Exploration of the Origin of Substrate Effects and Elastic Strain Properties during 
Thin Film Nanoindentation 
 
by 
 
Bo Zhou 
 
 
 
 
A dissertation submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Doctor of Philosophy 
 
Auburn, Alabama 
December 18, 2009 
 
 
 
 
Keywords: Nanoindentation, Thin Film, Young?s Modulus 
Substrate Effect 
 
 
Copyright 2009 by Bo Zhou 
 
 
Approved by 
 
Barton C. Prorok, Chair, Associate Professor of Materials Engineering  
Ruel A. Overfelt, Professor of Materials Engineering 
Dong-Joo Kim, Associate Professor of Materials Engineering 
Pradeep Lall, Thomas Walter Professor of Mechanical Engineering 
Minseo Park, Associate Professor of Physics 
 
 
 
 
 
 
 
 
 
Abstract 
 
 
 In this study, the main goal was to investigate the true origin of an influence of 
substrate effects during nanoindentation of thin films. The work examined the proper 
relation between the elastic strain intensities in the film and substrate, and modified the 
Doerner & Nix function used to first describe this situation. A universal mathematical 
model/formula was developed to better describe the nanoindentation Modulus?
Displacement curve. Furthermore, a physical explanation was suggested for the new 
function. In this project, 15 specimens (including 14 amorphous thin films and 1 
nanocrystalline film deposited on Si substrates) were tested using a MTS nanoindentation 
XP system. The films were tested with a continuous stiffness measurement (CSM), where 
flat regions were found in the early stage of the E-h curves that reflected the true film 
modulus.  
The flat region (the region without substrate effects) and critical indentation depth 
(h
cr
) was determined for each material. A unique non-linear trend was found between the 
critical indentation depth/film thickness ratio (h
cr
/t) and film/substrate modulus ratio 
(E
f
/E
s
). It was found parameter ? in Doerner & Nix function was not a constant as 
originally suggested. Instead, it changed with the indentation depth. To improve Doerner 
& Nix function, two parameters ?
1
 and ?
2
 were suggested in the equation instead of one, 
as the elastic strain growth is not continuous in the film and substrate. Based on the 
experimental investigation and analytical modeling, the two parameters were determined 
ii 
 
as the Poisson?s ratios of the film and substrate. The new function was found to be adept 
at closely matching all experimental data collected, which spanned both soft films on 
hard substrates and hard films on soft substrates. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 iii
 
 
 
 
 
Acknowledgments 
 
 
 I would like to thank Dr. Prorok for his support and suggestions during this work. 
I would like to thank Shakib, Nicole, Kevin, Sang, Joshua, Shankar, and Mike for their 
help in film synthesis, fabrication and finite element Analysis. I also want to thank Dr. 
Erdemir and Dr. Eryilmaze from Argonne National Lab for their assistance with DLC 
films deposition. I want to thank all my committee members for their guidance 
throughout this research. Finally, I want to thank my family for their love and support all 
these years. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 iv
 
 
 
 
 
Table of Contents 
 
 
Abstract............................................................................................................................... ii 
Acknowledgments.............................................................................................................. iv 
List of Tables .................................................................................................................... vii 
List of Figures.................................................................................................................. viii 
List of Abbreviations ........................................................................................................ xii 
Chapter 1 Introduction .........................................................................................................1 
 1.1 Synopsis .............................................................................................................1 
 1.2 Dissertation Structure.........................................................................................2 
Chapter 2 Literature Review................................................................................................5  
2.1 Instrumented Indentation Testing ......................................................................5 
 2.2 Continuous Stiffness Measurement .................................................................16 
 2.3 Thin Films Substrate Effects............................................................................20 
 2.4 Finite Element Analysis of Thin Film Nanoindentation..................................30 
 2.5 Pile-up and Sink-in Effects ..............................................................................34 
 2.6 Thin Film Microstructure Effects (Size Effects) .............................................41 
 2.7 Thin Film Deposition Mechanism ...................................................................42 
 2.8 X-ray Diffraction Technique............................................................................46 
Chapter 3 Experimental Setup ...........................................................................................48 
3.1 Thin Film Selection and Deposition ................................................................48 
 v
 3.2 Thin Film Characterizations ............................................................................55 
 3.3 Nanoindentation Setup.....................................................................................56 
 3.4 Finite Element Analysis Setup.........................................................................57 
Chapter 4 Results and Discussions ....................................................................................60 
4.1 Film Thickness Measurement & XRD Results................................................60 
 4.2 Indentation Impression (Sink-in & Pile-up) Characterizations .......................63 
 4.3 Film Young?s Modulus and Critical Indent Depth Determination ..................68 
 4.4 Doerner & Nix Function Investigation ............................................................74 
 4.5 Alpha (?) Parameter Determination.................................................................82 
 4.6 Physical Explanation of ?
1
 and a
2 
in Modified Doerner & Nix Function......103 
Chapter 5 Conclusions .....................................................................................................115 
References  ......................................................................................................................117 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 vi
 
 
 
 
 
