2nd-order polynomial ?tting. . . . . . . . . . . . . . . . . . . . . . . . . 85
4.23 Pressure dependence of the elastic constants of Rh2O3(II)-Al2O3. (a) US-
PP, (b) PAW. Discrete symbols represent directly calculated data. Curves
are from the 2nd-order polynomial ?tting. . . . . . . . . . . . . . . . . . . . 87
4.24 Pressure dependence of the elastic constants of pPV-Al2O3. (a) US-PP, (b)
PAW. Discrete symbols (except crosses) represent directly calculated data
from this work. Crosses denote GGA calculation from Stackhouse et al.36 at
136 GPa. Curves are from the 2nd-order polynomial ?tting. . . . . . . . . . 88
4.25 Phase transformation along TP1. (a) Unit cell of B4 structure, (b) Unit cell
of B1 structure, (c), (d) Transformation pattern of ?transition units? in B4
and B1 phases. Dashed lines denote primitive unit cell. (Origin shifted onto
the position of an atom for better illustration of the unit cell.) . . . . . . . 92
4.26 Phase transformation along TP2, TP3, TP4 and TP5. (a) Unit cell of B4
structure, (b) Unit cell of B1 structure, (c), (d) Transformation pattern of
?transition units? in B4 and B1 phases. (Origin shifted onto the position of
an atom for better illustration of unit cell.) . . . . . . . . . . . . . . . . . . 93
4.27 Four-atom ?transition unit? from B4 to B1 structure. . . . . . . . . . . . . 94
4.28 Cohesive energy per atom as a function of volume for B4 and B1-AlN. En-
thalpy variation of both phases as a function of pressure is also shown. . . . 96
xiv
4.29 Enthalpy as a function of transition parameter relative to B4 phase for ?ve
TPs at six pressures. (a) at 0 GPa, (b) at 5 GPa, (c) at the predicted
transition pressure 9.85 GPa, (d) at 15 GPa, (e) at 20 GPa and (f) at 30 GPa.100
4.30 Forward (B4-to-B1) and backward (B1-to-B4) activation barrier height (en-
thalpy) as a function of pressure for ?ve TPs. . . . . . . . . . . . . . . . . . 101
5.1 Polymorphs of Si3N4 and synthesis conditions . . . . . . . . . . . . . . . . . 104
5.2 Crystal structures of (a),(b) ?-, (c),(d) ?-, and (e),(f),(g) ?-Si3N4. In the
panel of ?- and ?-Si3N4, the ?rst graph illustrates the unit-cell model and
the second graph is the 2?2?1 supercell model viewed in the direction of c
axis. In the panel of ?-Si3N4, the ?rst graph shows the conventional cubic cell
of the spinel structure and the following two graphs show the fourfold and
sixfold coordinated Si units(SiN4 and SiN6) with tetrahedra and octahedra,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Energy-volume curves for ?-, ?- and ?-Si3N4 in the scale of per atom. ?
phase has an equilibrium energy of 3 meV lower than that of ? phase. E0 of
? phase is 93 meV higher than ? phase. . . . . . . . . . . . . . . . . . . . . 110
5.4 Phonon dispersion curves and vibrational density of states (VDOS) of (a) ?-,
(b) ?-, and (c) ?-Si3N4 at zero pressure. . . . . . . . . . . . . . . . . . . . . 112
5.5 Calculated dispersion curves (scattered circles) of mode Gr?uneisen parameter
of (a) ?-, (b) ?-, and (c) ?-Si3N4 at zero pressure. Red horizontal line is
present to separate the positive and negative values. . . . . . . . . . . . . . 113
5.6 Gibbs free energy of ?-Si3N4 relative to that of ? phase as a function of
temperature. Solid, dashed and dotted lines represent the pressure of 0, 5
and 10 GPa, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.7 T-P phase diagram of Si3N4. Solid curve denotes the phase boundary be-
tween ?- and ?-Si3N4. Dashed curve denotes the phase boundary between
?- and ?-Si3N4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.8 (a) Raman, (b) IR and (c) silent mode frequencies as a function of pressure
up to 60 GPa for ?-Si3N4. Experimental pressure dependence of Raman
modes up to 30 GPa is also presented in discrete symbols as a comparison200.
Solid squares denote measurements upon pressure increase and open squares
denote measurements upon pressure decrease. Several low-frequency modes
are found to decrease with increasing pressure. One Bu branch of silent
modes is found dropping to zero at about 60 GPa. . . . . . . . . . . . . . . 117
xv
5.9 Phonon dispersion of ?-Si3N4 at a pressure of 48 GPa. Two competing soft
phonon modes are found: one TA branch at M point and one optic branch
at ? point. No LOTO splitting correction is added for the interests of low-
frequency modes only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.10 The square of vibration frequency (?2) as a function of pressure for two
competing soft phonon branches: one TA branch at M point and one Bu
branch at ? point. Solid squares and circles represent data from calculation.
Solid and dashed lines are from a linear ?tting. . . . . . . . . . . . . . . . . 119
5.11 Ball-stick models of (a) P63/m, (b) P?6, (c) P21/m and (d) P3 structures
viewed along the c axis. Balls in dark color represent N atoms and Si atoms
are in light color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.12 The total energy of P63/m (?), P?6, P21/m and P3 structures as a function
of volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.13 Enthalpy landscape and its contour plot as a function of fxy and fz at the
transition pressure of 38.5 GPa. . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.14 Enthalpy barrier (relative to ? phase) as a function of linearly interpreted
transition parameter at 30 GPa (thinner) and the transition pressure of 38.5
GPa (thicker). Solid curves denote the ??P3??P3 path and the dashed
curves denote direct ??P3 path. Horizontal axis is de?ned as qualitative
structural similarity. The left end represents ? structure and the right end
represent P3 structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.15 Temperature dependence of volume thermal expansion coe?cient of bulk ?-
Si3N4 at zero pressure. Solid and dashed lines show present work with static
energies ?tted to the 2nd-order BM-EOS and 3rd-order BM-EOS, respectively,
and both thermal free energies ?tted to the 2nd-order BM-EOS. Discrete
symbols denote experimental data83?86. . . . . . . . . . . . . . . . . . . . . 127
5.16 Temperature dependence of bulk Gr?uneisen parameter of ?-Si3N4 at zero
pressure. Solid and dashed lines show present work with static energies ?tted
to the 2nd-order BM-EOS and 3rd-order BM-EOS, respectively, and both
thermal free energies are ?tted to the 2nd-order BM-EOS. Discrete symbols
denote experimental data85. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.17 Temperature dependence of volume thermal expansion coe?cient of bulk ?-
Si3N4 at pressures of 0, 10, 20 and 30 GPa. . . . . . . . . . . . . . . . . . . 129
5.18 Temperature dependence of isobaric heat capacity of ?-Si3N4 at zero pressure.
Solid and dashed lines show present work with static energies ?tted to the
2nd-order BM-EOS and 3rd-order BM-EOS, respectively, and both thermal
free energies are ?tted to the 2nd-order BM-EOS. Discrete symbols denote
experimental data85?88. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
xvi
5.19 Temperature dependence of volume thermal expansion coe?cient of ?-, ?-
and ?-Si3N4 at zero pressure. Discrete symbols represent experimental
data89,90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Crystal structure of monoclinic ?-Ga2O3 phase. Note that Ga3+ cations
(light color) occupy both tetrahedral and octahedral interstices within the
ccp lattice of O2? ions (dark color). . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Enthalpy di?erences relative to ? phase for ?, ? and Rh2O3(II)-Ga2O3 as a
function of pressure up to 60 GPa. The ? and ? curves cross at 0.5 GPa, the
? and Rh2O3(II) curves cross at 40.9 GPa. . . . . . . . . . . . . . . . . . . 137
6.3 T-P phase diagram for ?, ? and Rh2O3(II)-Ga2O3 polymorphs. . . . . . . . 140
6.4 LDA-calculated formation enthalpy ?H (i.e., Gibbs free energy of formation
?Gformation at zero temperature) as a function of pressure using (a) LDA
methods and (b) the GGA approach. The calculated positive ?H suggests
an endothermic formation. The solid plots are calculated assuming the oxyni-
trides are synthesized from ?-Ga2O3 and wurtzite GaN, and the dashed plots
are calculated assuming the oxynitrides are synthesized from ?-Ga2O3 and
wurtzite GaN. The solid plot and the dashed plot cross at the pressure of the
?-to-? phase transition in Ga2O3 (predicted to be 0.5 and 6.6 GPa by LDA
and GGA methods, respectively). . . . . . . . . . . . . . . . . . . . . . . . . 151
6.5 (a) Electronic band structure and (b) Electronic density of states for Ga3O3N
calculated using ?rst-principles (DFT) methods within the LDA. . . . . . . 155
6.6 Photoluminescence spectrum of the Ga2.8N0.64O3.24 sample obtained via
high-P,T synthesis in the multianvil experiments, using UV laser excitation
(325 nm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.7 (a) Raman spectrum collected for the Ga2.8N0.64O3.24 sample at an excita-
tion wavelength of 514.5 nm. The bold solid lines on the frequency scale
below indicate the positions of the Raman bands for the analogous spinel
form of ?-Ge3N4. (b) Phonon density of states (VDOS) calculated for the
R?3m pseudocubic Ga3O3N phase, predicted as ?model I? in the enthalpy
calculations (Table 6.7 ). The dashed lines indicate the Raman-active modes
for that phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
xvii
List of Tables
2.1 Point and space groups of Bravais lattices and crystal structures . . . . . . 17
4.1 Space groups, formula units per primitive unit cell Z and Wycko? sites of
four polymorphs of Al2O3: Corundum, Rh2O3-(II), Pbnm perovskite and
post-perovskite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Third-order BM-EOS parameters for Al2O3 polymorphs . . . . . . . . . . . 65
4.3 The theoretical data of Raman frequencies at zero pressure and their pressure
dependencies, and comparison with other reported experimental results on
corundum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 IR-active frequencies of corundum at zero pressure . . . . . . . . . . . . . . 69
4.5 Elastic constants of corundum at zero pressure and their pressure dependencies 86
4.6 Comparison of elastic moduli between corundum and Rh2O3(II) phase. The
corundum phase is treated as a monoclinic crystal with 20 atoms per unit cell. 88
4.7 Third-order BM-EOS parameters, zero pressure structural parameters for B4
and B1-AlN and B4-to-B1 transition pressure Pt. Parameters listed for B1
phase is for the 8-atom conventional cubic unit cell. . . . . . . . . . . . . . 95
4.8 Bond-preserving TPs for the B4-to-B1 phase transition. G denotes the space
group of two end structures and G? denotes the common subgroup. Z rep-
resents the number of formula units per unit cell along the TP. The trans-
formation matrices are given with respect to the primitive unit cell lattice
vectors and the fractional atomic coordinates are in terms of the setting of
G?. u ? 3/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1 Space groups, formula units per primitive unit cell and Wycko? sites of hexag-
onal ?-, ?-Si3N4 and cubic ?-Si3N4. . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Summary of calculated and measured crystal parameters of ?-, ?- and ?-
Si3N4. V0 is the equilibrium volume per atom, B is the bulk modulus and
B? is the ?rst-order pressure derivative. Measurements were made at room
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xviii
5.3 Summary of phase transition pressure and temperature for synthesizing ?-Si3N4115
6.1 Third-order BM-EOS parameters for Ga2O3 polymorphs . . . . . . . . . . . 136
6.2 Calculated and experimental zone-center Raman peak positions and
Gr?uneisen parameter for the ?-Ga2O3 phase . . . . . . . . . . . . . . . . . . 138
6.3 Calculated and experimental zone-center Raman peak positions for the ?-
Ga2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Estimated internal strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5 Raman mode frequencies and frequency shifts in ?-Ga2O3 nanowires with
the [40?1] and [110] growth directions. Overall, excellent agreement between
the observed and calculated shifts is seen for all mode frequencies except the
one marked with an *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.6 Third-order Birch-Murnaghan EOS parameters, zero pressure structural pa-
rameters for wurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.7 Parameters of the third-order Birch-Murnaghan equation of states of three
unit-cell-based atomic models of the spinel-structured Ga3O3N that have the
lowest energy calculated within the LDA. . . . . . . . . . . . . . . . . . . . 150
6.8 Analysis of local coordination ordering schemes within the three unit-cell
models of the spinel-structured Ga3O3N in terms of the ratio of various types
of AX4 tetrahedral and AX6 octahedral units. . . . . . . . . . . . . . . . . . 153
A.1 Value and meaning of ISIF tag . . . . . . . . . . . . . . . . . . . . . . . . . 181
B.1 Independent components of Cij for all the crystal classes . . . . . . . . . . . 184
B.2 Correction term to Cij due to pressure . . . . . . . . . . . . . . . . . . . . . 187
xix
Chapter 1
INTRODUCTION
1.1 Pressure-Induced Phase Transitions and Thermodynamic Properties of
Main-Group Oxides and Nitrides
Main group nitrides and oxides are important solid compounds with applications in
?elds ranging from structural ceramics to catalysts and electronic materials. In this disser-
tation, I report our studies of ?ve material systems of these groups: Al2O3, AlN, Si3N4,
Ga2O3, and Ga3O3N.
1.1.1 Al2O3
Aluminium oxide (Al2O3) is often referred to as alumina, sapphire or aloxite in the ce-
ramic, mining and materials science communities. It is commonly used as an abrasive due
to its hardness and as a refractory material due to its high melting point. The naturally-
occurring crystalline form of Al2O3 at ambient condition is primarily corundum. Rubies
(Cr+3 doped) and sapphires are gem-quality forms of corundum. In high-pressure experi-
ments, ruby usually serves as a standard pressure gauge (ruby scale) in diamond anvil cell
(DAC)1 and sapphire is used as window material in shock wave experiments2. Al2O3 is
also one of the major constituents of the Earth?s lower mantle. High-pressure behaviors
and thermal properties of alumina are important for both experimental research and better
understanding of the interior of the earth.
At ambient conditions, The crystalline form of Al2O3 is corundum (?-Al2O3, space
group R?3c). This structure is known to exist over a wide range of pressure and tem-
perature conditions. In the past twenty years theoretical3?7 and experimental8?11 studies
showed that ?-Al2O3 transforms into the Rh2O3(II) structure (space group Pbcn) around
1
90 GPa, and further transforms into the post-perovskite structure (space group Cmcm)
at ?130 GPa. Several calculations predicted a transition from Rh2O3(II) structure to the
orthorhombic perovskite (PV) structure (space group Pbnm) at even higher pressures5,12.
However, it has never been observed in experiments. Recent calculations showed that Pbnm
perovskite is not thermodynamically favored with respect to the corundum, Rh2O3(II) and
post-perovskite phases6,7. Here we report our LDA calculated equilibrium T-P phase di-
agram with both ultra-soft pseudopotentials (US-PP) and PAW method. The results are
compared with other reported calculations3?6,6,7,13.
In the corundum?Rh2O3(II) transformation, in situ heating is found necessary. At
room temperature, X-ray di?raction experiments showed that corundum phase is stable up
to 175 GPa14,15, which implies the existence of a large kinetic barrier for the corundum
to transform into the Rh2O3(II) phase. On the other hand, Lin et al. found that the
high-pressure Rh2O3(II) phase can be seen as low as 85 GPa (Pt = 96 GPa) on decompres-
sion after laser heating10, indicating the Rh2O3(II)?corundum transition is also sluggish.
No experiment has quenched the Rh2O3(II) phase to ambient conditions. We proposed a
transformation pathway for the corundum?Rh2O3(II) transition and evaluate the kinetic
barrier based on the proposed pathway. We further predict the meta-stability of two phases.
Thermodynamic properties, such as thermal expansion coe?cient (TEC), heat capac-
ity CP, entropy and adiabatic bulk modulus have long been studied experimentally16?25
. Almost a decade ago Hama et al. calculated the thermal properties of corundum phase
by extending the formalism of Thomsen and combining the results with the Vinet model
and the Debye model for lattice vibrtations26. To date, no ?rst-principles studies have
been reported to predict the thermodynamic properties of Al2O3. And, both experimental
and theoretical data of thermal properties of the high-pressure phases are lacking. We thus
present our calculated high-pressure TEC and bulk modulus for Rh2O3 (II) and pPV phases
as a function of temperature up to 3000 K.
Before 2004, experimental elastic constants of ?-Al2O3 were reported from as early as
1950s25,27?29. And many e?orts of ab initio calculations were made to predict the elastic
2
constants independently30,31. However, some calculations showed con?ict in the sign of
C1432. In 2004, Gladden et al.33 reported their reexamination of the elasticity of ?-Al2O3
using resonance ultrasound spectroscopy (RUS) and con?rmed that C14 is positive rather
than negative. Gladden?s conclusion was later con?rmed by both measurement with a
di?erent technique34 and ?rst-principles calculations35. Although measurements of Cij for
the high-pressure phases of Al2O3 are still lacking, Duan et al.31 and Stackhouse36 have
calculated the elastic constants of Rh2O3(II) (from 75 GPa to 300 GPa) and post-perovskite
phases (at 136 GPa), respectively. Here we report our predicted Cij of ?-, Rh2O3(II)- and
pPV-Al2O3 with pressure dependencies, which will be compared with other available results.
Elastic properties of both corundum and Rh2O3(II) phases have been studied, but a direct
comparison between the two phases is not accessible due to di?erent crystal classes. Here we
compare the elastic properties of corundum and Rh2O3(II) phase by treating both phases
as the common-subgroup monoclinic lattices.
1.1.2 AlN
AlN is an important semiconductor primarily due to its wide band-gap and thermal
properties. The structural changes and properties at high pressures are of special interests.
Experiments37?40 and calculations41?46 revealed that a pressure-induced ?rst-order phase
transition from wurtzite (B4) structure to rocksalt (B1) structure happens for AlN. On
the experimental side, without heating, the lowest pressure at which the rocksalt structure
started to show up is 14 GPa39 and the B4-to-B1 transition was observed to complete at
20-31.4 GPa39,40. Xia et al. also found that the rocksalt phase is quenchable to ambient
conditions39. However, theoretical calculations consistently predicted a transition pressure
that is lower than the experimental values. Most of the recent ?rst-principles predicted
static Pt is less than 10 GPa. The discrepancy implies the existence of a hysteresis for the
forward and backward transitions which is caused by the activation barrier at the transition.
To calculate the activation barrier, we ?rst examine the previously proposed transition
paths. Using the computer program COMSUBS 47, Stokes et al. proposed ?ve TPs by
3
preserving the nearest neighbors along the transition pathway and limiting the size of unit
cell to no more than four di?erent Wycko? positions48. He also found a common bilayer
sliding mechanism along ?ve paths, which had been pointed out by Zahn et al.49 and
Sowa50. Another work adopted the systematic approach is from Capillas et al.51, which was
performed using the databases and tools provided by the Bilbao Crystallographic Server52.
They proposed eight possible paths with di?erent orthorhombic and monoclinic symmetries
by setting the maximum k-index equal to 4, strain tolerance Stol < 0.15 and maximum
atomic displacement ?tol< 2 ?A. The intermediate structures along all the eight paths have
eight atoms per unit cell. The di?erence between these two reports is that the nearest
neighbours are not preserved along all the eight paths proposed by Capillas et al.
DFT calculations from Shimojo et al. showed that the enthalpy barrier of transforma-
tion is independent of the three paths (Cmc21, Pna21 and P21) for CdSe. Cai suggested,
without calculation, that the B4-to-B1 transition is characterized by the transformation of
the four-atom ?transition unit?, while the long-range pattern may be less important. On
the other hand, using MD simulations, Zahn et al. pointed out that the favored paths have
a tendency to avoid excess strains during the transformation.
It would be interesting to investigate the proposed TPs with another material via ?rst-
principles method. Previous studies53?55 suggested that the energetically favored TPs are
bond preserving. In this paper, we studied all ?ve bond-preserving TPs proposed by Stokes
et al. and one bond-breaking TP proposed by Capillas et al. from an energetic point of view
for the B4-to-B1 transition in AlN. The correlation of the enthalpy barrier with di?erent
TPs and strains will be discussed. We also relate the bilayer sliding mechanism to the
long-range patterns of ?transition units?, and Cai?s hypothesis will be examined.
1.1.3 Si3N4
For Si3N4, it is widely used in cutting tools and anti-friction bearings due to its excel-
lent mechanical properties, low mass density, and thermal stability56. It is also used as an
insulator layer or as an etch mask because of its dielectric properties and a better di?usion
4
barrier against impurities in microelectronics57. For its technological importance, the me-
chanical and thermal properties of silicon nitride at ambient pressure has been investigated
extensively by both experiment and theory56,57. In contrast, its properties at high-pressure
is less known.
? (P31c) and ? (P63/m) phases are the only two bulk polymorphs of Si3N4 known at
ambient pressure. Both phases can be synthesized by nitriding pure silicon. In 1999, ?-Si3N4
(or c-Si3N4, Fd?3m) with the cubic spinel structure was synthesized at high pressure and
high temperature58. Despite intensive research e?orts in searching the ?post-spinel? phases
in Group-IVB nitrides, the spinel structured ? phase remains as the only experimentally
identi?ed high-pressure phase.
Phase transitions in Si3N4 have drawn extensive attention for more than a decade. The
relative phase stability between ? and ? phases has been a topic of investigation for many
years. Direct measurements of energetics of Si3N4 were reported by Liang et al.59. However,
the di?erence in formation enthalpies between ?- and ?-Si3N4 was founded to be less than
the intrinsic experimental uncertainty of ?22 kJ/mol (?32.6 meV/atom). Nevertheless,
the ? phase is believed to be the ground state in Si3N4 because no ??? transition is
ever observed. The stability condition for ? phase has been experimentally studied at
temperatures of 1300??1800?C and pressures up to 60 GPa60?67. A solution-precipitation
mechanism was proposed for the ??? transformation67. The observed liquid phase on the
?-Si3N4 surfaces was believed to lower the activation energy of atomic transportation. The
stability of pristine ?-Si3N4 at high temperatures is ascribed to the extremely high value of
the activation energy with clean surfaces. On the theory side, several studies con?rmed that
the static bonding energy of ? phase is slightly higher than that of ? phase68?71. Wendel
et al.70 and Kuwabara et al.71 carried out statistical QHA calculations, and they both
found that the ? phase remains metastable in the temperature range from 0 to 2000 K at
ambient pressure. Yet, pressure e?ects on the relative thermodynamic stability between ?
and ? phases was not addressed in previous studies. Our study is to understand the relative
thermodynamic stability at high pressures.
5
The spinel structured ?-Si3N4 was ?rst synthesized by Zerr et al. with laser-heated
diamond anvil cell (LH-DAC)58. Later experiments showed that ?-Si3N4 can be obtained
from both ? and ?-Si3N4 upon compression and simultaneous in-situ heating58,72?75. We
predict the equilibrium phase boundaries for the ??? and ??? transitions. The transition
pressures for the ??? transition is about 0.5 GPa lower than that of ??? transition. The
? phase is quenchable to the ambient condition, and it remains stable at temperatures
ranging up to about 1670 K at ambient pressure76,77. When ?-Si3N4 ?decomposes? at
ambient pressure upon heating, the samples may consist of both ? and ?-Si3N476.
The in-situ heating to high temperature is found to be necessary to form the ?-Si3N4
at high pressures. At room temperature the ??? transition is, however, by-passed. Zerr
found that ?-Si3N4 exists up to 34 GPa and it then transforms into a new phase (labeled
as ?-phase) under further compression78. This phase transition was identi?ed by Raman
spectroscopy and energy dispersive X-ray powder di?raction (EDXD). But the structure of ?
phase was not fully determined. Zerr proposed three possible unit-cells based on the EDXD
pattern: two tetragonal and one orthorhombic. He further suggested that the ?-Si3N4 should
be considered as a metastable intermediate stage in the ??? transition. Kroll has proposed
a metastable willemite-II-Si3N4 phase which is an intermediate between ? and ?-Si3N4 in
both energetics and density79. However, the wII phase is unlikely to be the experimentally
observed unknown phase at high pressure and room temperature. Because 1) the wII
phase, which is structurally closely related to the spinel ?-Si3N4, has been shown to have
a signi?cantly lower activation barrier for the ??wII transformation, comparing to that
of ??? transformation79. Although the activation barrier of the ??wII transformation is
unknown, it is more likely to be high enough to exclude the room temperature transition.
2) The calculated Raman frequencies of wII-Si3N4 could not match many strong peaks
appeared in the measurements, e.g., two observed peaks at about 500 cm?1 and 550 cm?1
are absent in the calculation. A recent experimental work from McMillan et al. reproduced
Zerr?s ?ndings on ?-Si3N4, but excluded the wII cubic structure80.
6
Meanwhile, ?-Ge3N4 is found to transform into the metastable polymorph ?-Ge3N4
with hexagonal P3 symmetry at room temperature by Soignard et al.81 Ab initio calculation
from Dong et al. showed that a ??P?6?P3 transition sequence could occur in Ge3N4 at
the pressure of about 20 GPa and 28 GPa82, which are of second-order that driven by soft
phonons. If ?-Ge3N4 directly transforms into the P3 structure, the transition was predicted
to be ?rst-order and Pt = ?23 GPa. Dong also pointed out that the ??P?6 transition is
originated from a soft silent Bu mode. Room temperature experimental study by Soignard et
al. con?rmed the direct ??P3 transition associated with a 5-7% volume reduction81. The
Raman data they observed excludes the intermediate P?6 structure. Based on the density
consideration, Soignard et al. suggested that the new polymorph is a ?post-phenacite?
phase, in stead of ?post-spinel?. Comparison of the X-ray di?raction and Raman data
between Ge3N4 and Si3N4 shows similarity which may suggest a P3 structure for ?-Si3N4.
It is still unclear whether there are intrinsic di?erences between the HP-RT behaviors of
Si3N4 and Ge3N4, or the experimental results may be interpreted di?erently. Our study is
to theoretically investigate structural instabilities and possible metastable phase transitions
in ?- and ?-Si3N4 at high pressures and room temperature. We found no sign of dynamical
instability in the ? phase at high-pressure. On the other hand, we predicted a phonon-
softening related ?rst-order phase transition at about 38.5 GPa in ? phase. At this pressure,
the density of the proposed high-pressure phase is 4.16 g/cm3 which is larger than that of
?-Si3N4 (3.71 g/cm3), yet smaller than that of ?-Si3N4 (4.53 g/cm3, calculated). We further
estimated the kinetic barrier heights for our proposed ??P3 transition, which is only 67.23
meV/atom at 38.5 GPa. Despite being of ?rst-order phase transition, the small barrier
height suggests that the P3 phase is unlikely to be recovered below 38.5 GPa.
We also performed a series of systematical calculations of thermodynamic properties
of Si3N4, such as thermal expansion coe?cient (TEC), heat capacity and bulk Gr?uneisen
parameter, and compared our results with available experimental data83?90 and some pre-
vious calculations70,71,91,92. The overall good agreement with experiment validates the
7
adopted statistical quasi-harmonic approximation (QHA) and the Birch-Murnagahn equa-
tion of states (EOS) models. Our results support the prediction from Kuwabara et al.71 on
the negative TEC of ? and ? phases at temperatures below 100 K. We attributed the origin
of the negative TEC to the low-frequency phonon modes with the negative mode Gr?uneisen
ratios in the two phases.
1.1.4 Ga2O3
Monoclinic gallium oxide (Ga2O3) is usually known as a wide-band-gap semiconduc-
tor (Eg = 4.9 eV); however, the conductivity can be varied from insulating to conducting
behavior depending upon the preparation conditions93. It is well known that Ga2O3 can
exist in several forms, including ?, ?, ?, ?, and ? polymorphs that all have di?erent struc-
ture types94. Of these, the most stable form at ambient conditions is determined to be
?-Ga2O3. It is of great interest to determine the pressure-induced phase transformations
among Ga2O3 polymorphs in order to establish the stable and metastable phase relations
between di?erent crystalline modi?cations, and to evaluate their production under di?erent
synthesis conditions. The relative densities of ?- and ?-Ga2O3 are 5.94 and 6.48 g?cm?3,
respectively95, indicating that a ? ? ? transformation should occur at high pressure.
Nanocrystalline ?-Ga2O3 particles embedded in a glassy matrix were also studied at high
pressure using energy-dispersive x-ray di?raction96. In that work, a ?-to-? phase transfor-
mation was found to be initiated at 6 GPa, but the process was not completed by 15 GPa,
the highest pressure achieved in the study. However, it is known that the silica glass host
matrix undergoes important structural and density changes within this pressure range97,98,
so that it is not yet known if the structural changes are intrinsic to the ?-Ga2O3 material
presumably in?uenced by the nanocrystalline nature of the sample , or are promoted by
anomalous densi?cation among the SiO2 matrix. These results prompted us to theoretically
investigate the high-pressure behavior of the phase-pure bulk ?-Ga2O3.
One-dimensional nanostructured forms of ?-phase of gallium oxide (?-Ga2O3) such
as nanotubes, nanobelts, and nanowires, have attracted recent interest due to enhanced
8
optical properties99,100. Recently, Choi et al.101 synthesized ?-Ga2O3 nanowires (diameter
range of 15?45 nm) with a [001] growth direction using an arc-discharge method. Gao
et al.102 synthesized [40?1] ?-Ga2O3 nanowires with diameters ranging from ?10?100 nm
in a vertical radio-frequency furnace. Interestingly, the Raman mode frequencies of the
[001] ?-Ga2O3 nanowires coincide with the corresponding frequencies in bulk ?-Ga2O3101.
On the other hand102, the Raman mode frequencies of the [40?1] ?-Ga2O3 nanowires are
redshifted relative to corresponding frequencies in bulk ?-Ga2O3 by 4?23 cm?1. Using
plasma-enhanced chemical vapor deposition, Rao et al. have synthesized ?-Ga2O3 nanowires
whose growth is along the [110] direction103, and the Raman spectrum is signi?cantly
blueshifted in frequency104. Here we focus on the ?rst-principles calculations of the Raman
mode frequencies under internal strains. Our calculated Raman frequency shifts suggest
that the observed shifts in the nanowires with the [40?1] and [110] growth directions can
be explained in term of di?erent internal strains, in contrast to the previously suggested
quantum con?nement e?ects and defect-induced e?ects.
1.1.5 Ga3O3N
The group 13 oxynitride materials have many useful properties related to their elec-
tronic structure. ?-Ga2O3 with the corundum structure is conveniently alloyed with Al2O3
to provide selective reduction catalysts for gaseous NOx105, and various other Ga2O3 phases
have been proposed as gas sensors, and in nanoscale structures as electron emitters and mag-
netic memory materials106. Within the Al2O3?AlN system, several important AlxOyNz
ceramic alloys and compounds are known. At high AlN contents, layered forms based
on hexagonal/cubic intergrowths are present. As the Al2O3 content is increased, cubic
spinel-structured materials begin to appear. A large family of defect spinels (?-Al2O3,
AlxOyNz) contain vacancies on both cation and anion sites107. A stoichiometric oxynitride
spinel-structured compound is obtained at the Al3O3N composition, in which Al3+ ions
are present on the octahedral and tetrahedral sites, and O2? and N3? occupy tetrahedral
anion sites108,109. Among the related nitride compounds Si3N4 and Ge3N4, high-pressure
9
synthesis has recently resulted in formation of a new class of spinel structures, that contain
Si4+ and Ge4+ cations on both tetrahedral and octahedral sites110?114. Gallium oxyni-
tride (Ga3O3N) has been predicted to form a new spinel-structured compound within the
Ga2O3?GaN system, with potentially useful electronic properties115,116. It is predicted to
be a direct wide bandgap semiconductor, comparable with GaN115. There has previously
been an experimental report of a cubic gallium oxynitride phase with composition close to
Ga2.8O3.5N0.5, that formed metastably during GaN thin ?lm synthesis from chemical precur-
sors117,118. Here, we report our ?rst-principles theoretical study of the formation energetics,
stability, and electronic properties of the Ga3O3N spinel-structured phase, combined with
experiments using a combination of high pressure-high temperature techniques to establish
the formation and stability of spinel-structured Ga3O3N from Ga2O3+GaN mixtures, and
to determine the chemical composition, structure and properties of the resulting materials.
1.2 First-Principles Theoretical Studies
Simulation of macroscopic properties of a physical system typically involves solving an
ordinary or partial di?erential equation over large numbers of degrees of freedom. Even if a
precise mathematical theory is available, it is only in very few cases that analytical solutions
are possible. Computational physics is the study and implementation of numerical algorithm
to solve problems in physics where a quantitative theory exists. Before the prevalence of
powerful computers, empirical and semi-empirical approaches are often adopted, which rely
on the phenomenological model or parameters ?tted from measured data. Nowadays, ab
initio (or ?rst-principles) calculations are routinely performed in the ?elds of computational
physics and chemistry. Compared with the empirical approach, predictions from ?rst-
principles method can provide unbiased comparison and interpretation to the experimental
data. It is also advantageous in calculating properties of materials at conditions where
no or limited experimental data is available, and designing novel materials with promising
properties. As one of the ab initio approaches, density function theory (DFT) is extremely
successful in the electronic structure calculations for solids, which results from the work
10
of Hohenberg, Kohn and Sham119,120. Beside bulk systems, density functional theory has
also become popular for complex materials such as nanostructures. In this dissertation,
within the DFT, the many-electron exchange-correlation is approximated with the local
density approximation (LDA). For part of the study that is associated with small energy
di?erences, the LDA results are compared with calculations using the generalized gradient
approximation (GGA). To improve numerical e?ciency, core electrons were approximated
with ultrasoft pseudopotentials (US-PP)121. In the case of Al2O3, we also conducted a
parallel comparative study using Projector Augmented-Wave (PAW) method.
As mentioned above, the DFT based calculation is only possible and useful with the
rapid development of high performance computer technologies, especially the parallel com-
puting techniques. Considering the current computational e?ciency, the DFT calculation
can only deal with atomistic models of no more than a few hundred atoms on a single-
core CPU. Our calculations are performed on the departmental computational resource, a
distributed memory computer system (Beowulf cluster). This cluster is comprised of 96
AMD Athlon MP CPUs running Red Hat Enterprise Linux 5. A photo of our recently con-
structed cluster system is shown in Figure 1.1. We have implemented parallel algorithms
to distribute each job into a set of calculations. Each corresponds to one DFT calcula-
tion. The most computationally expensive part in this study is to calculate the real-space
force-constant matrix using supercell models of ?100-200 atoms.
I have been focused on adopting and further developing computational methods based
on density functional theory and statistical theory to study the behavior of materials, in
particular the pressure induced solid-solid phase transitions and thermodynamic properties,
over a wide range of temperature-pressure (T-P) conditions. To predict the equilibrium
transition pressure at a ?nite temperature and thermodynamic properties (such as thermal
expansion coe?cient (TEC), heat capacity, entropy and bulk modulus, etc.), the calculation
of free energy is required. We adopt quasiharmonic approximation (QHA) to model the
lattice dynamics. Our approach of calculating phonon frequencies belongs to a method of
the direct approach: a ?rst-principles real-space supercell force-constant (SC-FC) method
11
Figure 1.1: Photo of recently constructed departmental computer cluster.
which calculates the phonon frequencies from the forces obtained via the Hellmann-Feynman
theorem. This technique is proved to be e?cient and successful in predicting the full phonon
spectra of many materials122?129.
For pressure-induced phase transitions, we are interested in investigating both equi-
librium transition conditions and the kinetic process. Regarding reconstructive transitions
with no group-subgroup relation, the equilibrium phase boundary is determined by equat-
ing the Gibbs free energy of the two phases. However, in general, there exists a kinetic
barrier at the transition, which can be overcome by the thermal activation energy from
the environment (small barrier) or heating process (large barrier). Based on the pressure
dependencies of the forward and backward barrier heights, we can predict the metastability
of polymorphs. The kinetic barrier can be estimated from the knowledge of the micro-
scopic mechanism of the transition. For transformations with a group-subgroup relation,
a transition path (TP) can be easily de?ned by a set of continuous atomic displacements
12
and/or lattice strains that the system transforms from one phase into another. For some
simple reconstructive transformations with no group-subgroup relation, although nucleation
processes may occur, di?usionless collective atomic displacements can still characterize the
transformation on a local basis. The concept of transition pathway is then possible to
describe the transformation from the starting phase to the ending phase in a continuous
manner. Among the in?nite number of ways to transform one structure into another, the
theoretical studies are restricted only to those most possible paths, e.g., preservation of
bonds, less strains, etc.
The pressure dependence of phonon spectrum can also reveal the information of struc-
tural stability under compression. In the case when soft phonon happens, a vanishing
phonon frequency indicates the disappearance of the restoring forces related to the corre-
sponding normal mode, and the structure consequently undergoes a continuous transition
to a lower-symmetry phase, which can be found according to the vibrational pattern based
on the associated eigenvectors. However, the transition happened at room or higher tem-
peratures is often ?rst-order, where no phonon has become soft yet. But this ?rst-order
phase transition is driven by the phonon mode with softening tendency.
1.3 Outline of Dissertation
The rest of this dissertation is organized as follows. In Chapter Two I ?rst review the
fundamental theory of solid-solid phase transitions, which include the reconstructive phase
transitions and the soft-phonon driven continuous transitions. Then our computational ap-
proaches of investigating the transformation mechanism and phase metastability/instability
are introduced. In Chapter Three I discuss the ?rst-principles methodologies we adopted in
this study, i.e., total energy calculation based on density functional theory and our methods
to calculate the ?nite-temperature thermodynamic potentials. The in?uence of equation of
state models on the prediction of thermal properties is also presented. In Chapter Four
we show our ?rst-principles studies on the pressure-induced phase transitions in Al2O3 and
AlN. Our focus is on the microscopic mechanism of the transition from an energetic point
13
of view. The thermal and elastic properties of Al2O3 are also studied. In Chapter Five we
predict a metastable high-pressure transition for ?-Si3N4. We also predict the thermody-
namic properties of three known polymorphs of Si3N4. In Chapter Six I present our study of
pressure-induced phase transitions in bulk Ga2O3 and the blue-shifted Raman frequencies
in Ga2O3 nanowires, as well as the theoretical optimal synthesis condition, electronic prop-
erties and phonon spectrum of the spinel-structured gallium oxynitride (Ga3O3N). Finally
in Chapter Seven we summarize our key results and discuss the future works.
14
Chapter 2
THEORY OF PHASE TRANSITIONS IN SOLIDS
2.1 Phases and Crystal Symmetries
Before we discuss the phase transitions, it is necessary to elucidate the concept of phase.
The word ?phase? may have di?erent meanings in di?erent disciplines. In thermodynamics,
a phase of a macroscopic system refers to a type of internal structure of the system when
the composition of the system is speci?ed. Throughout a single phase, all the physical
properties of the system is uniform at the macroscopic level. These properties include
chemical composition, density, heat capacity, index of refraction and so on. For a material
system, the internal structure is simply its atomic con?guration if magnetic or electronic
degrees of freedom can be ignored. A macroscopic solid material is composed of numerous
atoms or molecules, usually in the order of ? 1023. The atoms or molecules that compose
the solid are closely packed together and the chemical bonds between them are relatively
strong. The short-range structural orders are largely controlled by the chemistry of the
constituent elements, while statistics a?ects the long-range structural order. According to
their long-range atomic order/disorder, solids are divided into two categories: crystals and
amorphous solids. Real solids contain imperfections, such as surfaces, grain boundaries,
and defects. In this dissertation, my study is limited to only ideal crystals, which are good
approximations to real bulk crystalline materials that contain only small amount of defects
and impurities.
Symmetry is an essential character of crystals. Atoms in a crystal vibrate around their
equilibrium positions, which form a periodical and ordered pattern in the three dimension
space. The pattern is made up of a group of equilibrium positions of atoms, which is called
the atomic basis. The collection of repetition of identical structural units with translational
15
symmetry is called Bravais lattice.
crystal structure = Bravais lattice + atomic basis (2.1)
The location of each Bravais lattice point, in another word, the origin of each repeated
unit in a crystal, can be described by
R = n1a1 + n2a2 + n3a3 (2.2)
where the 3-D vectors ai (i = 1,2,3) are linearly independent unit-cell vectors. n1, n2 and
n3 are integers. The smallest repeatable unit cell of a lattice is called primitive unit cell,
and the corresponding unit vectors are primitive unit-cell vectors. Note that the choice of
unit-cells in a crystal is not unique.
There are seven distinct crystal systems, i.e., cubic, hexagonal, rhombohedral (also
known as trigonal), tetragonal, orthorhombic, monoclinic, and triclinic. By considering ad-
ditional possible lattice points at body centers, face centers, or base centers, whereas not to
reduplicate, there are precisely 14 distinct Bravais lattices, e.g., the orthorhombic system
includes four di?erent Bravais lattices, i.e., simple orthorhombic, body-centered orthorhom-
bic, face-centered orthorhombic, and base-centered orthorhombic lattices; and for the cubic
system, there are three distinct Bravais lattices, i.e., simple cubic (sc), body-centered cubic
(bcc), and face-centered cubic (fcc) lattices. In the case of body-centered, face-centered, or
base-centered Bravais lattice, the primitive unit cell is di?erent from its conventional unit
cell. And accordingly, the primitive unit vectors are di?erent from those conventional unit
vectors. As an example, for face-centered cubic lattice (fcc), the conventional unit cell is
constructed by having atoms on the corners of a cube and an atom at the center of each
16
Table 2.1: Point and space groups of Bravais lattices and crystal structures
Bravais lattice Crystal structure
(Basis of spherical symmetry) (Basis of arbitrary symmetry)
Number of point group 7 crystal systems 32 crystallographic point group
Number of space group 14 Bravais lattices 230 space groups
face. The conventional unit vectors and primitive unit vectors are
?
????
?
???
??
a1 = (a,0,0)
a2 = (0,a,0)
a3 = (0,0,a)
and
?
????
?
???
??
a1 =parenleftbig0, a2, a2parenrightbig
a2 =parenleftbiga2,0, a2parenrightbig
a3 =parenleftbiga2, a2,0parenrightbig
(2.3)
respectively. The volume of the fcc primitive unit cell is 1/4 of the conventional cubic cell.
In addition to translational symmetry as described by the Bravais lattice, a crystal is
also invariant under a set of point group symmetry operations, such as rotation, re?ection
and inversion. There are totally 32 crystallographic point groups. Each point group can
be classi?ed into one of the 7 crystal systems. Point group is also called crystal class. The
combination of translational symmetry operations and point group operations in a crystal
is referred as the crystal?s space group symmetry. There are a grand total of 230 space
groups130. Table 2.1 shows the relation between point and space groups of Bravais lattices
and crystal structures.
The nomenclature of space group is not unique. One commonly adopted notation is
listed in the International Union of Crystallography, which assigns each space group with
a number (#1 to #230). The Hermann-Mauguin notation and Sch?on?ies notation are also
commonly used in crystallography community. For example, ?-Al2O3 belongs to the space
group #167, which can be equivalently noted as R3C or D63d.
Experimentally, the crystal structure can be determined by either atomic scale imag-
ing or di?raction techniques, such as X-ray di?raction (XRD). To understand the general
principles involved in solving di?raction data, it is necessary to introduce the concept of
the reciprocal lattice. The reciprocal lattice plays a fundamental role in studies of functions
17
with the periodicity of a Bravais lattice. Once a Bravais lattice in real space is given, one
can construct the reciprocal lattice correspondingly. The primitive unit vectors in reciprocal
space are de?ned as ?
???
??
???
??
b1 = 2? a2?a3a1?(a2?a3)
b2 = 2? a3?a1a2?(a3?a1)
b3 = 2? a1?a2a3?(a1?a2)
(2.4)
where ai (i = 1,2,3) are primitive unit vectors of real space Bravais lattice. Under this
de?nition, ai ?bj = 2??ij (i,j = 1,2,3). Any reciprocal lattice vector can be written as
G = k1b1 + k2b2 + k3b3 (2.5)
where ki (i = 1,2,3) are integers. Then
G?R = 2? (n1k1 + n2k2 + n3k3) (2.6)
and
eiG?R = 1 (2.7)
where R = n1a1 + n2a2 + n3a3 is any real space lattice vector. For any family of lattice
planes separated by a distance d, there are reciprocal lattice vectors normal to the planes.
The shortest one has a length of 2?/d. It is convenient to describe the orientation of lattice
planes based on this relation. The commonly used notation is Miller indices. The Miller
indices for a family of lattice planes are the integral coe?cients of the shortest reciprocal
lattice vector perpendicular to the planes in terms of the primitive unit vectors in reciprocal
space, e.g., the reciprocal lattice vector hb1 + kb2 + lb3 determines a family of planes with
Miller indices (h,k,l).
The di?raction pattern of X-rays incident on a crystal provides information on inter-
planar spacing, and ultimately the space group and structure of the crystal. Generally,
there are two equivalent ways to explain the phenomenon of X-ray di?raction by a perfect
18
crystal, i.e., the Bragg law and von Laue condition. W. L. Bragg simply treated the crystal
as parallel planes of atoms and the incident waves are specularly (mirror-like) re?ected from
these planes. The condition for a constructive interference leads to the famous Bragg law
2dsin? = n? (2.8)
where d is the distance between two adjacent parallel planes, ? is the angle measured from
the plane (90? minus angle of incidence), integer n is the order of interference and ? is the
wavelength of incident radiation. The total de?ected angle measured from the incident light
is 2?.
The von Laue approach regards the crystal as identical lattice points (a set of atoms)
which can reradiate the incident wave in all directions. Sharp peaks will be observed at
directions where constructive interference happens. The Laue condition can be written as
?k = G (2.9)
i.e., the change in wave vector equals to a reciprocal lattice vector. Under the assumption
of elastic scattering (the magnitude of the wave vector does not change), the Laue condition
yields Bragg law. The practical and detailed experimental methods to determine the crystal
structure are beyond the scope of this dissertation.
2.2 Equilibrium Thermodynamic Theory of Phase Transitions
2.2.1 Thermodynamic Stability
The occurrence of phase transitions can be interpreted synonymous to changes of atomic
structures of matters. This phenomenon has long been studied and many natural forms of
transitions are well observed in our everyday life, e.g., water freezes into ice below freezing
temperatures. Here we focus our attention to the solid-solid phase transitions, especially the
19
pressure-induced structural phase transitions. Within this limitation, some important as-
pects of the complete theory of phase transitions, such as critical phenomena, melting, glass
transition, ferromagnetism, superconductivity, etc., are not considered in this dissertation.
The ?rst set of questions are: (1) why does a phase transition happen? (2) when does
it happen? and (3) how does it happen? The ?rst question is related to the interactions
between a solid and its surrounding media. For simplicity, we ignore magnetic and electric
interactions, and focus only on mechanical and thermal interactions. According to the 1st
and 2nd law of thermodynamics, the most thermodynamically stable state of an isolated
state is the state that minimizes the energy (E). When the system is interacting with the
surrounding media through mechanical work, the system reaches a mechanical equilibrium
with the media when its pressure (P) equals the pressure of the media, and it becomes
thermodynamically stable against any spontaneous ?uctuations when its enthalpy (H), de-
?ned as H = E + PV , becomes minimized. Here, V represents the volume of the system.
Similarly, when the system exchanges its energy with the media through heating, the ther-
mal equilibrium is reached when the temperature (T) of the system equals the temperature
of the media, and it becomes thermodynamically stable when its Helmholtz free energy F,
de?ned as F = E ?TS, becomes minimized. Here, S is the entropy of the system.
In our studies, a solid is considered to be in a thermal equilibrium at temperature T
and a mechanical equilibrium at pressure P with the surrounding media. At a given (T,P)
condition, the thermodynamically stable state is reached when:
?E + P?V ?T?S ? 0 (2.10)
where ?E, ?V , and ?S are virtual variations of total internal energy, volume, and entropy
respectively. Given a su?ciently long period of time and allowing all possible ?uctuations,
a solid at a given (T,P) condition should reach its thermodynamically stable state, which
minimizes its Gibbs free energy G, de?ned as G = E + PV ? TS. Equation 2.10 is the
Gibbs-Duhem stability criterion which is equivalent to ?G ? 0.
20
A thermodynamically stable phase of a solid at a given (T,P) condition corresponds
to a global minimal of Gibbs free energy for all atomic con?gurations. Correspondingly,
metastable phases are referred to those associated local minimums of G. Altering T or P
will change the Gibbs free energy landscapes and may in turn shu?e the relative orders
among the local G minimums. The emergence of a new global minimal of G will lead to a
phase transition in the solid.
The answer of when and how a phase transition happens requires detailed knowledge of
not only the thermodynamic properties of individual phases but also the kinetic process that
connect the initial and ?nal phases. At equilibrium condition, a phase transition becomes
possible when a new phase has lower Gibbs free energy than the initial phase. The initial
phase then can be considered as a metastable phase at the new (T,P) condition. However,
the lifetime of a metastable state is controlled by its local kinetic barrier heights. In reality,
metastable states can exist for a long period of time and an equilibrium phase transition
can be hindered when there exist large kinetic barriers that against the ?uctuations. A
well-known example is the two crystalline forms of element carbon: diamond and graphite.
Graphite is known to be the ground state at ambient conditions whereas diamond is also
?stable? in a wide range of usual environment unless enough activation energy (such as
heating) is provided.
2.2.2 Phase Diagram and Classification of Phase Transitions
When two phases of a single-component system coexist at equilibrium, three conditions
must be satis?ed between two phases:
1. The temperatures of the two phases must be equal.
T1 = T2 (2.11)
21
2. The pressures in two phases must be equal.
P1 = P2 (2.12)
3. The chemical potentials of the two phases must be equal.
?1 = ?2 (2.13)
Since the chemical potential is the molar Gibbs free energy, we can rewrite the phase
equilibrium conditions in the following form if both phases have the same number of atoms.
G1 (T,P) = G2 (T,P) (2.14)
The basic concepts of an equilibrium phase diagram is sketched in Figure 2.1. Each
point in the two-dimensional (T,P) space denotes one equilibrium state of the system. In
most the T-P region of the phase diagram, only a single phase is thermodynamically stable.
At the boundary of two adjacent regions, two phases are equally stable along the T-P phase
boundary lines. Phase transitions happen when the phase boundaries are crossed. At triple
points, such as the (T1,P1)and (T2,P2) points in the plot, three phases coexist. For a single
component system, no more than three phases can reach simultaneous thermodynamic
equilibria. There is a special point C where the phase boundary curve between the liquid
and gas phases come to an end. This point is named as critical point, at and beyond which
the liquid and gas phases become identical. It should be mentioned that the critical point
can exist only for phases which are quantitatively di?erent, e.g., a liquid and a gas. On the
contrary, solid phases have certain internal symmetries. The di?erence between two solid
phases of a substance, or between the liquid and solid phases, are qualitatively di?erent.
During a quasi-static process of phase transition, the system may absorb or release the
so-called latent heat (L). The latent heat is associated with the change of entropy of the
22
P
T
C
Liquid
Gas
Solid 1
Solid 2
(T1,P1)
(T2,P2)
(TC,PC)
Figure 2.1: Phase diagram with two solid phases, one liquid phase and one gas phase. Two
triple points and one critical point are present.
system.
L = T?S (2.15)
where ?S is the entropy di?erence between the ?nal and initial phases. ?S may be zero
in continuous phase transitions.
If we di?erentiate both sides of equation 2.14 with respect to temperature, the Clapeyron-
Clausius equation can be derived.
dT
dP =
V2 ?V1
S2 ?S1 =
T?V
L (2.16)
Clapeyron-Clausius equation gives the slope of the phase boundary in the phase dia-
gram which is directly related to the change of volume and latent heat. We can estimate
the amount of heat being absorbed/released during the transition from equation 2.16, nev-
ertheless both ?V and L are zero for continuous phase transitions (see below).
23
As discussed above, while Gibbs free energy is always continuous during equilibrium
phase transitions, other thermodynamic variables might be discontinuous. I now introduce
the concept of order of a phase transition based on the Gibbs free energy.
1. First-order transitions are accompanied with a discontinuous change in the ?rst-order
derivatives of the thermodynamic potentials, such as volume and entropy. Thus a
density jump and a latent heat are presented during a ?rst-order phase transition.
Transitions of this type is referred as the ?rst kind or discontinuous phase transitions.
V =
parenleftbigg?G
?P
parenrightbigg
T
S = ?
parenleftbigg?G
?T
parenrightbigg
P
(2.17)
2. In second-order transitions, the ?rst-order derivatives of Gibbs free energies are con-
tinuous. Yet, the transitions are accompanied with discontinuity in second-order
derivatives of thermodynamic potentials, such as isobaric speci?c heat capacity CP
and isothermal compressibility ?T.
CP = T
parenleftbigg?S
?T
parenrightbigg
P
= ?T
parenleftbigg?2G
?2T
parenrightbigg
P
(2.18)
?T = ? 1V
parenleftbigg?V
?P
parenrightbigg
T
= ?1/V
parenleftbigg?2G
?2P
parenrightbigg
T
(2.19)
Higher order transitions can be speci?ed in a similar fashion. Within the framework of
Landau theory, second and higher order transitions belong to the second-kind (continuous).
For these transitions, system continuously passes from one phase to another without any
abrupt changes in volume, entropy and consequently no latent heat.
In the case of solid-solid phase transitions in crystals, reconstructive transitions reform
the crystal lattices from one type to a distinctly di?erent one without any direct structural
relations, e.g., no group-subgroup relations. This type of transitions are always ?rst-order.
On the other hand, during displacive phase transitions, atoms gradually deviate from their
original equilibrium positions to their corresponding new positions in the ?nal phase in
24
a collective fashion. Although the displacements might be small, crystal symmetry is al-
tered. Since the displacements happen in a continuous manner, one structure must has a
higher symmetry and the other structure has a lower symmetry which is a subgroup of the
spacegroup of the high-symmetry phase. The displacive transitions can be any order
2.3 Landau Theory for Phase Transitions of Second Kind
2.3.1 The Order Parameter and Landau Free Energy
Landau theory is a general phenomenological theory for phase transitions of second
kind. Landau introduced a quantity order parameter to describe the changes in structure
during the phase transition. The order parameter is zero in the high symmetry phase and
takes non-zero value in the low-symmetry phase. For displacive phase transitions, the order
parameter can be easily chosen as the atomic displacement from the equilibrium sites of
the initial high-symmetry phase. Usually the free energy is independent of the sign of the
order parameter that it only contains even-power terms.
G(T,P,?) = G0 + A?2 + B?4 + ??? (2.20)
where G0 is the free energy of the high symmetry phase. Apparently this expansion is
valid only for small values of the order parameter, i.e., close to the transition point. For
simplicity, we consider equation 2.20 up to the fourth-order term, the sign of coe?cient A
will determine whether it leads to a single minimum at ? = 0 or a local maximum at ? = 0.
Plot of the free energy as a function of order parameter is shown in Figure 2.2 for these two
cases. If A > 0, the global minimum is at ? = 0 which corresponds to the high symmetry
phase, while if A < 0, the high symmetry phase become energetically unstable and the
system is stable at non-zero values of the order parameter (??0).
In this dissertation, we have studied the pressure induced displacive phase transition
from phenacite to post-phenacite phase in Si3N4. Because the transition is largely pressure
driven, we neglect the temperature dependence of the landau free energy. Therefore at the
25
G(?)
A>0
A<0
-?0 ?0
?
Figure 2.2: Variation of the Landau free energy with positive and negative A coe?cients.
transition pressure the coe?cient A changes sign, so that it is positive for the high symmetry
phenacite phase below the transition pressure and is negative for the post-phenacite phase
above the transition pressure. The simplest form of this condition is
G(P,?) = G0 + a(Pt ?P)?2 + b?4 (2.21)
where a and b both have positive values. We assume A = a(Pt ?P) where Pt is the
transition pressure, and B = b. When P > Pt, the equilibrium condition ?G?? = 0 leads to
the non-zero value of the order parameter.
?0 =
radicalbigg
a(P ?Pt)
2b (2.22)
Note that the ?rst order derivative of Gibbs free energy with respect to pressure is
continuous at P = Pt.
26
2.3.2 Dynamic Lattice Stability and Soft Phonon Modes
As a phenomenological approach, Landau theory is mathematically simple and ?exible
as the theoretical background of many studies of phase transitions. However, from the ab
initio simulation point of view, the new low-symmetry phase is determined from a di?erent
approach. For instance, the low-symmetry structure can be derived by relaxing the high-
symmetry structure to equilibrium at the conditions where it is unstable. Usually a slight
distortion is needed to break the symmetry. For phonon instability (elastic instability is
less common), the distortion is a set of collective atomic displacements, which is associated
with the soft phonon theory.
At ?nite temperature, atoms inside a solid never ?freeze? at their equilibrium posi-
tions. Instead, they oscillate around their respective equilibrium positions. The atomic
displacements of atoms can be analyzed in term of the vibrational normal modes. The
quasi-particle representation of quantization in harmonic lattice vibration is phonon. The
squares of phonon (vibrational) frequencies are normally positive. A vanishing ?2 indicates
that the restoring forces related to that normal mode disappear. The corresponding vibra-
tion is called soft mode or soft phonon. A crystal structure containing soft phonon modes
are considered as dynamically unstable because atomic displacements along the eigenmode
pattern will lead to a new structure with lower energy. Note that the new structure belongs
to a lower symmetry whose space group is a subgroup of the original space group, and the
eigenmodes of soft phonons are often adopted as the order parameters in Landau theory
discussed above.
The simplest case of soft phonon modes is that only one phonon mode becomes softened.
If the soft mode drops to zero at ?-point (center of Brillouin zone), the periodicity of the
low symmetry structure will be of the same size of unit cell as the high symmetry phase. If
phonon softening happens at the zone boundary, the periodicity of the new phase will be
di?erent from the high symmetry phase. The unit cell of the low symmetry phase, in general,
is a super cell of the previous primitive unit cell. The size of the super cell is determined by
the symmetry of the system and the k-point that has soft phonon, e.g., if the soft phonon
27
happens at X-point (1,0,0), the new structure will have a double-sized primitive unit cell
vector along a1 compared to the previous phase. Real space displacements of atoms from
high symmetry equilibrium sites can be obtained based on the eigenvector of the soft mode.
By breaking the symmetry with atomic displacements (usually small), the low-symmetry
structure can be found by relaxing the distorted high-symmetry con?guration energetically
at the condition that it is unstable.
In the case that more than one mode (at di?erent k-points) involve soft phonons, in
principle, the real space atomic displacements that lead to the low-symmetry stable phase
can be derived from a linear combination of the eigenvectors of the soft modes. In practice,
we usually treat the soft phonon modes in serial. For each soft phonon, a new low-symmetry
structure can be derived from total energy calculation by imposing its symmetry. In the
next step, we calculate the phonon dispersion for the new phase. We repeat the process
until all soft phonon modes disappear. The energetically favored structure among these
stable or metastable phases is the one with the lowest energy, which can be determined
from the E-V curves.
2.4 Transition Paths of Reconstructive Phase Transitions
Many structural phase transitions in solids belong to the ?rst-order reconstructive
category, i.e., the symmetries of the two structures have no group-subgroup relation and
typically bonds breaking/forming process occurs during the transition. The dynamics of
reconstructive phase transitions is not yet well understood for most known solid-solid phase
transitions. It is known that for some transitions of this kind, an interface forms between
the initial and ?nal phases, and atoms go through a long range di?usion process during the
transformation. In our study, we examine the transformation mechanism from a di?erent
angle, assuming the initial and ?nal structures are related by a di?usionless transition path
(TP). Intermediate structures along the path can be given with a crystalline character ap-
proximately. Transition pathway studies are helpful for understanding the mechanism of
transitions at a microscopic level, and predicting the kinetic barrier height of the proposed
28
path(s) from the energy landscape. However, there are in principle an in?nite number
of pathways for one structure to transform into another. If neglecting defects and sur-
face e?ects that exist in reality, we assume a certain symmetry is maintained when atoms
are displaced collectively along the TP. For di?usionless reconstructive transitions, it is
known that the nucleation free path is de?ned by is a common subgroup of both end struc-
tures131?134. Although the number of common subgroups is in?nite, it is possible to obtain
a ?nite set of common subgroups with certain constrains, such as the size of unit cell, bonds
preserving, maximum strains etc. Among these TPs, the energetically favorable path(s)
is/are that/those associated with the lowest kinetic energy barrier(s).
From the theoretical point of view we could investigate the most possible paths and
?nd the one(s) with the least barrier height. Stokes and Hatch have implemented a com-
puter program called COMSUBS which systematically ?nds maximal common subgroups
of two structures. The major constraint for COMSUBS is the bond condition, since many
studies suggested that the most energetically favored TPs are those preserve the number of
bonds53?55. Other constrains that a?ect the number of common subgroups are the minimum
and maximum size of unit-cell, strain tolerance, maximum atomic displacement, minimum
distance allowed between nearest neighbors, etc. This method has been adopted in the stud-
ies of B1-to-B2 transitions in sodium chloride (NaCl) and lead sul?de (PbS)135, zinc-blende
to rocksalt transitions in silicon carbide (SiC)136, and wurtzite-to-rocksalt transition48.
For each possible transition path, in principle, the energy barrier height is determined
by the calculated n-dimensional potential-energy surface (PES), where n is the degree of
freedom of the intermediate state. For example, in the case of wurtzite-to-rocksalt transition
in AlN, the degree of freedom is 6 for the Pna21 path, i.e., there are 6 free parameters
that can be adjusted independently. The complete information of the wurtzite-to-rocksalt
transformation path can be revealed from a 6-dimensional PES. However, ?rst-principles
calculation of a 6D PES is not an easy task. To calculate the barrier height, two reasonable
approaches have been investigated. The ?rst approach is named as bow-function method,
as adopted in the study of zinc-blende to rocksalt transition in SiC136. The ?rst step within
29
this method is to calculate the enthalpy along a linear TP. Here ?linear? means that all the
structural parameters, including both external parameters (lattice parameters) and internal
coordinates, vary according to a single transition parameter.
xm = (1 ??)xmi + ?xmf (2.23)
where ? is the transition parameter which varies from 0 to 1, xmi and xmf are the initial and
?nal mth structural parameter, respectively. In many cases we have studied, the peak of the
barrier height locates at around ? = 0.5. In this way, although we obviously overestimate
the enthalpy barrier, it is e?cient for eliminating some TPs that give much larger barriers.
With the number of TPs being reduced, we further develop a numerical algorithm to possibly
lower the barrier height. On the basis of previous linear TP, we assume the saddle point
(which gives the enthalpy barrier height) of the ?true? TP is not far from the maximum
point of the linear one. By adding a quadratic term to equation 2.23 we change the linear
function to a bow function.
xm = (1 ??)xmi + ?xmf + wm? (1 ??) (2.24)
where wm is the ?weight? of the bow function for the mth structural parameter. Then we
minimize the enthalpy barrier height with respect to wm. This is done by searching for the
lowest height of the peak with di?erent weights (positive or negative) for each structural
parameter. If more than one peak are present, we will track the highest peak across the
entire TP. If the change on wm decreases the barrier height, we keep the change, otherwise
we undo the change. This process is done when no changes on wm will further lower the
height of the peak. At this situation, saddle point of the true TP is found and so for the true
barrier height. We should stress that our aim is to ?nd the barrier height and our proposed
TP only matches the saddle point and two end points of the true TP. The quadratic bow
function that links these three points is generally di?erent from the true TP.
30
However, the bow-function algorithm may fail if the landscape of the enthalpy function
is complex, e.g., if the saddle point is surrounded by peaks and the true path is o? the linear
path, the barrier height found by bow function can be higher than the saddle point. In all
the concerns our estimated enthalpy barrier height will always be an upper limit compared
to the true barrier.
Besides the bow function method, sometimes a reconstructive phase transition can be
characterized by only a few structural parameters. In this case a proper choice of the struc-
tural parameter is essential. Other ?unimportant? parameters will be fully relaxed during
the DFT calculation. In the simplest case, if only one structural parameter is primarily
responsible during the transition, the transition parameter is de?ned based on the changes
of this parameter. If two equally important structural parameters both cause signi?cant
changes in enthalpy from one structure to the other, we can calculate the enthalpies as
a function of two transition parameters (a 2D PES). Calculations involve more than two
transition parameters can be cumbersome in the sense of both computational load and
analysis.
Let?s take the single parameter case as an example to illustrate the procedure of tran-
sition pathway calculations. The particular structural parameter can be found either by
comparison of two end structures or by a testing calculation as follows. If we choose the
transition parameter at values which are uniform samplings between two end phases (equa-
tion 2.23), the saddle point normally locates at ? = 0.5 or nearby (this testing is not
applicable to the case where a metastable state exists around ? = 0.5). A series of to-
tal energy calculation can be done by ?xing only one structural parameter at a time and
fully relaxing the others starting with ? = 0.5. If the parameter we ?xed is unimportant,
the resulting energy will be close to the energy of one end phase, whereas the important
structural parameter will yield an energy apparently di?erent from the end phases. Next
we sample the transition parameter in the [0,1] region and calculate the energies at each
sampling point by ?xing the chosen parameter and relaxing the other parameters. To get
the enthalpy under a given pressure, we preform the energy calculations at several volume
31
points and obtain the enthalpy based on the equation of states, H = E + PV . Finally a
relative enthalpy as a function of transition parameter can be plotted and the barrier height
is estimated from that. We emphasize that the calculated barrier height is approximate and
should be understood as an upper limit of the true one.
For technical reason, it is easy to ?x the internal parameter(s), however, di?cult to do
so for the external ones. So the second approach currently only works for transitions that
characterized by changes in the internal structural parameter(s).
32
Chapter 3
FIRST-PRINCIPLES CALCULATIONS OF THERMODYNAMIC
PROPERTIES OF CRYSTALS
3.1 First-Principles Total Energy Theory
3.1.1 Density Functional Theory (DFT)
Atomic-scale theory and simulation of solids rely on accurate evaluation of the total
energy of an N-atom system with a speci?ed atomic con?guration. Many theories, evolving
from simple empirical models in early years to quantum Monte Carlo method, have been
developed to estimate the total electronic energy of a given system. Bene?ting from the
increasing power of computational technology and to our purpose, in this study, we adopt
density functional theory (DFT). DFT is one of the most successful ?rst-principles quan-
tum mechanical theories for atoms, molecules and solids. Nowadays, DFT calculations are
routinely performed in the ?elds of materials physics and chemistry.
In quantum mechanics, a system is described by the wave function ? of its Hamiltonian.
Within Born-Oppenheimer approximation137 (adiabatic approximation), one assumes that
the electrons are in their ground state for the instantaneous ionic con?guration at any
moment. Therefore, we are able to treat ionic motion with classical mechanics, and only
the electronic Hamiltonian is treated with quantum mechanics. A stationary electronic
state of a N-electron Hamiltonian is described by a wave function ?(r1,...,rN) satisfying
the Schr?odinger equation:
?H? =bracketleftBig?T + ?V + ?UbracketrightBig? =
?
?
Nsummationdisplay
i
? planckover2pi1
2
2m?
2
i +
Nsummationdisplay
i
v(ri) +
summationdisplay
i<j
U(ri,rj)
?
?? = E? (3.1)
33
where U(ri,rj) is the electron-electron coulomb interaction. Exact solution to this Schr?odinger
equation even for a system with 100 atoms is computationally forbidden. DFT provides an
exact way to map the many-body problem onto an e?ective single-body problem. This is
done by replacing the many-body electronic wavefunction with the electron density as the
basic quantity.
The predecessor to DFT is the simple Thomas-Fermi model138 which neglects the ex-
change and correlation energy. Hohenberg and Kohn119 proved two celebrated theorems
(HK theorems) stating that (1) the nondegenerate ground-state wave function is a unique
functional of the ground-state electron density and (2) the ground-state electron density
minimizes the total energy of the system. According to HK theorems, the system?s wave-
function only depends on the particle density which is a function of three spatial variables
whereas a many-body wavefunction is dependent on 3N variables (three degrees of freedom
for each of the N electrons). Since the electron density n(r) is given by:
n(r) = N
integraldisplay
d3r2
integraldisplay
d3r3 ???
integraldisplay
d3rN??(r,r2,...,rN)?(r,r2,...,rN) (3.2)
The ground-state wavefunction is a unique functional of n0, i.e., ?0 = ?0 [n0]. Thus
all the observables are also functionals of n0, in particular the ground-state energy.
E0 = E [n0] =
angbracketleftBig
?0 [n0]
vextendsinglevextendsingle
vextendsingle?T + ?V + ?U
vextendsinglevextendsingle
vextendsingle?0 [n0]
angbracketrightBig
(3.3)
In practice, for a speci?c system (v(r) is known), all one need to do is to minimize the
sum of kinetic, potential and interaction energies.
Ev [n] = T [n] + U [n] + V [n] = T [n] + U [n] +
integraldisplay
d3rn(r)v(r) (3.4)
Exact form of E [n] functional is unknown. Most commonly adopted approach in DFT
is the Kohn-Sham method120. The kinetic energy functional of interacting electrons T [n]
34
can be treated as a summation of two parts.
T [n] = TS [n] + TC [n] (3.5)
where S and C represent ?single-particle? and ?correlation?. TS [n] is the kinetic energy
density functional of noninteracting particles and TC [n] is the remainder. TS [n] is an ex-
plicit function of the single-particle wavefunctions ?i (r), where ?i (r) is a density functional,
so TS [n] is an implicit functional of the electron density.
TS [n] = TS [{?i [n]}] (3.6a)
TS [n] = ? planckover2pi1
2
2m
Nsummationdisplay
i
integraldisplay
d3r??i (r)?2?i (r) (3.6b)
Without any approximation, the energy can be written as:
E [n] = T [n] + U [n] + V [n] = TS [{?i [n]}] + UH [n] + Exc [n] + V [n] (3.7)
where UH = q22 integraltext d3rintegraltext d3r?n(r)n(r?)|r?r?| , which is the electrostatic interacting Hartree energy of
the charge distribution n(r). And Exc [n] is the exchange-correlation (xc) energy which has
contributions from T ?TS and U ?UH. This term can be decomposed as Exc = Ex + Ec,
where Ex is the exchange energy due to Pauli principle and Ec is the correlation energy.
An e?ective potential energy can be de?ned as:
VS (r) = V (r) + UH (r) + Exc (r) (3.8)
Consequently, the Schr?odinger equation is:
bracketleftbigg
? planckover2pi1
2
2m?
2 + VS (r)
bracketrightbigg
?i (r) = ?i?i (r) (3.9)
By solving the charge density of this equation of noninteracting single-body system
in potential VS (r), one can obtain the density of the interacting many-body system in
35
potential V (r).
n(r) = nS (r) =
Nsummationdisplay
i
fi|?i (r)|2 (3.10)
where fi is the occupation of the ith orbital.
The above three equations are called Kohn-Sham (KS) equations. The purpose of
KS equations is to solve a noninteracting Schr?odinger equation to replace the problem of
minimizing the total energy E [n]. With an initial guess of the density n(r), an iterative
process is performed until convergence is reached (e.g., convergence in energy, density, or
some other observables). In numerical calculations, the KS orbital is typically expanded
using a set of basis functions. There are di?erent ways to construct a suitable basis function.
Our DFT based calculations are implemented in the Vienna ab initio Simulation Package
(VASP)139, which uses a planewave expansion140,141.
Because the core electrons are bounded to the nuclei and their orbitals typically do
not change much, it is a good approximation that the Hartree and xc terms in VS (r) are
evaluated only for the valence electron density nv, and the core electrons are treated with
a pseudopotential (PP) V PPext . The pseudopotentials that are adopted in our VASP calcu-
lations are Vanderbilt Ultra-Soft Pseudopotentials (US-PP)121,142 or Projector Augmented
Wave (PAW)143,144 method.
Further approximation is needed to evaluate the exchange-correlation functional Exc [n].
Unless stated otherwise, this study is performed with local density approximation (LDA)145,146
which is a local functional. A semilocal, gradient dependent functional, generalized gradient
approximation (GGA)147?151 is sometimes used as a comparison.
3.2 Statistical Theory of Bulk Crystals
3.2.1 Phonon Theory of Lattice Dynamics
Equilibrium positions of atoms in a crystal are determined by the minimum total
energy. However, the atoms are not ?xed at their equilibrium positions. Because the
interatomic chemical bonds are not in?nitely strong, nor do the atoms have in?nite masses,
36
the atoms may oscillate about their equilibrium locations at a ?nite temperature. According
to statistical mechanics, the equilibrium energy at a given temperature is the ensemble
average over all the possible atomic con?gurations of the system. If we denote by Ei the
static energy of the ith stationary con?guration of the crystal and ? = 1/kBT, the total
energy can be written as:
E =
summationtext
i
Eie??Ei
summationtext
i
e??Ei (3.11)
However, exact evaluation of the total energy is a formidable task. Here we make
two assumptions which are valid for most crystals at equilibrium conditions below melting
temperature:
1. We assume the mean equilibrium position of each atom is a Bravais lattice site and
the atom oscillates about this position.
2. We assume the deviation from the equilibrium position is small compared with the
interatomic distance.
The potential energy of a crystal is a function of the instantaneous atomic positions. If
we let u?,i (?) denote the ? cartesian component of the displacement of the ith atom in the
?th primitive unit cell (x(?) = ?1a1 + ?2a2 + ?3a3 , where ? collectively denotes ?1, ?2 and
?3) from its equilibrium position, under the two assumptions, we may expand the potential
energy in power series of these components:
V = V0 +
summationdisplay
?,i,?
parenleftbigg ?V
?u?,i (?)
parenrightbigg
0
u?,i (?)
+12
summationdisplay
?,i,?
summationdisplay
??,j,?
parenleftbigg ?2V
?u?,i (?)?u?,j (??)
parenrightbigg
0
u?,i (?)u?,jparenleftbig??parenrightbig
+16
summationdisplay
?,i,?
summationdisplay
??,j,?
summationdisplay
???,k,?
parenleftbigg ?3V
?u?,i (?)?u?,j (??)?u?,k (???)
parenrightbigg
0
u?,i (?)u?,jparenleftbig??parenrightbigu?,kparenleftbig???parenrightbig
+??? (3.12)
37
In this expansion, V0 is the potential energy of the static crystal when all the atoms are
at their equilibrium positions. The subscript ?0? indicates that the derivatives are evaluated
with atoms at the equilibrium positions. Since there is no net force on any atom in the
equilibrium con?guration, the ?rst order derivative
parenleftBig
?V
?u?,i(?)
parenrightBig
0
vanishes and the second
term in the expansion is zero. It will be convenient to use di?erent symbols to replace those
derivatives as follows:
??i,?jparenleftbig?,??parenrightbig =
parenleftbigg ?2V
?u?,i (?)?u?,j (??)
parenrightbigg
0
(3.13a)
A?i,?j,?kparenleftbig?,??,???parenrightbig =
parenleftbigg ?3V
?u?,i (?)?u?,j (??)?u?,k (???)
parenrightbigg
0
(3.13b)
B?i,?j,?k,?lparenleftbig?,??,???,????parenrightbig =
parenleftbigg ?4V
?u?,i (?)?u?,j (??)?u?,k (???)?u?,l (????)
parenrightbigg
0
(3.13c)
The ?rst nonvanishing contribution to the constant equilibrium static potential energy
is the quadratic term and this is the only term being kept in the harmonic approximation.
Those higher order terms are known as anharmonic terms which are considerably responsible
for many physical properties. Typically they are treated as perturbations to the dominant
harmonic term. If we denote p?,i (?) the ? cartesian component of the momentum of the
ith atom in the ?th primitive unit cell and mi is the mass of the ith atom in the primitive
unit cell, the total kinetic energy of the system is:
T =
summationdisplay
?,i,?
p2?,i (?)
2mi (3.14)
Consequently the vibrational Hamiltonian can be written as:
H = H0 + HA (3.15)
38
where
H0 =
summationdisplay
?,i,?
p2?,i (?)
2mi +
1
2
summationdisplay
?,i,?
summationdisplay
??,j,?
??i,?jparenleftbig?,??parenrightbigu?,i (?)u?,jparenleftbig??parenrightbig (3.16a)
HA =
?summationdisplay
n=3
1
n!
summationdisplay
?1,i1,?1
???
summationdisplay
?n,in,?n
parenleftbigg ?nV
?u?1,i1 (?1)????u?n,in (?n)
parenrightbigg
0
u?1,i1 (?1)u?n,in (?n)
(3.16b)
The Hamiltonian H0 is the harmonic part of the vibrational Hamiltonian. The Hamil-
tonian HA is the anharmonic part of the vibrational Hamiltonian. In the harmonic approx-
imation, the anharmonic contribution has been neglected. In order to solve the vibrational
energy, it is convenient to introduce the normal coordinates.
Q? (q) = 1?N
summationdisplay
?,i,?
?m
ie??,i (q,?) ?u?,i (?) e?iq?x(?) (3.17)
P? (q) = 1?N
summationdisplay
?,i,?
1?
mie?,i (q,?) ?p?,i (?) e
iq?x(?)
where ? = 1,2,??? ,3n that labels the index of branch and n is the number of atoms in a
primitive unit cell. N is the total number of atoms in the crystal. q is the wave vector
in the reciprocal space and it has s discrete values which is the number of primitive unit
cells in the crystal (n ? s = N). Usually a bulk crystal consists of a very large number of
primitive unit cells which allows q to change quasi-continuously. And e?,i (q,?) is the ?
cartesian component of the eigenvector of the ith atom at reciprocal space point q from the
?th branch. It has the property as follows:
summationdisplay
i,?
e?,i (q,?) ?e??,i (q,?) = ??? (3.18)
Derivation shows that with normal coordinates the kinetic energy and the harmonic
potential energy can be expressed as summations of square terms, which transform the
39
Hamiltonian H0 into the Hamiltonian of 3n independent harmonic oscillators.
H0 = 12
summationdisplay
?
P2? (q) + 12
summationdisplay
?
?2? (q)Q2? (q) (3.19)
where ?? (q) is determined by solving the following secular equation
summationdisplay
j,?
D?i,?j (q) ?e?,j (q,?) = ?2? (q) ?e?,i (q,?) (3.20)
In equation 3.20 D (q) is often called dynamical matrix which can be obtained from
the real space force constant matrix.
D?i,?j (q) = 1?m
imj
summationdisplay
h
??i,?j (0,h) ? e?iq?h (3.21)
where h = x(??) ? x(?). Without losing generality, it is convenient to set x(?) as the
reference lattice point. The vibrational frequency ? is a function of the wave vector q and
this dependence is often called the dispersion relation. For each q, there are 3n normal
modes, and the normal modes are identical for wave vectors di?er by reciprocal lattice
vectors. For crystals have one atom in its primitive unit cell (n = 1), only three acoustic
modes exist. Otherwise (n greaterorequalslant 2) there are three acoustic modes and 3n? 3 optical modes.
Within the harmonic approximation, the hamiltonian is expressed as 3N independent
oscillators, whose energy levels are well known in quantum mechanics. The total vibrational
energy is simply the summation of the energies of individual oscillators
Evib =
summationdisplay
q?
parenleftbigg
n? (q) + 12
parenrightbigg
planckover2pi1?? (q) (3.22)
where n? (q) is the occupation number of the normal mode, which takes the values of
0,1,2,??? . 12planckover2pi1?? (q) is the zero-point vibrational energy of the normal mode. And equiva-
lently, one usually says there are n? (q) phonons with wave vector q in the ? branch instead
40
of the n? (q)th excited state of the normal mode. Using equation 3.11, one can show that
n? (q) = 1e?planckover2pi1?
?(q) ? 1
(3.23)
The calculation of phonon dispersion relation is the major part of our computation and
is necessary to further study the thermodynamic properties. Our techniques of calculating
the phonon dispersion and the corresponding vibrational density of state are described in
the next subsection.
3.2.2 Statistical Harmonic Approximation
We have discussed the equilibrium vibrational energy of a harmonic crystal in Section
3.2.1. Based on statistical ensemble theory, the canonical partition function is Z = summationtext
i
e??Ei
where i represent any possible state of the system. The total energy within harmonic
approximation is:
E = E0 +
summationdisplay
q?
parenleftbigg
n? (q) + 12
parenrightbigg
planckover2pi1?? (q) (3.24)
where E0 is the static equilibrium energy and the second term is the vibrational energy
which has been shown earlier in equation 3.22. Thus the partition function can be written
as:
Z =
?summationdisplay
n?(q)=0
e
??
parenleftBigg
E0+summationtext
q?
(n?(q)+12)planckover2pi1??(q)
parenrightBigg
= e??E0
productdisplay
q?
?
?
?summationdisplay
n?(q)=0
e?(n?(q)+12)planckover2pi1??(q)
?
?
= e??E0
productdisplay
q?
e12?planckover2pi1??(q)
e?planckover2pi1??(q) ? 1 (3.25)
41
From the partition function, we can further derive all the thermodynamic potentials
and their derivatives which are measurable thermal properties.
E = ?? lnZ?? = E0 +
summationdisplay
q?
1
2planckover2pi1?? (q) +
summationdisplay
q?
planckover2pi1?? (q)
e?planckover2pi1??(q) ? 1 (3.26a)
F = ?kBT lnZ = E0 +
summationdisplay
q?
bracketleftbigg1
2planckover2pi1?? (q) + kBT ln
parenleftbigg
1 ? e
?planckover2pi1??(q)
kBT
parenrightbiggbracketrightbigg
(3.26b)
S = ?
parenleftbigg?F
?T
parenrightbigg
V
(3.26c)
In equation 3.26, summation over all the q points and phonon branches is needed to
evaluate each thermal quantity. Although the q points in principle take discrete values, the
typical size of a real crystal consists of a great number of primitive unit cells, which will
make the q points quasi-continues. Using the concept of phonon density of state (DOS),
the Helmholtz free energy can be calculated from the Brillouin zone integration
F (T,V ) = E0 (V ) +
integraldisplay ?
0
bracketleftbigg1
2planckover2pi1?? (q) + kBT ln
parenleftbigg
1 ? e
?planckover2pi1??(q)
kBT
parenrightbiggbracketrightbigg
g (?)d? (3.27)
where g (?) is the phonon DOS. There is an upper limit of the phonon frequencies, above
which the phonon DOS is zero. As a good approximation, we can also evaluate the vibra-
tional free energy as a summation and being normalized to a primitive unit cell if the q
point sampling is dense enough.
Fvib = 1N
q
summationdisplay
q?
bracketleftbigg1
2planckover2pi1?? (q) + kBT ln
parenleftbigg
1 ? e
?planckover2pi1??(q)
kBT
parenrightbiggbracketrightbigg
(3.28)
3.2.3 First-Principles Phonon Calculations
In ab initio calculations, there are basically two approaches to obtain the phonon
spectrum, namely the linear response theory 152?154 (density-functional perturbation theory,
DFPT) and the direct method. The DFPT approach assumes that the interatomic force-
constant matrix is determined by the linear response of electron density to the periodic
42
lattice perturbation. The dynamical matrix may be obtained from the inverse dielectric
matrix. The major advantage of DFPT method is one can calculate phonon frequencies
at arbitrary q point without using supercell. It also directly provides the Born effective
charge (BEC) and dielectric constant which are necessary in predicting the LO-TO splitting
for ionic crystals. However, The implementation of DFPT calculation requires complex
coding and the dielectric matrix is obtained from the eigenfunctions and eigenvalues of the
unperturbed system, which demands summations over unoccupied conduction bands. The
computational cost for DFPT calculation is in the order of 3N4 ? R3IFC, where N is the
number of atoms and RIFC is the range of interatomic force constants.