List of Tables 
 
 
Table 2.1 Geometries of Different Indenter Tips...............................................................15 
Table 3.1 Deposition Information for All the Specimens..................................................51 
Table 3.2 Selected Sputtering Parameters for Amorphous CrO
x
.......................................52 
Table 4.1 h
cr
, E
f
, Ef, E
f
/E
s
, and h
cr
/t for All Tested Specimens..........................................71 
Table 4.2 h
cr
/t and E
f
/E
s
 Data of NFC films with Different Thickness..............................73 
Table 4.3 List of ?
1
 and ?
2
 Values for Each thin film with the Film Poisson?s Ratio.......94 
Table 4.4 ? in the Doerner & Nix Model and corresponding K and R values.................108 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 vii
 
 
 
 
 
List of Figures 
 
 
Figure 2.1a Schematic of the Basic Components of a Nanoindenter ..................................6 
Figure 2.1b Image of a Nanoindenter XP System ...............................................................7 
Figure 2.2   Schematic of Load-Displacement Curve Obtained by Nanoindentation .........8 
Figure 2.3   Load-Displacement Curve of Bulk <111> Silicon.........................................11 
Figure 2.4   Load-Displacement Curve of Bulk Aluminum ..............................................11 
Figure 2.5   A Load-Displacement Curve Showing a Nonlinear Unloading Portion ........14 
Figure 2.6   Schematic of a Nanoindentation Impression..................................................15 
Figure 2.7   Schematic of Continuous Stiffness Measurement..........................................17 
Figure 2.8   Load-Displacement Curve from a CSM Test.................................................17 
Figure 2.9   CSM modulus-displacement curve for bulk fused silica................................19 
Figure 2.10 Modulus-Displacement Curve for a SiO
2
 Thin Film on the Si Substrate.......19 
Figure 2.11 Modulus-Displacement Curve for a Si
3
N
4
 Thin Film on the Si Substrate .....20 
Figure 2.12 Compliance vs. Reciprocal Contact Depth for Tungsten Film on Si .............23 
Figure 2.13 Comparison of Doerner & Nix and Gao?s Weighting Factors.......................23 
Figure 2.14 DLC E-h Data with Curves from Doerner&Nix and Gao?s Functions ..........24 
Figure 2.15 Alpha as a Function of Normalized Punch Size from King?s Work ..............25 
Figure 2.16 Indentations from a Flat Punch and the Effective Flat Tip for a Berkovich ..26 
Figure 2.17 Experimental and the Repression Values for the TbDyFe/Si Film on Si.......29 
Figure 2.18 Center of a Typical 2D Axisymmetric Mesh for Thin Film Indentation .......32 
 viii
Figure 2.19 Experimental and FEA Modulus vs. Displacement Curves of a-tC Film ......33 
Figure 2.20 Sink-in and Pile-up Profiles during Nanoindentation ....................................35 
Figure 2.21 SEM Image of a Pile-up Indentation Impression ...........................................37 
Figure 2.22 Schematic of Pile-up and Sink-in Contact Areas during Indentation.............38 
Figure 2.23 AFM (a) Image and (b) Profile of a 750 Indentation on Al/Glass .................38 
Figure 2.24 Schematic Representation of the Projected Contact Area due to Pile-up ......39 
Figure 2.25 (P/S
2
) vs. Indentation Depth for Fused Silica Film on Si Substrate...............40 
Figure 2.26 Various Deposition Systems for Thin Films ..................................................45 
Figure 2.27 Schematic of RF PECVD and Plasma Sheath................................................46 
Figure 3.1   Image of a Denton Sputtering System............................................................53 
Figure 3.2   Schematic of the RF PECVD System at Argonne National Lab....................54 
Figure 3.3   Image of a Jeol 5000 Atomic Force Microscope............................................55 
Figure 3.4   Image of a Jeol 7000F Filed Emission Scanning Electron Microscope.........56 
Figure 3.5   Schematic of the Finite Element Analysis of the Nanoindentation................59 
Figure 4.1a Cross-sectional SEM Image of ta-C:H Film on Si Substrate .........................61 
Figure 4.1b Composition SEM Image of the Cross-section of NFC/Si.............................62 
Figure 4.2   X-ray Spectrum ..............................................................................................63 
Figure 4.3   SEM Image of an Indentation Impression on NFC/Si....................................65 
Figure 4.4   AFM Images of a Sink-in Impression of 300 nm Indentation on NFC/Si .....66 
Figure 4.5   AFM Images of a Sink-in impression of 500 nm Indentation on ta-C:H/Si ..66 
Figure 4.6   AFM Images of (a) 300 nm and (b) 500nm Indentation on a-Si Film ...........67 
Figure 4.7   CSM Modulus-Displacement Curves of (a) SiO
2
 and (b) Si
3
N
4 
Films ..........70 
Figure 4.8   h
cr
/t vs. E
f
/E
s
 for All Tested Specimens..........................................................72 
 ix
Figure 4.9   Positions of the NFC Films with Different Thickness in h
cr
/t vs. E
f
/E
s
.........74 
Figure 4.10 Experimental and Theoretical Curves of h
cr
/t-E
f
/E
s
.......................................76 
Figure 4.11 SiO
2
 Modulus-Displacement Curve with Theoretical Curves........................77 
Figure 4.12 Plot of ? Value at Critical Indentation Depth vs. E
f
/E
s
..................................79 
Figure 4.13 Plot of ? vs. h for NFC Film Contrasted with the E-h Curve.........................80 
Figure 4.14 Plot of ? vs. h for SiO
2
 Film Contrasted with the E-h Curve.........................81 
Figure 4.15 Plot of ? vs. h for ta-C:H Film Contrasted with the E-h Curve......................82 
Figure 4.16 A Suggested Power Function of h for Alpha..................................................83 
Figure 4.17 Schematic of Certain Degree of the Elastic Strain Growth in the Specimen .85 
Figure 4.18 FEA Images of the Elastic Strain Distribution in NFC/Si Specimen.............87 
Figure 4.19 FEA Image of the Elastic Strain Distribution in ta-C:H/Si ............................90 
Figure 4.20 FEA Result of Indenting 20 nm into (a) UNCD/Si and (b) Si/UNCD...........91 
Figure 4.21 Comparison of ?
1 
-h data for SiO
2
 with the Doerner & Nix function ............93 
Figure 4.22 E-h Curve of SiO
2
 Film with the new, Doerner&Nix, and Gao Models........95 
Figure 4.23 E-h Curve of a-Si Film with the new, Doerner&Nix, and Gao Models.........95 
Figure 4.24 E-h Curve of FeB Film with the new, Doerner&Nix, and Gao Models.........96 
Figure 4.25 E-h Curve of TiO
x
 Film with the new, Doerner&Nix, and Gao Models........96 
Figure 4.26 E-h Curve of AlN Film with the new, Doerner&Nix, and Gao Models ........97 
Figure 4.27 E-h Curve of NFC Film with the new, Doerner&Nix, and Gao Models........97 
Figure 4.28 E-h Curve of FeO
x
 Film with the new, Doerner&Nix, and Gao Models .......98 
Figure 4.29 E-h Curve of AlO
x
 Film with the new, Doerner&Nix, and Gao Models .......98 
Figure 4.30 E-h Curve of CrO
x
 Film with the new, Doerner&Nix, and Gao Models .......99 
Figure 4.31 E-h Curve of SiC Film with the new, Doerner&Nix, and Gao Models .........99 
 x
Figure 4.32 E-h Curve of Si
3
N
4
 Film with the new, Doerner&Nix, and Gao Models ....100 
Figure 4.33 E-h Curve of ta-C:H Film with the new, Doerner&Nix, and Gao Models ..100 
Figure 4.34 E-h Curve of Metglas Film with the new, Doerner&Nix, and Gao Models 101 
Figure 4.35 E-h Curves of (a) Polymer and (b) UNCD Films with the New Model.......102 
Figure 4.36 Effective Elastic Strain Volume Developed during Nanoindentation..........104 
Figure 4.37 Important Parameters for the Spherical Segment.........................................105 
Figure 4.38 Comparison of the D&N Factor and the Analytical Model from Eq. (22)...107 
Figure 4.39 Plot of (a) 1/K vs. ? and (b) R vs. ? for a 10 nm Indentation .......................109 
Figure 4.40 Comparison between 1-? and 2-? Models....................................................110 
Figure 4.41 Radius of the Elastic Strain Boundary vs. Alpha1 & Poisson?s Ratio .........111 
Figure 4.42 Schematic of an Indentation Impression in the Film....................................113 
Figure 4.43 Elastic Strain Distributions in the Films with Different Poisson?s Ratios ...114 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 xi
 