The direct approach falls into two classes. In frozen-phonon method155, the phonon
frequencies can be calculated from energy di?erences between displaced system and relaxed
system. This method is limited as wave vector q must be the reciprocal lattice vector of
the supercell, but too large supercell is not computationally a?ordable. Our calculations
belong to another method of the direct approach: a ?rst-principles real-space supercell force-
constant (SC-FC) method which calculates the phonon frequencies from the forces obtained
via the Hellmann-Feynman theorem. This technique is proved to be e?cient and successful
in predicting the full phonon spectrum of many materials122?129. The major limitation of
this method comes from the range of the interatomic force constants RIFC. This type of
error can be reduced by using a large supercell. The computational cost of a complete
interatomic force constants calculation is in the order of N ? R9IFC. For systems with a
relatively small RIFC, SC-FC method will be e?cient by using a suitable size of supercell
to balance the accuracy and computational workload. To predict the phonon dispersion
for ionic solids, which involves induced Born e?ective charges Z? due to lattice vibration,
a separate calculation proposed by Kunc and Martin156 need to be performed to take care
of the LO-TO splitting as a correction.
Start from here I will describe the detailed procedure of our phonon calculation. Within
this subsection, we are restricted to the harmonic approximation. At a ?xed volume, a
relaxed structure model is provided by the previous static total energy calculation, i.e., no
43
net force on each atom at this con?guration. Then a ?nite small displacement is made
to a single atom and all the atomic positions are not allowed to change. According to
Hellmann-Feynman theorem, ? component of the force on the ith atom in the ?th unit cell
is:
F?,i (?) = ? ?E?u
?,i (?)
= ?
angbracketleftbigg
?
vextendsinglevextendsingle
vextendsinglevextendsingle ?H
?u?,i (?)
vextendsinglevextendsingle
vextendsinglevextendsingle?
angbracketrightbigg
(3.29)
where H is the total hamiltonian and ? is the electronic ground-state wavefunction. In
the Born-Oppenheimer approximation, ionic displacement is the parameter. Assuming the
amount of single displacement is ?, using equation 3.16 and keeping up to the 4th order
term in the hamiltonian expansion, the Hellmann-Feynman (H-F) forces is
F?,i (?) = ???i,?jparenleftbig?,??parenrightbig??? 12A?i,?j,?kparenleftbig?,??,???parenrightbig??2? 16B?i,?j,?k,?lparenleftbig?,??,???,????parenrightbig??3 (3.30)
Note that all the other atoms are held in their equilibrium positions. The reason to
keep up to the 4th order term is to better estimate the harmonic force constant ??i,?j (?,??)
from the ?rst-principles calculated H-F forces. A simple displacement scheme is applied in
order to eliminate the anharmonic terms. If we calculate the H-F forces for the jth atom
deviates in the ? direction by ? from its equilibrium position, we also do calculations with
the displacements of ??, 2?, and ?2?, where the negative sign implies the displacement
is in the opposite direction. Provided an label to each of the four forces, we have
?
???
????
?
???
???
??
F+?,i (?) = ???i,?j (?,??) ? ? ? 12A?i,?j,?k (?,??,???) ? ?2 ? 16B?i,?j,?k,?l (?,??,???,????) ? ?3
F??,i (?) = ??i,?j (?,??) ? ? ? 12A?i,?j,?k (?,??,???) ? ?2 + 16B?i,?j,?k,?l (?,??,???,????) ? ?3
F2+?,i (?) = ?2??i,?j (?,??) ? ? ? 2A?i,?j,?k (?,??,???) ? ?2 ? 43B?i,?j,?k,?l (?,??,???,????) ? ?3
F2??,i (?) = 2??i,?j (?,??) ? ? ? 2A?i,?j,?k (?,??,???) ? ?2 + 43B?i,?j,?k,?l (?,??,???,????) ? ?3
(3.31)
It is straightforward to show that
??i,?jparenleftbig?,??parenrightbig = ?8F
+
?,i (?) + 8F
?
?,i (?) + F
2+
?,i (?) ?F
2?
?,i (?)
12? (3.32)
44
We currently use a simpli?ed version to calculate the FC matrix that we ignore the
typically small 4th order term and cancel all the odd order anharmonic terms.
??i,?jparenleftbig?,??parenrightbig? ?12
parenleftBiggF+
?,i (?)
? +
F??,i (?)
??
parenrightBigg
(3.33)
In this way we can further reduce the computational cost by half. If each atom needs
to be displaced in one direction (including positive and negative directions) by 4 times,
i.e., 4 total energy calculations, a supercell with N atoms requires 12N calculations. The
typical size of the supercell in phonon calculation is in the order of ?100 atoms. However,
the number of direct calculations can be greatly reduced based on the crystal symmetry.
First we go through the full point group operations and possible gliding vectors to
?nd all the spacegroup symmetry operations for a fully relaxed crystal structure (code
FindSG.x). Then an one-on-one mapping is established to ?nd the independent and depen-
dent atoms (code Find1on1.x). The mapping and operations relate them are saved in the
?le named as 1on1map.dat. The next code is called FindIR.x which reads in 1on1map.dat
and ?nd the irreducible directions of independent atoms. We call it ?moves? to specify
which atom to displace, in which direction to displace, and by what amount. It should
be mentioned that the direction of displacement can be in terms of either the cartesian
coordinates or the primitive unit vectors which is called ?direct? format. We also save
the information to obtain the H-F forces of the dependent moves from the results of the
independent ones.
In our calculations an absolute amount of ? = 0.015 ?A is usually chosen for cartesian
moves and a relative amount of ? = 0.002 for direct moves. The amount of displacement
should be applied with care. Too large deviation will include more anharmonic e?ect while
too small deviation may worsen the results due to numerical errors. For the irreducible
moves, the H-F forces are calculated directly and the corresponding FC matrix elements
are derived from these forces. For other reducible moves, the FC matrix element can be
obtained by applying the one-on-one spacegroup symmetry operation to the irreducible
45
ones. Both cartesian and direct transformation matrices are saved so we can handle either
case. With the fully reconstructed FC matrix, we can get the dynamical matrix based on
equation 3.21. Note that in equation 3.21, the subscript of the phase factor is ?iq?h, where
h is the di?erence of real-space lattice vectors. Another implementation is by using the
phase factor of e?iq?(xj(h)?xi(0)) where xi (0) and xj (h) are the positions of the ith atom
in the reference primitive unit cell and the jth atom in the primitive unit cell with a lattice
vector h relative to the reference unit cell, respectively. These two variants are equivalent
with a di?erence in the phase factor. The corresponding dynamical matrices are related by
a unitary transformation so that there is no in?uence on the phonon frequencies.
By solving the secular equation 3.20, phonon frequencies at arbitrary q point can be
calculated. To illustrate the dispersion relation, the q points are generated along some high
symmetry directions. The frequency changes in certain LO modes caused by BEC in the
lattice vibration will be further considered for ionic crystals.
Phonon density of state (DOS) is imperative for Brillouin zone integrations. We
adopted the tetrahedron method157,158 for DOS calculation. Using a uniformly generated
coarse grids in the reciprocal space, we calculate the phonon frequencies at these q points.
In the tetrahedron method, the ?rst Brillouin zone is divided into small tetrahedrons and
density of state is evaluated within each tetrahedron. The integration of DOS over the full
phonon spectrum is normalized to 3n where n is the number of atoms per primitive unit
cell. Figure 3.1 is an example of our calculated phonon dispersion and density of state,
together with experimental data as a comparison.
3.3 Thermodynamic Properties
3.3.1 Quasi-Harmonic Approximation
Many properties of real crystals, such as thermal expansion and transport properties,
can only be explained with lattice anharmonicity. If the real crystal is purely harmonic, the
phonon frequencies will be the same at any volume, which is not the case. To illustrate the
46
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s32s101s120s112s116s46
s32s116s104s101s111s114s121
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s40
s99
s109
s45
s49
s41
s65s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s90s32s32s32s32s32s32s32s32s32s32s32s32s32s32s68s32s32s32s32s32s32s32s32s32s32s32s32s32s32
s40s97s41s32s68s105s115s112s101s114s115s105s111s110s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s40s98s41s32s86s68s79s83s32
s32
s32
s97s114s98s46s32s117s110s105s116s115
s45s65s108
s50
s79
s51
s85s83s45s80s80
s80s114s101s115s115s117s114s101s58s32s48s71s80s97
Figure 3.1: LDA (with US-PP) calculated (a) phonon dispersion relation, (b) vibrational
density of states of ?-Al2O3 at zero pressure. High symmetry points are A: parenleftbig0, 12,0parenrightbig, ?:
(0,0,0) Z:parenleftbig12, 12, 12parenrightbig, and D:parenleftbig12,0, 12parenrightbig. Lines denote theoretical spectrum and discrete squares
denote experimental data.
anharmonic e?ect, a qualitative plot of the potential energy as a function of the interatomic
distance is shown in Figure 3.2.
The interatomic distance gets smaller with decreasing volume. At a volume less than
the equilibrium volume V0 one can see that the true potential curve is steeper than the
harmonic curve, which provides a higher phonon frequencies. And, at a volume larger than
V0, the true curve is ?atter than the harmonic one, which makes the phonon frequencies
relatively lower. For a volume that is in the vicinity of V0, the anharmonic e?ect is compar-
atively small. In this case the anharmonicity can be approximated with the quasi-harmonic
approximation (QHA). Within this approximation, we treat the crystal harmonically at
a ?xed volume, whereas the phonon frequencies are volume dependent. A quantity ?q,?,
known as the mode Gr?uneisen parameter, describe this dependence, i.e.,
?? (q) = ? V?
? (q)
??? (q)
?V = ?
? (ln?? (q))
? (lnV ) (3.34)
47
U(r)
r
harmonic approximation
a
Figure 3.2: Potential energy vs. interatomic distance curve. Dashed line denotes harmonic
potential curve. Minimum potential energy is provided at r = a
and the overall Gr?uneisen parameter is de?ned as:
? =
summationtext
q,?
?? (q)cv,? (q)
summationtext
q,?
cv,? (q) (3.35)
where cv,? (q) is the mode speci?c heat capacity at constant volume. The mode Gr?uneisen
parameter can be calculated via Hellmann-Feynman theorem.
?? (q) = ? V2?2
? (q)
angbracketleftbigg
e(q,?)
vextendsinglevextendsingle
vextendsinglevextendsingle?D(q)
?V
vextendsinglevextendsingle
vextendsinglevextendsinglee(q,?)
angbracketrightbigg
(3.36)
where D(q) is the dynamical matrix with wave vector q and e(q,?) is the eigenvector of
the ?th normal mode at reciprocal lattice point q. The ?rst order derivative of D (q) with
respect to the volume can be approximately evaluated using the ?nite di?erence technique.
Typically thermodynamic properties, instead of thermodynamic potentials, are mea-
surable quantities. The thermal properties of a system can be determined from certain
thermodynamic potential according to the given ensemble, e.g., for a system with temper-
ature and pressure as independent variables, the related potential is the Gibbs free energy.
48
At temperatures far below melting point, the total energy and free energy can be evaluated
within the harmonic approximation, according to equation 3.26. However, some thermal
properties, such as heat capacity at constant volume and entropy, are derived from the ?rst
order derivatives of the thermodynamic potentials with respect to temperature at constant
volume. Other thermal properties that are not de?ned at constant volume, such as thermal
expansion and isothermal bulk modulus, can be evaluated via the equation of state (EOS),
which will be discussed in the next section.
Heat capacity is the ?rst order derivative of the total energy with respect to temperature
at constant volume (neglecting the temperature dependence of phonon frequency), and
entropy can be derived in a very similar way as the ?rst order derivative of free energy.
CV =
parenleftbigg?E
?T
parenrightbigg
V
= kBN
q
summationdisplay
q?
parenleftBigplanckover2pi1?
?(q)
kBT
parenrightBig2
e
planckover2pi1??(q)
kBT
parenleftbigg
e
planckover2pi1??(q)
kBT ? 1
parenrightbigg2 (3.37)
S = ?
parenleftbigg?F
?T
parenrightbigg
V
= ?kBN
q
summationdisplay
q?
?
?ln
parenleftbigg
1 ? e
?planckover2pi1??(q)
kBT
parenrightbigg
+
planckover2pi1??(q)
kBT
1 ? e
planckover2pi1??(q)
kBT
?
? (3.38)
At a ?xed volume, these two quantities can be calculated directly from phonon frequen-
cies and the corresponding phonon density of state. For the reason that it is impractical
to perform ?rst-principles calculations at a large number of volume points in order to nu-
merically evaluate the derivatives, EOS is of special importance for interpolation with the
limited number of volume points. With the adopted EOS, we are able to obtain the rela-
tion among pressure, volume and temperature. Thus we can predict the thermal expansion
coe?cient with the ?nite di?erence method.
? = 1V
parenleftbigg?V
?T
parenrightbigg
P
(3.39)
With the adopted equation of states and previously calculated quantities, the Gibbs
free energy, isothermal bulk modulus, isobaric heat capacity, and adiabatic bulk modulus
49
can be calculated as follow:
G(T,P) = F(T,V ) + PV (3.40a)
BT = ?V
parenleftbigg?P
?V
parenrightbigg
T
(3.40b)
CP = T
parenleftbigg?S
?T
parenrightbigg
P
(3.40c)
BS = CPC
V
?BT (3.40d)
The bulk Gr?uneisen parameter is related to the thermal expansivity, volume, bulk
modulus and heat capacity:
? = ?VBTC
V
(3.41)
3.3.2 Equation of State (EOS) Models
In previous sections, we have discussed the calculation of Helmholtz free energy at
a given volume. The temperature e?ect is considered within the framework of QHA and
a further correction from the lowest order anharmonic contribution is introduced in Ap-
pendix C. To predict the thermoelastic and thermodynamic properties such as bulk mod-
ulus, thermal expansion coe?cient, Gr?uneisen parameter etc., equation of state (EOS)
is necessary. From the experimental point of view, the EOS is usually the Pressure-
Volume-Temperature (P ?V ?T) relationship, but on the theory side, the Energy-Volume-
Temperature (F ?V ?T) relationship is often convenient. The roles that EOS plays are
mainly in the following aspects
1. It extends the direct calculated free energies (high computational cost) from a few
volume points to a continuous range.
2. The interpolation with EOS from directly calculated points may reduce the random
type of numerical error.
3. It provides an analytical form of relationship which is helpful in deriving other quan-
tities.
50
4. Some physical quantities, such as equilibrium volume, equilibrium energy, bulk mod-
ulus at zero pressure and its pressure derivative, can be extracted by ?tting data to
the EOS.
Although many EOS models have been proposed in the past, we con?ne our study
to those most commonly adopted ones, namely the 2nd-order, 3rd-order Birch-Murnaghan
EOS (BM-EOS)159,160 and the Vinet EOS161,162.
Birch-Murnaghan EOS is derived from the ?nite strain theory based upon a Taylor
expansion of the free energy in terms of powers of small volumetric strains. There are
two types of strains that are di?erentiated from the de?nition of reference con?guration.
Eulerian strain takes the deformed state as the reference con?guration, and Lagrangian
strain chooses the undeformed state as the reference state. Both strains can be expressed
as a function of the volume ratio.
? = 12
bracketleftBigg
1 ?
parenleftbiggV
0
V
parenrightbigg2/3bracketrightBigg
(3.42a)
? = 12
bracketleftBiggparenleftbigg
V
V0
parenrightbigg2/3
? 1
bracketrightBigg
(3.42b)
where ? and ? are Eulerian and Lagrangian strains, respectively. Eulerian strain is employed
in the BM-EOS for the reason of better convergence and the free energy is in the form of
F =
nmsummationdisplay
n=1
An?n (3.43)
where An is the nth power coe?cient and nm determines up to which order to terminate.
With nm = 2, equation 3.43 leads to the second order BM-EOS.
F (T,V ) = F0 + 98K0V0
bracketleftBiggparenleftbigg
V0
V
parenrightbigg2/3
? 1
bracketrightBigg2
(3.44)
where F0 and K0 are the equilibrium free energy and the bulk modulus at zero pressure
condition. Note that F0, V0, and K0 are temperature dependent implicitly. The hydrostatic
51
pressure can be derived as the negative of the ?rst-order partial derivative of free energy
with respect to volume.
P (T,V ) = ?
parenleftbigg?F
?V
parenrightbigg
= 32K0
bracketleftBiggparenleftbigg
V0
V
parenrightbigg7/3
?
parenleftbiggV
0
V
parenrightbigg5/3bracketrightBigg
(3.45)
Third order BM-EOS can be obtained from equation 3.43 with nm = 3.
F (T,V ) = F0 + 98K0V0
bracketleftBiggparenleftbigg
V0
V
parenrightbigg2/3
? 1
bracketrightBigg2
?
braceleftBigg
1 + 12 parenleftbig4 ?K?0parenrightbig
bracketleftBigg
1 ?
parenleftbiggV
0
V
parenrightbigg2/3bracketrightBiggbracerightBigg
(3.46)
P (T,V ) = 32K0
bracketleftBiggparenleftbigg
V0
V
parenrightbigg7/3
?
parenleftbiggV
0
V
parenrightbigg5/3bracketrightBigg
?
braceleftBigg
1 + 34 parenleftbig4 ?K?0parenrightbig
bracketleftBigg
1 ?
parenleftbiggV
0
V
parenrightbigg2/3bracketrightBiggbracerightBigg
(3.47)
where K?0 is the pressure derivative of K0 at P = 0 parenleftbigK?0 = parenleftbigdKdPparenrightbigP=0parenrightbig. With K?0 = 4, the
third-order BM-EOS reduces to the second-order BM-EOS.
Vinet EOS is derived from an empirical interatomic potential and resulting in:
F (T,V ) = F0 + 2K0V0(K?
0 ? 1)
2 ?
braceleftBigg
2 ?
bracketleftBigg
5 + 3K?0
parenleftBiggparenleftbigg
V
V0
parenrightbigg1/3
? 1
parenrightBigg
?3
parenleftbiggV
V0
parenrightbigg1/3bracketrightBigg
? e
3
2(K
?0?1)
parenleftbiggparenleftBig
V
V0
parenrightBig1/3
?1
parenrightbiggbracerightBigg
(3.48)
P (T,V ) = 3K0
parenleftbiggV
V0
parenrightbigg?2/3bracketleftBigg
1 ?
parenleftbiggV
V0
parenrightbigg1/3bracketrightBigg
e
3
2(K
?
0?1)
parenleftbigg
1?
parenleftBig
V
V0
parenrightBig1/3parenrightbigg
(3.49)
Numerous studies have been done for comparing models of EOSs on all types of solids.
Stacey et al.163 and Anderson164 have pointed out that the convergence of the BM-EOS
from polynomial expansion terms may not be good for large strains if K?0 negationslash= 4. This is due
to the fact that the coe?cient of the 4th order term is larger than that of the 3rd order
term. In this sense, the second order BM-EOS is capable dealing with rather low pressures
52
where K?0 is close to 4 and does not vary much. At very high pressures up to several
megabars, the third order BM-EOS will surpass the second order for better accounting the
pressure dependence of K?. In the extremely high pressure case (of the order terapascals),
the EOS models derived from ?nite strains will in general fail due to large compression. The
Vinet EOS derived from an empirical potential is comparatively good in the low pressure
conditions and it ?ts better with large compressions.
Temperature is another factor that introduces strains. It is convenient to separate the
free energy into static and thermal part.
F (T,V ) = E0 (V ) + Fvib (3.50)
where E0 (V ) is the static energy and Fvib is the free energy due to lattice vibrations, which
also includes the zero-point energy. Only the second term has temperature dependence
and therefor we separate the overall ?tting (to EOS) into two parts: ?tting to static EOS
and isothermal EOS. For the static energy part, considering the nature of studies in this
dissertation, the pressures will not exceed 2 megabars and temperatures are typically well
below the melting points, we adopt the BM-EOS over the Vinet EOS because of its wide
acceptance and simplicity. In practical calculations, a set of our ?rst-principles calculated
E-V data is ?tted to the third order BM-EOS (equation 3.46) with least-square-?tting
algorithm. E0, V0, K0 and K?0 can be obtained as ?tting parameters. For a better result,
the volume points are chosen to be evenly spaced with V0 (the minimum of the E-V curve)
around the middle.
For the thermal part, we usually choose the 2nd order BM-EOS which has been shown
to be e?ective. Meanwhile we have done a case study in MgO to investigate several EOS
models for ?tting to the thermal free energies.
53
3.3.3 Case Study: MgO
MgO (periclase) is an important mineral of the Earth?s lower mantle and it has a very
high melting point that allows us to study its thermal properties in a wide temperature
range. The motivation of this study is to verify the in?uence of EOS model on the thermal
expansion coe?cient as a function of temperature, and to ?nd the model that has a better
agreement with published experimental data. Within the framework of QHA, we have
calculated the thermal Helmholtz free energies (Fvib) at 17 volume points. These volume
points are evenly spaced and cover the pressure range at which our study is concerned.
Based on equation 3.28, we evaluate the thermal free energy at a temperature step of 5 K
from 0 K to 3000 K. Figure 3.3 shows our calculated data at temperatures of 0 K and 2000
K. At the ?rst glance, two data sets almost resemble each other. However, the vertical scales
are di?erence in these two plots, thus the temperature has more e?ect in lowering the free
energy of larger volumes. Also, the numerical uncertainties in our calculation contributes
to the unsmoothness of the curves.
Besides the 2nd and 3rd order BM-EOS, based on the appearance of the curve in Figure
3.3, we proposed a model which assumes linear volume dependence of Fvib. Furthermore, we
add an additional term to our linear model to address the nonlinear e?ect for small volumes.
This additional term has been tested with two variants: one with a lnV dependence and
the other with an inverse volume dependence. Next our data at each temperature point
are ?tted to these ?ve EOS schemes using the least-square-?tting algorithm. As most of
the discrepancies of thermal expansion happen at the high temperature region, we plot our
?tting results from di?erent models at T = 2000 K and zero pressure in Figure 3.4. In the
region where directly calculated data points are available, all models ?t the data similarly
well with minor di?erences, whereas the di?erences become apparent immediately outside
data region. This indicates the necessity for providing data sets in a wide volume range.
With each thermal EOS model and the common 3rd order BM-EOS for static en-
ergy, we have calculated the thermal expansion and compared with experimental results,
as shown in Figure 3.5. Most experimental data are measured from 300 K to 3000 K. At
54
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
6 7 8 9 10 11
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
(a)
 
 
F vib
 (e
V/a
tom
)
MgO
T=0K
(b) MgO
T=2000K
 
 
F vib
 (e
V/a
tom
)
Volume (?3/atom)
Figure 3.3: LDA calculated thermal free energy at several volume points. (a) At T = 0 K.
(b) At T = 2000 K.
4 5 6 7 8 9 10 11 12 13-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
 Calculated data
 2nd-order BM EOS
 3rd-order BM EOS
 Fvib=a+bV
 Fvib=a+bV+clnV
 Fvib=a+bV+c/V
 
 
Vi
bra
tio
nal
 fre
e e
ner
gy
 (e
V/
ato
m)
Volume (?3/atom)
MgO
Figure 3.4: LDA calculated Fvib ?V data at T = 2000 K are ?tted to several EOS models.
55
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48 s50s53s48s48 s51s48s48s48
s48
s50
s52
s54
s56
s49s48
s84s104s105s115s32s119s111s114s107s58
s32s50
s110s100
s32s111s114s100s101s114s32s66s77s32s69s79s83
s32s51
s114s100
s32s111s114s100s101s114s32s66s77s32s69s79s83
s32s70
s118s105s98
s61s97s43s98s86
s32s70
s118s105s98
s61s97s43s98s86s43s99s108s110s86
s32s70
s118s105s98
s61s97s43s98s86s43s99s47s86
s32
s32
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115
s105
s111
s110
s32
s40
s49
s48
s45
s53
s32
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s77s101s97s115s117s114s101s109s101s110s116s115s58
s32s78s105s101s108s115s101s110s32s38s32s76s101s105s112s111s108s100s32s40s49s57s54s51s41
s32s83s97s120s101s110s97s32s38s32s90s104s97s110s103s32s40s49s57s57s48s41
s32s82s101s101s98s101s114s32s101s116s46s32s97s108s46s44s32s40s49s57s57s53s41
s32s68s117s98s114s111s118s105s110s115s107s121s32s38s32s83s97s120s101s110s97s32s40s49s57s57s55s41
s32s70s105s113s117s101s116s32s101s116s46s32s97s108s46s44s32s40s49s57s57s57s41
s77s103s79
s80s61s48s32s71s80s97
Figure 3.5: Thermal expansion of MgO at zero pressure. Lines represent this work from
LDA calculation. Discrete symbols are reported measurements.
low temperatures from 300 K to 600 K, calculated thermal expansions of all models have
a good agreement with most measurements except the linear model is slightly lower than
the majority. At temperatures above 600 K, the 3rd BM-EOS and two modi?ed linear
models lead to an overestimation and have the tendency to diverge. Thermal expansion
from the linear model at high temperatures does not diverge as much as those three models
but is still apparently higher than all measured values. Considering its discrepancy at low
temperatures, this model does not meet the satisfaction. Despite the di?erences among
experimental data, the 2nd order BM-EOS is signi?cantly better than the other models.
Regarding the good ?tting results in Figure 3.4, we attribute the discrepancies in thermal
expansion to the its sensitivity to minor volume changes. Fiquet et al. has pointed out
that a volume error of 1% translates into di?erences of up to 20% in thermal expansion
coe?cients21. This could explain why 3rd order BM-EOS is worse than 2nd order since it
picks up more random errors from the data sets.
For the numerical error caused overestimation mentioned above, we also tried the Debye
model to predict the thermal expansion since only one parameter, the Debye temperature,
is needed to handle the temperature e?ect which may even out the previously magni?ed
56
unsmoothness at high temperatures (Figure 3.3). We ?rst calculate the Debye temperatures
for all volume points from vibrational free energy at zero temperature.
Fvib (T = 0K) = 98kBTD (3.51)
where TD is the Debye temperature. Based on the appearance of (TD,V ) data sets, we ?tted
them to a TD = a + blnV model using least square ?tting algorithm. Thermal pressure as
a function of T and V can be expressed in term of Debye temperature.
Pth =
bracketleftBigg
27
parenleftbigg T
TD
parenrightbigg4integraldisplay TD/T
0
lnparenleftbig1 ? e?xparenrightbigx2dx? 98
?9
parenleftbigg T
TD
parenrightbigg
ln
parenleftBig
1 ? e?TD/T
parenrightBigbracketrightbigg
?kB bV (3.52)
With equation 3.52 as our thermal EOS, we predict the thermal expansion coe?cient
as a function of temperature at zero pressure in Figure 3.6. Unfortunately result from
Debye model diverges even faster at high temperatures. This, on one hand, recon?rms the
sensitivity of thermal expansion to EOSs, and on the other hand implies that an accurate
density of state can improve the result.
57
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48 s50s53s48s48
s48
s50
s52
s54
s56
s49s48
s49s50
s77s101s97s115s117s114s101s109s101s110s116s115s58
s32s78s105s101s108s115s101s110s32s38s32s76s101s105s112s111s108s100s32s40s49s57s54s51s41
s32s83s97s120s101s110s97s32s38s32s90s104s97s110s103s32s40s49s57s57s48s41
s32s82s101s101s98s101s114s32s101s116s46s32s97s108s46s44s32s40s49s57s57s53s41
s32s68s117s98s114s111s118s105s110s115s107s121s32s38s32s83s97s120s101s110s97s32s40s49s57s57s55s41
s32s70s105s113s117s101s116s32s101s116s46s32s97s108s46s44s32s40s49s57s57s57s41
s84s104s105s115s32s119s111s114s107s58
s32s68s101s98s121s101s32s109s111s100s101s108
s32
s32
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115
s105
s111
s110
s32
s40
s49
s48
s45
s53
s32
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s77s103s79
s80s32s61s32s48s71s80s97
Figure 3.6: Thermal expansion of MgO at zero pressure. Solid line denotes our calculation
with Debye model. Discrete symbols are reported measurements.
58
Chapter 4
PRESSURE-INDUCED PHASE TRANSITIONS IN ALUMINIUM OXIDE AND
ALUMINIUM NITRIDE
4.1 Aluminium Oxide: Al2O3
4.1.1 Introduction
Aluminium oxide (Al2O3) is often referred to as alumina, sapphire or aloxite in the ce-
ramic, mining and materials science communities. It is commonly used as an abrasive due
to its hardness and as a refractory material due to its high melting point. The naturally-
occurring crystalline form of Al2O3 at ambient condition is primarily corundum. Rubies
(Cr+3 doped) and sapphires are gem-quality forms of corundum. In high-pressure experi-
ments, ruby usually serves as a standard pressure gauge (ruby scale) in diamond anvil cell
(DAC)1 and sapphire is used as window material in shock wave experiments2. Al2O3 is
also one of the major constituents of the Earth?s lower mantle. High-pressure behaviors
and thermal properties of alumina are important for both experimental research and better
understanding of the interior of the earth.
At ambient conditions, The crystalline form of Al2O3 is corundum (?-Al2O3, space
group R?3c). This structure is known to exist over a wide range of pressure and tem-
perature conditions. In the past twenty years theoretical3?7 and experimental8?11 studies
showed that ?-Al2O3 transforms into the Rh2O3(II) structure (space group Pbcn) around
90 GPa, and further transforms into the post-perovskite structure (space group Cmcm)
at ?130 GPa. Several calculations predicted a transition from Rh2O3(II) structure to the
orthorhombic perovskite (PV) structure (space group Pbnm) at even higher pressures5,12.
However, it has never been observed in experiments. Recent calculations showed that Pbnm
perovskite is not thermodynamically favored with respect to the corundum, Rh2O3(II) and
59
post-perovskite phases6,7. Here we report our LDA calculated equilibrium T-P phase di-
agram with both ultra-soft pseudopotentials (US-PP) and PAW method. The results are
compared with other reported calculations3?6,6,7,13.
In the corundum?Rh2O3(II) transformation, in situ heating is found necessary. At
room temperature, X-ray di?raction experiments showed that corundum phase is stable up
to 175 GPa14,15, which implies the existence of a large kinetic barrier for this transition. On
the other hand, Lin et al. found that the high-pressure Rh2O3(II) phase can be seen as low
as 85 GPa on decompression after laser heating10, indicating the Rh2O3(II)?corundum
transition is also sluggish. No experiment has recovered the Rh2O3(II) phase to ambi-
ent conditions upon decompression. Here We proposed a transformation pathway for the
corundum?Rh2O3(II) transition and evaluate the kinetic barrier based on the proposed
pathway. We further predict the meta-stability of two phases.
In addition to X-ray di?raction studies, ?-point vibrational excitations are commonly
used to identify the phase transformations. Their pressure dependencies also provide infor-
mation on the structural stability under compression. Raman165?167 and IR168?170 spectra
of the ?-Al2O3 were subjected to several experimental studies. The dependence of Raman
frequencies on uniaxial stress has also been measured171. Besides ?-point phonons, the
phonon dispersion curves along some high symmetry directions were measured using inelas-
tic neutron scattering by Schober et al 172. On the theoretical side, Heid et al. calculated the
dispersion curves and vibrational density of state by adopting LDA and norm-conserving
pseudopotentials within the frame of density functional perturbation theory173. The phonon
dispersion was also calculated from suppressLodziana et al. by means of DFT approach with GGA
and ultrasoft pesudopotentials174. Montanari et al. investigated the ?-point vibrational
modes using all-electron Gaussian-type basis set with three di?erent exchange-correlation
functionals175. Presently no measurements has been reported for the two high-pressure
phases: Rh2O3(II) and pPV. Ono et al. has computed Raman frequencies of the pPV-Al2O3
at 130 GPa using LDA and non-local Troullier-Martins pseudopotentials with partial core
corrections. We have calculated the phonon dispersion and the corresponding VDOS for the
60
four polymorphs. Phonon dispersion, vibrational density of state for ?-Al2O3 and pressure
dependence of ?-point phonon frequencies for ?, Rh2O3(II) and pPV phases will be shown.
Thermodynamic properties, such as thermal expansion coe?cient (TEC), heat capac-
ity CP, entropy and adiabatic bulk modulus have long been studied experimentally16?25
. Almost a decade ago Hama et al. calculated the thermal properties of corundum phase
by extending the formalism of Thomsen and combining the results with the Vinet model
and the Debye model for lattice vibrtations26. To date, no ?rst-principles studies have
been reported to predict the thermodynamic properties of Al2O3. And, both experimental
and theoretical data of thermal properties of the high-pressure phases are lacking. We thus
present our calculated high-pressure TEC and bulk modulus for Rh2O3 (II) and pPV phases
as a function of temperature up to 3000 K.
Before 2004, experimental elastic constants of ?-Al2O3 were reported from as early as
1950s25,27?29. And many e?orts of ab initio calculations were made to predict the elastic
constants independently30,31. However, some calculations showed con?ict in the sign of
C1432. In 2004, Gladden et al.33 reported their reexamination of the elasticity of ?-Al2O3
using resonance ultrasound spectroscopy (RUS) and con?rmed that C14 is positive rather
than negative. Gladden?s conclusion was later con?rmed by both measurement with a
di?erent technique34 and ?rst-principles calculations35. Although measurements of Cij for
the high-pressure phases of Al2O3 are still lacking, Duan et al.31 and Stackhouse36 have
calculated the elastic constants of Rh2O3(II) (from 75 GPa to 300 GPa) and post-perovskite
phases (at 136 GPa), respectively. Here we report our predicted Cij of ?-, Rh2O3(II)- and
pPV-Al2O3 with pressure dependencies, which will be compared with other available results.
Elastic properties of both corundum and Rh2O3(II) phases have been studied, but a direct
comparison between the two phases is not accessible due to di?erent crystal classes. Here we
compare the elastic properties of corundum and Rh2O3(II) phase by treating both phases
as the common-subgroup monoclinic lattices.
61
4.1.2 Crystal Structures, Total Energies and Vibrational Properties
The space group, formula units per primitive unit cell Z and Wycko? sites of the
above four polymorphs are summarized in Table 4.1. Their structures are shown in Figure
4.1. As a denser phase, high-pressure structure usually has a larger average coordination
number than the low-pressure structure. However, for these four polymorphs of Al2O3, all
the Al atoms are six coordinated and all the O atoms are four coordinated. Corundum has
R?3c rhombohedral symmetry which can also be viewed as a hexagonal lattice with Z = 6
formula units per cell. As illustrated in the ball-stick model of Figure 4.1(a), O atoms form
a lightly distorted hexagonal close-packed (hcp) sublattice with Al atoms occupying 2/3 of
the octahedral interstices. Each AlO6 octahedron has 1 face-sharing, 3 edge-sharing and 9
corner-sharing with other octahedra, as shown in Figure 4.1(a) (three ?gures on the right).
The centered octahedron is highlighted in yellow color. Viewing from c axis, each ?gure
shows the neighboring environment of the centered polyhedron with one octahedron layer (in
the a-b plane). For clarity purpose, the centered polyhedron is not shown in the last ?gure
as it is overlapped with the octahedron underneath. Rh2O3-(II) structure (Figure 4.1(b))
is closely related to the corundum structure which will be shown later. Each octahedron in
Rh2O3-(II) structure has 1 face-sharing, 2 edge-sharing and 11 corner-sharing with other
octahedra. Di?erent from corundum and Rh2O3-(II), two types of polyhedra are presented
in the Pbnm perovskite structure (Figure 4.1(c)). Each Al atom still forms six bonds with
neighboring O atoms, whereas the 4b and 4c Al atoms form AlO6 octahedra and triangular
prism, respectively. Each octahedron is surrounded by 6 corner-sharing octahedra and 4
edge-sharing, 4 corner-sharing prisms. For the triangular prisms, each is neighboring with 2
edge-sharing, 2 corner-sharing prisms and 4 edge-sharing, 6 corner-sharing octahedra. The
Cmcm post-perovskite structure also has both AlO6 octahedra (4a site Al) and triangular
prisms (4c site Al). The octahedron shares edges with 2 octahedra and 2 prisms, also shares
corners with 2 octahedra and 8 prisms. The prisms share their triangular faces with two
neighboring prisms along the a axis, and has 2 edge-sharing, 8 corner-sharing octahedra
neighbors.
62
Table 4.1: Space groups, formula units per primitive unit cell Z and Wycko? sites of four
polymorphs of Al2O3: Corundum, Rh2O3-(II), Pbnm perovskite and post-perovskite.
Phase Space group Z Species Wycko? site
? R?3c 2 O 6e
Al 4c
Rh2O3-(II) Pbcn 4 O 4c, 8d
Al 8d
Perovskite Pbnm 4 O 4c, 8d
Al 4b, 4c
Post-perovskite Cmcm 2 O 4c, 8f
Al 4a, 4c
Figure 4.1: Crystal structures of four polymorphs of Al2O3: (a) Corundum, (b) Rh2O3-(II),
(c) Pbnm perovskite and (d) post-perovskite.
63
Total energy calculations were performed for ?-, Rh2O3(II)-, PV- and pPV-Al2O3 by
imposing their symmetries. We adopted density functional theory (DFT) with a plane wave
basis set and ultrasoft pseudopotentials (US-PP) (or PAW method), which is implemented
in the VASP code139. The exchange and correlation functional is treated with local density
approximation (LDA). The plane wave basis functions with energies up to 395.7 eV (or 400.0
eV) were used for US-PP (or PAW method). Total energy change of 10?9 eV was chosen
as the convergence criterion for the self-consistent iterations. The k-point sampling for
Brillouin zone integration in our total energy calculations was carried out by the Monkhorst-
Pack method with grids of 6 ? 6 ? 6, 6 ? 8 ? 8, 6 ? 6 ? 4 and 6 ? 6 ? 4 for ?-, Rh2O3(II)-,
PV- and pPV-Al2O3, respectively.
The calculated static energy at various volumes are ?tted to the 3rd-order Birch-
Murnaghan EOS. Fitting parameters, E0, V0, K and K?, are listed in Table 4.2, together
with reported experimental and theoretical results. As the measurements are usually made
at room temperature, our predicted parameters at 300K from quasi-harmonic approxima-
tion are also listed. The present work is consistent with other calculations and experiments,
and PAW method provides a slightly better agreement on bulk modulus than US-PP.
To calculate the phonon frequencies, we have constructed 120-atom, 160-atom, 80-
atom and 120-atom supercells for ?-, Rh2O3(II)-, PV- and pPV-Al2O3 respectively and
adopted the real-space supercell force constant method. The size of the supercell is large
enough to approximately neglect the interaction between one atom in the cell with its
image outside the cell. Figure 4.2 shows the phonon spectra and VDOS of ?-Al2O3 at
zero pressure. Experimental data is also presented for comparison. As an ionic crystal,
lattice vibrations of optic modes will induce dipole-dipole interactions which in turn a?ect
the phonon frequencies around the ? point (k = 0). The interaction causes the LO-TO
splitting. However, this e?ect has not been taken into account in the VASP code. We thus
manually calculate the LO-TO splitting from a correction to the dynamical matrix. The
overall agreements for both US-PP and PAW are good and within the typical accuracy of ab
initio calculations, but dispersion using PAW method ?ts the measured data slightly better.
64
Table 4.2: Third-order BM-EOS parameters for Al2O3 polymorphs
Source V0 (?A3/atom) B (GPa) B?
Corundum
Static calculation
LDA+US-PP (this work) 8.430 249.1 3.540
LDA+PAW (this work) 8.357 259.9 3.967
Calculation5 8.441 258.9 4.01
Calculation7 8.10 248 4.13
Calculation6 8.486 252.6 4.237
300 K
LDA+US-PP (this work) 8.519 241.3 3.540
LDA+PAW (this work) 8.454 250.4 3.999
Calculation13 8.498 251.0 4.04
Experiment176 8.484 254.4 4.275
Rh2O3(II)
Static calculation
LDA+US-PP (this work) 8.275 246.5 3.736
LDA+PAW (this work) 8.173 259.0 3.967
Calculation5 8.254 261.8 3.93
Calculation7 7.93 252 4.07
Calculation6 8.284 258.2 4.140
300 K
LDA+US-PP (this work) 8.353 237.5 3.735
LDA+PAW (this work) 8.272 250.3 3.967
Perovskite
Static calculation
LDA+US-PP (this work) 8.292 217.2 4.030
LDA+PAW (this work) 8.206 236.3 3.963
Calculation5 8.322 235.0 3.98
Calculation7 7.986 223 4.22
Calculation6 8.324 229.2 4.286
300 K
LDA+US-PP (this work) 8.404 207.8 4.031
LDA+PAW (this work) 8.304 229.8 3.963
Post-Perovskite
Static calculation
LDA+US-PP (this work) 8.056 225.3 4.189
LDA+PAW (this work) 7.975 239.6 4.185
Calculation7 7.756 231 4.38
Calculation6 8.058 241.6 4.464
Calculation13 7.985 251.6 4.11
300 K
LDA+US-PP (this work) 8.171 214.6 4.190
LDA+PAW (this work) 8.086 230.6 4.185
Experiment11 7.92 249 4 (?xed)
65
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s69s120s112s101s114s105s109s101s110s116
s32s83s99s104s111s98s101s114s32s101s116s32s97s108
s84s104s101s111s114s121
s32s116s104s105s115s32s119s111s114s107
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s40
s99
s109
s45
s49
s41
s65s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s90s32s32s32s32s32s32s32s32s32s32s32s32s32s32s68s32s32s32s32s32s32s32s32s32s32s32s32s32s32
s40s97s41s32s32s32s32s32s32s32s32s32s32s32s32s32s68s105s115s112s101s114s115s105s111s110s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s86s68s79s83s32
s32
s32
s97s114s98s46s32s117s110s105s116s115
s45s65s108
s50
s79
s51
s85s83s45s80s80
s80s32s61s32s48s32s71s80s97
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s69s120s112s101s114s105s109s101s110s116
s32s83s99s104s111s98s101s114s32s101s116s32s97s108
s84s104s101s111s114s121
s32s116s104s105s115s32s119s111s114s107
s32
s32
s32
s65s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s90 s68s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s97s114s98s46s32s117s110s105s116s115
s40s98s41s32s32s32s32s32s32s32s32s32s32s32s32s32s32s68s105s115s112s101s114s115s105s111s110s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s86s68s79s83s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s40
s99
s109
s45
s49
s41
s32
s32
s32
s45s65s108
s50
s79
s51
s80s65s87
s80s32s61s32s48s71s80s97
Figure 4.2: LDA calculated phonon dispersion curves and vibrational density of state of ?-
Al2O3 at zero pressure using (a) US-PP and (b) PAW. Discrete squares denote experimental
data172.
Among the calculated 30 phonon branches, the low-frequency ones have better agreement
and the largest di?erence is given by the top branch which is 7% smaller for US-PP and 5%
smaller for PAW. Heid et al.173 pointed out two pseudogaps near 43 and 87 meV as well
as a strong peak at about 92 meV in their calculated VDOS. In the VDOS we calculated,
two pseudogaps are found to be 42, 86 meV for US-PP and 43, 87 meV for PAW, while
the strong peak is located at 91 meV and 93 meV for US-PP and PAW, respectively. The
agreement is excellent. Our calculation shows no soft phonon modes in ?-Al2O3 at least up
to ?185 GPa. And for Rh2O3(II), there is no soft phonon up to 212 GPa. However, the PV
phase is dynamically unstable (soft phonon) below ?135 GPa and the pPV phase shows
the tendency of phonon softening at low pressure. The phonon instability of pPV-Al2O3
can be related to the fact that it is not quenchable to low pressures11.
?-point phonon frequencies are of special interests since the measurable Raman and
infrared (IR) spectra are commonly used to identify the phase transformations. And the
pressure dependence can provide information on the structural instability. From group
theory analysis, the irreducible representation for ?-point phonons of ?-Al2O3 yields seven
66
Raman-active and six IR-active modes.
?Raman = 2A1g + 5Eg
?IR = 2A2u + 4Eu
Figure 4.3 shows our calculated Raman frequencies of ?-Al2O3 as a function of pressure
up to 40 GPa, for both US-PP and PAW. One can see that the frequency does not increase
linearly with pressure in the range from 0 to 40 GPa. And the pressure dependencies
reported from experiments are up to 1 GPa in Watson et al.166 and Shin et al.?s171 works,
and up to 20 GPa in Xu et al.?s167 work. In order to compare with the measurements, we
perform linear ?tting only to data sets which are below 20 GPa and expand the ?tting lines
to the full pressure range. It is clear that the high-pressure Raman modes exhibit decreasing
slopes against pressure. However, the PAW result is more ?linear? than the US-PP one.
The experimental and our calculated zero-pressure Raman frequencies ((?i)0) and pressure
dependencies (??i/?P) of ?-Al2O3 are listed in Table 4.3. For (?i)0, the PAW result
reproduces the experimental data but the US-PP data has an constant underestimation of
3-4%. For ??i/?P, PAW result is also in better agreement except two Eg modes.
s48 s49s48 s50s48 s51s48 s52s48
s51s53s48
s52s53s48
s53s53s48
s54s53s48
s55s53s48
s56s53s48
s32s69
s103
s32s65
s49s103
s32
s32
s82
s97
s109
s97
s110
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s45s65s108
s50
s79
s51
s85s83s45s80s80
s40s97s41
s48 s49s48 s50s48 s51s48 s52s48
s51s53s48
s52s53s48
s53s53s48
s54s53s48
s55s53s48
s56s53s48
s32s69
s103
s32s65
s49s103
s32
s32
s45s65s108
s50
s79
s51
s80s65s87
s40s98s41
s82
s97
s109
s97
s110
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
Figure 4.3: Calculated Raman-active frequencies of ?-Al2O3 as a function of pressure. (a)
US-PP, (b) PAW. Solid and dashed lines represent Eg and A1g modes from linear ?tting
with data sets below 20 GPa.
67
Table 4.3: The theoretical data of Raman frequencies at zero pressure and their pressure
dependencies, and comparison with other reported experimental results on corundum
Modes (cm?1 for (?i)0 and cm?1/GPa for ??i/?P)
Source A1g A1g Eg Eg Eg Eg Eg
(?i)0
This work (US-PP) 403.4 625.9 366.0 421.1 433.3 555.1 733.2
This work (PAW) 417.8 643.1 373.4 434.4 445.8 575.9 751.5
Experiment165 418 645 378 432 451 578 751
Experiment166 417.4 644.6 378.7 430.2 448.7 576.7 750.0
??i/?P
This work (US-PP) 1.47 3.44 1.29 1.81 2.18 2.31 4.06
This work (PAW) 1.52 3.64 0.85 1.85 2.24 2.68 4.44
Experiment171 1.7?0.1 5.0?0.4 2.3?0.2 1.8?0.1 1.0?0.2 2.7?0.3 2.5?0.3
Experiment166 2.11?0.06 ? 1.37?0.06 2.95?0.08 1.66?0.1 2.77?0.12 4.8?0.2
Experiment167 1.703 3.481 1.335 2.794 ? 2.760 4.218
The calculated IR-active modes of ?-Al2O3 as a function of pressure are shown in
Figure 4.4. The TO and LO modes are plotted separately since they split at ?-point. In
general, (?i)0 and ??i/?P calculated using US-PP are smaller than the PAW data, which
is consistent with the Raman results. And ??i/?P also decreases with pressure for all
the IR modes. No measurements have been done to obtain the pressure dependence of IR
frequencies. Table 4.4 shows the IR frequencies at zero pressure from experiments and the
present calculation. Again the PAW data shows better agreement than US-PP.
At present, no experimental data is available for the high-pressure phases of Al2O3.
For Rh2O3(II)-Al2O3 there are thirty Raman-active modes (7Ag + 8B1g + 7B2g + 8B3g) and
twenty IR-active modes (7B1u + 6B2u + 7B3u) out of the sixty ?-point vibrational modes.
Our predicted Raman frequencies of Rh2O3(II) phase as a function of pressure up to 160
GPa is presented in Figure 4.5. One can see that ??i/?P continuously decreases with
respect to pressure for every mode. For pPV?Al2O3, the ?-point irreducible representation
shows that there are twelve Raman modes (4Ag +3B1g +B2g +4B3g) and thirteen IR modes
(5B1u + 5B2u + 3B3u). The calculated Raman frequencies of pPV phase are plotted from
70 GPa to 270 GPa, together with Ono et al.?s11 theoretical data at 130 GPa. For each
mode, Raman frequency increases with increasing pressure but the slope becomes smaller.
Our PAW predicted frequencies are larger than the US-PP result, and agree with Ono et
68
s51s53s48
s52s53s48
s53s53s48
s54s53s48
s55s53s48
s51s53s48
s52s53s48
s53s53s48
s54s53s48
s55s53s48
s48 s49s48 s50s48 s51s48
s51s53s48
s52s53s48
s53s53s48
s54s53s48
s55s53s48
s56s53s48
s57s53s48
s48 s49s48 s50s48 s51s48 s52s48
s51s53s48
s52s53s48
s53s53s48
s54s53s48
s55s53s48
s56s53s48
s57s53s48
s32
s73
s82
s32
s102
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s32s32s32s84s79s32s109s111s100s101s115
s32s69
s117
s32s65
s50s117
s32
s32
s40s97s41s32 s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s32s32s32s84s79s32s109s111s100s101s115
s32s69
s117
s32s65
s50s117
s40s98s41s32 s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s32
s32s32s32s76s79s32s77s111s100s101s115
s32s69
s117
s32s65
s50s117
s40s99s41s32 s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s32
s32
s32s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s32
s32s32s32s76s79s32s109s111s100s101s115
s32s69
s117
s32s65
s50s117
s40s100s41s32 s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s32
Figure 4.4: Calculated IR-active frequencies of ?-Al2O3 as a function of pressure. (a) TO
modes with US-PP, (b) TO modes with PAW, (c) LO modes with US-PP, (d) LO modes
with PAW. Solid and dashed curves denote Eu and A2u modes from 2nd order polynomial
?tting.
Table 4.4: IR-active frequencies of corundum at zero pressure
(?i)0 (cm?1)
Source A2u A2u Eu Eu Eu Eu
TO modes
This work (US-PP) 384 564 378 425 557 613
This work (PAW) 395 593 389 440 579 633
Experiment168 400 583 385 442 569 635
Experiment169 399.5 584 384.6 439.3 569.5 635
Experiment170 397.5 582.4 385.0 439.1 569.0 633.6
LO modes
This work (US-PP) 511 830 380 475 609 867
This work (PAW) 526 848 393 486 626 884
Experiment168 512 871 388 480 625 900
Experiment169 514 886.5 387.7 482 630.5 908
Experiment170 510.9 881.1 387.6 481.7 629.5 906.6
69
al.?s result very well. As better agreements are consistently obtained from calculation using
PAW method by comparing with other calculations and experiments, we conclude that LDA
calculation with PAW method is more suitable than using US-PP for predicting properties
of Al2O3.
4.1.3 T-P Phase Diagram
Static enthalpies of four polymorphs (Figure 4.7) show the sequence of stable phases at
increasing pressure, and the perovskite phase is unfavored at all pressures. The corundum-
to-Rh2O3(II) transition pressure is predicted to be 93.8 GPa from US-PP and 84.0 GPa
from PAW. Both of them are consistent with results from experiments8?10 (79 ? 100 GPa)
and other calculations4?6 (78 ?105 GPa). Our calculated Rh2O3(II)?pPV transition pres-
sure is 147.8 GPa (136.8 GPa) from US-PP (PAW), which is in accordance with reported
calulations6,7,13 (131 ? 156 GPa) and Ono et al.?s observation (130 GPa)11.
Within the quasi-harmonic approximation, the T-P phase diagram is calculated and
shown in Figure 4.8. The corundum?Rh2O3(II) transition pressure by US-PP (PAW) is
94.2 (82.9) GPa at 0 K, 94.5 (83.0) GPa at 300 K, 95.2 (82.2) GPa at 1000 K and 95.5
(79.6) GPa at 2000 K, which means the equilibrium Pt is not a?ected by the temperature.
And for transformation from Rh2O3(II) to post-perovskite phase, the calculated pressure
by US-PP (PAW) is 146.9 (134.4) GPa at 0 K, 146.5 (133.7) GPa at 300 K, 143.4 (128.5)
GPa at 1000 K and 136.3 (117.5) GPa at 2000 K. The Rh2O3(II)?pPV phase boundary
has negative Clapeyron slope, i.e., Pt decreases with increasing temperature. The phase
diagram calculated by Oganov et al.6 and Umemoto et al.13 (LDA) show Pt of 79, 90
GPa at 0 K and 73, 88 GPa at 2000 K for the ?rst transition, and 128, 133 GPa at 0
K and 114, 120 GPa at 2000 K for the second transition. These are consistent with this
study. The stability zone of perovskite phase (bounded with dashed lines) is calculated
based on the volume sets without soft phonon (above 135 GPa) and extrapolated to the
low pressure region. However the PV phase only has the lowest Gibbs free energy below
70
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s48 s53s48 s49s48s48 s49s53s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s48 s53s48 s49s48s48 s49s53s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s32
s32
s32
s40s97s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s65
s103
s32s77s111s100s101s115
s32
s40s98s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s65
s103
s32s77s111s100s101s115
s40s99s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s32
s82
s97
s109
s97
s110s32s70
s114
s101
s113s117s101
s110s99
s121s32s40
s99
s109
s45
s49
s41
s32
s32
s66
s49s103
s32s77s111s100s101s115
s40s100s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s32
s66
s49s103
s32s77s111s100s101s115
s40s101s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s32
s32
s66
s50s103
s32s77s111s100s101s115
s40s102s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s32
s32
s66
s50s103
s32s77s111s100s101s115
s40s103s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s32
s32
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s66
s51s103
s32s77s111s100s101s115
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s40s104s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s32
s66
s51s103
s32s77s111s100s101s115
Figure 4.5: Calculated Raman-active frequencies of Rh2O3(II)-Al2O3 as a function of pres-
sure. (a), (c), (e), (g) US-PP, (b), (d), (f), (h) PAW. Solid curves are from 2nd order
polynomial ?tting.
71
s55s48 s49s50s48 s49s55s48 s50s50s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s55s48 s49s50s48 s49s55s48 s50s50s48 s50s55s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s40s97s41
s32s79s110s111s32s101s116s32s97s108s32s40s50s48s48s54s41
s84s104s105s115s32s119s111s114s107
s32s65
s103
s32s66
s49s103
s32s66
s50s103
s32s66
s51s103
s32
s32
s82
s97
s109
s97
s110
s32
s102
s114
s101
s113
s117
s101
s110
s99
s121
s40
s99
s109
s45
s49
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s112s80s86s45s65s108
s50
s79
s51
s85s83s45s80s80
s40s98s41
s112s80s86s45s65s108
s50
s79
s51
s80s65s87
s32s79s110s111s32s101s116s32s97s108s32s40s50s48s48s54s41
s84s104s105s115s32s119s111s114s107
s32s65
s103
s32s66
s49s103
s32s66
s50s103
s32s66
s51s103
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
Figure 4.6: Calculated Raman-active frequencies of pPV-Al2O3 as a function of pressure.
(a) US-PP, (b) PAW. Solid, dashed, dotted and dash-dotted curves represent Ag, B1g, B2g
and B3g modes from 2nd order polynomial ?tting.
135 GPa. Without taking the high temperature anharmonic e?ect into account, we think
it is unlikely for the PV phase to be stable at the predicted conditions.
4.1.4 Transition Pathways in the ?-to-Rh2O3(II) Transition
In order to investigate the transformation pathway, we found that the rhombohedral
corundum and the orthorhombic Rh2O3(II) structure are related by the common monoclinic
lattice, which has 20 atoms per primitive unit cell. If we reinterpret the corundum structure
(10-atom primitive unit cell) as this monoclinic lattice, it is a slightly distorted orthorhombic
lattice, which is structurally close to the Rh2O3(II) structure. The 20-atom monoclinic cell
can be derived from the rhombohedral setting in the following way. In terms of the 30-
atom hexagonal lattice vectors, the unit vectors of the primitive rhombohedral cell can be
expressed as: ?
??
??
?
ar
br
cr
?
??
??
?
=
?
??
??
?
1
2ah
?3
6 ah
1
3ch
?12ah
?3
6 ah
1
3ch
0 ?
?3
3 ah
1
3ch
?
??
??
?
(4.1)
72
s45s48s46s50
s45s48s46s49
s48s46s48
s48s46s49
s48s46s50
s48s46s51
s48 s53s48 s49s48s48 s49s53s48 s50s48s48
s45s48s46s51
s45s48s46s50
s45s48s46s49
s48s46s48
s48s46s49
s48s46s50
s48s46s51
s32
s32
s32
s32s67s111s114s117s110s100s117s109
s32s82s104
s50
s79
s51
s40s73s73s41
s32s80s111s115s116s45s80s101s114s111s118s115s107s105s116s101
s32s80s101s114s111s118s115s107s105s116s101
s40s97s41s32s85s83s45s80s80
s72
s32
s40
s101
s86
s47
s97
s116
s111
s109
s41
s57s51s46s56s32s71s80s97
s49s52s55s46s56s32s71s80s97
s32
s32
s32s67s111s114s117s110s100s117s109
s32s82s104
s50
s79
s51
s40s73s73s41
s32s80s111s115s116s45s80s101s114s111s118s115s107s105s116s101
s32s80s101s114s111s118s115s107s105s116s101
s40s98s41s32s80s65s87
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s56s52s46s48s32s71s80s97
s49s51s54s46s56s32s71s80s97
Figure 4.7: Static enthalpies of Al2O3 polymorphs as a function of pressure relative to
that of corundum phase using (a) US-PP and (b) PAW. The static ? ?Rh2O3(II) and
Rh2O3(II)?pPV transition pressure are present. The PV phase is metastable at all pres-
sures considered among the four polymorphs.
73
s48
s53s48s48
s49s48s48s48
s49s53s48s48
s50s48s48s48
s50s53s48s48
s51s48s48s48
s48
s53s48s48
s49s48s48s48
s49s53s48s48
s50s48s48s48
s50s53s48s48
s48 s53s48 s49s48s48 s49s53s48 s50s48s48
s32
s84
s101
s109
s112
s101
s114
s97
s116
s117
s114
s101
s32
s40
s75
s41
s67s111s114s117s110s100s117s109
s82
s104
s50
s79
s51
s40
s73
s73
s41
s112
s111
s115
s116
s45
s112
s101
s114
s111
s118
s115
s107
s105
s116
s101
s80s86
s65s108
s50
s79
s51
s85s83s45s80s80
s40s97s41
s40s98s41
s32
s32
s65s108
s50
s79
s51
s80s65s87
s84
s101
s109
s112
s101
s114
s97
s116
s117
s114
s101
s32
s40
s75
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s67s111s114s117s110s100s117s109
s82
s104
s50
s79
s51
s40
s73
s73
s41
s112
s111
s115
s116
s45
s112
s101
s114
s111
s118
s115
s107
s105
s116
s101
s80s86
Figure 4.8: Calculated T-P phase diagram of Al2O3. (a) US-PP, (b) PAW. Dashed lines
denote possible phase boundaries.
where ar, br and cr are three unit vectors of the rhombohedral primitive cell, ah and ch
are magnitudes of lattice vectors of the 30-atom hexagonal cell. The monoclinic cell can be
obtained by applying the following transforation matrix
?
??
??
?
am
bm
cm
?
??
??
?
=
?
??
??
?
1 1 ?1
1 ?1 0
0 0 ?1
?
??
??
?
?
??
??
?
ar
br
cr
?
??
??
?
(4.2)
And the monoclinic lattice vectors can be written as
?
??
??
?
am
bm
cm
?
??
??
?
=
?
??
??
?
am 0 0
0 bm 0
cm ? cos? 0 cm ? sin?
?
??
??
?
(4.3)
74
Figure 4.9: Comparison of structures of (a) corundum and (b) Rh2O3(II)-Al2O3. The
corundum phase is viewed from an angle in which it appears ?orthorhombic?-like. The
di?erences between two polymorphs are highlighted.
here am =
radicalBig
4
3a
2
h +
1
9c
2
h, bm = ah, cm =
radicalBig
1
3a
2
h +
1
9c
2
h and ? = cos
?1bracketleftbig(2
3a
2
h ?
1
9c
2
h)/(am ?cm)
bracketrightbig.
The monoclinic unit cell has 20 atoms which is the same as the Rh2O3(II) phase. After this
transformation, the two structures (Figure 4.9) are closely related and one-to-one correspon-
dence of each atom can be found. The lattice parameters of the ?monoclinic? corundum
structure, taking the volume at transition pressure (84.0 GPa) as an example, are am =
6.481 ?A, bm = 4.426 ?A, cm = 4.732 ?A and ? = 95.26?. For the orthorhombic Rh2O3(II)
phase at this pressure, ao = 6.475 ?A, bo = 4.438 ?A, co = 4.599 ?A and ? = 90?. The in-
termediate structures link the two end phases belong to monoclinic symmetry with space
group P2/c, which is a common subgroup of R?3c (corundum phase) and Pbcn (Rh2O3(II)
phase). The O atoms occupy 2e, 2f, I4g and II4g Wycko? sites and Al atoms occupy I4g
and II4g sites. During the phase transformation from corundum to Rh2O3(II) structure,
besides the evolution of the external parameters (a, b, c and ?), the major changes on the
internal coordinates are the z coordinates of I4g site O and I4g site Al, which are bonded
to each other. Changes on other internal coordinates are relatively much smaller.
75
To transform from corundum to Rh2O3(II) structure there are eighteen degrees of
freedom for the intermediate phase, i.e., a, b, c, ? and fourteen free parameters from inter-
nal coordinates. The complete information of the corundum-to-Rh2O3(II) transformation
path can be revealed from a 18-dimensional potential-energy surface (PES). However, ?rst-
principles calculation of the 18D PES is not an easy task. Considering the similarity between
corundum and Rh2O3(II) structure, as can be seen from Figure 4.9, the phase transforma-
tion is mainly characterized by the displacements of the z coordinates of the I4g O and Al
atoms. Using the 20-atom unit cells, we constructed intermediate structures whose internal
and external coordinates vary linearly from corundum to Rh2O3(II) phase. Transition pa-
rameter (tp), which varies from 0 (corundum) to 1 (Rh2O3(II)), is de?ned to describe the
intermediate state, z = zi + (zf ?zi) ?tp. By ?xing the z coordinates of the I4g O atoms
only, LDA total energy calculations were performed for 11 tp points and several volumes
at each tp, with all the other internal and external parameters being fully relaxed. Figure
4.10 shows the variation of Al-O bond lengths at the corundum-to-Rh2O3(II) transition
pressure (84.0 GPa). For both corundum and Rh2O3(II) structures, all the O atoms are
four coordinated and all the Al atoms are six coordinated. Among the 12 distinct bonds,
only one Al (I4g)-O(I4g) bond breaks and reforms during the transformation while the rest
remain almost unchanged. We denote this one bond breaking and reforming mechanism
simply as OB-BAR. The OB-BAR mechanism happens for 1/3 of the O atoms (I4g) and
1/2 of the Al atoms (I4g). The largest Al-O distance of the breaking bond corresponds to
tp = 1/2.
The volume variation during the transformation is investigated (Figure 4.11) at dif-
ferent pressures. At low pressure, if the transition could occur, the volume would expand
slightly up to tp = 0.5 and monotonically decrease to the volume of Rh2O3(II) phase. As
pressure increases, the volume expansion is gradually depressed and the volume variation
takes place mainly between tp = 0.4 and 0.8. Relative to the corundum phase, the volume
variation during the transition is 0.151 ?A3/atom (2.3% reduction) at 84.0 GPa and 0.132
?A3/atom (2.2% reduction) at 160 GPa.
76
0.0 0.2 0.4 0.6 0.8 1.01.6
2.0
2.4
2.8
3.2
 
 
Bo
nd
 len
gth
 (?
)
Transition parameter
P = Ptr= 84.0 GPa
Corun
dum Rh2 O
3 (II)
Figure 4.10: Al-O bond lengths as a function of the transition parameter at 84.0 GPa. As
the transition parameter changes from 0 to 1, corundum structure transforms smoothly to
the Rh2O3(II) phase. There are 12 distinct bonds and only one bond breaks and reforms
during the transformation.
Figure 4.11: Volume per atom for the corundum?Rh2O3(II) transformation at several
pressures. Same volume ranges are used to illustrate the pressure e?ect.
77
s48s46s48 s48s46s50 s48s46s52 s48s46s54 s48s46s56 s49s46s48
s45s48s46s48s57
s45s48s46s48s54
s45s48s46s48s51
s48s46s48s48
s48s46s48s51
s48s46s48s54
s48s46s48s57
s48s46s49s50
s48s46s49s53
s32
s32
s69
s110
s116
s104
s97
s108
s112
s121
s32
s68
s105
s102
s102
s101
s114
s101
s110
s99
s101
s32
s40
s101
s86
s47
s97
s116
s111
s109
s41
s84s114s97s110s115s105s116s105s111s110s32s112s97s114s97s109s101s116s101s114
s32s80s32s61s32s48s32s71s80s97
s32s80s32s61s32s52s48s32s71s80s97
s32s80s32s61s32s80
s116s114
s61s56s52s32s71s80s97
s32s80s32s61s32s49s50s48s32s71s80s97
s32s80s32s61s32s49s54s48s32s71s80s97
s65s108
s50
s79
s51
Figure 4.12: Enthalpies for the corundum?Rh2O3(II) transformation at several pressures.
For the corundum?Rh2O3(II) transition at room temperature (no heating in experi-
ments), it is a good approximation to use enthalpy instead of Gibbs free energy to investigate
the phase transition. We calculate the enthalpy along the transformation path at several
pressures, as shown in Figure 4.12 (relative to the corundum phase). One can see that
the forward barrier height (corundum?Rh2O3(II)) is not pressure sensitive (Figure 4.13),
whereas the backward barrier height (Rh2O3(II)?corundum) decreases signi?cantly on de-
compression. The forward barrier is about 140 meV/atom at and above Pt. The ?constant?
forward barrier height is in agreement with the observation that corundum phase is stable
up to 175 GPa without heating. The backward barrier has a magnitude of 144 meV/atom
at Pt and it drops to 43 meV/atom at 0 GPa. The decreasing backward barrier height is
consistent with Lin?s ?ndings that the high-pressure Rh2O3(II) phase is not quenchable to
low pressures10. And, our estimated barrier height should be considered as an upper limit
of the real barrier. It is very likely that the small backward barrier can be overcome by the
room temperature thermal energy at low pressures.
We also calculated the full phonon spectra to investigate the dynamical stability of
corundum phase at high pressure (Figure 4.14) and Rh2O3(II) phase at low pressure (Figure
4.15). Both are dynamically stable, which means that the meta-stability of each phase is
due to kinetic reasons.
78
s48 s53s48 s49s48s48 s49s53s48
s48s46s48s52
s48s46s48s54
s48s46s48s56
s48s46s49s48
s48s46s49s50
s48s46s49s52
s48s46s49s54
s48s46s49s56
s48s46s50s48
s48s46s50s50
s32
s32
s66
s97
s114
s114
s105
s101
s114
s32
s72
s101
s105
s103
s104
s116
s32
s40
s101
s86
s47
s97
s116
s111
s109
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s32 s45s116s111s45s82s104
s50
s79
s51
s40s73s73s41
s32s82s104
s50
s79
s51
s40s73s73s41s45s116s111s45
Figure 4.13: Enthalpy barrier height for the corundum?Rh2O3(II) and
Rh2O3(II)?corundum transformations as a function of pressure.
s48
s51s48s48
s54s48s48
s57s48s48
s49s50s48s48
s49s53s48s48
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s70s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s84s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s76s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s84
s67s111s114s117s110s100s117s109
s80s32s61s32s50s48s52s32s71s80s97
Figure 4.14: Calculated phonon dispersion curves of corundum-Al2O3 at 204 GPa.
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s90s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s32s32s88s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s83s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s82s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s89
s82s104
s50
s79
s51
s40s73s73s41
s80s32s61s32s48s32s71s80s97
Figure 4.15: Calculated phonon dispersion curves of Rh2O3(II)-Al2O3 at 0 GPa.
79
4.1.5 Thermal Properties
Within the QHA, thermodynamic properties are calculated as described in Section 3.3.
Here, in the case of Al2O3, the static energy is ?tted to the 3rd-order Birch-Murnaghan
equation of state (BM-EOS) and the thermal free energy is ?tted to the 2nd-order BM-
EOS. F (T,V ) = Estatic (V ) + Fvib (T,V ). Our calculated zero pressure thermal expansion
coe?cient (TEC) of ?-Al2O3 as a function of temperature using either US-PP or PAW
method are compared with former measurements. First, no negative TEC is found at low
temperatures which is consistent with the positive pressure dependencies of Raman and IR
frequencies (Figure 4.3 and 4.4). Below 300 K, our US-PP and PAW results are coincident
and both agree well with Wachtman et al.16 and Schauer17, while slight underestimation
appears with increasing temperature. Above 300 K, the experimental data are widely spread
over the temperature range from 300 K to 2400 K. Our US-PP data is in good agreement
with Amatuni et al.?s result18 measured from 300 K to 2000 K, and with Aldebert et al.?s
result20 at temperatures above 1000 K. Our PAW calculation predict a slightly larger TEC
than that from using US-PP above room temperature. Our PAW data lies in the middle
of the experimental data and agrees with Schauer?s data below 700 K and Wachtman et
al.?s data above 1200 K. At 2500 K, our predicted TEC from US-PP is 2.8?10?5 K?1, and
3.0?10?5 K?1 from PAW method.
At elevated pressures, the TEC decreases as shown in Figure 4.17. One can see that the
TEC predicted by PAW method is larger than that from US-PP at any pressure. Presently
no measurement is reported for TEC of corundum at high pressures. Hama and Suito26 have
calculated the thermal expansivity of corundum using empirical model for several pressures
(same pressures as ours) as a function of temperature up to 2000 K. Their calculated TEC
at 2000 K are 3.0?10?5 K?1, 2.5?10?5 K?1, 2.1?10?5 K?1, 1.6?10?5 K?1 and 1.1?10?5
K?1 at 0, 10, 20, 50 and 100 GPa. The TEC we predict from US-PP (PAW) at the same
temperature are 2.7 ?10?5 K?1 (2.9?10?5 K?1), 2.3?10?5 K?1 (2.5?10?5 K?1), 2.1?10?5
K?1 (2.2?10?5 K?1), 1.5?10?5 K?1 (1.6?10?5 K?1) and 1.0?10?5 K?1 (1.1?10?5 K?1)
80
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48 s50s53s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s51s46s53
s52s46s48
s84s104s101s111s114s121
s32s84s104s105s115s32s119s111s114s107s32s40s85s83s45s80s80s41
s32s84s104s105s115s32s119s111s114s107s32s40s80s65s87s41
s69s120s112s101s114s105s109s101s110s116
s32s87s97s99s104s116s109s97s110s32s101s116s32s97s108s32s40s49s57s54s50s41
s32s83s99s104s97s117s101s114s32s40s49s57s54s53s41
s32s65s109s97s116s117s110s105s32s101s116s32s97s108s32s40s49s57s55s54s41
s32s65s108s100s101s98s101s114s116s32s38s32s84s114s97s118s101s114s115s101s32s40s49s57s56s52s41
s32s70s105s113s117s101s116s32s101s116s32s97s108s32s40s49s57s57s57s41
s32s87s104s105s116s101s32s97s110s100s32s82s111s98s101s114s116s115s32s40s49s57s56s51s41
s45s65s108
s50
s79
s51
s80s32s61s32s48s71s80s97
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115
s105
s111
s110
s32
s99
s111
s101
s102
s102
s105
s99
s105
s101
s110
s116
s32
s40
s49
s48
s45
s53
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s32
s32
Figure 4.16: Comparison of the present theoretical calculation with measured thermal ex-
pansion coe?cients of ?-Al2O3 as a function of temperature at zero pressure. Solid and
dashed lines represent our LDA calculations using US-PP and PAW method, respectively.
Discrete symbols are reported experimental data.16?21
at 0, 10, 20, 50 and 100 GPa, respectively. Our PAW result is in very good agreement with
Hama and Suito?s results.
In Figure 4.18, the calculated temperature dependence of adiabatic bulk modulus at
zero pressure are compared with experimental values. Below 200 K, BS does remains
nearly unchanged with respect to temperature, which is the case for both calculation and
measurements. Our PAW BS at 0 K is in consistence with Chung et al.24 and Te?et?s23
data and our US-PP BS at 0 K has an underestimation of about 4%. The temperature
dependence predicted by US-PP and PAW are about -0.0122 GPa/K, which is larger than
the experimental slope of -0.020 GPa/K. At 1800 K, the US-PP (PAW) calculated BS
overestimates about 1.8% (5.4%) compared with Goto et al.?s data25. As shown in Figure
4.19, our calculated isobaric hear capacity CP and entropy S reproduce the zero pressure
experimental result from Furukawa et al.22 from 0 K to 1200 K. The CP and S calculated
using US-PP and PAW yield very similar values.
For the two high-pressure polymorphs, i.e., Rh2O3(II) and pPV, the TEC are calculated
as a function of temperature at several pressures, as shown in Figure 4.20 and 4.21. US-PP
81
s48 s49s48s48s48 s50s48s48s48 s51s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s51s46s53
s48 s49s48s48s48 s50s48s48s48 s51s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s51s46s53
s49s48s48s32s71s80s97
s53s48s32s71s80s97
s50s48s32s71s80s97
s49s48s32s71s80s97
s32
s32
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115s105
s111
s110
s32
s99
s111
s101
s102
s102
s105
s99
s105
s101
s110
s116
s32
s40
s49
s48
s45
s53
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s40s97s41 s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s48s32s71s80s97
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s40s98s41s32 s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s48s32s71s80s97
s49s48s32s71s80s97
s50s48s32s71s80s97
s53s48s32s71s80s97
s49s48s48s32s71s80s97
Figure 4.17: Theoretical thermal expansion coe?cients of ?-Al2O3 as a function of temper-
ature at several pressures from using (a) US-PP, (b) PAW.
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s50s49s48
s50s50s48
s50s51s48
s50s52s48
s50s53s48
s50s54s48
s32
s32
s65
s100
s105
s97
s98
s97
s116
s105
s99
s32
s98
s117
s108
s107
s32
s109
s111
s100
s117
s108
s117
s115s32
s40
s71
s80
s97
s41
s45s65s108
s50
s79
s51
s80s32s61s32s48s71s80s97
s84s104s101s111s114s121
s32s84s104s105s115s32s119s111s114s107s32s40s85s83s45s80s80s41
s32s84s104s105s115s32s119s111s114s107s32s40s80s65s87s41
s69s120s112s116s46
s32s67s104s117s110s103s32s38s32s83s105s109s109s111s110s115s32s40s49s57s54s56s41
s32s71s111s116s111s32s101s116s32s97s108s32s40s49s57s56s57s41
s32s84s101s102s102s101s116s32s40s49s57s54s54s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
Figure 4.18: Comparison of the present LDA predicted adiabatic bulk modulus as a function
of temperature of ?-Al2O3 with Voigt-Ruess-Hill data by Te?et23, by Goto et al.25 and the
polycrystalline data by sound velocity measurements by Chung and Simmons24.
82
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s51s46s53
s48
s49
s50
s51
s52
s53
s54
s55
s56
s32
s32
s69
s110
s116
s114
s111
s112
s121
s58
s32
s83
s32
s40
s107
s66
s47
s97
s116
s111
s109
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s67
s80
s83
s32
s32 s32
s45s65s108
s50
s79
s51
s80s32s61s32s48s32s71s80s97
s73
s115s111
s98
s97
s114
s105
s99
s32
s104
s101
s97
s116
s32
s99
s97
s112
s97
s99
s105
s116
s121
s32
s40
s107
s66
s47
s97
s116
s111
s109
s41
s84s104s101s111s114s121
s32s84s104s105s115s32s119s111s114s107s32s40s85s83s45s80s80s41
s32s84s104s105s115s32s119s111s114s107s32s40s80s65s87s41
s69s120s112s116s46
s32s70s117s114s117s107s97s119s97s32s101s116s32s97s108s32s40s49s57s53s54s41
Figure 4.19: Comparison of calculated isobaric heat capacity and entropy as a function
of temperature of ?-Al2O3 with experimental data at zero pressure. Discrete open circles
denote measured data from Furukawa22.
and PAW provide comparable prediction at both low pressure and high pressure for the
Rh2O3(II) phase, whose calculated TEC are larger than those of corundum phase at any
pressure. The TEC di?erence between two phases decreases with increasing pressure. Since
the pPV phase is unstable at low pressures, the temperature dependence of TEC of pPV-
Al2O3 is plotted at pressures from 80 GPa to 200 GPa. TEC from US-PP is smaller than
that from PAW. Above 100 GPa, the TEC of corundum, Rh2O3(II) and pPV are predicted
to be comparable. Experimental data of TEC for the high-pressure phases are desired to
make further comparison.
4.1.6 High Pressure Elasticity
In this section, the athermal elastic constants of ?-, Rh2O3(II)- and pPV-Al2O3 are de-
termined from the strain-energy density relation, as described in Appendix B. The elasticity
calculation adopted same models and settings as the static energy-volume calculation, i.e.,
primitive unit cells, same Brillouin zone k-point samplings and plane-wave cut-o? energies.
Strain parameter ? = 2.5 % is used for the trigonal corundum phase while ? = 1.5% for the
orthorhombic Rh2O3(II) and pPV phases.
83
s48 s49s48s48s48 s50s48s48s48 s51s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s51s46s53
s52s46s48
s48 s49s48s48s48 s50s48s48s48 s51s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s51s46s53
s52s46s48
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115s105
s111
s110
s32
s99
s111
s101
s102
s102
s105
s99
s105
s101
s110
s116
s32
s40
s49
s48
s45
s53
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s40s97s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s85s83s45s80s80
s48s32s71s80s97
s49s48s32s71s80s97
s50s48s32s71s80s97
s53s48s32s71s80s97
s49s48s48s32s71s80s97
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s40s98s41s32s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s32s32s32s32s32s80s65s87
s48s32s71s80s97
s49s48s32s71s80s97
s50s48s32s71s80s97
s53s48s32s71s80s97
s49s48s48s32s71s80s97
Figure 4.20: LDA calculated thermal expansion coe?cients of Rh2O3(II)-Al2O3 at several
pressures as a function of temperature up to 3000 K. (a) US-PP, (b) PAW
s48 s49s48s48s48 s50s48s48s48 s51s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s48 s49s48s48s48 s50s48s48s48 s51s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s48s48s32s71s80s97
s49s54s48s32s71s80s97
s49s50s48s32s71s80s97
s40s98s41s32s80s111s115s116s45s112s114s111s118s115s107s105s116s101s32s65s108
s50
s79
s51
s32s32s32s32s32s32s80s65s87
s32
s32
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115s105
s111
s110
s32
s99
s111
s101
s102
s102
s105
s99
s105
s101
s110
s116
s32
s40
s49
s48
s45
s53
s32
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s40s97s41s32s80s111s115s116s45s112s114s111s118s115s107s105s116s101s32s65s108
s50
s79
s51
s32s32s32s32s32s32s85s83s45s80s80
s56s48s32s71s80s97
s56s48s32s71s80s97
s49s50s48s32s71s80s97
s49s54s48s32s71s80s97
s50s48s48s32s71s80s97
s32
s32
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
Figure 4.21: LDA calculated thermal expansion coe?cients of pPV-Al2O3 at several pres-
sures as a function of temperature up to 3000 K. (a) US-PP, (b) PAW
84
s48 s53s48 s49s48s48 s49s53s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s48 s53s48 s49s48s48 s49s53s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48s32s67
s49s49
s32s67
s49s50
s32s67
s49s51
s32s67
s49s52
s32s67
s51s51
s32s67
s52s52
s32
s32
s69
s108
s97
s115s116
s105
s99
s32
s99
s111
s110
s115s116
s97
s110
s116
s115s32
s40
s71
s80
s97
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s45s65s108
s50
s79
s51
s85s83s45s80s80
s40s97s41
s45s65s108
s50
s79
s51
s80s65s87
s32s67
s49s49
s32s67
s49s50
s32s67
s49s51
s32s67
s49s52
s32s67
s51s51
s32s67
s52s52
s32
s32
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s40s98s41
Figure 4.22: Pressure dependence of the elastic constants of corundum. (a) US-PP, (b)
PAW. Discrete symbols represent directly calculated data. Curves are from the 2nd-order
polynomial ?tting.
There are six independent non-zero elastic constants for corundum, i.e., C11, C33, C44,
C12, C13, C14. Cij as a function of pressure using US-PP and PAW are shown in Figure
4.22. Second order polynomial ?tting is applied since ?Cij/?P decreases with pressure. Our
calculated zero pressure elastic moduli and their pressure derivatives at 0 GPa are listed
in Table 4.5, together with previous calculations and experimental data. In 2004, Gladden
et al.33 corrected a long existing incorrect sign of C14, which should be positive. Excellent
agreement is obtained between our PAW results with the current accepted measured data,
whereas the US-PP results show underestimation of 3-5%. The pressure dependence we
predicted is in reasonable agreement with the measurement. PAW result is still better
than US-PP but ?C11/?P, ?C33/?P are noticeably smaller than the experimental values.
This could be due to the fact that measurements were taken at room temperature and at
pressures up to 1 GPa. We think ?C14/?P should be negative since the previous sign of C14
need to be reversed. C11 and C33 of corundum are comparable up to 160 GPa which indicate
the crystal has similar compressibility along c axis and in the a-b plane. C14 decreases with
increasing pressure and changes sign at about 70 GPa (US-PP) or 84 GPa (PAW).
85
Table 4.5: Elastic constants of corundum at zero pressure and their pressure dependencies
Source C11 C33 C44 C12 C13 C14
Cij (GPa)
This work (US-PP) 483 486 139 156 110 21
This work (PAW) 503 503 142 164 117 21
Calculation30 540 455 157 157 130 -48
Calculation31 502 501 157 161 125 -19
Calculation (LDA)35 497 493 154 165 130 19
Experiment28 497 498 147 164 111 -24
Experiment29 498 502 147 163 117 -23
Experiment33 498 503 147 163 116 23
?Cij/?P
This work (US-PP) 4.62 3.83 1.64 2.86 3.34 -0.40
This work (PAW) 4.82 3.96 1.90 3.20 3.50 -0.20
Calculation30 5.78 4.36 1.62 3.44 3.56 0.18
Calculation31 5.52 5.10 2.03 3.09 3.57 0.19
Experiment29 6.17 5.00 2.24 3.28 3.65 0.13
Nine independent elastic constants exist for orthorhombic crystals, which are C11, C22,
C33, C12, C13, C23, C44, C55, and C66. For Rh2O3(II) phase, with increasing pressure (Figure
4.23), the relative magnitudes of C11 and C33 (largest) remain stable but C22 becomes
smaller than C11 at high pressures. Consequently the c axis is the least compressible and
the b axis is getting softer compared with a axis. The shear moduli are similar in their
magnitudes at low pressure and diverge under compression. Again, our PAW result shows
slightly better agreement with former LDA calculation from Duan et al.31.
In Section 4.1.4, we showed that corundum structure can be viewed as a monoclinic
lattice which is closely related to the orthorhombic Rh2O3(II) phase. To directly compare
the elasticity between the corundum and Rh2O3(II) phase, we calculated the elastic con-
stants of both phases at 0, 84.0 GPa (Pt) and 160 GPa and the results are listed in Table
4.6. Common 20-atom monoclinic unit cell with unique axis b was used and thirteen inde-
pendent elastic constants C11, C22, C33, C12, C13, C23, C44, C55, C66, C15, C25, C35, C46
were calculated. At zero pressure, Cij of corundum are similar to those of Rh2O3(II) phase
except C33, which can be ascribed to the major di?erence between two structures (internal
86
s48 s53s48 s49s48s48 s49s53s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s49s52s48s48
s48 s53s48 s49s48s48 s49s53s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s49s52s48s48
s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s85s83s45s80s80
s67
s49s49
s67
s50s50
s67
s51s51
s67
s49s50
s67
s49s51
s67
s50s51
s32
s32
s69
s108
s97
s115s116
s105
s99
s32
s99
s111
s110
s115s116
s97
s110
s116
s115s32
s40
s71
s80
s97
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s67
s52s52
s67
s53s53
s67
s54s54
s82s104
s50
s79
s51
s40s73s73s41s45s65s108
s50
s79
s51
s80s65s87
s40s97s41 s40s98s41
s32
s32
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s67
s49s49
s67
s50s50
s67
s51s51
s67
s49s50
s67
s49s51
s67
s50s51
s67
s52s52
s67
s53s53
s67
s54s54
Figure 4.23: Pressure dependence of the elastic constants of Rh2O3(II)-Al2O3. (a) US-
PP, (b) PAW. Discrete symbols represent directly calculated data. Curves are from the
2nd-order polynomial ?tting.
coordinates along c axis). At high pressures, the elastic constants of corundum are still
di?er with Rh2O3(II) phase mostly by C33. Larger C33 predicted from Rh2O3(II) phase
implies its less compressibility along the c axis.
Pressure dependence of Cij of pPV phase using US-PP (PAW) are shown in Figure 4.24.
Previous GGA calculated elastic constants by Stackhouse et al.36 at 136 GPa is generally
consistent with our predictions. They agree with our US-PP data except their C23 and C66
are larger, and their predicted C11, C22, C33, and C23 are smaller when compared with our
PAW result. Magnitudes of Cij of pPV phase is not signi?cantly larger than those of ?
and Rh2O3(II) phase. One major di?erence is that a axis is the least compressible in pPV
phase. Currently no elastic properties have been measured for the high-pressure phases.
4.1.7 Conclusions
We have systematically investigated the pressure-induced phase transformations of
Al2O3 using density functional theory within the local density approximation (LDA). The
sequence of transitions under compression, i.e., corundum?Rh2O3(II)?pPV, and T-P
87
Table 4.6: Comparison of elastic moduli between corundum and Rh2O3(II) phase. The
corundum phase is treated as a monoclinic crystal with 20 atoms per unit cell.
0 GPa 84.0 GPa 160 GPa
corundum Rh2O3(II) corundum Rh2O3(II) corundum Rh2O3(II)
C11 450 424 956 976 1235 1288
C22 490 501 844 851 1073 1085
C33 455 541 852 1013 1213 1320
C12 180 170 420 369 639 541
C13 161 163 358 374 566 571
C23 125 109 412 418 680 692
C44 126 134 246 261 303 331
C55 185 170 230 218 207 204
C66 184 182 226 260 234 287
C15 -3 0 7 0 -5 0
C25 16 0 19 0 10 0
C35 9 0 21 0 24 0
C46 2 0 -14 0 -25 0
s57s48 s49s50s48 s49s53s48 s49s56s48 s50s49s48
s48
s51s48s48
s54s48s48
s57s48s48
s49s50s48s48
s49s53s48s48
s49s56s48s48
s49s50s48 s49s53s48 s49s56s48 s50s49s48 s50s52s48
s48
s51s48s48
s54s48s48
s57s48s48
s49s50s48s48
s49s53s48s48
s49s56s48s48
s32s67
s49s49
s32s67
s50s50
s32s67
s51s51
s32
s32
s69
s108
s97
s115s116
s105
s99
s32
s99
s111
s110
s115s116
s97
s110
s116
s115s32
s40
s71
s80
s97
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s32s67
s52s52
s32s67
s53s53
s32s67
s54s54
s32s67
s49s50
s32s67
s49s51
s32s67
s50s51
s40s97s41s32s112s80s86s45s65s108
s50
s79
s51
s32s32s32s32s32s32s85s83s45s80s80
s40s98s41s32s112s80s86s45s65s108
s50
s79
s51
s32s32s32s32s32s32s80s65s87
s32
s32
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s32s67
s49s49
s32s67
s50s50
s32s67
s51s51
s32s67
s49s50
s32s67
s49s51
s32s67
s50s51
s32s67
s52s52
s32s67
s53s53
s32s67
s54s54
s32s83s116s97s99s107s104s111s117s115s101s32s101s116s32s97s108 s32s83s116s97s99s107s104s111s117s115s101s32s101s116s32s97s108
Figure 4.24: Pressure dependence of the elastic constants of pPV-Al2O3. (a) US-PP, (b)
PAW. Discrete symbols (except crosses) represent directly calculated data from this work.
Crosses denote GGA calculation from Stackhouse et al.36 at 136 GPa. Curves are from the
2nd-order polynomial ?tting.
88
phase diagram we obtained are consistent with previous theoretical and experimental stud-
ies. Zero pressure ?-point phonon frequencies and their pressure dependencies are calculated
for three stable polymorphs. Comparison with measured data of corundum and theoretical
Raman frequencies of pPV phase suggest that, within LDA, predictions using PAW method
is better than US-PP. This conclusion is also con?rmed by the study of elastic properties.
We show that the rhombohedral corundum phase can also be interpreted as a slightly
distorted orthorhombic structure with monoclinic symmetry, which is closely related to the
Rh2O3(II) phase. A microscopic transformation mechanism is proposed for the corundum-
to-Rh2O3(II) transition, which is characterized by one bond breaking-and-reforming (OB-
BAR) for 1/3 of the four-coordinated O atoms (I4g site) and half of the six-coordinated Al
atoms (I4g site). Our calculated forward (C-to-R) enthalpy barrier height is 144 meV/atom
at the equilibrium transition pressure and it is not pressure sensitive. On the other hand,
the backward (R-to-C) barrier height decreases signi?cantly under decompression, which
indicates that the Rh2O3(II) phase is unlikely to be quenchable to ambient conditions.
The temperature dependencies of thermal expansion coe?cient from low to high pres-
sures have been predicted for ?-, Rh2O3(II)-, and pPV-Al2O3. For corundum at zero
pressure, our US-PP data is in good agreement with Amatuni et al.?s result measured from
300 K to 2000 K, and with Aldebert et al.?s result at temperatures above 1000 K. Our PAW
data lies in the middle of the experimental data and agrees with Schauer?s data below 700
K and Wachtman et al.?s data above 1200 K. Our calculated heat capacity CP, entropy and
adiabatic bulk modulus of corundum phase also agree well with measured results.
We have calculated the elastic constants and their pressure dependencies for corundum
and two high-pressure phases. PAW calculated Cij yield better agreement with accepted
experimental data of ?-Al2O3. We con?rmed that the sign of C14 is positive. In addition,
the elastic constants of the corundum and Rh2O3(II) phase are found comparable, except
for C33. The larger C33 of Rh2O3(II) phase means that its c axis is less compressible than
that of corundum.
89
4.2 Aluminum Nitride: AlN
4.2.1 Introduction
AlN is an important semiconductor primarily due to its wide band-gap and thermal
properties. The structural changes and properties at high pressures are well studied. Ex-
periments37?40 and calculations41?46 have revealed that a pressure-induced ?rst-order phase
transition from wurtzite (B4) to rocksalt (B1) structure happens for AlN. On the experimen-
tal side, without heating, the lowest pressure at which the rocksalt structure started to show
up is 14 GPa39 and the B4-to-B1 transition was observed to complete at 20-31.4 GPa39,40.
Xia et al. also found that the rocksalt phase is quenchable to ambient conditions39. On
the other hand, theoretical calculations consistently predicted a transition pressure that is
lower than the experimental values. Most of the recent ?rst-principles predicted static Pt is
less than 10 GPa43,46. The discrepancy implies the existence of a hysteresis for the forward
and backward transitions that caused by a large kinetic barrier in this transition.
AlN is a good example material to study the microscopic mechanisms of the ?rst-
order solid-solid phase transformations with no group-subgroup relation. In its wurtzite to
rocksalt transition, the coordination number changes from four to six. The wurtzite and
rocksalt structures belong to the P63mc and Fm?3m symmetry, respectively. There are four
atoms per primitive unit cell for wurtzite phase and two atoms for rocksalt phase. The
B4-to-B1 transition has been studied intensively for many III-V and II-VI semiconductors,
such as AlN, GaN, InN, ZnO, CdS, and CdSe, which crystallize in wurtzite (or zincblende)
structure at ambient conditions and transform into the rocksalt structure at high pressures.
Because of the interesting properties and applications of these semiconductors, numerous
studies have been conducted from both experiments and calculations that are intended to
understand the transition mechanism at the atomic level. Knowledge from these studies
can help to predict and possibly control transitions and transition related properties.
Although the reconstructive phase transitions may involve nucleation processes, di?u-
sionless collective atomic displacements can still characterize the transformation on a local
90
basis. The concept of transition pathway (TP) is then possible to describe the transfor-
mation from the starting phase to the ending phase in a continuous manner. Among the
in?nite number of ways for one structure to transform into another, the theoretical studies
are restricted only to those most possible paths, e.g., preservation of bonds, less strains, etc.
For the B4-to-B1 transition there has been, in principle, two di?erent approaches to propose
transition pathways. The ?rst approach adopted a systematic procedure with certain re-
strictions to ?nd the maximum number of pathways which belong to common subgroups of
the two end structures. Using the computer program COMSUBS 47, Stokes et al. proposed
?ve paths (Figure 4.25 and 4.26) by preserving the nearest neighbors along the transition
pathway and limiting the size of unit cell to no more than four di?erent Wycko? positions48.
He also found a common bilayer sliding mechanism along ?ve paths, which had been pointed
out by Zahn et al.49 and Sowa50. Another work adopted the systematic approach is from
Capillas et al.51, which was performed using the databases and tools provided by the Bilbao
Crystallographic Server 52. They proposed eight possible paths with di?erent orthorhombic
and monoclinic symmetries by setting the maximum k-index equal to 4, strain tolerance
Stol < 0.15 and maximum atomic displacement ?tol< 2 ?A. The intermediate structures
along all the eight paths have eight atoms per unit cell. The di?erence between these two
reports is that the nearest neighbors are not preserved along all the eight paths proposed
by Capillas et al.
The second approach is less systematic and most of the proposed models can be sum-
marized as two paths which are characterized by di?erent intermediate structures, i.e., the
?hexagonal? path and the ?tetragonal? path. These two paths have been proposed and
investigated intensively by experimental177?180 and theoretical44,46,49,50,53?55,181?183 works,
and they both belong to the Cmc21 path in the systematic approach. Cai et al. has studied
the ?hexagonal? and ?tetragonal? paths with LDA calculations, and concluded that the
?hexagonal? path is favored for AlN46. The Pna21 and P21 paths have also been described
in previous works49,50,55.
91
Figure 4.25: Phase transformation along TP1. (a) Unit cell of B4 structure, (b) Unit cell
of B1 structure, (c), (d) Transformation pattern of ?transition units? in B4 and B1 phases.
Dashed lines denote primitive unit cell. (Origin shifted onto the position of an atom for
better illustration of the unit cell.)
DFT calculations from Shimojo et al. showed that the enthalpy barrier of transforma-
tion is independent of the three paths (Cmc21, Pna21 and P21) for CdSe. Cai suggested,
without calculation, that the B4-to-B1 transition is characterized by the transformation
of the four-atom ?transition unit? (Figure 4.27), while the long-range pattern may be less
important. On the other hand, using MD simulations, Zahn et al. pointed out that the
favored paths have a tendency to avoid excess strains during the transformation.
It would be interesting to investigate the proposed TPs with another material via ?rst-
principles method. Previous studies53?55 suggested that the energetically favored TPs are
bond preserving. In this paper, we studied all ?ve bond-preserving TPs proposed by Stokes
et al. and one bond-breaking TP proposed by Capillas et al. from an energetic point of view
for the B4-to-B1 transition in AlN. The correlation of the enthalpy barrier with di?erent
TPs and strains will be discussed. We also relate the bilayer sliding mechanism to the
long-range patterns of ?transition units?, and Cai?s hypothesis is examined.
92
Figure 4.26: Phase transformation along TP2, TP3, TP4 and TP5. (a) Unit cell of B4
structure, (b) Unit cell of B1 structure, (c), (d) Transformation pattern of ?transition units?
in B4 and B1 phases. (Origin shifted onto the position of an atom for better illustration of
unit cell.)
93
Figure 4.27: Four-atom ?transition unit? from B4 to B1 structure.
4.2.2 Total Energy and Equilibrium Phase Transition
We ?rst calculated the total energies of the two end phases, i.e., wurtzite and rocksalt-
AlN. The calculation was within the frame of density functional theory (DFT) and adopted
Vanderbilt?s ultrasoft pseudopotentials (US-PP)121, as implemented in the VASP code139.
The exchange and correlation functional is treated with local density approximation (LDA).
Wave functions of valence electrons were expanded with plane-wave basis set up to the cut-
o? kinetic energy of 348 eV. Total energy change of 10?9 eV was chosen as the convergence
criterion for the self-consistent iterations. The k-point sampling for Brillouin zone integra-
tion in our total energy calculations was carried out by the Monkhorst-Pack method with
grids of 12 ? 12 ? 8 and 16 ? 16 ? 16 for wurtzite and rocksalt-AlN, respectively.
The calculated static energies at various volumes are ?tted to the 3rd-order Birch-
Murnaghan EOS159,160. General agreement of ?tting parameters, B0 and B?, and zero
pressure structural parameters is achieved between our calculation with other reported
values (Table 4.7). The zero pressure bulk modulus we predicted is slightly less than those
from other LDA calculations (<3%). This could be caused by the di?erent pseudopotentials
adopted.
94
Table 4.7: Third-order BM-EOS parameters, zero pressure structural parameters for B4
and B1-AlN and B4-to-B1 transition pressure Pt. Parameters listed for B1 phase is for the
8-atom conventional cubic unit cell.
Source B0 (GPa) B? a (?A) c/a u Pt (GPa)
Wurtzite (B4)
This work 202 3.61 3.084 1.601 0.382 9.85
Calculation184 209 5.58 3.057 1.617 0.380
Calculation43 209 3.7 3.061 1.600 0.382 9.2
Calculation185 208 3.87 3.100 1.6 0.382 21
Calculation46 ? ? 3.06 1.606 0.382 9.8
Experiment38 207.9?6.3 6.3?0.9 1.60 22.9
Experiment39 185.0?5.0 5.7?1.0 3.19 1.626 14-20
Experiment40 20-31.4
Rocksalt (B1)
This work 266 3.76 4.02
Calculation43 272 3.8 3.978
Calculation185 275 4.02 4.031
Calculation46 4.02
Experiment39 221.0?5.0 4.8?1.0 4.043
Experiment40 295?17 3.5?0.4 4.046
The static transition pressure can be determined from the common tangent of the E-V
curves (Figure 4.28) of B4 and B1 phases, or equivalently from the intersection of the H-P
curves, which gives Pt = 9.85 GPa. Our prediction is consistent with most of the recent
LDA calculations43,46, however, apparently less than those observed37?40. Despite of the
uncertainty of DFT calculations, we attribute the discrepancy to the relatively large kinetic
barrier along the phase transformation, which will be discussed in the next Section. The
quenchable high-pressure B1 phase also indicates the existence of the backward (B1-to-B4)
barrier at ambient conditions39.
4.2.3 Microscopic Mechanism for the Wurtzite-to-Rocksalt Transition
Five bond-preserving TPs proposed by Stokes et al. are summarized in Table 4.8.
Stokes et al. found that each TP is characterized by a sequence of bilayer sliding
mechanism, as described in their paper? . Each bilayer moves ?a/2 (hexagonal lattice
95
6 7 8 9 10 11
-8.2
-8.0
-7.8
-7.6
-7.4
0 5 10 15 20
-8.2
-8.0
-7.8
-7.6
-7.4
-7.2
-7.0
 Wurtzite (B4)
 Rocksalt (B1)
 
 
En
erg
y (
eV
/at
om
)
Volume (?3/atom)
 
 
En
tha
lpy
 (e
V/a
tom
)
Pressure (GPa)
Ptr = 9.85 GPa
Figure 4.28: Cohesive energy per atom as a function of volume for B4 and B1-AlN. Enthalpy
variation of both phases as a function of pressure is also shown.
Table 4.8: Bond-preserving TPs for the B4-to-B1 phase transition. G denotes the space
group of two end structures and G? denotes the common subgroup. Z represents the number
of formula units per unit cell along the TP. The transformation matrices are given with
respect to the primitive unit cell lattice vectors and the fractional atomic coordinates are
in terms of the setting of G?. u ? 3/8
TP G? G Z Transformation matrices Coordinates (Al) Coordinates (N)
1 Cmc21 (#36) B4 4 (1, 1, 0), (-1, 1, 0), (0, 0, 1) (0, 2/3, 0) (0, 2/3, u)
B1 4 (1, -1, 1), (1, -1, -1), (1, 1, -1) (0, 3/4, 0) (0, 3/4, 1/2)
2 Pna21 (#33) B4 4 (1, 2, 0), (-1, 0, 0), (0, 0, 1) (1/3, 0, 0) (1/3, 0, u)
B1 4 (-2, 2, 0), (0, 0, 1), (1, 1, -1) (3/8, -1/4, 0) (3/8, -1/4, 1/2)
3 P21 (#4) B4 6 (-1, 0, 0), (0, 0, 1), (0, 3, 0) (2/3, 0, 8/9) (2/3, u, 8/9)
(1/3, 1/2, 7/9) (1/3, u+1/2, 7/9)
(2/3, 0, 5/9) (2/3, u, 5/9)
B1 6 (0, 0, 1), (1, 1, -1), (-3, 3, 1) (1/3, 0, 11/12) (1/3, 1/2, 11/12)
(0, 1/2, 3/4) (0, 1, 3/4)
(2/3, 0, 7/12) (2/3, 1/2, 7/12)
4 Pna21 (#33) B4 8 (2, 4, 0), (-1, 0, 0), (0, 0, 1) (19/24, 3/4, 0) (19/24, 3/4, u)
(1/24, 1/4, 0) (1/24, 1/4, u)
B1 8 (-4, 4, 0), (0, 0, 1), (1, 1, -1) (13/16, 1/4, 0) (13/16, 1/4, 1/2)
(1/16, 1/4, 0) (1/16, 1/4, 1/2)
5 P21 (#4) B4 8 (-1, 0, 0), (0, 0, 1), (0, 4, 0) (2/3, 0, 11/12) (2/3, u, 11/12)
(1/3, 1/2, 5/6) (1/3, u+1/2, 5/6)
(2/3, 0, 5/12) (2/3, u, 5/12)
(1/3, 1/2, 1/3) (1/3, u+1/2, 1/3)
B1 8 (0, 0, 1), (1, 1, -1), (-4, 4, 1) (5/16, 0, 15/16) (5/16, 1/2, 15/16)
(-1/16, 1/2, 13/16) (-1/16, 1, 13/16)
(13/16, 0, 7/16) (13/16, 1/2, 7/16)
(7/16, 1/2, 5/16) (7/16, 1, 5/16)
96
parameter) in the ?[100] direction (in term of the B4 structure) relative to the adjacent
bilayers. Besides the relative sliding of atomic bilayers , two other independent atomic
displacements occur during the transformation, i.e., (1) N atoms with z = u ?3/8 (and
u ? 1/2) in the B4 phase displace in the [001] direction into the z = 1/2 (and 0). (2)
Spacings of bilayers increase until layers along [120] direction (in term of the B4 structure)
are equally spaced apart. Illustration of ?ve TPs listed in Table 4.8 are shown in Figure
4.25 and Figure 4.26. The small dark spheres represent N atoms and the large light spheres
represent Al atoms. Referring to Stokes et al.?s description, the repeatable bilayer sliding
patterns for the listed TPs are +, +?, + +?, + +??, and + + +? for TP1 through TP5,
respectively. Where + means the bilayer slides +a/2 relative to the previous bilayer and ?
means the bilayer slides ?a/2 relative to the previous bilayer.
We ?rst examined the ?ve bond-preserving TPs proposed by Stokes et al 48. The B4-
to-B1 transformation along TP1 can be described by ?ve independent parameters, i.e.,
(a,b,c,u,v). The parameters a, b, and c are the orthorhombic cell edges, u is the relative
position of two sublattices in [001] direction, and v is the relative displacements of atoms
of alternating (?110) planes in the direction of [?110]. For TP2 and TP4, there are six degree
of freedom, i.e., (a,b,c,?,u,s). Here ? represents the relative displacements of atoms of
alternating B4 (010) planes in the [100] (unique-b axis) direction (in the setting of G?), and
s represents the atomic displacements in the sliding direction of bilayers. For TP3 and TP5,
due to monoclinic symmetry, there is one more free parameter, angle ? between a and c. In
Figure 4.25 and Figure 4.26, for clarity, we illustrate the unit cells with origins happen at
the positions of atoms.
The complete information of the transition pathway requires computation of the po-
tential energy as functions of all the degrees of freedom, i.e., a multi-dimension potential
energy surface (PES). However, DFT calculation of 5-7 dimension PES is computationally
expensive. In this dissertation, our main purpose is to obtain the activation barrier height
for di?erent TPs. And considering the hexagonal path suggested by Cai et al. for AlN46,
we think the one dimensional PES can yield the correct kinetic barrier height. The only
97
parameter being controlled is v for TP1 or s for other TPs, which we believe is the param-
eter that characterizes for the structural transformation. Our calculated barrier height of
TP1 will be compared with the 2D PES result of Cai et al. to ensure the validity of our
simpli?ed approach.
For the ?ve TPs discussed above, we calculated the total energy as a function of the
transition parameter. The transition parameter is de?ned to vary from 0 (B4 phase) to 1
(B1 phase) for the controlled free parameter, which is the y coordinates for TP1, TP2, TP4
and x coordinates for TP3, TP5. DFT calculation was performed by ?xing the transition
parameter and relaxing all the other external and internal coordinates. The initial structural
parameters of the intermediate phases are generated from linear interpolation of the two
end structures. x = xi + (xf ?xi) ?tp where x denotes any internal or external parameter,
xi and xf denote x values of the starting (B4) and ending (B1) structures, respectively.
The k-point sampling for Brillouin zone integration was carried out by the Monkhorst-Pack
method with grids of 6?4?4, 4?6?4, 6?4?4, 4?6?6 and 6?4?4 for TP1 through
TP5, respectively. At each transition parameter, the calculation is made for several volumes
which were later ?tted to the 2nd-order BM-EOS in order to obtain the enthalpy-pressure
relation.
The information of fractional coordinate u and angle ? (between a and b) at 0 GPa and
the transition pressure are extracted after the calculation of TP1. For ? larger than ? 70?,
u becomes 0.5 after relaxation, which is the value of B1 phase. So the enthalpy barrier we
obtained is consistent with Cai et al.?s 2D PES. The calculated enthalpy as a function of
the transition parameter for ?ve TPs at di?erent pressures from 0 to 30 GPa are shown in
Figure 4.29. All ?ve TPs yield comparable results, while the di?erence in the shape and
position of the maximum enthalpy between TP1 and other TPs could be ascribed to the
di?erent transition parameters adopted. The discrepancies at tp = 1 should be due to the
di?erent size of unit cells and errors from ?tting to equation of states. As can be seen
from Figure 4.30, The forward (B4-to-B1) barrier height decreases with compression and
the backward (B1-to-B4) barrier height decrease with decompression for all TPs. This can
98
explain the larger transition pressure observed in room temperature experiments compared
with equilibrium transition pressure predicted by calculations. At 20 GPa, the forward
activation barrier is about 80 meV/atom and continuously drops below 50 meV/atom upon
compression up to 30 GPa. The low activation barrier height greatly increases the possibility
of the occurrence of the B4-to-B1 phase transformation. On the other hand, the backward
barrier height remains about 100 meV/atom even at 0 GPa. This is in agreement with the
fact that the rocksalt phase can be quenchable to ambient conditions39. The activation
barrier plot we obtained is very similar to the Figure 8 (a) of Cai et al.?s paper. Comparing
our TP1 result with their calculation, the forward barrier heights we predicted are 0.199
eV/atom at 0 GPa, 0.128 eV/atom at 9.85 GPa, 0.102 eV/atom at 15 GPa and theirs are
0.219 eV/atom at 1 bar (10?4 GPa), 0.144 eV/atom at 9.8 GPa, 0.109 eV/atom at 15 GPa.
We attribute the small discrepancy to the di?erent symmetry and size of unit cell in both
calculations. The volume sets used for ?tting of EOS can also play a role. Considering the
uncertainties involved in our calculation, the magnitudes of the barrier height of di?erent
TPs should be concluded as comparable at pressures from 0 to 30 GPa.
Since no apparent di?erence is found between TP1 and other TPs, we con?rmed Cai
et al.?s hypothesis, i.e., the four-atom ?transition unit? is responsible for the B4-to-B1
transforation and di?erent long-range arrangements of ?transition units? are less important.
From another point of view, Zahn et al.49 pointed out that the favored TPs have a tendency
to avoid excess strains during the transformation. TP2 and TP4, which have equal numbers
of bilayer sliding in opposite directions, yield less strains. Our calculation show no clear
preference of these two TPs over the rest. It is likely that the di?erent strains involved in
TPs may not be an important factor for the B4-to-B1 transformation.
We also calculated the #4 path proposed by Capillas et al., which involves bond break-
ing during the phase transformation. The forward barrier heights are 0.956 meV/atom at 0
GPa and 0.976 meV/atom at 15 GPa, which are much larger than those calculated from the
bond-preserving paths. So the bond-breaking path is less likely to be energetically favored
if there are bond-preserving paths available.
99
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s32
s32
s40s102s41 s51s48s32s71s80s97
Figure 4.29: Enthalpy as a function of transition parameter relative to B4 phase for ?ve
TPs at six pressures. (a) at 0 GPa, (b) at 5 GPa, (c) at the predicted transition pressure
9.85 GPa, (d) at 15 GPa, (e) at 20 GPa and (f) at 30 GPa.
100
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s49
Figure 4.30: Forward (B4-to-B1) and backward (B1-to-B4) activation barrier height (en-
thalpy) as a function of pressure for ?ve TPs.
4.2.4 Conclusions
In summary, we show that the ?ve bond-preserving paths can be interpreted as transfor-
mation of di?erent long-range patterns of the ?transition units? (two di?erent orientations).
The transformation of ?transition unit? is equivalent to the path along TP1 (with Cmc21
symmetry). Our calculated kinetic barriers are comparable for all ?ve paths at pressures
from 0 GPa to 30 GPa, which indicate that the wurtzite-to-rocksalt transition is charac-
terized by the transformation of the ?transition unit?, while the long-range pattern is less
important. And the di?erence in strains of di?erent TPs is not a major factor for at least
the transition from wurtzite to rocksalt phase in AlN. In addition, the bond-breaking path
is not energetically favored compared with the bond-preserving paths.
Besides the similarity from ?ve bond-preserving TPs, our estimated forward and back-
ward barrier heights are consistent with experimental observation and previous calculations.
101
Chapter 5
STUDY OF HIGH-PRESSURE PHASE TRANSITIONS IN SILICON NITRIDE
5.1 Introduction
Silicon nitride (Si3N4) is widely used in cutting tools and anti-friction bearings due
to its excellent mechanical properties, low mass density, and thermal stability56. It is
also used as an insulator layer or as an etch mask because of its dielectric properties and
a better di?usion barrier against impurities in microelectronics57. For its technological
importance, the mechanical and thermal properties of silicon nitride at ambient pressure
has been investigated extensively by both experiment and theory56,57. In contrast, its
properties at high-pressure is less known.
? (P31c) and ? (P63/m) phases are the only two bulk polymorphs of Si3N4 known at
ambient pressure. Both phases can be synthesized by nitriding pure silicon. In 1999, ?-Si3N4
(or c-Si3N4, Fd?3m) with the cubic spinel structure was synthesized at high pressure and
high temperature58. Despite intensive research e?orts in searching the ?post-spinel? phases
in Group-IV(B) nitrides, the spinel structured ? phase remains as the only experimentally
identi?ed high-pressure phase. The space groups and Wycko? sites for these three phases
are summarized in Table 5.1.
Table 5.1: Space groups, formula units per primitive unit cell and Wycko? sites of hexagonal
?-, ?-Si3N4 and cubic ?-Si3N4.
Phase Space group Formula units per Species Wycko? site
primitive unit cell
? P31c 4 N 2a, 2b, 6c, 6c
Si 6c, 6c
? P63/m 2 N 2c, 6h
Si 6h
? Fd?3m 2 N 32e
Si 8a, 16d
102
Phase transitions in Si3N4 have drawn extensive attention for more than a decade.
The temperature-pressure (T-P) conditions for obtaining various observed and hypothe-
sized polymorphs are sketched in Figure 5.1. The relative phase stability between ? and ?
phases has been a topic of investigation for many years. Direct measurements of energetics
of Si3N4 were reported by Liang et al.59. However, the di?erence in formation enthalpies
between ?- and ?-Si3N4 was founded to be less than the intrinsic experimental uncertainty
of ?22 kJ/mol (?32.6 meV/atom). Nevertheless, the ? phase is believed to be the ground
state in Si3N4 because no ??? transition is ever observed. The stability condition for ?
phase has been experimentally studied at temperatures of 1300??1800?C and pressures up
to 60 GPa60?67. A solution-precipitation mechanism was proposed for the ??? transforma-
tion67. Pure single-crystal ?-Si3N4 shows no sign of transformation at temperatures up to
1820??2200?C67,186. Suematsu et al. discovered that the ??? transformation occurs with
annealing in the presence of Y2O3 or other oxides. The catalyst oxides ?rst form a liquid
phase with Si3N4 on the surface at high temperatures. Then, through atomic transportation
in the liquid, small particles of ?-Si3N4 emerge. The observed liquid phase on the ?-Si3N4
surfaces was believed to lower the activation energy of atomic transportation. The stability
of pristine ?-Si3N4 at high temperatures is ascribed to the extremely high value of the ac-
tivation energy with clean surfaces. On the theory side, several studies con?rmed that the
static bonding energy of ? phase is slightly higher than that of ? phase68?71. Wendel et al.70
and Kuwabara et al.71 carried out statistical QHA calculations, and they both found that
the ? phase remains metastable in the temperature range from 0 to 2000 K at ambient pres-
sure. Yet, pressure e?ects on the relative thermodynamic stability between ? and ? phases
was not addressed in previous studies. The ?rst goal of the this chapter is to understand
the relative thermodynamic stability at high pressures. The relative Gibbs free energies
between two phases suggest that ? phase, compared with ? phase, is thermodynamically
stable at temperatures up to 2000 K and pressures up to 10 GPa. We further predicted the
equilibrium phase boundaries for the ??? and ??? transitions. The transition pressures
for the ??? transition is about 0.5 GPa lower than that of ??? transition.
103
?
? (phenacite)
? (spinel) post-spinel
?
P, high
 T
P, high
 T
P
P, RT
Hi
gh
 T
Figure 5.1: Polymorphs of Si3N4 and synthesis conditions
The spinel structured ?-Si3N4 was ?rst synthesized by Zerr et al. with laser-heated
diamond anvil cell (LH-DAC)58. Later experiments show that ?-Si3N4 can be obtained
from both ?- and ?-Si3N4 upon compression and simultaneous in-situ heating58,72?75. The ?
phase is quenchable to the ambient condition, and it remains stable at temperatures ranging
up to about 1670 K at ambient pressure76,77. When ?-Si3N4 ?decomposes? at ambient
pressure upon heating, the samples may consist of both ?- and ?-Si3N476. Previous ab initio
studies also calculated the ??? transition pressure (Pt) at adiabatic static condition58,79,
as well as at high temperatures71,187. The Clapeyron slope is found to be positive and Pt
varies only slightly over temperature. Furthermore, a new high-pressure post-spinel phase
was proposed by ?rst-principles method188,189. The new phase is predicted to be CaTi2O4
type and the calculated Pt from ? to post-spinel phase is over 160 GPa. The predicted new
phase has not yet been observed.
The in-situ heating to high temperature is found to be necessary to form the ?-Si3N4
at high pressures. At room temperature the ??? transition is, however, by-passed. Zerr
found that ?-Si3N4 exists up to 34 GPa and it then transforms into a new phase (labeled
as ?-phase) under further compression78. This phase transition was identi?ed by Raman
spectroscopy and energy dispersive X-ray powder di?raction (EDXD). But the structure of ?
phase was not fully determined. Zerr proposed three possible unit-cells based on the EDXD
pattern: two tetragonal and one orthorhombic. The ?rst hypothetical tetragonal unit-cell
would have a density of 4.5 g/cm3 at 42.6 GPa, which is smaller than that of ?-Si3N4 (4.50
104
g/cm3). At the same pressure, the second tetragonal and the orthorhombic structures are
proposed to have densities of 4.56 g/cm3 and 5.16 g/cm3, respectively, which are larger
than that of ? phase. The later two structures are considered as ?post-spinel? phases. Zerr
further suggested that the ?-Si3N4 should be considered as a metastable intermediate stage
in the ??? transition. Kroll has proposed a metastable willemite-II-Si3N4 phase which is
an intermediate between ?- and ?-Si3N4 in both energetics and density79. However, the
wII phase is unlikely to be the experimentally observed unknown phase at high pressure
and room temperature. Because 1) the wII phase, which is structurally closely related
to the spinel ?-Si3N4, has been shown to have a signi?cantly lower activation barrier for
the ??wII transformation, comparing to that of ??? transformation79. Although the
activation barrier of the ??wII transformation is unknown, it is more likely to be high
enough to exclude the room temperature transition. 2) The calculated Raman frequencies
of wII-Si3N4 could not match many strong peaks appeared in the measurements, e.g., two
observed peaks at about 500 cm?1 and 550 cm?1 are absent in the calculation. A recent
experimental work from McMillan et al. reproduced Zerr?s ?ndings on ?-Si3N4, but excluded
the wII cubic structure80.
Meanwhile, ?-Ge3N4 is found to transform into the metastable polymorph ?-Ge3N4
with hexagonal P3 symmetry at room temperature by Soignard et al.81. Ab initio calcula-
tion from Dong et al. showed that a ??P?6?P3 transition sequence could occur in Ge3N4
at the pressure of about 20 GPa and 28 GPa82, which are of second-order that driven by soft
phonons. If ?-Ge3N4 directly transforms into the P3 structure, the transition was predicted
to be ?rst-order and Pt = ?23 GPa. Dong also pointed out that the ??P?6 transition is
originated from a soft silent Bu mode. Room temperature experimental study by Soignard
et al. con?rmed the direct ??P3 transition associated with a 5-7% volume reduction81.
The Raman data they observed excludes the intermediate P?6 structure. Based on the den-
sity consideration, Soignard et al. suggested that the new polymorph is a ?post-phenacite?
phase, in stead of ?post-spinel?. Comparison of the X-ray di?raction and Raman data be-
tween Ge3N4 and Si3N4 shows similarity which may suggest a P3 structure for ?-Si3N4. It is
105
still unclear whether there are intrinsic di?erences between the HP-RT behaviors of Si3N4
and Ge3N4, or the experimental results may be interpreted di?erently. The second goal of
our study is to theoretically investigate structural instabilities and possible metastable phase
transitions in ?- and ?-Si3N4 at high pressures and room temperature. We found no sign
of dynamical instability in the ? phase at high-pressure. On the other hand, we predicted a
phonon-softening related ?rst-order phase transition at about 38.5 GPa in ? phase. At this
pressure, the density of the proposed high-pressure phase is 4.16 g/cm3 which is larger than
that of ?-Si3N4 (3.71 g/cm3), yet smaller than that of ?-Si3N4 (4.53 g/cm3, calculated). We
further estimated the kinetic barrier heights for our proposed ??P3 transition, which is
only 67.23 meV/atom at 38.5 GPa. Despite being of ?rst-order phase transition, the small
barrier height suggests that the P3 phase is unlikely to be recovered below 38.5 GPa.
Finally, we performed a series of systematical calculations of thermodynamic properties
of Si3N4, such as thermal expansion coe?cient (TEC), heat capacity and bulk Gr?uneisen
parameter, and compared our results with available experimental data83?90 and some pre-
vious calculations70,71,91,92. The overall good agreement with experiment validates the
adopted statistical quasi-harmonic approximation (QHA) and the Birch-Murnagahn equa-
tion of states (EOS) models. Our results support the prediction of Kuwabara et al.71 on
the negative TEC of ? and ? phases at temperatures below 100 K. We further attributed
the origin of the negative TEC to the low-frequency phonon modes with the negative mode
Gr?uneisen ratios in the two phases.
5.2 Crystal Structures, Static Binding Energies, and Vibrational Spectra
Atomic structures of ?-, ?-, and ?-Si3N4 are shown in Figure 5.2. Both ? and ?-
Si3N4 have hexagonal symmetry, and they contain similar local bonding: each Si atom
is tetrahedrally bonded to four N atoms (Si-N4) and each N atom has a threefold trigonal
coordinates (N-Si3). All the SiN4 tetrahedra are slightly distorted and connected by corner-
sharing. The di?erence between these two phases can be characterized by the stacking
sequence along c axis. The periodicity of ?- and ?-Si3N4 in that direction can be described
106
as ABCDABCD.... and ABAB.... stacking, respectively. From another point of view,
?-Si3N4 can be interpreted as a complex network formed with nonplanar six-membered (6-
atom) rings, whereas ?-Si3N4 is composed of non-planar six-, eight- and twelve-membered
rings. There are two types of trigonal N-Si3 units: those with N atoms at the 2a and 2b sites
of ?-Si3N4 or the 2c site of ?-Si3N4 locate at the basal plane perpendicular to the c axis,
while the rest N-Si3 units are in the vertical or near-vertical orientations. Most basal N-Si3
units are perfectly planar with three equal-length bonds and three 120? Si-N-Si bond angles,
except that the N-Si3 units with N at the 2b sites of ?-Si3N4 form triangular pyramids (i.e.
three bonds still have equal length, but the bond angles are less than 120?). The vertical
N-Si3 units are distorted in bond lengths and bond angles which yield distorted pyramidal
units. The ? phase has a distinctively di?erent structure, in which Si atoms occupy both
tetrahedral (1/3 of Si atoms, 8a sites) and octahedral (2/3 of Si atoms, 16d sites) sites, and
all the N atoms are tetrahedrally bonded. This is consistent with the fact that ?-Si3N4 is the
high pressure phase which has a larger coordination number. The spinel structure is named
after the mineral MgAl2O4 which has a fcc lattice with space group Fd?3m. For ?-Si3N4,
there are two formula units in the primitive unit cell and eight units in the conventional
cubic cell.
In this study, the equilibrium T-P phase diagram and thermodynamic properties are
predicted using the ?rst-principles calculated thermodynamic potentials. As an insulator,
the Helmholtz free energy of a bulk crystalline silicon nitride system consists of two parts:
F (T,V ) = Estatic (V ) + Fvib (T,V ) (5.1)
where Estatic (V ) is the static binding energy of the system and Fvib (T,V ) is the vibrational
free energy. Free energy associated with the electronic thermal excitation is neglected.
Estatic (V ) for ?-, ?-, and ?-Si3N4 are calculated with unit-cell models of respective crystal
symmetries. We adopted density functional theory (DFT) with a plane wave basis set and
ultrasoft pseudopotentials (US-PP)121, which is implemented in the VASP code139. The
107
Figure 5.2: Crystal structures of (a),(b) ?-, (c),(d) ?-, and (e),(f),(g) ?-Si3N4. In the panel
of ?- and ?-Si3N4, the ?rst graph illustrates the unit-cell model and the second graph is
the 2? 2?1 supercell model viewed in the direction of c axis. In the panel of ?-Si3N4, the
?rst graph shows the conventional cubic cell of the spinel structure and the following two
graphs show the fourfold and sixfold coordinated Si units(SiN4 and SiN6) with tetrahedra
and octahedra, respectively.
108
exchange and correlation functional is treated with local density approximation (LDA). The
plane wave basis functions with energies up to 347.9 eV were used. Total energy change
of 10?9 eV per unit cell was chosen as the convergence criterion for the self-consistent
iterations. The Brillouin zone integration in our total energy calculations was approximated
with the Monkhorst-Pack method, with grids of 4 ? 4 ? 6, 4 ? 4 ? 12 and 6 ? 6 ? 6 for ?-,
?- and ?-Si3N4, respectively. The calculated total energies at several chosen volumes were
?tted to the third-order Birch-Murnaghan equation of state (BM-EOS)159,160 by the least
square ?tting algorithm.
The calculated E-V data sets of ?-, ?- and ?-Si3N4 are shown in Figure 5.3, and the
corresponding ?tting parameters from the third-order BM-EOS (E0, V0, B and B?) are
listed in Table 5.2, together with reported experimental and other theoretical results. As
the measurements are usually made at room temperature, our predicted parameters at 300
K within QHA are also presented. Our calculation has a good overall agreement with other
theoretical and experimental results. Compared to the experiments, our calculated static
equilibrium volumes are consistently underestimated by about 1 to 3%, and the calculated
bulk moduli are within the range of reported experimental data, which contain about 5 to
15% di?erences among di?erent reports. The predicted thermal equation of states at 300 K
are slightly closer to the measurement. Our results are within the typical accuracy of LDA
calculation and they are consistent with the fact that LDA tends to slightly underestimate
the equilibrium volume and overestimate the bulk modulus by a few percent.
Our static total energy calculation shows that ? phase is only slightly energetically
more stable (i.e., about 3 meV/atom lower) than ? phase at their respective static equi-
librium volumes. Such a small energy di?erence is consistent with the fact that both ?
and ? phases are found to be co-exist during di?erent synthesis routes. Also, in agree-
ment with experiment, we ?nd that the calculated ? phase has larger density and lower
compressibility comparing to the ? phase. This suggests that ? phase is even less favored
thermodynamically at higher pressure relative to ? phase. The relative stability between
109
6 7 8 9 10 11 12
-9.1
-9.0
-8.9
-8.8
-8.7
-8.6
-8.5
-8.4
-8.3
 ? phase
 ? phase
 ? phase
 
 
En
erg
y (
eV
/at
om
)
Volume (?3/atom)
Figure 5.3: Energy-volume curves for ?-, ?- and ?-Si3N4 in the scale of per atom. ? phase
has an equilibrium energy of 3 meV lower than that of ? phase. E0 of ? phase is 93 meV
higher than ? phase.
these two phases will be further examined in later text with the consideration of tempera-
ture and pressure e?ects. For the ? phase, our calculation yields a static equilibrium energy
which is 93 meV/atom higher than that of the ? phase, and a static equilibrium volume
of 2 ?A3/atom smaller than that of the ? phase. These results agree with the fact that the
spinel structured ? phase is a high pressure phase in Si3N4.
Figure 5.4 shows the phonon dispersion curves and VDOS plots of the ?-, ?- and
?-Si3N4 at their respective static equilibrium volumes. All three phases studied here are
dynamically stable, i.e., no soft phonon modes. The ? and ? phases have very similar
VDOS which re?ects the similarity in their crystal structures and Si-N bonding. On the
other hand, the spinel structured ?-Si3N4 shows some distinctively di?erent characters in
its VDOS, comparing with those of ? and ? phases. High-pressure phases usually have
higher vibration frequencies. Yet, we ?nd that the top phonon branches in ? phase have
frequencies which are signi?cantly lower than those of ? or ? phase.
Mode Gr?uneisen ratios along some high symmetry directions are shown in Figure 5.5.
Although there are many similarities in the mode Gr?uneisen ratios between the ? and ?
phases, for example, their low-frequency phonon modes are found to have negative mode
110
Table 5.2: Summary of calculated and measured crystal parameters of ?-, ?- and ?-Si3N4.
V0 is the equilibrium volume per atom, B is the bulk modulus and B? is the ?rst-order
pressure derivative. Measurements were made at room temperature.
?-Si3N4
Source V0
parenleftBig?
A3/atom
parenrightBig
B (GPa) B?
LDA (this work, static) 10.260 232 2.583
LDA (this work, 300K) 10.328 226 2.576
LDA71 10.325 240 4.0 (?xed)
LDA190 10.237 257
OLCAO69 10.542 257
Force ?elds (300K)70 10.806 246
Experiment191 10.455
Experiment192 10.445
Experiment193 10.465 223.4 (?15) 4.5 (?1.3)
?-Si3N4
Source V0
parenleftBig?
A3/atom
parenrightBig
B (GPa) B?
LDA (this work, static) 10.199 241 3.439
LDA (this work, 300K) 10.267 237 3.440
LDA71 10.268 252 4.0 (?xed)
GGA194 237.2-241.5
LDA190 10.183 225
Force ?elds (300K)70 10.661 266
Experiment195 10.396
Experiment196 10.411 270 (?5) 4.0 (?1.8)
Experiment197 10.452 232.7
Experiment68 10.356
?-Si3N4
Source V0
parenleftBig?
A3/atom
parenrightBig
B (GPa) B?
LDA (this work, static) 8.140 308 3.898
LDA (this work, 300K) 8.220 297 3.898
LDA71 8.137 320 4.0 (?xed)
OLCAO69 8.595 280
Experiment198 8.270 290 (?5) 4.9 (?0.6)
Experiment74 8.286 308 4.0
Experiment76 8.261
Experiment197 300 (?10) 3.0 (?0.1)
Experiment58 8.474 (?0.26)
111
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s49s48s48s48
s49s50s48s48
s40s98s41s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s80s104s111s110s111s110s32s100s105s115s112s101s114s115s105s111s110s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s86s68s79s83
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s65s32s32s32s32s32s32s32s32s32s32s32s76s32s32s32s32s32s32s32s32s32s32s77s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s65s32s32s32s32s32s32s32s32s32s32s72s32s32s32s32s32s32s32s32s32s32s32s75s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s97s114s98s46s32s117s110s105s116s115
s45s83s105
s51
s78
s52
s80s32s61s32s48s71s80s97
s32 s32
s32
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s40s99s41s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s80s104s111s110s111s110s32s100s105s115s112s101s114s115s105s111s110s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s86s68s79s83
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s88s32s32s32s32s32s32s75s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s76s32s32s32s32s32s32s32s32s32s32s97s114s98s46s32s117s110s105s116s115
s45s83s105
s51
s78
s52
s80s32s61s32s48s71s80s97
s32
s32
Figure 5.4: Phonon dispersion curves and vibrational density of states (VDOS) of (a) ?-,
(b) ?-, and (c) ?-Si3N4 at zero pressure.
112
Gr?uneisen parameters while all the high-pressure modes have positive ratios with the upper
limit of about 1.5, there are some noticeable di?erences for phonons around the M-point
transverse acoustic (TA) mode and the ?-point optic Bu mode. The phonons close to these
two modes in ? phase are found to have large negative Gr?uneisen ratios, which suggest
possible structural instability of ? phase upon compression. On the other hand, the ?-
Si3N4 shows no negative mode Gr?uneisen ratios at all, and the values of its mode Gr?uneisen
ratios range from 0.24 to 1.66 at zero pressure..
-5
-4
-3
-2
-1
0
1
2(a)
 
 
 
Gr
?n
eis
en 
par
am
ete
r
A            L            M            ?            A            H            K            ?
?-Si3N4
P = 0 GPa -5
-4
-3
-2
-1
0
1
2(b)
 
 
 
Gr
?n
eis
en 
par
am
ete
r
A            L            M            ?            A            H            K            ?
?-Si3N4
P = 0 GPa
-5
-4
-3
-2
-1
0
1
2
Gr
?n
eis
en 
par
am
ete
r
 
?-Si3N4
P = 0 GPa
 
 
?                            X        K                              ?                            L
(c)
Figure 5.5: Calculated dispersion curves (scattered circles) of mode Gr?uneisen parameter
of (a) ?-, (b) ?-, and (c) ?-Si3N4 at zero pressure. Red horizontal line is present to separate
the positive and negative values.
113
5.3 Equilibrium Thermodynamic Stability and Phase Transitions
To illustrate the relative thermodynamic stability between the ? and ? phase, we plot
the LDA calculated Gibbs free energy di?erences between the two phases at 0, 5 and 10
GPa in Figure 5.6. A positive value of ?G??? means that ?-Si3N4 is thermodynami-
cally metastable. At isobaric conditions, the calculated ?G??? are almost constant over
the temperature range from 0 K to 2000 K. At zero pressure, our calculated ?G??? is
2.8 meV/atom at 0 K which agrees with Kuwabara?s (DFT+PAW+LDA) ?F??? of 2.6
meV/atom at 0 K. At 2000 K, our ?G??? is 2.6 meV/atom, while Kuwabara?s ?F???
decreases to 1.3 meV/atom. The results of Wendel et al. were based on empirical force ?eld
models and they gave opposite trend of temperature dependence, 0.1 meV/atom at 300 K
and 0.7 meV/atom at 2000 K. At elevated pressures, we predict an increasing ?G???. At
5 GPa and 10 GPa, ?G??? is about 4.6 meV/atom and 5.9 meV/atom, respectively. We
do not predict ?G??? at pressures higher than 10 GPa because the ? phase starts to show
signs of structural instability (see discussion in later text). We conclude that ? phase is
metastable compared to ? phase in the temperature range from 0 K to 2000 K and at least
up to 10 GPa.
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s48
s49
s50
s51
s52
s53
s54
s55
s32
s32
s32s80s32s61s32s48s32s71s80s97
s32s80s32s61s32s53s32s71s80s97
s32s80s32s61s32s49s48s32s71s80s97
s71
s32
s40
s109
s101
s86
s47
s97
s116
s111
s109
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
Figure 5.6: Gibbs free energy of ?-Si3N4 relative to that of ? phase as a function of temper-
ature. Solid, dashed and dotted lines represent the pressure of 0, 5 and 10 GPa, respectively.
114
Upon compression, both the ground state ? phase and meta-stable ? phase transform
into the ? phase. Our predicted equilibrium T-P phase boundaries are shown in Figure 5.7.
The Clapeyron slopes for the ??? (solid line) and ??? (dashed line) transitions are both
positive, which suggests that the high-pressure ? phase has a lower vibrational entropy.
Consequently, the transition pressure (Pt) increases with temperature. The predicted Pt of
the ??? transition is 7.5 GPa at 300 K, and it increases to 9.0 GPa at 2000 K. The Pt of
the ??? transition is about 0.5 GPa lower than that of ??? transition. Togo et al.187 and
Kuwabara et al.71 also predicted a positive Clapeyron slope for the ??? transition. The
large calculated Clapeyron slopes (dT/dP) means that the transitions are primarily volume
driven and the equilibrium Pt is not sensitive to the temperature. For example, Pt changes
by less than 2 GPa when temperature rises from 300 K to 2000 K. On the experimental
side, the transition pressures are scattered from 10 GPa to 36 GPa (Table 5.3). This could
be ascribed to the di?erent compositions/impurities of the starting samples being used.
Nonetheless, in situ heating is required for the synthesis of ?-Si3N4 in all experiments. This
is a clear indication that large kinetic barriers exist. For better comparison between theory
and experiment, we only list here the theoretical results at T = 2000 K.
Table 5.3: Summary of phase transition pressure and temperature for synthesizing ?-Si3N4
Method Starting material Pt (GPa) Temperature (K)
Experiment
diamond cell58 Si, amorphous Si3N4 and poly-
crystalline ? + ?
15 2100
Shock compression73 ?+2 wt% (Nd2O3+Y2O3)? 36 1990
Diamond anvil cell199 ? + 1%? 17.5 -
Multi-anvil72 ? + ? 17 2100
Shock wave75 ? 10 2073
Theory
PAW+GGA187 ? 13 2000
PAW+LDA71 ? 6.3 2000
USPP+LDA (this work) ? 8.5 2000
USPP+LDA (this work) ? 9.0 2000
115
s54 s55 s56 s57 s49s48
s53s48s48
s49s48s48s48
s49s53s48s48
s50s48s48s48
s32
s32
s84
s101
s109
s112
s101
s114
s97
s116
s117
s114
s101
s32
s40
s75
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
Figure 5.7: T-P phase diagram of Si3N4. Solid curve denotes the phase boundary between
?- and ?-Si3N4. Dashed curve denotes the phase boundary between ?- and ?-Si3N4.
5.4 Phonon-Softening Induced Structural Instability in ?-Si3N4 at High Pres-
sures
Although ?-Si3N4 transforms into the ? phase at high pressures and temperatures,
the ? phase is stable at the room temperature up to at least 30 GPa. To investigate
the structural stability of the ? phase, we calculated the pressure dependence of lattice
vibration. First, we examined the phonon modes at the zone center ?-point. There are in
total 42 vibrational modes for ?-Si3N4 with space group P63/m. Using group theory, the
irreducible representation for ?-point phonon modes is
?acoustic = Au + E1u (5.2)
?optic = 4Ag + 2Au + 3Bg + 4Bu + 2E1g + 5E2g + 4E1u + 2E2u (5.3)
For the optic modes, 11 modes (4Ag+2E1g+5E2g) are Raman active, 6 modes (2A2u+4E1u)
are infrared (IR) active, and the rest (3Bg + 4Bu + 2E2u) are silent modes, among which
Raman and IR spectra can be detected in experiments. Figure 5.8 shows our calculated
Raman, IR and silent modes of the ? phase as a function of pressure up to 60 GPa. Experi-
mental pressure dependencies up to 30 GPa are presented for comparison. For the measured
116
Raman modes from Zerr et al.200, one Ag mode is missing, possibly due to the weak in-
tensity. The rest Raman modes match well with our calculation. Our prediction tends to
underestimate the frequencies by about 2%-4% , which is typical for calculations of this
type. The calculation shows a clear pattern that all low frequency modes ( 400 cm?1 and
below) have zero or negative pressure dependencies. The lowest Bu silent mode decreases
much faster than the others and eventually vanishes at about 60 GPa. The predicted neg-
ative pressure dependence in these modes is consistent with the calculated negative mode
Gr?uneisen ratios (Figure 5.5b). The calculated phonon softening pattern in ?-Si3N4 is in
agreement with our previous results for ?-Ge3N482.
s48 s49s48 s50s48 s51s48 s52s48 s53s48 s54s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s40s97s41
s69
s50s103
s65
s103
s69
s49s103
s32
s32
s82
s97
s109
s97
s110
s32
s102
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s69
s50s103
s65
s103
s69
s50s103
s65
s103
s69
s50s103
s69
s49s103
s65
s103
s69
s50s103
s45s83s105
s51
s78
s52
s48 s49s48 s50s48 s51s48 s52s48 s53s48 s54s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s40s98s41
s69
s49s117
s45s83s105
s51
s78
s52
s32
s32
s73
s82
s32
s102
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s69
s49s117
s65
s117
s69
s49s117
s69
s49s117
s65
s117
s48 s49s48 s50s48 s51s48 s52s48 s53s48 s54s48
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s66
s117
s66
s103
s69
s50s117
s66
s103
s66
s117
s66
s117
s45s83s105
s51
s78
s52
s32
s32
s83
s105
s108
s101
s110
s116
s32
s102
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s66
s117
s66
s103
s69
s50s117
s40s99s41
Figure 5.8: (a) Raman, (b) IR and (c) silent mode frequencies as a function of pressure up
to 60 GPa for ?-Si3N4. Experimental pressure dependence of Raman modes up to 30 GPa is
also presented in discrete symbols as a comparison200. Solid squares denote measurements
upon pressure increase and open squares denote measurements upon pressure decrease.
Several low-frequency modes are found to decrease with increasing pressure. One Bu branch
of silent modes is found dropping to zero at about 60 GPa.
Next, we extended our study to all the phonon modes in the reciprocal space. Our
calculated phonon dispersion curve of ?-Si3N4 at 48 GPa (Figure 5.9) shows that two
low-frequency branches decrease dramatically upon compression, i.e., one TA branch along
the ?-M direction and lowest optic Bu branch. The TA mode goes soft at the Brillouin
zone boundary M point, i.e. k = 2?a
parenleftBig
1?
3,0,0
parenrightBig
and the optic mode goes soft at the zone
center ? point, i.e. k = (0,0,0). A vanishing phonon frequency results from the vanishing
117
s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s49s50s48s48
s32
s32
s32
s70
s114
s101
s113
s117
s101
s110
s99
s121
s32
s40
s99
s109
s45
s49
s41
s45s83s105
s51
s78
s52
s80s32s61s32s52s56s32s71s80s97
s65s32s32s32s32s32s32s32s32s32s32s32s32s76s32s32s32s32s32s32s32s32s32s32s32s32s77s32s32s32s32s32s32s32s32s32s32s32s32 s32s32s32s32s32s32s32s32s32s32s32s32s65s32s32s32s32s32s32s32s32s32s32s32s32s72s32s32s32s32s32s32s32s32s32s32s32s32s75s32s32s32s32s32s32s32s32s32s32s32s32
Figure 5.9: Phonon dispersion of ?-Si3N4 at a pressure of 48 GPa. Two competing soft
phonon modes are found: one TA branch at M point and one optic branch at ? point. No
LOTO splitting correction is added for the interests of low-frequency modes only.
restoring force against the atomic displacement for the corresponding vibrational mode.
Consequently, the crystal structure may undergo a displacive transition to reach a new
minimal-energy con?guration with lower symmetry. Our calculated ?2 of the two most
signi?cant soft modes as a function of pressure are shown in Figure 5.10. The two ?2
are found to exhibit linear pressure dependencies. Comparing to the M-point TA mode,
the softening Bu mode has a higher frequency at ambient pressure, yet it decreases much
faster with the increase of pressure. Phonon frequencies of both softening modes reach zero
at ?60 GPa. Although the frequency of the M-point TA phonon vanishes before the Bu
branch, the predicted di?erence is, however, small. We thus consider both softening phonon
modes as two competing mechanisms that may be responsible for the structural instability
of ?-Si3N4 at high pressures. It is worth to point out that ?-Si3N4 does not show any signs
of structural instability in our calculation, which is consistent with the di?erences we have
pointed out for the calculated mode Gr?uneisen ratios (Figure 5.5(a)).
The atomic displacements according to the soft M-point TA mode are in the x-y plane
and the symmetry of the unit-cell is reduced from hexagonal P63/m to monoclinic P21/m
after the distortion. One vector of the P21/m primitive unit cell is about twice of the
118
s48 s49s48 s50s48 s51s48 s52s48 s53s48 s54s48
s49
s50
s51
s52
s53
s54
s32s77s45s112s111s105s110s116s32s115s111s102s116s32s109s111s100s101
s32 s45s112s111s105s110s116s32s115s111s102s116s32s109s111s100s101
s32
s32
s50
s32
s40
s49
s48
s52
s32
s99
s109
s45
s50
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s45s83s105
s51
s78
s52
Figure 5.10: The square of vibration frequency (?2) as a function of pressure for two
competing soft phonon branches: one TA branch at M point and one Bu branch at ? point.
Solid squares and circles represent data from calculation. Solid and dashed lines are from
a linear ?tting.
a vector of the original hexagonal unit-cell. Constrained with the P21/m symmetry, we
calculated the total energies of the distorted structure for several volumes by allowing further
relaxation of both unit-cell shapes and internal coordinates. The calculated E-V curve is
shown in Figure 5.12. At volumes larger than 8.75 ?A3/atom, the P21/m structure relaxes
back to the original ? structure with energy minimization calculation. Yet at volumes
smaller than 8.75 ?A3/atom, the P21/m phase yields a lower energy. The relaxed structure
with a volume of 8.25 ?A3/atom is shown in Figure 5.11(b). After relaxation, for volume
8.25 ?A3/atom, the lattice parameters a = 13.912 ?A, b = 6.674 ?A and c = 2.777 ?A. The
length of a is slightly larger than twice that of b. The angle between a and b becomes
116.4? from the original 120? in ? phase. The c/b ratio is getting larger compared to that in
?-Si3N4. This is consistent with the fact that it becomes more di?cult to compress along
c axis than in the x-y plane after the structural distortion. The displacements of internal
coordinates can be described in terms of N atoms. Around each 2c N atom in ?-Si3N4
there are three nearest 6h N atoms which are in the same basal plane. The displacements
of those 6h N atoms are in a way that it causes the previous planar vertical N-Si3 units
to pucker. The puckering pattern can be seen in Figure 5.11(b). In P21/m structure, two
119
of the nearest N atoms become closer to the ?centered? 2c N atom (no longer 2c site in
the P21/m symmetry, but it is convenient to label it consistently) but the third one moves
away from it. Consequently the ?centered? 2c N atom is ?pushed? away by the two closer
N atoms, which breaks the hexagonal symmetry and causes the three Si-N-Si bond angles
to be distorted from the perfect 120?. More importantly, the interatomic distance between
the Si atoms at the 2e site and one of their 2nd nearest neighbor deceases rapidly upon
compression. At the volume of 8.25 ?A3/atom, this distance is only 1.988 ?A which is slightly
larger than that of previous Si-N bonds (less than 1.7 ?A). This tendency of forming an extra
bond may help to stabilize the distorted structure under high pressures. The new P21/m
phase is dynamically stable at pressures up to 75 GPa.
A similar distortion calculation was performed for the soft Bu mode at ? point. The
atomic displacements based on the corresponding vibrational pattern yields a new structure
which has a hexagonal P?6 symmetry. The size of the primitive unit cell is the same as ?-
Si3N4 and the displacements are still within the x-y plane. The E-V curve and data points
of P?6 phase is shown in Figure 5.12 as the (red) dash line. Its structure returns to ?
phase after fully relaxation for volumes larger than 8.75 ?A3/atom. Its energy is slightly
lower than ? phase at a smaller volume, however, it is higher than that of the P21/m
phase. Figure 5.11(c) shows the relaxed P?6 structure at the volume of 8.25 ?A3/atom. The
c/a ratio increase slightly compared to ?-Si3N4 and this may again be ascribed to the
less compressibility along c axis. The structure of P?6 can be interpreted in terms of the
puckering pattern of 6h N atoms. Unlike the P21/m structure, as shown in Figure 5.11(c),
three ?in-plane? 6h N atoms move clockwise and become closer to one of the ?centered? 2c
N atom, which has a z coordinate of 34 in term of c in our case. However, the other three
6h N atoms move counterclockwise and become away from the other ?centered? 2c N atom
(z = 14).
We further calculated the phonon spectrum of the P?6 structure and discovered an
optic soft phonon mode happens at its ? point. Based on the corresponding eigenvector,
we obtained a new structure with hexagonal P3 symmetry. The primitive unit cell is in
120
the same size as the ? phase, i.e., two formula Si3N4 units per cell. The E-V relation of
P3 phase is shown in Figure 5.12 as the (green) dash-dotted line. Its structure returns to
? phase beyond a volume of 8.75 ?A3/atom and remains stable at a smaller volume. The
total energy of the P3 phase is lower than the other three phases below the volume of
8.37 ?A3/atom. Between 8.37 ?A3/atom and 8.75 ?A3/atom, the P21/m phase has the lowest
energy. Structure model of P3 phase at the volume of 8.25 ?A3/atom is shown in Figure
5.11(d). The c/a ratio of P3 structure is very close to that of ? phase. This structure can be
understood as a further distortion of the P?6 phase. Relative to the P?6 structure, the major
di?erence in P3 phase is the z coordinate of the ?centered? 2c N atom which is surrounded
by three closer N atoms. This ?centered? 2c N atom, denoted thereafter as the puckering 2c
N, is ?pushed? up or down by three approaching N atoms. As the volume getting smaller,
the puckering 2c N will be ?pushed? by the three closer 6h N atoms eventually to the middle
of two ?closer-N-atoms? layers (z = 14) and become six coordinated. The other ?centered?
2c N atom remains its z coordinate because there is no ?push? e?ect. For volume between
8.00 ?A3/atom and 8.75 ?A3/atom, which is before the puckering 2c N atom reach its ?nal
position (z = 14), the z coordinates of other atoms deviate slightly from their previous
values. However, these z coordinates recover their previous values perfectly (z = 14 and 34)
when the puckering 2c N atom is stabilized at z = 14. Using the same criterion to verify the
formation of bonds, there are six extra bonds being formed within a primitive unit cell, i.e.,
3 extra bonds per formula unit. And for P21/m phase, it is only 12 extra bonds per formula
unit.
5.5 Room Temperature Metastable P3 Phase
Based on the E-V curves shown in Figure 5.12, the transition from ?-Si3N4 to one of
the three candidates is determined by the common tangent line between them. The smallest
magnitude of the slope (negative) is corresponding to the lowest transition pressure, and this
is made by the P3 phase. Both P21/m and P?6 phases are likely bypassed. The transition
pressure is estimated to be 38.5 GPa, which is comparable to the experimentally observed
121
Figure 5.11: Ball-stick models of (a) P63/m, (b) P?6, (c) P21/m and (d) P3 structures
viewed along the c axis. Balls in dark color represent N atoms and Si atoms are in light
color.
7.5 8.0 8.5 9.0 9.5 10.0-9.1
-9.0
-8.9
-8.8
-8.7
-8.6
-8.5
-8.4
-8.3
 P63/m (? phase)
 P6
 P21/m
 P3
 
 
En
erg
y (
eV
/at
om
)
Volume (?3/atom)
Figure 5.12: The total energy of P63/m (?), P?6, P21/m and P3 structures as a function
of volume.
122
35 GPa for the unidenti?ed ?-phase78. The transition pressure is much smaller than 60
GPa, at which one phonon frequency becomes zero in the ?-Si3N4. The predicted ?-to-P3
transition is a ?rst-order phase transition, and the predicted volume reduction is about
10.8%.
To estimate the kinetic barrier height in the ??P3 transition, we calculated the en-
thalpy landscape in terms of the atomic displacements in the x-y plane and of the z coor-
dinate of the puckering 2c N atom. At the transition pressure, we took the ? phase as the
starting structure and the P3 phase as the ending structure. Two transition parameters,
fx?y and fz, are used to linearly interpret the phase transition. Initial internal coordinates
of the intermediate structure can be expressed as
x = xi + (xf ?xi)fx?y
y = yi + (yf ?yi)fx?y
z = zi + (zf ?zi)fz
(5.4)
where the subscript ?i? and ?f? denote the starting (initial) and ending (?nal) structures,
respectively. Both fx?y and fz range from 0 to 1, and they can be set independently.
10?10 uniform grids were adopted for the intermediate structures. In the total energy
calculation of each structure, by ?xing the internal coordinates, we allowed the external
parameters to relax. Because this transition is observed to occur at room temperature, it
is a good approximation to use enthalpy instead of Gibbs free energy to investigate the
phase transition. The enthalpy landscape and its contour plot as functions of fx?y and fz
at 38.5 GPa are shown in Figure 5.13. Two minimum points correspond to ? (0,0) and
P3 (1,1) structures. The transition path is given by the gradient curve connecting the two
minimum points. It will pass the saddle point which provides the transition barrier height.
The pathway we predict is close to the linear path that fx?y and fz vary at similar paces.
The calculated saddle point locates at (0.6,0.5) and the corresponding enthalpy barrier is
67.23 meV/atom. To overcome this barrier, certain activation temperature is necessary to
123
stimulate the atomic vibrations to a level that is comparable to ?H. Using Dulong and
Petit law E = 3kBT, the ?threshold? activation temperature is estimated to be 260 K,
which is lower than the room temperature. Since all the internal coordinates are ?xed in
our calculation, the activation temperature should be considered as an upper limit to its
actual value. The ??P3 transformation should be interpreted as a low-barrier transition
induced by softening phonon modes.
s48s46s48
s48s46s50
s48s46s52
s48s46s54
s48s46s56
s49s46s48
s48
s49s48s48
s50s48s48
s51s48s48
s48s46s48
s48s46s50
s48s46s52
s48s46s54
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s49s46s48
s72
s32
s40
s109
s101
s86
s47
s97
s116
s111
s109
s41
s102 s122
s102
s120
s45
s121
s48
s54s48s46s48s48
s49s50s48s46s48
s49s56s48s46s48
s50s52s48s46s48
s51s48s48s46s48
Figure 5.13: Enthalpy landscape and its contour plot as a function of fxy and fz at the
transition pressure of 38.5 GPa.
It is interesting to point out that the new P3 phase is dynamically stable above the
transition pressure (i.e., 38.5 GPa). However, one of its TA branch shows a tendency to
vanish at K point below the transition pressure. The atomic displacements according to
the K-point softening mode suggest a structure which still belongs to P3 symmetry, but
the unit cell is three times larger than the previous P3 phase, i.e., six formula units per
primitive unit cell. To distinguish with the previous P3-Si3N4, we will denote this second
P3 structure as P3? phase later on. Taking the P3? structure as the initial structure,
we performed total energy calculations with both internal and external parameters being
full relaxed. P3? phase is found to exist only between the volume of 8 ?A3/atom and 8.75
?A3/atom. Its structure relaxes back to the ? structure for volumes larger than 8.75 ?A3/atom
and becomes P3 phase for volumes smaller than 8 ?A3/atom. Its energy is slightly lower than
124
that of P3 phase by merely a few meV/atom. The structure of P3? phase is very similar to
P3 phase except the z coordinates of each P3 Wycko? site split into three di?erent values
with small deviations. In another word, the P3 phase is a special case of the P3? structure.
As indicated in the calculated enthalpy landscape shown in Figure 5.13, the transition path
is close to the linear path along which fxy and fz vary cooperatively. If we take P3? phase as
an intermediate state connecting ? and P3 structures, the enthalpy barriers at 30 GPa and
38.5 GPa are shown in Figure 5.14 together with the barriers from direct ??P3 transition
as a comparison. The barrier heights along two paths are very comparable. The ?H in
??P3??P3 path is lower than the ??P3 path by only 5.6 meV/atom at 38.5 GPa and
9.8 meV/atom at 30 GPa.
s48
s50s48
s52s48
s54s48
s56s48
s49s48s48
s80
s116s114
s32s61s32s51s56s46s53s32s71s80s97
s32s32s32 s32 s45s80s51
s32s32s32 s32 s45s80s51s39s45s80s51
s80s32s61s32s51s48s32s71s80s97
s32s32s32 s32 s45s80s51
s32s32s32 s32 s45s80s51s39s45s80s51
s80s51s39
s80s51s39
s32
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s40
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s116
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s109
s41
s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s32s80s51
Figure 5.14: Enthalpy barrier (relative to ? phase) as a function of linearly interpreted
transition parameter at 30 GPa (thinner) and the transition pressure of 38.5 GPa (thicker).
Solid curves denote the ??P3??P3 path and the dashed curves denote direct ??P3 path.
Horizontal axis is de?ned as qualitative structural similarity. The left end represents ?
structure and the right end represent P3 structure.
Our predicted P3 phase has a hexagonal symmetry which is di?erent from what Zerr
proposed based on the EDXD pattern78. However, the interplanar spacings for the six
peaks he observed could also be assigned to a crystal system with hexagonal symmetry. A
supportive evidence is that Soignard et al.81 observed a similar ??P3 metastable transition
125
in Ge3N4. They claimed that Zerr?s ?-Si3N4 is likely to be analogous to their observed ?-
Ge3N4 based on comparison of the X-ray di?raction and Raman data. More experimental
works are needed to con?rm the structure of ? phase.
5.6 Thermodynamic Properties
As a direct by-product of our thermodynamic potential calculation, we derived ther-
mal properties for ?-, ?-, and ?-Si3N4 over a wide T-P range. In our vibrational Helmholtz
free energy calculations, at each temperature, the data points of eight volumes (from 9.25
?A to 11.0 ?A) were ?tted to the second-order BM-EOS by the least square ?tting algorithm.
Note that the static EOS Estatic(V ) is ?tted with the 3rd-order BM-EOS models. Fig-
ure 5.15 shows the temperature dependence of the volume thermal expansion of ?-Si3N4
at zero pressure. The experimental data are widely scattered in both low T and high T
regions which may be attributed to the lack of good-quality single crystal samples. There-
fore ?rst-principles calculated thermal properties of bulk ?-Si3N4 can serve as a guide for
interpretation of experiments. To illustrate the numerical uncertainties associated with
the choices of di?erent thermal BM-EOS models, we also include the results that are de-
rived from the static energies ?tted with 2nd-order BM-EOS. The two set of calculations
both agree reasonably with experimental data. Below room temperature the two calculated
curves are almost identical, and they gradually split as temperature increases. At T = 2000
K, the TEC is 1.19 ? 10?5 K?1 for EOS-I (Estatic ?tted with 2nd-order BM-EOS and Fvib
?tted with 2nd-order BM-EOS) and 1.11?10?5 K?1 for EOS-II (Estatic ?tted with 3rd-order
BM-EOS and Fvib ?tted with 2nd-order BM-EOS). The thermal expansivity of EOS-I ?ts
slightly better to the experimental results. For temperatures lower than 150 K our calcula-
tion shows negative thermal expansion, and this has also been pointed out by Kuwabara et
al. in their calculations71. At low temperatures of T < 500 K, our prediction agrees well to
the measured data except Reeber?s, which is apparently di?erent from the others. For tem-
peratures T > 800 K, Henderson?s data has a nearly linear temperature dependence which
126
is questionable. In most temperature range, our calculated thermal expansion coe?cient
from EOS-I is consistent with Bruls? results.
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s48s46s48
s48s46s50
s48s46s52
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s32
s32
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s104
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s109
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s108
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s105
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s32
s75
s45
s49
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s69s120s112s101s114s105s109s101s110s116
s32s72s101s110s100s101s114s115s111s110s32s40s49s57s55s53s41
s32s66s114s117s108s101s115s32s40s50s48s48s49s41
s32s82s101s101s98s101s114s32s40s50s48s48s53s41
s32s83s99s104s110s101s105s100s101s114s32s40s49s57s57s52s41
s84s104s105s115s32s119s111s114s107
s32s69s79s83s45s73
s32s69s79s83s45s73s73
Figure 5.15: Temperature dependence of volume thermal expansion coe?cient of bulk ?-
Si3N4 at zero pressure. Solid and dashed lines show present work with static energies
?tted to the 2nd-order BM-EOS and 3rd-order BM-EOS, respectively, and both thermal free
energies ?tted to the 2nd-order BM-EOS. Discrete symbols denote experimental data83?86.
The negative thermal expansion coe?cient below 150 K can be related to the negative
bulk Gr?uneisen parameter ? in that temperature range. ? = ?CV /(BTV ). Figure 5.16
shows our calculated Gr?uneisen parameter of ?-Si3N4 together with reported experimental
data. The present temperature dependence of bulk Gr?uneisen parameter is in good agree-
ment with Bruls? measured data in the temperature range between 300 K and 1300 K.
The estimated percentage di?erence between experiment and calculation is within 10% for
300 K< T <500 K and the di?erence is gradually reduced to about 2% at T = 1300 K.
Also, one can notice that the Gr?uneisen parameter is not sensitive to the two EOS schemes
we used. The EOS-I scheme yield a larger thermal expansivity and larger volume at high
temperatures, while the EOS-II scheme yield a larger isothermal bulk modulus, and both
EOS schemes give very similar CV curves at all temperatures. The cancelation e?ect leads
to the similarity in the bulk Gr?uneisen parameter.
The bulk Gr?uneisen parameter is the weighted average of mode Gr?uneisen ratios. At
low temperature, only low-frequency phonons, which mostly have negative mode Gr?uneisen
127
s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s45s49s46s48
s45s48s46s56
s45s48s46s54
s45s48s46s52
s45s48s46s50
s48s46s48
s48s46s50
s48s46s52
s48s46s54
s48s46s56
s49s46s48
s69s120s112s101s114s105s109s101s110s116
s32s66s114s117s108s115s32s40s50s48s48s49s41
s84s104s105s115s32s119s111s114s107
s32s69s79s83s45s73
s32s69s79s83s45s73s73
s71
s114
s252
s110
s101
s105
s115
s101
s110
s32
s112
s97
s114
s97
s109
s101
s116
s101
s114
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
Figure 5.16: Temperature dependence of bulk Gr?uneisen parameter of ?-Si3N4 at zero
pressure. Solid and dashed lines show present work with static energies ?tted to the 2nd-
order BM-EOS and 3rd-order BM-EOS, respectively, and both thermal free energies are
?tted to the 2nd-order BM-EOS. Discrete symbols denote experimental data85.
ratios, are thermally excited. This yields the negative overall bulk Gr?uneisen parameters
and consequently it leads to the negative thermal expansion coe?cients. The two branches
that corresponding to the most negative mode Gr?uneisen parameters are found to be the
softening M-point TA and ?-point Bu modes, which are responsible for the instability of
?-Si3N4 at high pressures.
Figure 5.17 shows the calculated TEC of ?-Si3N4 based on EOS-II as a function of
temperature at several pressures up to 30 GPa. As pressure increases from 0 to 30 GPa, the
TEC decreases from 1.11 ? 10?5 to 0.69 ? 10?5 K?1 at 2000 K temperature. The negative
TEC range extends from below 150 K at 0 GPa to 220 K at 30 GPa. The most negative
TEC value also decreases from ?3.11 ? 10?7 to ?5.09 ? 10?7 K?1. This pressure e?ect of
TEC in ?-Si3N4 is in agreement with the calculated pressure e?ect on low frequency phonon
modes and the soft-phonon associated structural instability discussed in earlier sections.
Isobaric heat capacity per atom as a function of temperature at zero pressure is shown
in Figure 5.18. First, both EOS schemes yield very alike curves throughout the plotted tem-
perature region, which implies that CP is not sensitive to the equation of states adopted.
128
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s48s46s48
s48s46s50
s48s46s52
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s48s46s56
s49s46s48
s49s46s50
s49s46s52
s49s46s54
s32s80s32s61s32s32s32s48s32s71s80s97
s32s80s32s61s32s49s48s32s71s80s97
s32s80s32s61s32s50s48s32s71s80s97
s32s80s32s61s32s51s48s32s71s80s97
s32
s32
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s84
s104
s101
s114
s109
s97
s108
s32
s101
s120
s112
s97
s110
s115s105
s111
s110
s32
s40
s49
s48
s45
s53
s32
s75
s45
s49
s41
Figure 5.17: Temperature dependence of volume thermal expansion coe?cient of bulk ?-
Si3N4 at pressures of 0, 10, 20 and 30 GPa.
Below room temperature our calculations are in excellent agreement with all shown exper-
imental results, and the agreement is still good above room temperature except Reeber?s
data. Note that Reeber?s data are considerably di?erent from other experimental results
and theoretical calculations in both thermal expansivity and isobaric heat capacity. At the
same time, we ?nd a persistent agreement with the Bruls? measurements for both CP and
TEC.
Figure 5.19 shows the temperature dependencies of TEC for ?-, ?- and ?-Si3N4 at
zero pressure. The equation of state scheme we adopted is EOS-II. We also present the
experimental TEC of ?-Si3N4 in the same plot. Although the two experimental works do
not agree well with each other, our prediction is consistent with Paszkowicz et al. below
room temperature and the agreement is still reasonable in the temperature range from
300 K to 1000 K. More high quality measurements are required to assess the TEC of ?-
Si3N4. Thermal expansion coe?cients of ?- and ?-Si3N4 are similar in magnitude over all
temperatures and their di?erence at 2000 K is less than 2?10?6 K?1. In Kuwabara et al.?s
calculation, TEC of ? phase was predicted to be slightly smaller than that of ? phase. This
might be ascribed to the sensitive nature of TEC upon the numeric ?uctuation of thermal
data. Both ? and ? phases present negative TEC at low temperature which is consistent
129
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s48s46s48
s48s46s53
s49s46s48
s49s46s53
s50s46s48
s50s46s53
s51s46s48
s32
s32
s67
s80
s32
s40
s107
s66
s47
s97
s116
s111s109
s41
s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s69s120s112s101s114s105s109s101s110s116
s32s75s117s114s105s121s97s109s97s32s40s49s57s55s56s41
s32s75s111s115s104s99s104s101s110s107s111s32s40s49s57s56s50s41
s32s66s114s117s108s115s32s40s50s48s48s49s41
s32s82s101s101s98s101s114s32s40s50s48s48s53s41
s84s104s105s115s32s119s111s114s107
s32s69s79s83s45s73
s32s69s79s83s45s73s73
Figure 5.18: Temperature dependence of isobaric heat capacity of ?-Si3N4 at zero pressure.
Solid and dashed lines show present work with static energies ?tted to the 2nd-order BM-
EOS and 3rd-order BM-EOS, respectively, and both thermal free energies are ?tted to the
2nd-order BM-EOS. Discrete symbols denote experimental data85?88.
with the ?rst-principles calculation from Kuwabara et al. The lowest TEC for the ? phase
is about ?1.5 ? 10?7 K?1 at 90 K, and for the ? phase it is ?3.1 ? 10?7 K?1 at 100 K.
Unlike ?- and ?-Si3N4, TEC of ? phase is about twice as large as the that of ?- or ?-Si3N4
in the temperature range from room temperature to 2000 K. At 2000 K, we predict a TEC
of 2.3 ? 10?5 K?1 which is in good agreement with Kuwabara et al.?s ab initio calculated
2.2 ? 10?5 K?1. Paszkowicz et al. pointed out that their measured TEC tends to vanish
for T < 100 K90. However, our calculation is not clearly supportive. Similarly, two other
?rst-principles calculations from Paszkowicz et al. and Kuwabara et al. do not show the
negative or vanishing feature either.
5.7 Conclusions
In summary, we have theoretically studied phase transitions in silicon nitride (Si3N4)
at high pressure using a ?rst-principles density functional theory method. We ?nd that
?-Si3N4 remains as a metastable phase at temperatures up to 2000 K and pressures up
to 10 GPa. The equilibrium ??? transition pressure is predicted as 7.5 GPa at 300K
130
s48 s53s48s48 s49s48s48s48 s49s53s48s48 s50s48s48s48
s48s46s48
s48s46s53
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s32
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s108
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s84s101s109s112s101s114s97s116s117s114s101s32s40s75s41
s69s120s112s101s114s105s109s101s110s116s32s40 s45s83s105
s51
s78
s52
s41 s84s104s101s111s114s121s32s40s116s104s105s115s32s119s111s114s107s41
s32s72s105s110s116s122s101s110s32s40s50s48s48s51s41 s32 s32s112s104s97s115s101
s32s80s97s115s122s107s111s119s105s99s122s32s40s50s48s48s52s41 s32 s32s112s104s97s101
s32 s32s112s104s97s115s101
Figure 5.19: Temperature dependence of volume thermal expansion coe?cient of ?-, ?- and
?-Si3N4 at zero pressure. Discrete symbols represent experimental data89,90.
and it increases to 9.0 GPa at 2000K. Both ?- and ?-Si3N4 are dynamically stable at low
pressure. However, two competing phonon-softening mechanisms are found in the ? phase
at high pressures. At room temperature, ?-Si3N4 is predicted to undergo a ?rst-order
??P3 transition above 38.5 GPa, while ?-Si3N4 shows no signs of structural instability.
The predicted metastable high-pressure P3 phase is structurally related to ?-Si3N4. The
enthalpy barrier height is estimated as only 67.23 meV/atom. Our LDA predicted thermal
expansion coe?cient, heat capacity and bulk Gr?uneisen parameter are in good agreement
with Bruls? measured results. We ?nd relatively large discrepancies between our calculation
with experimental data from Reeber. And we attribute the cause of predicted negative
TEC at low temperatures in ? and ?-Si3N4 to the low-frequency phonon modes that have
negative mode Gr?uneisen ratios.
131
Chapter 6
FIRST-PRINCIPLES STUDY OF GALLIUM OXIDE AND GALLIUM
OXYNITRIDE
6.1 Gallium Oxide: Ga2O3
6.1.1 Introduction
The oxides of group 13 elements (Al, Ga, In) are important solid-state compounds with
applications in ?elds ranging from structural ceramics to catalysts and electronic materi-
als201. Monoclinic gallium oxide (Ga2O3) is usually known as a wide-band-gap semiconduc-
tor (Eg = 4.9 eV) ; however, the conductivity can be varied from insulating to conducting
behavior depending upon the preparation conditions93. Due to its tunable optical and elec-
tronic properties, ?-Ga2O3 is being developed for use in a wide variety of applications, for
instance, as optical windows202, in high-temperature chemical gas sensors203, as a mag-
netic memory material204, and for dielectric thin ?lms205. Recently, considerable e?ort has
been devoted to the study of low-dimensional Ga2O3 materials, and ?-Ga2O3 nanowires
have been obtained through physical evaporation and arc-discharge methods101. ?-Ga2O3
has also attracted recent interest as a phosphor host material for applications in thin ?lm
electroluminescent displays206,207. Due to its chemical and thermal stability, ?-Ga2O3 may
emerge as a useful alternative to sul?de based phosphors208.
It is well known that Ga2O3 can exist in several forms, including ?, ?, ?, ?, and ? poly-
morphs that all have di?erent structure types94. Of these, the most stable form at ambient
conditions is determined to be ?-Ga2O3 (monoclinic C2/m, Figure 6.1)94. However, other
metastable varieties can be prepared and they have been characterized at ambient pressure
and temperature. This is an important observation, because the di?erent forms have dra-
matically di?erent optoelectronic properties. For example, the band gap of the ?-Ga2O3
132
Figure 6.1: Crystal structure of monoclinic ?-Ga2O3 phase. Note that Ga3+ cations (light
color) occupy both tetrahedral and octahedral interstices within the ccp lattice of O2? ions
(dark color).
polymorph that is isostructural with corundum (?-Al2O3) is 2.41 eV, much narrower than
that of ?-Ga2O3209.
It is of great interest to determine the pressure-induced phase transformations among
Ga2O3 polymorphs in order to establish the stable and metastable phase relations between
di?erent crystalline modi?cations, and to evaluate their production under di?erent synthe-
sis conditions. It is particularly important to understand the role of di?erential mechanical
stresses that are present in creation of nanoparticles or nanowires, in promoting the forma-
tion of speci?c polymorphic forms.
The high-pressure behavior of Al2O3 compounds has been studied extensively, par-
ticularly the corundum-structured ?-Al2O3 phase, because of its importance as a mineral
structure within the deep Earth and also due to the widespread use of ruby (Cr3+-doped
?-Al2O3) as a luminescent pressure gauge for in situ high-pressure experiments in the di-
amond anvil cell1. Cr3+-doped ?-Ga2O3 has likewise been proposed as a pressure gauge
material. The R1 luminescence line in this phase shows a pressure shift nearly three times
that of ruby, indicating that it would make a more sensitive pressure sensor that is especially
133
useful in the lower pressure range210. In situ high-pressure and high-temperature measure-
ments on ?-Al2O3 using synchrotron x-ray di?raction in a diamond anvil cell, combined
with ab initio theory predictions, have now been used to characterize a transition into the
Rh2O3-II structure occurring at P ? 100 GPa and T >? 1000 K5,8.
The high-pressure behavior of Ga2O3 has received much less attention. The vari-
ous low-density Ga2O3 structures encountered at low pressure contain the Ga3+ cations in
tetrahedral coordination (i.e., GaO4 species). The thermodynamically stable ?-Ga2O3 poly-
morph is isomorphous with the metastable ?-Al2O3 structure, which represents a key phase
achieved during metastable transformations among various partially dehydrated ?transi-
tional? aluminas as they evolve towards corundum211. ?-Al2O3 constitutes an intermediate
structure between the cubic close packing of anions achieved within the low-temperature
metastable aluminas, and hexagonally close-packed ?-Al2O3 corundum (isomorphous with
?-Ga2O3).
In a recent study using synchrotron energy-dispersive x-ray di?raction techniques in
the diamond anvil cell, it was reported that a sample of ??-Ga2O3? transformed to a
structure assigned to be tetragonal at a pressure of approximately 13.3 GPa.212 However,
the x-ray di?raction pattern of the starting material most strongly resembled that of ?-
Ga2O3, rather than the ?-form, and a mixture of phases was present. Commercial Ga2O3
samples usually consist mainly of ?-Ga2O3, along with some ?-Ga2O3; that phase can
be removed by heat treatment94. The relative densities of ?- and ?-Ga2O3 are 5.94 and
6.48 g?cm?3, respectively95, indicating that a ??? transformation should occur at high
pressure. Nanocrystalline ?-Ga2O3 particles embedded in a glassy matrix were also studied
at high pressure using energy-dispersive x-ray di?raction96. In that work, a ?-to-? phase
transformation was found to be initiated at 6 GPa, but the process was not completed
by 15 GPa, the highest pressure achieved in the study. However, it is known that the
silica glass host matrix undergoes important structural and density changes within this
pressure range97,98, so that it is not yet known if the structural changes are intrinsic to the
?-Ga2O3 material presumably in?uenced by the nanocrystalline nature of the sample , or
134
are promoted by anomalous densi?cation among the SiO2 matrix. These results prompted
us to theoretically investigate the high-pressure behavior of the phase-pure bulk ?-Ga2O3,
accompanied with an experimental study using Raman spectroscopy and high-resolution
synchrotron x-ray di?raction angle dispersive techniques.
Recently, two further pressure-induced phase transitions have been con?rmed from
both ?rst-principles calculations and high-pressure x-ray di?raction measurements using
laser-heated diamond-anvil cell (LHDAC). The further transition sequence found in Ga2O3
is the same as that in Al2O3, i.e., ??Rh2O3(II)?postperovskite (Cmcm CaIrO3-type).
The experimentally determined Pt for these two transitions are ?37 GPa at 2000?100 K
and 164 GPa at 1300?500 K.
One-dimensional nanostructured forms of ?-phase of gallium oxide (?-Ga2O3) such
as nanotubes, nanobelts, and nanowires, have attracted recent interest due to enhanced
optical properties99,100. Recently, Choi et al.101 synthesized ?-Ga2O3 nanowires (diameter
range of 15?45 nm) with a [001] growth direction using an arc-discharge method. Gao
et al.102 synthesized [40?1] ?-Ga2O3 nanowires with diameters ranging from ?10?100 nm
in a vertical radio-frequency furnace. Interestingly, the Raman mode frequencies of the
[001] ?-Ga2O3 nanowires coincide with the corresponding frequencies in bulk ?-Ga2O3101.
On the other hand102, the Raman mode frequencies of the [40?1] ?-Ga2O3 nanowires are
redshifted relative to corresponding frequencies in bulk ?-Ga2O3 by 4?23 cm?1. Using
plasma-enhanced chemical vapor deposition, Rao et al. have synthesized ?-Ga2O3 nanowires
whose growth is along the [110] direction103, and the Raman spectrum is signi?cantly
blueshifted in frequency104. Here we focus on the ?rst-principles calculations of the Raman
mode frequencies under internal strains. Our calculated Raman frequency shifts suggest
that the observed shifts in the nanowires with the [40?1] and [110] growth directions can
be explained in term of di?erent internal strains, in contrast to the previously suggested
quantum con?nement e?ects and defect-induced e?ects.
135
Table 6.1: Third-order BM-EOS parameters for Ga2O3 polymorphs
Source Phase V0 (?A3/atom) B (GPa) B?
This work ? 10.134 177 3.84
Calculation214 10.228 142 4.1
Experiment213 202(7) 2.4(6)
Experiment96 191(5) 8.3(9)
Experiment214 10.462 134(12) 4
This work ? 9.375 226 3.971
Calculation214 9.448 243 4.0
Experiment213 ?250
Experiment214 9.632(1) 223(2) 4
This work Rh2O3(II) 9.130 234 3.98
Calculation214 9.188 244 4.3
Experiment214 9.244(4) 271(10) 4
6.1.2 Total Energy and Vibrational Properties
During the course of the experimental studies from our collaborators104,213, we began
a parallel theoretical investigation of the structures, relative energetics, and properties of
Ga2O3 phases. Our calculations are based on the ?rst-principles density functional theory
within the local density approximation (LDA). The calculations were carried out with the
VASP codes139?142, using planewave basis sets and ultrasoft pseudopotentials (US-PP)121.
In this study, both valence (4s4p) and semi-core (3d) electrons in Ga atoms were treated
explicitly, while the core electrons were approximated with the US-PP. The energy cuto? of
the plane-wave basis was chosen as 396 eV. The Brillouin zone integration of total energy
of the unit cells was carried out using 6 ? 6 ? 6 grids for both phases.
Equilibrium volume V0, bulk moduli B and B? for ?, ? and Rh2O3(II) phases are
obtained from the least-square ?tting to the 3rd-order Birch-Murnaghan equation of state.
Fitting parameters V0, B and B?, are listed in Table 6.1. Our LDA results agree reasonably
with other calculated and experimental data. Compared with the experimental values of
V0, our prediction has an underestimation of less than 3%. The static enthalpies of three
polymorphs relative to the ? phase are plotted in Figure 6.2 as a function of pressure. It
136
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s45s48s46s49s48
s45s48s46s48s53
s48s46s48s48
s48s46s48s53
s48s46s49s48
s48s46s49s53
s48s46s50s48
s48s46s50s53
s69
s110
s116
s104
s97
s108
s112
s121
s32
s68
s105
s102
s102
s101
s114
s101
s110
s99
s101
s32
s40
s101
s86
s47
s97
s116
s111
s109
s41
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s32 s32s112s104s97s115s101
s32 s32s112s104s97s115s101
s32s82s104
s50
s79
s51
s40s73s73s41
s71s97
s50
s79
s51
Figure 6.2: Enthalpy di?erences relative to ? phase for ?, ? and Rh2O3(II)-Ga2O3 as a
function of pressure up to 60 GPa. The ? and ? curves cross at 0.5 GPa, the ? and
Rh2O3(II) curves cross at 40.9 GPa.
can be seen that ? phase is predicted to transform into ? phase at 0.5 GPa and ? phase
further transforms into Rh2O3(II) phase at 40.9 GPa.
The phonon dynamical matrices Dij (k) were constructed at the Brillouin zone center
(?-point: k = 0) using a (realspace) force constant matrix ?ij (r) by calculating forces on
each atom as it is slightly displaced from its equilibrium position (e.g., by 0.015 ?A). Further
approximations were adopted to calculate the Dij (k) matrix at a general k-point. In the
case of ?-Ga2O3, we ?rst obtained the real space ?ij (r) matrix using a 120-atom supercell
model. Because of the large size of the supercell model, and the fact that the material is
insulating wide-gap semiconducting , we can safely neglect interatomic interactions between
atoms separated by >50% of the supercell lattice constants, and thus obtain Dij (k) by
Fourier transformation of the real-space matrix elements ?ij (r).
Fifteen Raman-active modes are expected for the ?-Ga2O3 structure (point symmetry
C32h) from symmetry analysis:
?Raman = 10Ag + 5Bg (6.1)
137
Table 6.2: Calculated and experimental zone-center Raman peak positions and Gr?uneisen
parameter for the ?-Ga2O3 phase
Frequency (cm?1) Gr?uneisen ratio
calculated measured
Mode This work Empirical This work
symmetry (LDA) calculation215 Machon et al.213 Rao et al.104 (LDA) (measured213)
Ag 104 113 110.2 1.39
Bg 113 114 113.6 -0.7
Bg 149 152 144.7 144 1.53 1.97(8)
Ag 165 166 169.2 169 1.00 0.35(3)
Ag 205 195 200.4 200 1.30 0.98(2)
Ag 317 308 318.6 317 1.13 0.95(1)
Ag 346 353 346.4 344 1.83 1.52(1)
Bg 356 360 1.47
Ag 418 406 415.7 416 0.58 0.78(4)
Ag 467 468 472 1.26
Bg 474 474 473.5 1.14 1.27(9)
Ag 600 628 629 1.70
Bg 626 644 628.7 0.8 1.54(3)
Ag 637 654 652.5 654 1.39 1.39(2)
Ag 732 760 763.9 767 1.23 1.11(1)
The calculated frequencies and their mode Gr?uneisen parameters are reported in Ta-
ble 6.2. The Raman-active modes of ?-Ga2O3 can be classi?ed into three groups: high-
frequency stretching and bending of GaO4 tetrahedra (?770?500 cm?1), midfrequency de-
formation of Ga2O6 octahedra (?480?310 cm?1) , and lowfrequency libration and transla-
tion (below 200 cm?1) of tetrahedra-octahedra chains215. The listed experimental values of
Machon et al. are from an unpolarized Raman spectrum of ?-Ga2O3 recorded from pow-
dered material obtained by annealing a commercial sample213. Our calculated frequencies
agree well with the observed values, to within 0.1%?6%, which is typical for LDA calcula-
tions (Table 6.2). The Bg mode predicted at 356 cm?1 was likely unresolved from the Ag
mode at 346 cm?1 in the measured Raman spectrum213; however, a weak peak at this fre-
quency was recorded by Dohy et al.215. The band observed at 474 cm?1 also likely contains
contributions from the calculated Ag and Bg modes at 467 and 474 cm?1. Surprisingly,
we found no experimental evidence for the predicted Ag mode at 600 cm?1. We have no
explanation for that observation. Symmetry analysis indicates that seven Raman active
modes (2A1g + 5Eg) are expected for the corundum structure. Our theoretically calculated
frequencies agree to within 0.7%?5.5% with the observed values (Table 6.3).
138
Table 6.3: Calculated and experimental zone-center Raman peak positions for the ?-Ga2O3
Frequency (cm?1)
Mode symmetry (calculated) (measured)
A1g 215 217.4
Eg 239 240.8
Eg 281 286.1
Eg 344 328.7
Eg 410 432.2
A1g 551 573
Eg 680 688.1
6.1.3 T-P Phase Diagram
The monoclinic ?-Ga2O3 structure is the stable polymorph at ambient pressure and
temperature. In this phase, the O2? anions form a slightly distorted fcc lattice and cations
occupy tetrahedral and octahedral interstices (Figure 6.1). This structure is quite di?erent
from that of the ?-Ga2O3 phase (corundum structure), which is based on a distorted hcp
O2? sublattice with 2/3 of the octahedral interstices occupied by Ga3+ ions . The ???
transition is expected to result from increasing the pressure, from the observed density
relationships between the two phases. The transformation involves a change in the O2?
packing from cubic to hexagonal, accompanied by a shift in Ga3+ ions between tetrahedral
and octahedral sites. The reconstructive nature of the transition indicates that it is ther-
modynamically of the ?rst order, and it might be expected to involve a large activation
energy, which gives rise to slow transformation kinetics at low temperature. In addition,
the high-density structure has signi?cant possibilities for disorder among the Ga3+ positions
on octahedral sites within the hcp O2? sublattice, which might not be readily detected by
x-ray di?raction8.
?-? and ?-Rh2O3(II) phase boundaries of Ga2O3 are plotted in Figure 6.3. The cal-
culated transition pressure from ? to ? phase is 0.3 GPa at 0 K and 1.6 GPa at 2000
K. The Clapeyron slope for this transition is positive which has the value of about +0.6
MPa/K at 1000 K. Our predicted Pt is consistent with reported LDA calculation from Yusa
et al 214. On the experimental side, ?-Ga2O3 was synthesized from ?-Ga2O3 at 4.4 GPa
139
s48
s53s48s48
s49s48s48s48
s49s53s48s48
s50s48s48s48
s48 s49s48 s50s48 s51s48 s52s48 s53s48 s54s48
s32
s32
s80s114s101s115s115s117s114s101s32s40s71s80s97s41
s84
s101
s109
s112
s101
s114
s116
s117
s114
s101
s32
s40
s75
s41
s82s104
s50
s79
s51
s40s73s73s41
s71s97
s50
s79
s51
Figure 6.3: T-P phase diagram for ?, ? and Rh2O3(II)-Ga2O3 polymorphs.
and 1000 ?C95,216. However, the room temperature compression experiments obtain much
higher transition pressures. Tu et al. have reported a ?-to-? phase transition in Ga2O3 at
13.3 GPa212. But there is some uncertainty in the nature of the starting material used in
that work. Recently, x-ray di?raction and Raman scattering results from Machon et al.213
on ?-Ga2O3 clearly indicate that a pressure-induced phase transformation occurs within
the P = 20 ? 22 GPa range, and perhaps as low as P = 18.5 GPa. Analysis of the x-ray
di?raction data suggest that the high-density phase corresponds to corundum-structured
?-Ga2O3. However, broadening observed both in the x-ray di?raction peaks and in the
Raman spectra indicate that the material is structurally disordered. The large discrepancy
on Pt may be caused by a large kinetic barrier, which is signi?cantly lowered under com-
pression. Our calculated ?-to-Rh2O3(II) transition pressure is 40 GPa at 0 K and 37 GPa
at 2000 K, which is in excellent agreement with the experimental value (about 37 GPa at
2000?100 K)214. The phase boundary we predict shows negative Clapeyron slope which is
about -1.65 MPa/K at 1000 K. Tsuchiya et al. reported their calculated Clapeyron slope
of the ?-Rh2O3(II) boundary, which is -2.2 MPa/K at 1000 K217. The discrepancy may be
ascribed to the adopted computational methodologies.
140
6.1.4 Blue-Shifted Raman Scattering in Gallium Oxide Nanowires
Rao et al.104 reported the Raman and Fourier transform infrared spectra of ?-Ga2O3
nanowires with [110] growth direction which is blueshifted relative to the bulk spectra by
10?40 cm?1. The blueshift in phonon frequencies of low-dimensional materials are often
attributed to the size-con?nement e?ect218,219. However, the average diameter of the ?-
Ga2O3 nanowires is around 25 nm. It is unlikely that the quantum size con?nement at
this length scale is signi?cant enough to cause the phonon shifts as large as 50 cm?1.
Furthermore, three distinctly di?erent shift patterns have been experimentally observed for
the ?-Ga2O3 nanowires of di?erent growth directions. In contrast to the blueshift in the
Raman and FTIR spectra reported by Rao et al., Choi et al. showed that their Fourier
transform Raman spectrum of [001] ?-Ga2O3 nanowires is identical to that of bulk ?-
Ga2O3101, while Gao et al. exhibited a redshift of 4?23 cm?1 in the Raman peak frequencies
of their [40?1] ?-Ga2O3 nanowires relative to the corresponding Raman frequencies in bulk
?-Ga2O3102. The size con?nement e?ect is clearly insu?cient to explain the diversity of
the observed shift patterns.
On the other hand, the redshift in the phonon frequencies has also been attributed to
the presence of impurities and defects, such as point defects, twins, and stacking faults220.
These defects are also likely to be responsible for additional vibrational modes observed
in the Raman spectra (and to a small extent in the FTIR spectrum) of nanowires. From
a detailed high-resolution transmission electron microscopy (HRTEM) study, Gao et al.
con?rmed the presence of twins and edge dislocations in their nanowires102. Dai et al.221
also proposed that the O vacancies and the stacking faults caused an abnormality in the
Ga?O bond vibration and led to redshift in the Raman frequencies. Although this simple
hypothesis is plausible, there is one obvious weakness, i.e., lack of close correlation between
defect types and the growth directions. Presumably, similar defects might exist in the
nanowires with di?erent growth directions. It is also not clear which types of defects will lead
to a blueshift in vibrational frequencies. Moreover, di?erent regions in the nanowires contain
di?erent defects which would imply that di?erent shifts in the Raman and/or IR spectra
141
should be observed when di?erent regions of the same nanowire are probed. However, this
does not seem to be the case and instead overall distinct blueshifts or redshifts have been
observed for a given nanowire. Therefore, alternative models that are capable of describing
these diverse peak-shift patterns in a consistent fashion are needed.
Based on a ?rst-principles calculation which we describe next, we propose that the
phonon frequencies in di?erent ?-Ga2O3 nanowires are shifted as a result of internal strains
in the nanowire. The basic assumption of our model is the presence of non-negligible internal
strains in the nanowires due to their large surface/volume ratio. Di?erent growth direc-
tions will cause di?erent surface reconstruction, and consequently lead to internal strains
of di?erent magnitudes and directions. This model provides a consistent explanation for all
three aforementioned Raman spectra.
Direct ?rst principles calculations of phonon frequencies of 25-nm-diam nanowires is a
computationally challenging task as large supercell models (of at least tens of thousands of
atoms) are needed. Instead, our current computation study focuses on providing a quan-
titative estimation of the internal strains which can account for the observed blue- and
redshifts in the Raman frequencies for [110] and [40?1] ?-Ga2O3 nanowires, respectively.
We have calculated the strain dependencies of the bulk ?-Ga2O3 using a density functional
theory (DFT) method. The internal strains of the nanowires were estimated based on the
least-squares ?tting of the experimentally observed Raman frequency shifts with theoret-
ically predicted linear strain coe?cients d?/d?ij, where ? and ?ij are Raman frequencies
and components of strain tensors, respectively.
The ?-point phonons of bulk ?-Ga2O3 were calculated with a real-space ?nite displace-
ment technique129. Because of its C2h space-group symmetry, the LO-TO splitting in the
optic modes of ?-Ga2O3 only exist for the infrared (Au and Bu) phonon modes, not the
Raman active (Ag and Bg) phonon modes. Therefore, all our Raman frequency calculations
of ?-Ga2O3 were carried out with 10-atom base-centered monoclinic unit-cell model without
the correction for the macroscopic interaction. As shown in Table 6.2, the theoretical data
for bulk ?-Ga2O3 matches well with 13 out of the 15 Raman active (10Ag + 5Bg) modes
142
Table 6.4: Estimated internal strains
[110] [40?1]
Strain nanowire nanowire
?11 -0.0077 0.0029
?22 0.0180 -0.0064
?33 -0.0311 0.0106
?13 0.0233 -0.0256
?V/V -0.0208 0.0071
observed experimentally104, as well as those of the previous study of Dohy et al 215. In both
cases of the unobserved Raman modes, there is another Raman active mode in the close
proximity. For example, our LDA calculations predicted two Raman modes at 469 and 474
cm?1, and two Raman modes at 601 and 629 cm?1. This suggests that it is possible that
the two ?missing? Raman modes are hidden by the stronger adjacent Raman modes.
The strain tensor of this monoclinic crystal has six independent elements, ?11, ?22,
?33, ?23, ?13 and ?12. For simplicity, we restricted this study to linear e?ects, i.e., ? (?) ?
?0 +summationtext(d?/d?ij)??ij. This approximation is valid for small strains. We further neglected
the strain of ?23 or ?12 because their d?/d?ij coe?cients are zeroes due to the monoclinic
lattice symmetry. For each of four remaining types of strains (?11, ?22, ?33, and ?13), the
Raman frequencies were calculated for ?ve ?nite strain values between -0.02 and +0.02.
The calculated frequencies were then ?tted with a polynomial function to obtain the linear
strain coe?cients.
Fitting the experimental data within our strain-induced phonon shifts model, we predict
the internal strains in nanowires which showed the three distinct Raman spectra. The results
of the [40?1] and [110] nanowires are listed in Table 6.4, and our model predicts the strain
tensor for the [001] nanowires contain non-negligible ?11, ?22, ?33, and ?13 components. As
shown in Table 6.5, we obtain overall excellent ?ts for both the redshifted and blueshifted
Raman spectra, with exception of the 134 cm?1 Bg mode in the [40?1] nanowire (Gao et al.).
Our calculation shows that the [110] nanowire is compressed along its a and c axis, and
stretched along its b axis. The strain in the [40?1] nanowire exhibits a contrasting pattern
and its strain magnitude is only about 1/3 of that evaluated for the [110] nanowire. In both
143
Table 6.5: Raman mode frequencies and frequency shifts in ?-Ga2O3 nanowires with the
[40?1] and [110] growth directions. Overall, excellent agreement between the observed and
calculated shifts is seen for all mode frequencies except the one marked with an *.
Gao et al.102 This work
[40?1] growth direction [110] growth direction
Frequency Calculated Frequency Calculated
Bulk Nanowire shifts frequency Bulk Nanowire shifts frequency
frequency frequency ?? shifts frequency frequency ?? shifts
(cm?1) (cm?1) (cm?1) (cm?1) (cm?1) (cm?1) (cm?1) (cm?1)
-4.8 12.0
1.7 -3.1
142 134 -8* -1.0 144 3.8
167 160 -7 -7.2 169 180 +11 10.6
198 194 -4 -6.5 200 213 +13 13.5
320 -3.9 317 9.5
344 332 -12 -6.7 344 16.8
-1.7 0.7
415 409 -6 -5.3 416 428 +12 12.3
473 -4.4 472 492 +20 21.1
-6.2 18.5
627 -5.7 629 645 +16 17.8
-0.3 3.6
651 641 -10 -14.8 654 697 +43 36.4
765 742 -23 -20.6 767 810 +43 47.0
cases, the a axis has the smallest change (Table 6.4). The strain-induced volume changes
are predicted to be -2% and 0.7% for the [110] and [40?1] nanowires, respectively. Seo et
al.222 studied the internal strains of GaN nanowires using x-ray measurements and they
reported the strains of ?xx=2.3%, ?yy=-0.734%, and ?zz=-0.4% based on their experimental
x-ray measurement. The magnitudes of our predicted strains of ?-Ga2O3 are comparable
to those of GaN nanowires.
6.1.5 Conclusions
For bulk Ga2O3, we have calculated the Raman frequencies and their mode Gr?uneisen
parameters for ? and ? phases. Good agreement is achieved between our results with
experiments and other calculations. We also predict the equilibrium T-P phase diagram
of Ga2O3 consists of ?, ? and Rh2O3(II) phases. Our LDA calculated transition pressure
from ? phase to ? phase is 0.3 GPa at 0 K and 1.6 GPa at 2000 K. The Clapeyron slope
144
for this transition is positive which has the value of about +0.6 MPa/K at 1000 K. This is
consistent with previous calculations. However, the experimental Pt at room temperature
is much larger (20?22 GPa). The discrepancy can be attributed to the existence of a large
kinetic barrier, which may be signi?cantly lowered under compression. Our calculated ?-to-
Rh2O3(II) transition pressure is 40 GPa at 0 K and 37 GPa at 2000 K, which is in excellent
agreement with the experimental value (about 37 GPa at about 2000 K).
In the case of nanowires, based on a comparison of the experimental Raman mode
frequencies with our ?rst-principles calculations, we ?nd compelling evidence for growth
direction induced internal strains in ?-Ga2O3 nanowires which signi?cantly in?uence the
vibrational mode frequencies. Within the linear model approximation, the observed blue
and redshifts of peak frequencies in the micro-Raman spectra of the ?-Ga2O3 nanowires with
di?erent growth directions can be attributed to two small anisotropic internal strains: one
compressive strain of 2% volume change, and the other tensile strain of 0.7% volume change.
The overall high quality of the ?tted models to available experimental data suggests a strong
correlation between the shifts in Raman mode frequencies and the growth direction-induced
internal strains in the Ga2O3 nanowires.
6.2 Gallium Oxynitride: Ga3O3N
6.2.1 Introduction
Semiconducting materials with a wide bandgap are of interest for applications in high-
temperature electronics and in optoelectronic devices, particularly when the gap is direct.
The oxides and nitrides of gallium are wide-gap semiconductors that give rise to materials
that are useful in the blue to UV range at short wavelength. The cubic (sphalerite) and
hexagonal (wurtzite) forms of GaN have bandgap energies of 3.3223 and 3.4 eV224, respec-
tively. When alloyed with In and Al, hexagonal GaN-based materials with a direct gap
have been developed for use in light-emitting diodes and lasers at wavelengths extending
145
from the blue to the ultraviolet range225. The thermodynamically stable ?-Ga2O3 poly-
morph has a direct gap of 4.7 eV, and it has also been proposed for development as a
solid-state LED material for UV applications226. The group 13 oxynitride materials have
other useful properties related to their electronic structure. ?-Ga2O3 with the corundum
structure is conveniently alloyed with Al2O3 to provide selective reduction catalysts for
gaseous NOx105, and various other Ga2O3 phases have been proposed as gas sensors, and in
nanoscale structures as electron emitters and magnetic memory materials106. Within the
Al2O3?AlN system, several important AlxOyNz ceramic alloys and compounds are known.
At high AlN contents, layered forms based on hexagonal/cubic intergrowths are present.
As the Al2O3 content is increased, cubic spinel-structured materials begin to appear. A
large family of defect spinels (?-Al2O3, AlxOyNz) contain vacancies on both cation and
anion sites107. A stoichiometric oxynitride spinel-structured compound is obtained at the
Al3O3N composition, in which Al3+ ions are present on the octahedral and tetrahedral sites,
and O2? and N3? occupy tetrahedral anion sites108,109.
Among the related nitride compounds Si3N4 and Ge3N4, high-pressure synthesis has
recently resulted in formation of a new class of spinel structures, that contain Si4+ and Ge4+
cations on both tetrahedral and octahedral sites110?114. The new solid-state compounds are
recoverable to ambient conditions, and they possess high hardness and low compressibility,
comparable with materials such as Al2O3-corundum107. The new group 14 nitride spinels
are also predicted to be wide direct band semiconductors, with band gaps calculated to lie
within the range 2.2-4.0 eV82,227. The analogous spinel-structured compound Sn3N4 has
also been made at ambient pressure228,229, raising the possibility of future preparation of
?-(Si,Ge)3N4 ?lms via metastable synthesis routes, such as chemical vapor deposition, to
yield materials compatible with optoelectronics applications, for example.
Gallium oxynitride (Ga3O3N) has been predicted to form a new spinel-structured com-
pound within the Ga2O3?GaN system, with potentially useful electronic properties115,116.
It is predicted to be a direct wide bandgap semiconductor, comparable with GaN115. There
146
has previously been an experimental report of a cubic gallium oxynitride phase with com-
position close to Ga2.8O3.5N0.5, that formed metastably during GaN thin ?lm synthesis
from chemical precursors117,118. Here, we report our ?rst-principles theoretical study of the
formation energetics, stability, and electronic properties of the Ga3O3N spinel-structured
phase, combined with experiments using a combination of high pressure-high tempera-
ture techniques to establish the formation and stability of spinel-structured Ga3O3N from
Ga2O3+GaN mixtures, and to determine the chemical composition, structure and proper-
ties of the resulting materials.
6.2.2 Total Energy Calculations of GaN and Ga3O3N
In order to obtain the Gibbs free energy of formation of gallium oxynitride, Gibbs free
energies of Ga2O3, GaN and Ga3O3N are required. As an approximation, assuming a can-
celation e?ect on the vibrational entropies between Ga2O3+GaN and Ga3O3N, the static
formation enthalpy ?H (P) is adopted as an estimation of ?G(T,P). Enthalpies of Ga2O3
polymorphs as a function of pressure has been shown in section 6.1.2. For GaN, previous
studies showed that the ground wurtzite structure undergoes a phase transition into the
rocksalt structure at high pressure. The experimentally observed transition pressure is at
37-52 GPa230?232 and the theoretically calculated Pt is 35-52 GPa43,233?235. In this study,
the synthesis pressure of Ga3O3N (several GPa) is much lower than the WZ-to-RS Pt of
GaN. Therefore the only polymorph we considered for GaN is the wurtzite phase, which
is one of the starting materials (99.99% Ga2O3, containing a mixture of ?+? phases, and
99.99% pure GaN) in the experimental work236. Here, we ?rst present our total energy
calculation of wurtzite-GaN and spinel-Ga3O3N. The calculation was on the basis of ?rst-
principles density functional theory (DFT), implemented by the VASP code139?142. The
many-electron exchange-correlation interaction was approximated within the local density
approximation (LDA). For parts of the study that were associated with small energy dif-
ferences, we compared the LDA results with calculations using the generalized gradient
approximation (GGA). To improve numerical e?ciency, core electrons were approximated
147
Table 6.6: Third-order Birch-Murnaghan EOS parameters, zero pressure structural param-
eters for wurtzite GaN
Source B0 (GPa) B? a (?A) c/a u
This work (LDA) 198 4.01 3.144 1.631 0.376
This work (GGA) 175 4.15 3.187 1.633 0.375
Calculation43 196 4.3 3.180 1.632 0.376
Calculation184 172 5.11 1.632 0.376
Experiment230 245 0.377
Experiment231 188 3.2 3.191 1.626
Experiment232 237 4.3 3.191 1.627 0.377
with ultrasoft pseudopotentials (US-PP)121, and only the s and p valence electrons in these
elements and the semicore 3d electrons in Ga were treated explicitly. The wave functions of
the valence and semi-core electrons were expanded using a planewave basis, with a kinetic
energy cut-o? set at 348 eV for GaN and 396 eV for Ga3O3N.
The calculated static energies at various volumes are ?tted to the 3rd-order Birch-
Murnaghan EOS. General agreement of ?tting parameters, B0 and B?, and zero pressure
structural parameters is achieved between our calculation with other reported values (Table
6.6). GGA calculated bulk modulus is about 12% less than that of the LDA value, which
is also the case from other calculations. The experimental B0 is scattered from 188 to 245
GPa, as well. Despite of the underestimation of B0, the lattice parameter a by GGA gives
a better agreement with other reported values.
Results for Ga-O-N phases relevant to the synthesis of the spinel-structured Ga3O3N
oxynitrides and the stability and properties of that compound are described below.
6.2.3 Theoretical Study of the Synthesis, Structure, and Stability of Ga3O3N
Spinel
We performed ab initio calculations of the atomic structures and energetic properties
of the Ga3O3N systems to understand the formation and thermodynamic stability of the
spinel-structured phase. The ideal spinel crystal has an A3X4 stoichiometry (A and X
represent cations and anions, respectively) with two molecular equivalents per primitive unit
148
cell corresponding to the fcc structure. Assuming a Ga:O:N ratio of 3:3:1 that corresponds
to the ideal stoichiometry , a solid of n Ga3O3N molecular units contains n tetrahedrally
coordinated Ga atoms (labeled as IV Ga), 2n octahedrally coordinated Ga atoms (V IGa),
3n O atoms, and n N atoms. The Gibbs free energy of formation of the oxynitride from a
mixture containing the corresponding oxide and nitride is de?ned as
?Gformation (T,P) = GGa3O3N (T,P) ? [GGa2O3 (T,P) + GGaN (T,P)] (6.2)
Here G represents the Gibbs free energies for each molecular unit, and a negative ?Gformation
corresponds to a driving force for the formation of the oxynitride. Because the oxynitride
spinel can have O or N atoms distributed among the anion sites, it is likely that there
will be a large con?gurational entropy term contained within ?Gformation. A complete
statistical modeling of the Gibbs free energy of Ga3O3N material that might contain such
oxygen/nitrogen compositional disorder requires calculating a very large number of atomic
con?gurations using supercell models. In the ?rst stage of our theoretical investigation, we
performed the ab initio energetic calculations with a limited number of atomic con?gura-
tions using (pseudo) face-centered-cubic unitcell models. We interpreted our data using a
simpli?ed model that breaks the Gibbs energy into two terms:
GGa3O3N (T,P) = Hground (P) + Galloy (T,P) (6.3)
Here Hground is the temperature-independent enthalpy of the ground-state (lowest energy)
con?guration, which allows us to gain insights into the energetically favored local coordina-
tion states and their O/N ordering, and to study pressure e?ects on the Gibbs free energy
of formation. The temperature e?ects are then described by the second term, Galloy, which
models the contributions related to the O/N disorder. In this study, we ignored the entropy
contributions due to the lattice vibrations.
We ?rst studied the pressure e?ects on formation of Ga3O3N by setting T = 0 K; the
Gibbs free energy of formation ?Gformation is thus identical to the enthalpy of formation
149
Table 6.7: Parameters of the third-order Birch-Murnaghan equation of states of three unit-
cell-based atomic models of the spinel-structured Ga3O3N that have the lowest energy
calculated within the LDA.
E0 V0 a0 B0 B?
model (eV/Ga3O3N) (?A3/Ga3O3N) (?A) (GPa)
I -48.109 (0.000) 69.5424 8.2246 210 4.13
II -47.966 (0.143) 69.6928 8.2357 208 4.17
III -47.854 (0.255) 69.6181 8.2275 209 4.14
?H (P) = ?E + P?V . We ?rst carried out a search for the lowest energy unit-cell con-
?guration of the spinel-structured Ga3O3N. Among the 8!/(6!2!) = 28 con?gurations, there
are three crystallographically distinct O/N arrangements for (Ga3O3N)2. We used the LDA
to calculate lattice parameters at P = 1 atm and also the V (P) relations for these three
models: the calculated third-order Birch-Murnaghan equation of state parameters for each
of these three unit-cell models are listed in Table 6.7. The con?guration labeled as model
I has a rhombohedral symmetry (R?3m). This has the lowest equilibrium energy, and it is
considered to provide the ground-state con?guration of Ga3O3N in our theoretical study.
The rhombohedral distortion from the ideal cubic structure is very small; i.e., the angles
between the pseudocubic lattice vectors are 89.33? within the equilibrium con?guration.
The calculated equilibrium volume and the bulk modulus of this ground-state con?guration
compare favorably with the experimental measurements (see below). At zero pressure, our
LDA calculations predict an endothermic enthalpy of formation, with ?H = +272 meV per
Ga3O3N unit (26.2 kJ/mol) with respect to Ga2O3 (monoclinic ? phase) + GaN (hexag-
onal wurtzite structure). The LDA calculations predict a negative d?H/dP slope (solid
line in Figure 6.4a), so the Gibbs free energy of formation becomes negative at P > 17
GPa. However, Ga2O3 does not always remains in the monoclinic ? phase at high pressure.
According to our (static) LDA calculations, a ?-to-? phase transition in Ga2O3 takes place
at P ? 0.5 GPa. More importantly, the slope of d?H/dP becomes positive when Ga2O3
is in the ? phase (dashed line plot, Figure 6.4a). The experimental phase transition has
been reported to occur between 0.1 MPa and 4.4 GPa237,238; however, a direct transition
pressure was not recorded in those studies213. Recently, Tu et al. have reported a ?-to-?
150
Figure 6.4: LDA-calculated formation enthalpy ?H (i.e., Gibbs free energy of formation
?Gformation at zero temperature) as a function of pressure using (a) LDA methods and
(b) the GGA approach. The calculated positive ?H suggests an endothermic formation.
The solid plots are calculated assuming the oxynitrides are synthesized from ?-Ga2O3 and
wurtzite GaN, and the dashed plots are calculated assuming the oxynitrides are synthesized
from ?-Ga2O3 and wurtzite GaN. The solid plot and the dashed plot cross at the pressure
of the ?-to-? phase transition in Ga2O3 (predicted to be 0.5 and 6.6 GPa by LDA and GGA
methods, respectively).
phase transition in Ga2O3 at 13.3 GPa212. However, there is some uncertainty in the nature
of the starting material used in that work, and Machon et al. repeated the study: he found
that the ?-to-? transition occurs at P = 22 GPa213, without heating. It is obvious that the
nature of the Ga2O3 starting material, and how it transforms under high-P, T conditions,
will a?ect the energetics of the high-pressure synthesis experiment.
Within the present study, we repeated our calculations within the GGA (Figure 6.4b).
The GGA calculations predict a phase transition pressure for Ga2O3 of 6.6 GPa. The
GGA formation enthalpy for the Ga3O3N synthesis reaction is predicted as +370 meV
(35.6 kJ/mol) at P = 1 atm, and this value is reduced to its minimum of +245 meV (23.6
kJ/mol) at the Ga2O3 transition pressure. Despite the quantitative di?erences between
the two sets of calculations, the GGA results agree qualitatively with the ?ndings from the
LDA calculations. Both the LDA and GGA studies predict that the optimal pressure for
151
synthesizing the spinel-structured Ga3O3N from Ga2O3 and GaN mixtures is around that
of the ?-to-? phase transition in Ga2O3 (P ? 6.6 GPa, according to GGA calculations).
Next, we investigated the temperature e?ects on the stability of the oxynitrides systems.
A simple ideal solution model has been previously adopted to estimate the e?ects of O/N
disorder in spinel-structured oxynitrides116,239. Such a simple statistical model is valid
only in the cases where all the atomic con?gurations have very similar energies, and it
approximates the additional Gibbs free energy term with a contribution due to the alloy
disorder with a pure entropic term:
Galloy (T,P) = ?4kBT [xlnx + (1 ?x) ln (1 ?x)] (6.4)
(here the factor 4 is due to the presence of four possible di?erent anion sites in the molecular
unit that are assumed to be equally accessible). It is known that such simple models can
signi?cantly underestimate the alloy formation temperature if some of the atomic con?g-
urations are energetically inaccessible at the temperature of the experiment. To obtain a
better estimate of the entropic contribution to the Gibbs free energy that is more relevant
to the experimental results, we studied the correlation between the energetic properties of
the oxynitride materials and their local atomic ordering schemes among the three unit-cell
models (Table 6.8). The spinel structure is described as a packing of AX4 tetrahedra and
AX6 octahedra present in a 1:2 ratio. In the case of Ga3O3N, there are ?ve possible types of
AX4 tetrahedra, i.e., IV GaO4, IV GaO3N, IV GaO2N2, IV GaON3, and IV GaN4. Similarly,
there exist seven types of AX6 octahedra: V IGaO6, V IGaO5N, V IGaO4N2, V IGaO3N3,
V IGaO2N4, VIGaON5, and VIGaN6. The distribution of various types of AX4 and AX6
units within the three models studied is listed in Table 6.8. No IV GaON3, IV GaN4,
V IGaON5, or V IGaN6 species were considered to simplify the statistical analysis; such
N-rich AX4 or AX6 species are expected to have low concentration because the average
O:N ratio is 3:1. The LDA calculations show a clear energetic preference for the structure
containing IV GaO3N sites over those with the combination 50% IV GaO4 + 50% IV GaO2N2
152
Table 6.8: Analysis of local coordination ordering schemes within the three unit-cell models
of the spinel-structured Ga3O3N in terms of the ratio of various types of AX4 tetrahedral
and AX6 octahedral units.
tetrahedral Ga (%) octahedral Ga (%)
model IV GaO4 IV GaO3N IV GaO2N2 VIGaO6 V IGaO5N V IGaO4N2
I 100 25 75
II 100 50 50
III 50 50 50 50
and the 25% V IGaO6 +75% VIGaO4N2 combination over that with 50% V IGaO5N + 50%
V IGaO4N2.
We then constructed a three-energy-level model to investigate the consequences of this
anion site ordering on the formation energetics of the oxynitride spinel, using the ground-
state energies of the three lowest energy models found above (Table 6.7):
?Gformation (T,P) = ?H (P) ? 12kBT ln
bracketleftBig
4 + 12e?2??1/kBT + 12e?2??2/kBT
bracketrightBig
(6.5)
Here ?H (P) is the static formation enthalpy as discussed above, the values 4, 12, and 12
are the degeneracies of the levels corresponding to each model, and ??1 and ??2 are the
energies of the two ?excited? states (i.e., models II and III) compared to the ground-state
energy (as listed in column 2 of Table 6.7). At a given pressure, the alloy formation tem-
perature corresponds to that at which ?Gformation becomes zero. On the basis of our LDA
calculations, this condition occurs at Talloy = 2800 K. This result means that, according to
our model, a stoichiometric Ga3O3N spinel phase would become thermodynamically stabi-
lized with respect to other ordering schemes and could be synthesized from Ga2O3 + GaN
mixtures above P ? 6?7 GPa and T = 2800 K. This temperature estimate is considerably
higher than that used experimentally to synthesize an oxynitride spinel-structured material
from the component oxides and nitrides (synthesis temperatures as low as 1200 ?C at 5
GPa; see below). The main reasons for the discrepancy are that the oxynitride materials
obtained experimentally contain vacancies on the Ga3+ and perhaps also on the anion sites
and the O:N ratio obtained is larger than the ideal stoichiometry that was modeled. Both
153
considerations will have a large e?ect on the relative energies of ground-state and exper-
imentally accessible ?excited-state? models. Also, it is not yet clear that it is justi?ed to
exclude high-N-content environments such as V GaON3, IV GaN4, V IGaON5, or V IGaN6
on purely statistical grounds from the thermodynamic treatment. Such species could have
special stability due to local bonding environments, including bond valence constraints128.
That will have to be tested in future investigations of local coordinations in GaxOyNz mate-
rials using appropriate experimental probes of the local structural environments (e.g., X-ray
absorption spectroscopy/EXAFS, NMR, etc.).
6.2.4 Electronic Properties
Using ab initio LDA methods, we calculated the electronic band structure within the
LDA on the basis of our atomic model of the ground-state con?guration of Ga3O3N (model
I). The LDA-predicted band dispersion is plotted in Figure 6.5a from the center of the Bril-
louin zone (the ? point) along three directions to the F, T, and L points at the zone bound-
aries. The electronic density of states function is shown in (b). Our calculations indicate
that spinel-structured gallium oxynitrides are direct wide band gap semiconductors, with
optoelectronic properties that are similar to those of wurtzite- and sphalerite-structured
GaN and gallium oxides. Because of well-known limitations of the LDA for such electronic
structure calculations, the predicted magnitude of the band gap (2.1 eV) is likely to be
underestimated in this study. The same LDA methods underestimate the band gaps of the
GaN (wurtzite) and ?-Ga2O3 phases by 1.3 and 2.3 eV, respectively; we thus expect the
experimental band gap of Ga3O3N (spinel) to lie around 4 eV. Our collaborator obtained
room-temperature photoluminescence spectra of the Ga2.8N0.64O3.24 sample obtained by
high-P,T synthesis, using 325 nm laser excitation (Figure 6.6). The onset of the photolu-
minescence signal begins just below 2.5 and extends to 1.5 eV. Because the experimentally
synthesized material contains a large quantity of defects on the Ga3+ sites, and also per-
haps on the anion sites, along with O/N disorder, it is unlikely that the photoluminescence
feature corresponds to excitations across the band gap. Instead, the observed PL band is
154
Figure 6.5: (a) Electronic band structure and (b) Electronic density of states for Ga3O3N
calculated using ?rst-principles (DFT) methods within the LDA.
likely to arise from defect related transitions between states mainly within the gap, so that
the intrinsic band gap for a stoichiometric ordered material would lie considerably above
2.5 eV, as predicted by theory.
6.2.5 Phonon Spectrum
The Raman spectrum of the new oxynitride spinel phase contains several broad bands
(Figure 6.7), indicating substantial disorder among the O and N atoms on the 32e sites in the
spinel structure and/or the presence of cation (Ga3+) or anion vacancies. The broad bands
have maxima near 700 and 800 cm?1, and also near 300 cm?1, that correspond generally
to the positions of Raman-active modes within the analogous spinel-structured compound
?-Ge3N4240. To aid in the interpretation of our experimentally obtained Raman spectra
of GaxOyNz phases, we calculated the ? point phonon frequencies of the 14-atom rhombo-
hedral unit cell of Ga3O3N (model I structure) using ?rst-principles LDA methods. The
same technique was previously used to predict the positions of the Raman-active vibrational
modes in ?-Ge3N4114,240. The pseudocubic R?3m model structure for Ga3O3N is predicted
to have nine Raman-active modes (4A1g + 5Eg), with zone-center frequencies calculated
155
Figure 6.6: Photoluminescence spectrum of the Ga2.8N0.64O3.24 sample obtained via high-
P,T synthesis in the multianvil experiments, using UV laser excitation (325 nm)
at 213, 219, 367, 379, 499, 512, 634, 647, and 782 cm?1. These calculated frequencies are
denoted by dashed lines in Figure 6.7. Within an anion-disordered structure, as expected
for the real GaxOyNz spinel, he predicted zone-center modes act as poles for interpreting
the broadened spectra that approach the full vibrational density of states (VDOS). The
expected mode frequencies are grouped around the ?ve frequency values (i.e., 216, 373, 506,
640, and 782 cm?1) that are associated with the ideal spinel structure. By analogy with
the Raman spectrum of ?-Ge3N4, we expect the lowest frequency peak (216 cm?1) and the
two highest frequency peaks (640 and 782 cm?1) to have the strongest intensities, whereas
the two intermediate frequency peaks (373 and 506 cm?1) are relatively weak. We cannot
yet directly calculate the e?ects of the O/N disorder on the broadening patterns observed
in our Raman spectra. However, it is likely that the observed Raman spectrum provides a
?rst view of the VDOS functions of Ga3O3N and also ?-Ge3N4 spinels.
To provide a semiquantitative estimation of the widths of the broad peaks, we cal-
culated the full phonon dispersion using a 112-atom supercell model. The Born e?ective
charge induced LO-TO splitting in the ionic compounds are corrected on the basis of the
interplanar force constant model proposed by Kunc and Martin156. As shown in Figure 6.7,
156
Figure 6.7: (a) Raman spectrum collected for the Ga2.8N0.64O3.24 sample at an excitation
wavelength of 514.5 nm. The bold solid lines on the frequency scale below indicate the
positions of the Raman bands for the analogous spinel form of ?-Ge3N4. (b) Phonon density
of states (VDOS) calculated for the R?3m pseudocubic Ga3O3N phase, predicted as ?model
I? in the enthalpy calculations (Table 6.7 ). The dashed lines indicate the Raman-active
modes for that phase.
the highest frequency strong Raman peak (i.e., the Ag mode near 782 cm?1) is expected to
be the sharpest one. The other strong peak near 640 cm?1 is expected to have a broader
width (40-60 cm?1).
6.2.6 Conclusions
Our theoretical study showed that the most stable structure for Ga3O3N corresponds
to a rhombohedral distortion of the ideal spinel structure. The formation of Ga3O3N is
endothermic at ambient pressure and low temperature, and the optimal synthesis pressure
is predicted to lie close to that for the ?-to-? phase transition in Ga2O3 (around 6.6 GPa
according to our GGA calculations). The calculated direct band gap energy for a stoichio-
metric oxynitride spinel was estimated to be around 4 eV. This value is larger than that
obtained from photoluminescence data collected on the experimentally synthesized sample,
which likely contains Ga3+ vacancies and other structural defects. The synthesis of this new
Ga-O-N phase makes contact with the important optoelectronic materials known to exist
157
in the (Ga,Al,In)N system that provide light-emitting diodes and solid-state lasers in the
blue to UV range. A cubic Ga3O3N material similar to the compound synthesized here has
recently been prepared in thin ?lm form via chemical precursor techniques241. That result
indicates that the new materials could be developed for use within novel optoelectronic
devices.
158
Chapter 7
CONCLUSIONS AND FUTURE WORK
In this dissertation, I have adopted and further developed a series of ?rst-principles
computational techniques to theoretically investigate the pressure-induced phase transitions
and thermodynamic properties for several representative main-group oxides and nitrides.
For Al2O3, we have systematically investigated the pressure-induced phase transforma-
tions. The sequence of transitions under compression, i.e., corundum?Rh2O3(II)? pPV,
and the T-P phase diagram we obtained are consistent with previous theoretical and ex-
perimental studies. Results using US-PP and PAW are presented. By ?nding a transition
path that links the corundum and Rh2O3(II) phases with monoclinic intermediate struc-
tures which has a space group P2/c, we proposed a single-bond breaking-and-reforming
(SB-BAR) mechanism to describe the transformation between corundum and Rh2O3(II)
phase. Total energy calculation using PAW method shows that the enthalpy barrier height
of the forward corundum-to-Rh2O3(II) transition is around 130 meV/atom and remains
unchanged with varied pressures. However, the barrier height of the backward Rh2O3(II)-
to-corundum transition decreases signi?cantly under decompression, which indicates that
the Rh2O3(II) phase may not be quenchable to ambient conditions. In addition, great simi-
larity is found in the elastic constants between corundum and Rh2O3(II) phase, except C33.
The larger C33 of Rh2O3(II) phase means that its c axis is less compressible than that of
corundum. Zero pressure ?-point phonon frequencies and their pressure dependencies are
calculated for three stable polymorphs. Comparison with measured data of corundum and
theoretical Raman frequencies of pPV phase suggest that, within LDA, predictions using
PAW method have better results than US-PP. This conclusion is also con?rmed from the
study of elastic properties. We have calculated the elastic constants and their pressure
dependencies for corundum and two high-pressure phases. PAW calculated Cij yield better
159
agreement with available experimental data of ?-Al2O3. The temperature dependencies
of thermal expansion coe?cient from low to high pressures have been predicted for ?-,
Rh2O3(II)-, and pPV-Al2O3. For corundum at zero pressure, our US-PP data is in good
agreement with Amatuni et al.?s result measured from 300 K to 2000 K, and with Aldebert
et al.?s result at temperatures above 1000 K. Our PAW data lies in the middle of the ex-
perimental data and agrees with Schauer?s data below 700 K and Wachtman et al.?s data
above 1200 K. Our calculated heat capacity CP, entropy and adiabatic bulk modulus of
corundum phase also agree well with measured results.
For AlN, we presented our ab initio calculation of activation barriers of the B4-to-B1
transition for ?ve TPs proposed by Stokes et al. We showed that the ?ve bond-preserving
paths can be interpreted as transformation of di?erent long-range patterns of the ?transition
units? (two di?erent orientations). The transformation of ?transition unit? is equivalent to
the path along TP1 (with Cmc21 symmetry). Our calculated kinetic barriers are comparable
for all ?ve paths at pressures from 0 GPa to 30 GPa, which indicate that the wurtzite-to-
rocksalt transition is characterized by the transformation of the ?transition unit?, while the
long-range pattern is less important. And the di?erence in strains of di?erent TPs is not a
major factor for at least the transition from wurtzite to rocksalt phase in AlN. In addition,
the bond-breaking path is not energetically favored compared with the bond-preserving
paths. Besides the studies of di?erent TPs, our estimated forward and backward barrier
heights are consistent with experimental observation and previous calculations.
For Si3N4, in summary, we have theoretically studied phase transitions in silicon nitride
(Si3N4) at high pressure using a ?rst-principles density functional theory method. We ?nd
that ?-Si3N4 remains as a metastable phase at temperatures up to 2000 K and pressures
up to 10 GPa. The equilibrium ??? transition pressure is predicted as 7.5 GPa at 300K
and it increases to 9.0 GPa at 2000K. Both ?- and ?-Si3N4 are dynamically stable at low
pressure. However, two competing phonon-softening mechanisms are found in the ? phase
at high pressures. At room temperature, ?-Si3N4 is predicted to undergo a ?rst-order
??P3 transition above 38.5 GPa, while ?-Si3N4 shows no signs of structural instability.
160
The predicted metastable high-pressure P3 phase is structurally related to ?-Si3N4. The
enthalpy barrier height is estimated as only 67.23 meV/atom. Our LDA predicted thermal
expansion coe?cient, heat capacity and bulk Gr?uneisen parameter are in good agreement
with Bruls? measured results. We ?nd relatively large discrepancies between our calculation
with experimental data from Reeber. And we attribute the cause of predicted negative
TEC at low temperatures in ? and ?-Si3N4 to the low-frequency phonon modes that have
negative mode Gr?uneisen ratios.
For bulk Ga2O3, we have calculated the Raman frequencies and their mode Gr?uneisen
parameters for ? and ? phases. Good agreement is achieved between our results with
experiments and other calculations. We also predict the equilibrium T-P phase diagram
of Ga2O3 consists of ?, ? and Rh2O3(II) phases. Our LDA calculated transition pressure
from ? to ? phase is 0.3 GPa at 0 K and 1.6 GPa at 2000 K. The Clapeyron slope for
this transition is positive which has the value of about +0.6 MPa/K at 1000 K. This is
consistent with previous calculations. However, the experimental Pt is much larger. The
large discrepancy can be attributed to a large kinetic barrier, which may be signi?cantly
lowered under compression. Our calculated ?-to-Rh2O3(II) transition pressure is 40 GPa
at 0 K and 37 GPa at 2000 K, which is in excellent agreement with experimental value
(about 37 GPa at about 2000 K). In the case of nanowires, based on a comparison of
the experimental Raman mode frequencies with our ?rst-principles calculations, we ?nd
compelling evidence for growth-direction-induced internal strains in ?-Ga2O3 nanowires
which signi?cantly in?uence the vibrational mode frequencies. Within the linear model
approximation, the observed blue and redshifts of peak frequencies in the micro-Raman
spectra of the ?-Ga2O3 nanowires with di?erent growth directions can be attributed to two
small anisotropic internal strains: one compressive strain of 2% volume change, and the
other tensile strain of 0.7% volume change. The overall high quality of the ?tted models to
available experimental data suggests a strong correlation between the shifts in Raman mode
frequencies and the growth-direction-induced internal strains in the Ga2O3 nanowires.
161
For gallium oxynitride, our theoretical study showed that the most stable structure
for Ga3O3N corresponds to a rhombohedral distortion of the ideal spinel structure. The
formation of Ga3O3N is endothermic at ambient pressure and low temperature, and the
optimal synthesis pressure is predicted to lie close to that for the ?-to-? phase transition in
Ga2O3 (around 6.6 GPa according to our GGA calculations). The calculated direct band
gap energy for a stoichiometric oxynitride spinel was estimated to be around 4 eV. This
value is larger than that obtained from photoluminescence data collected on the experimen-
tally synthesized sample, which likely contains Ga3+ vacancies and other structural defects.
The synthesis of this new Ga-O-N phase makes contact with the important optoelectronic
materials known to exist in the (Ga,Al,In)N system that provide light-emitting diodes and
solid-state lasers in the blue to UV range. A cubic Ga3O3N material similar to the com-
pound synthesized here has recently been prepared in thin ?lm form via chemical precursor
techniques. That result indicates that the new materials could be developed for use within
novel optoelectronic devices.
The above results show that DFT calculations within the frame of quasi-harmonic
approximation (QHA) is successful in calculating the ?nite-temperature thermodynamic
potentials for hard materials. However, to improve the prediction of thermodynamic prop-
erties for normal materials at high temperatures or strongly anharmonic crystals even at
low temperatures, we are looking forward to developing and implementing an algorithm
based on the perturbation theory to add anharmonic correction to the QHA vibrational
free energy, as shown in Appendix C. The ?rst order correction is contributed from the
3rd and 4th order lattice anharmonicity tensors. With these anharmonicity tensors, the
temperature dependence of elasticity and phonon frequencies are also within the scope.
And, for the metastable P3 phase we found in Si3N4, our collaborator who did ex-
periments showed that there is a mismatch between the observed X-ray di?raction (XRD)
pattern after phase transition at high pressure and the calculated XRD pattern from P3
structure at the same pressure. But the unknown ? phase is believed to be structurally close
162
to the P3 phase. We will look at the low-frequency phonon modes with softening tendency
for ?, P?6, P3 and P21/m phases, which may suggest other possible structures.
163
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178
241. I. Kinski, G. Miehe, G. Heymann, R. Theissmann, R. Riedel, and H. Huppertz, Z.
Naturforsch., B: Chem. Sci. 60, 831 (2005).
242. URL http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html.
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Publishing Company American Elsevier Publishing Company, Inc., 1974).
179
Appendices
180
Appendix A
Total Energy Calculation with VASP
The VASP package242 that we adopt in our study is implemented with ultra-soft pseu-
dopotentials (US-PP) or the projector-augmented wave (PAW). The four essential input
?les needed for a VASP calculation are INCAR, POTCAR, KPOINTS and POSCAR. The
INCAR ?le contains a large number of parameters and most of these parameters should be
left at their default values in our calculations. One important parameter worth mentioning
here is the ISIF-tag. ISIF controls which degree of freedom (ions, cell volume, cell shape)
are allowed to change. With di?erent crystal symmetries and purposes of calculations, one
can choose a proper value. Table A.1 shows the meaning of each ISIF tag. Taking B1 NaCl
as an example, which has cubic symmetry, if we are interested in a constant-volume total
energy calculation, ISIF=0 is the choice. The POTCAR ?le contains the pseudopotential
of each type of atoms. The KPOINTS ?le contains the k-point sampling scheme. And the
POSCAR ?le basically includes information of cell geometry and the atomic coordinates.
Table A.1: Value and meaning of ISIF tag
ISIF calculate
force
calculate stress
tensor
relax ions change cell
shape
change cell
volume
0 yes no yes no no
1 yes trace only yes no no
2 yes yes yes no no
3 yes yes yes yes yes
4 yes yes yes yes no
5 yes yes no yes no
6 yes yes no yes yes
7 yes yes no no yes
Although VASP allows relaxation of cell volume, it is not recommended in an energy-
volume calculation. The incompleteness of the plane wave basis set with respect to changes
of volume will cause Pulay stress, which will underestimate the equilibrium volume unless
a large plane wave cuto? is used. Nonetheless, if only volume conserving relaxations are
carried out, the Pulay stress can often be neglected. Furthermore, errors from Pulay stress
can also be reduced and reliable lattice constant, bulk modulus and other elastic properties
can be yielded by doing calculations at di?erent volumes with the same energy cuto? for
each calculation and then ?tting the total energies to an equation of state. But the interval
of volume changes should be in the order of 5%-10% to average out the errors due to the
basis set incompleteness.
181
Appendix B
Elasticity Calculation
A solid responds to stress by changing its shape. Elasticity is the physical property to
represent the ability of a material to recover when the stress is released. ?Elastic? implies
that the solid will restore to its original equilibrium shape if the stress is released.
?ij = Sijkl?kl (B.1a)
?ij = Cijkl?kl (B.1b)
where ?ij and ?kl are strain and stress tensors, Sijkl and Cijkl are called the elastic com-
pliance constants and elastic sti?ness constants (or simply called elastic constants, sti?ness
constants). ?ij is dimensionless. ?ij has the dimension of [force]/[area] or [energy]/[volume].
Because each su?x runs from 1 to 3 (x, y, z), there are totally 81 elastic constants or com-
pliance constants. Elastic constants are measurable and characteristic properties of a solid.
Without considering any temperature e?ect (entropy is zero), the change of the internal
energy density (energy per unit volume) purely caused by strain is a quadratic function of
?ij:
E/V = 12
summationdisplay
ijkl
Cijkl?ij?kl (B.2)
Since ?ij and ?ji always appear together, we can de?ne a symmetrical strain tensor ?
as follows:
?ij = 12 (?ij + ?ji) (B.3)
where ?ij = ?ji and the energy density is still in the previous form.
E/V = 12
summationdisplay
ijkl
Cijkl?ij?kl (B.4)
From the above equation it is easy to prove that Cijkl = Cijlk and Cijkl = Cjikl. These
two conditions reduce the number of independent elastic constants from 81 to 36. However,
it is cumbersome to deal with a fourth-rank tensor. Fortunately, the symmetry makes it
possible to use a matrix form. The most commonly accepted way of translation is Voigt?s
notation, by reducing the ?rst two su?xes to one and the last two su?xes in the same way.
Accordingly, the stress and strain tensor (second-order) components will be represented
182
with a single su?x.
?
?
?11 ?12 ?13
?21 ?22 ?23
?31 ?32 ?33
?
? ?
?
?
?1 ?6 ?5
?6 ?2 ?4
?5 ?4 ?3
?
? (B.5a)
?
?
?11 ?12 ?13
?21 ?22 ?23
?31 ?32 ?33
?
? ?
?
?
?1 12?6 12?5
1
2?6 ?2
1
2?41
2?5
1
2?4 ?3
?
? (B.5b)
The purpose of introducing a factor of 12 to the o?-diagonal strain components is to keep
equation B.4 in the same format. Besides the su?xes abbreviation, the elastic compliance
constants are changed by a factor of 1, 2, or 4 in the following way:
??
?
Sijkl = Smn if m and n are 1, 2 or 3
2Sijkl = Smn if either m or n are 4, 5 or 6
4Sijkl = Smn if both m and n are 4, 5 or 6
(B.6)
While at the same time, for Cijkl there are no such factors changes.
Cijukl = Cmn (B.7)
where i,j,k,l = 1,2,3 and m,n = 1,??? ,6. And the energy density due to strain is written
as:
E/V = 12
summationdisplay
ij
Cij?i?j (B.8)
In equation B.8, exchanging of su?x i and j does no e?ect, which leads to Cij = Cji.
Thus the number of independent Cij is further reduced to 21, and this is the case for a
crystal with triclinic symmetry, which is the lowest symmetry of all. Any material with
higher symmetry has even less number of independent elastic constants. Results for all the
crystal classes are given in table B.1.
183
Table B.1: Independent components of Cij for all the crystal classes
Crystal class Independent Cij
Triclinic All 21
Monoclinic unique axis b C11, C22, C33, C12, C13 , C23, C44, C55, C66, C15, C25, C35, C46
unique axis c C11, C22, C33, C12, C13, C23 , C44, C55, C66, C16, C26, C36, C45
Orthorhombic C11, C22, C33, C12, C13, C23, C44, C55, C66
Cubic C11, C12, C44
Tetragonal Classes 4, ?4, 4/m C11, C33, C12, C13, C44, C66, C16
Classes 4mm, ?42m, 422, 4/mmm C11, C33, C12, C13, C44, C66
Trigonal Classes 3, ?3 C11, C33, C12, C13, C44, C14, C15
Classes 32, ?3m, 3m C11, C33, C12, C13, C44, C14
Hexagonal C11, C33, C12, C13, C44
Isotropic C11, C12
18
4
Equation B.8 is the foundation for evaluating the elastic constants. The procedure
of our calculation can be summarized as three steps. 1) Based on the symmetry of the
material, we generate a series of strained POSCAR ?les. 2) We calculate the total energy
for each strained POSCAR. 3) Obtain the independent elastic constants.
Diagonal and o?-diagonal terms of Cij are treated di?erently. Taking cubic system for
illustration, there are three independent elastic constants: C11, C12 and C44. For C11 and
C44, we introduce single strain ?1 and ?4, respectively, to the unstrained system. Then total
energies of the strained systems are calculated. Because C11 and C44 are both diagonal
terms, they are obtained in exactly the same way. Let?s take C11 as an instance. First,
assuming optimized CONTCAR ?le at a given volume is acquired in previous static total
energy calculations, it serves as the unstrained POSCAR in the elasticity calculation. When
single strain is applied to the system, the total energy will be changed due to strain and
can be expanded as a Taylor?s series of strain, provided the strain is small (usually between
0.5% and 1.5%).
E = a + b? + 12c?2 + 16d?3 + 124e?4 + ??? (B.9)
where a, b, c, d and e are coe?cients of the ?rst ?ve terms in Taylor?s series. We keep up
to the ?fth term as a good approximation.
C11 = c (B.10)
C11 is the coe?cient of the quadratic term because the elastic energy due to strain ?1 is
U = 12C11?21. We have implemented a ?six-point? method to get this coe?cient. Six single
?1-strains with the amount of 3?, 2?, ?, ??, ?2?, ?3? are applied to the original POSCAR
and total energy calculations are carried out for each of them.
?
???
???
?
???
???
?
E1 = a + 3b? + 92c?2 + 92d?3 + 278 e?4
E2 = a + 2b? + 2c?2 + 43d?3 + 23e?4
E3 = a + b? + 12c?2 + 16d?3 + 124e?4
E4 = a?b? + 12c?2 ? 16d?3 + 124e?4
E5 = a? 2b? + 2c?2 ? 43d?3 + 23e?4
E6 = a? 3b? + 92c?2 ? 92d?3 + 278 e?4
(B.11)
By canceling the terms of no interests, we have
C11 = c = ?3E1 + 16E2 ? 13E3 ? 13E4 + 16E5 ? 3E624?2 (B.12)
For C12, we introduce two strains ?1 and ?2, at the same time, to the unstrained system.
The related elastic energy is
U = C12?1?2 + C11?21 + C22?22 (B.13)
C12 is the coe?cient in the front of the ?1?2 term. With pair strains, the total energy
of the system can also be expressed in a Taylor?s series, being cut at the fourth-order terms
185
as an approximation.
E = a + b?1 + c?2 + d?1?2 + 12e?21 + 12f?22 + 16g?31 + 12h?21?2 + 12i?1?22
+16j?32 + 124k?41 + 16l?31?2 + 14m?21?22 + 16n?1?32 + 124o?42 (B.14)
where C12 = d. To eliminate the unrelated coe?cients, an ?eight-point? algorithm is
implemented. The eight pair strains for (?1,?2) we apply are: (2?,2?), (2?,?2?), (?2?,?2?),
(?2?,2?), (?,?), (?,??), (??,??) and (??,?).
?
???
???
????
????
???
????
????
???
???
???
????
????
???
????
????
???
E1 = a + 2b?1 + 2c?2 + 4d?1?2 + 2e?21 + 2f?22 + 43g?31 + 4h?21?2 + 4i?1?22
+43j?32 + 23k?41 + 83l?31?2 + 2m?21?22 + 83n?1?32 + 23o?42
E2 = a + 2b?1 ? 2c?2 ? 4d?1?2 + 2e?21 + 2f?22 + 43g?31 ? 4h?21?2 + 4i?1?22
?43j?32 + 23k?41 ? 83l?31?2 + 2m?21?22 ? 83n?1?32 + 23o?42
E3 = a? 2b?1 ? 2c?2 + 4d?1?2 + 2e?21 + 2f?22 ? 43g?31 ? 4h?21?2 ? 4i?1?22
?43j?32 + 23k?41 + 83l?31?2 + 2m?21?22 + 83n?1?32 + 23o?42
E4 = a? 2b?1 + 2c?2 ? 4d?1?2 + 2e?21 + 2f?22 ? 43g?31 + 4h?21?2 ? 4i?1?22
+43j?32 + 23k?41 ? 83l?31?2 + 2m?21?22 ? 83n?1?32 + 23o?42
E5 = a + b?1 + c?2 + d?1?2 + 12e?21 + 12f?22 + 16g?31 + 12h?21?2 + 12i?1?22
+16j?32 + 124k?41 + 16l?31?2 + 14m?21?22 + 16n?1?32 + 124o?42
E6 = a + b?1 ?c?2 ?d?1?2 + 12e?21 + 12f?22 + 16g?31 ? 12h?21?2 + 12i?1?22
?16j?32 + 124k?41 ? 16l?31?2 + 14m?21?22 ? 16n?1?32 + 124o?42
E7 = a?b?1 ?c?2 + d?1?2 + 12e?21 + 12f?22 ? 16g?31 ? 12h?21?2 ? 12i?1?22
?16j?32 + 124k?41 + 16l?31?2 + 14m?21?22 + 16n?1?32 + 124o?42
E8 = a?b?1 + c?2 ?d?1?2 + 12e?21 + 12f?22 ? 16g?31 + 12h?21?2 ? 12i?1?22
+16j?32 + 124k?41 ? 16l?31?2 + 14m?21?22 ? 16n?1?32 + 124o?42
(B.15)
To solve for C12, we do addition or subtraction with these equations and ?nd that
C12 = d = ?E1 + E2 ?E3 + E4 + 16E5 ? 16E6 + 16E7 ? 16E848?2 (B.16)
So far we have discussed the derivation and computation procedure of the elastic con-
stants for an unstressed solid. However, elastic constants will change under stress. Here we
follow Barron & Klein?s derivation and care only about the case of isotropic stress (hydro-
static pressure). When pressure is applied to the solid, Barron showed that the strain-energy
density relation becomes:
Cijkl = 1V ?E??
ij??kl
+ 12P (2?ij?kl ??il?jk ??ik?jl) (B.17)
where V is the volume at that pressure, P is the pressure and ?ij is the Kronecker delta
(?ij = 1 when i = j; ?ij = 0 when i negationslash= j). The ?rst term in equation B.17 is obtained from
the strain-energy density relation as if the solid is not under stress. The second term is
a correction because of pressure. In Voigt?s notation, the correction terms for each elastic
constant is shown in table B.2.
186
Table B.2: Correction term to Cij due to pressure
Cij Correction term Cij Correction term Cij Correction term
C11 0 C13 P C26 0
C22 0 C14 0 C34 0
C33 0 C15 0 C35 0
C44 ?12P C16 0 C36 0
C55 ?12P C23 P C45 0
C66 ?12P C24 0 C46 0
C12 P C25 0 C56 0
Again, if we use cubic crystal as an example, and let ?C represent the ?rst term in
equation B.17, the three independent elastic constants are:
?
?
?
C11 = ?C11
C12 = ?C12 + P
C44 = ?C44 ? 12P
(B.18)
At the current stage, we only calculate the elastic constant and its pressure dependence
at zero temperature condition.
187
Appendix C
Beyond Harmonic Approximation: Perturbation Theory of Lattice
Anharmonicity
The quasi-harmonic approximation (QHA) has achieved a great success in predicting
thermodynamic properties for many hard materials at temperatures not too close to the
melting point. However, to improve the prediction of thermodynamic properties of normal
materials at high temperatures or strongly anharmonic crystals even at low temperatures,
it is desired to add the anharmonicity contribution to the free energy as a next level ap-
proximation. This correction is usually small compare to the harmonic contribution. With
the third and fourth order anharmonicity, we can evaluate the lowest order of anharmonic
contribution to the harmonic free energy via perturbation theory. Within this theory, we
treat the anharmonic terms in the hamiltonian as perturbations. The total hamiltonian can
be written as:
H = HH + HA
= H0 + H2 + H3 + H4 + ??? (C.1)
where HH = H0 + H2 which is the harmonic part and HA = summationtext?n=3 Hn is the anharmonic
part. Hn is the nth order term in the expansion of hamiltonian. As the anharmonic e?ect is
more important at high temperatures, we deal this problem within the classical limit. Then
the partition function of the system is:
Z = h?3N
integraldisplay
dx1 ???
integraldisplay
dx3N
integraldisplay
dp1 ???
integraldisplay
dp3N ? e??(HH+HA)
= ZH ?
integraltext dx
1 ???
integraltext dx
3N
integraltext dp
1 ???
integraltext dp
3N ? e??HH ? e??HAintegraltext
dx1 ???integraltext dx3N integraltext dp1 ???integraltext dp3N ? e??HH
= ZH ?ZA (C.2)
where ZH = h?3N integraltext dx1???integraltext dx3N integraltext dp1 ???integraltext dp3N ?e??HH is the partition function of the
harmonic hamiltonian and ZA is the ensemble average of the quantity e??HA over the
188
unperturbed harmonic system. Expanding e??HA into Taylor series, we have
ZA =
angbracketleftBig
e??HA
angbracketrightBig
0
=
angbracketleftBigg
1 +
?summationdisplay
n=1
(?1)n
n! ?
n ?Hn
A
angbracketrightBigg
= 1 +
?summationdisplay
n=1
(?1)n
n! ?
n ??Hn
A?0 (C.3)
Since we have expressed the total partition function as a product of the harmonic
partition function and ZA, the anharmonic contribution to the Helmholtz free energy is:
FA = ?kBT lnZA
= ???1 ln
parenleftBigg
1 +
?summationdisplay
n=1
(?1)n
n! ?
n ??Hn
A?0
parenrightBigg
(C.4)
Assuming the anharmonic terms are small compared to unity, the logarithm function
in equation C.4 can be expanded into series following ln (1 + x) = x? x22 + x33 ???? when
?1 < x < 1. The expansion is expressed in terms of the (??)nn! and the coe?cient of (??)nn!
is called the nth cumulant ?HnA?0,C
FA = ???1
?summationdisplay
n=1
(?1)n
n! ?
n ??Hn
A?0,C (C.5)
Comparing equation C.5 with equation C.4, we have
?HA?0,C = ?HA?0 (C.6a)
angbracketleftbigH2
A
angbracketrightbig
0,C =
angbracketleftbigH2
A
angbracketrightbig
0 ? (?HA?0)
2 (C.6b)
angbracketleftbigH3
A
angbracketrightbig
0,C =
angbracketleftbigH3
A
angbracketrightbig
0 ? 3
angbracketleftbigH2
A
angbracketrightbig
0?HA?0 + 2 (?HA?0)
3 (C.6c)
According to pairing theorem243, the lowest order anharmonic contribution from ?HA?0,C
term is ?H4?0. The lowest order anharmonic contribution from angbracketleftbigH2Aangbracketrightbig0,C term is angbracketleftbigH23angbracketrightbig0 ?
(?H4?0)2. Other higher order terms are ignored as an approximation. The anharmonicity
correction to the free energy under this approximation is:
FA = ?H4?0 ? ?2
bracketleftBigangbracketleftbig
H23angbracketrightbig0 ? (?H4?0)2
bracketrightBig
(C.7)
189
Using normal coordinates, the third and fourth order anharmonic terms of hamiltonian
can be written as:
H3 =
summationdisplay
q,q?,q??
summationdisplay
?,??,???
W3
parenleftbigg q,q?,q??
?,??,???
parenrightbigg
Q? (q)Q??parenleftbigq?parenrightbigQ???parenleftbigq??parenrightbig (C.8a)
H4 =
summationdisplay
q,q?,q??,q???
summationdisplay
?,??,???,????
W4
parenleftbigg q,q?,q??,q???
?,??,???,????
parenrightbigg
Q? (q)Q??parenleftbigq?parenrightbigQ???parenleftbigq??parenrightbigQ????parenleftbigq???parenrightbig
(C.8b)
where W3 and W4 are de?ned as
W3
parenleftbigg q,q?,q??
?,??,???
parenrightbigg
= 16
summationdisplay
???
summationdisplay
ijk
N?12?q+q?+q??,G?A?i,?j,?kparenleftbigq?,q??parenrightbig
?e?,i (q,?)e?,jparenleftbigq?,??parenrightbige?,kparenleftbigq??,???parenrightbig (C.9a)
W4
parenleftbigg q,q?,q??,q???
?,??,???,????
parenrightbigg
= 124
summationdisplay
????
summationdisplay
ijkl
N?1?q+q?+q??+q???,G?B?i,?j,?k,?lparenleftbigq?,q??,q???parenrightbig
?e?,i (q,?)e?,jparenleftbigq?,??parenrightbige?,kparenleftbigq??,???parenrightbige?,lparenleftbigq???,????parenrightbig (C.9b)
In equation C.9, A?i,?j,?k (q?,q??) is the third order anharmonic dynamical tensor, which
can be derived from the real-space third order force-constant tensor through a Fourier
transformation. B?i,?j,?k,?l (q?,q??,q???) is the fourth order anharmonic dynamical tensor.
A?i,?j,?kparenleftbigq?,q??parenrightbig = 1?m
imjmk
summationdisplay
h,h?
A?i,?j,?kparenleftbig0,h,h?parenrightbig? ei(q??h+q???h?) (C.10a)
B?i,?j,?k,?lparenleftbigq?,q??,q???parenrightbig = 1?m
imjmkml
summationdisplay
h,h?,h??
B?i,?j,?k,?lparenleftbig0,h,h?,h??parenrightbig? ei(q??h+q???h?+q????h??)
(C.10b)
With the pairing theorem we can evaluate the averaged hamiltonian over the unper-
turbed canonical ensemble.
?H4?0 = 3?3
summationdisplay
q1,q2
summationdisplay
?1,?2
W4
parenleftbigg q
1,?q1,q2,?q2
?1,?1,?2,?2
parenrightbigg
?2?1 (q1)?2?2 (q2) (C.11a)
angbracketleftbigH2
3
angbracketrightbig
0 =
6
?3
summationdisplay
q1,q2,q3
summationdisplay
?1,?2,?3
W3
parenleftbigg q
1,q2,q3
?1,?2,?3
parenrightbigg
W3
parenleftbigg ?q
1,?q2,?q3
?1,?2,?3
parenrightbigg
?2?1 (q1)?2?2 (q2)?2?3 (q3)
+ 9?3
summationdisplay
q1,q2,q3
summationdisplay
?1,?2,?3
W3
parenleftbigg q
1,?q1,q3
?1,?1,?3
parenrightbigg
W3
parenleftbigg q
2,?q2,?q3
?2,?2,?3
parenrightbigg
?2?1 (q1)?2?2 (q2)?2?3 (q3)
(C.11b)
190
where ?? (q) is the phonon frequency of the ?th normal mode at reciprocal lattice point q.
We have proposed an algorithm to evaluate the third order force-constant tensor
A?i,?j,?k (?,??,???). In order to get all the tensor elements, we apply a pair of displace-
ments to the system. Here we keep up to the 4th order term of the hamiltonian and the
amount of each displacement is ? which is much smaller than the interatomic distance.
The pair-displacement scheme we adopt is to deviate the jth atom in ? direction and the
kth atom in ? direction by (?,?), (?,??), (??,?), and (??,??), respectively. The ?
component of the H-F forces on the ith atom are:
F++?,i (?) = ?bracketleftbig??i,?jparenleftbig?,??parenrightbig+ ??i,?kparenleftbig?,???parenrightbigbracketrightbig? ?
?12 bracketleftbigA?i,?j,?jparenleftbig?,??,??parenrightbig+ A?i,?k,?kparenleftbig?,???,???parenrightbig+ 2A?i,?j,?kparenleftbig?,??,???parenrightbigbracketrightbig? ?2
?16
bracketleftbigg B
?i,?j,?j,?j (?,??,??,??) + B?i,?k,?k,?k (?,???,???,???)
+2B?i,?j,?j,?k (?,??,???,????) + 2B?i,?j,?k,?k (?,??,???,????)
bracketrightbigg
? ?3
(C.12a)
F+??,i (?) = ?bracketleftbig??i,?jparenleftbig?,??parenrightbig? ??i,?kparenleftbig?,???parenrightbigbracketrightbig? ?
?12 bracketleftbigA?i,?j,?jparenleftbig?,??,??parenrightbig+ A?i,?k,?kparenleftbig?,???,???parenrightbig? 2A?i,?j,?kparenleftbig?,??,???parenrightbigbracketrightbig? ?2
?16
bracketleftbigg B
?i,?j,?j,?j (?,??,??,??) ?B?i,?k,?k,?k (?,???,???,???)
?2B?i,?j,?j,?k (?,??,???,????) + 2B?i,?j,?k,?k (?,??,???,????)
bracketrightbigg
? ?3
(C.12b)
F?+?,i (?) = ?bracketleftbig???i,?jparenleftbig?,??parenrightbig+ ??i,?kparenleftbig?,???parenrightbigbracketrightbig? ?
?12 bracketleftbigA?i,?j,?jparenleftbig?,??,??parenrightbig+ A?i,?k,?kparenleftbig?,???,???parenrightbig? 2A?i,?j,?kparenleftbig?,??,???parenrightbigbracketrightbig? ?2
?16
bracketleftbigg ?B
?i,?j,?j,?j (?,??,??,??) + B?i,?k,?k,?k (?,???,???,???)
+2B?i,?j,?j,?k (?,??,???,????) ? 2B?i,?j,?k,?k (?,??,???,????)
bracketrightbigg
? ?3
(C.12c)
F???,i (?) = +bracketleftbig??i,?jparenleftbig?,??parenrightbig+ ??i,?kparenleftbig?,???parenrightbigbracketrightbig? ?
?12 bracketleftbigA?i,?j,?jparenleftbig?,??,??parenrightbig+ A?i,?k,?kparenleftbig?,???,???parenrightbig+ 2A?i,?j,?kparenleftbig?,??,???parenrightbigbracketrightbig? ?2
+16
bracketleftbigg B
?i,?j,?j,?j (?,??,??,??) + B?i,?k,?k,?k (?,???,???,???)
+2B?i,?j,?j,?k (?,??,???,????) + 2B?i,?j,?k,?k (?,??,???,????)
bracketrightbigg
? ?3
(C.12d)
From equation C.12, simple derivation yields
A?i,?j,?kparenleftbig?,??,???parenrightbig = ?F
++
?,i (?) + F
+?
?,i (?) ?F
?+
?,i (?) + F
??
?,i (?)
4?2 (C.13)
For a 3N?3N?3N tensor, there are C26N +6N = 18N2+3N H-F forces required. This
number can be reduced by noticing that when (?,j,??) = (?,k,???), A?i,?j,?j (?,????) can be
obtained from the single-displacement calculations which have been done when acquiring
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the second order force-constant matrix.
A?i,?j,?jparenleftbig?,??,??parenrightbig = F
+
?,i (?) + F
?
?,i (?) ?F
2+
?,i (?) ?F
2?
?,i (?)
3?2 (C.14)
However, there are still C26N forces needed which is not an easy task to calculate for a
supercell. Again, the crystal symmetry plays an important role to greatly reduce the number
of direct calculations. A code named as Moves Analysis.x has been implemented to ?gure
out the irreducible number of pair-moves. Let us take the 120-atom supercell of ?-Al2O3
as an example. The total number of pair-moves is 258840 initially. After the symmetry
analysis, the number of independent pair-moves is reduced to 5168. Although this number
has been greatly deducted, it is still impractical to carry out ?rst-principles calculations
for such a large number. At this stage, an approximation could be made that the tensor
element is typically around zero if two atoms in the atomic indices of A?i,?j,?k (?,??,???) are
far away from each other. If we drop all the calculations involving the distance between the
jth and kth atoms that beyond the second nearest neighbor, we can reduce the number of
direct pair-move calculations down to 632, which is now a feasible job.
After the direct calculations, other tensor elements can be either reconstructed from
the irreducible ones or equal to zero.
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