 
 
 
 
List of Abbreviations 
 
 
DLC Diamond Like Carbon    
NFC Near Frictionless Carbon 
CSM Continuous Stiffness Measurement 
UNCD       Ultrasmall Nano-Crystalline Diamond 
 
 
 
 
 
 
 
 xii
 1 
 
 
 
 
 
CHAPTER 1 
INTRODUCTION 
1.1.  Synopsis 
In the last three decades, Instrumented Indentation Testing (IIT), also known 
as depth sensing indentation, has become one of the most important methods for the 
mechanical characterizations of bulk and thin film materials [1-66]. As the world has 
been more and more inspired by the nano-scale materials and devices, instrumented 
indentation, inexorably, followed this ?expanding? trend. From micro-scale to the 
nano-scale regime, nanoindentation has been widely used to determine the Young?s 
modulus, hardness, fracture strength, residual stress and other important mechanical 
properties of extremely small systems, such as ultra-thin films, nanowires, 
nanoparticles, NEMS, and bio-nanomaterials [20,29,37,47,50]. However, several 
key issues have limited the use of nanoindentation, or more precisely, prevented it 
from obtaining accurate information of the target materials [10,16,67-69]. For 
instance, thin film substrate effects have always complicated measurements as the 
indentation is influenced by m both the film and the substrate. Eliminating substrate 
effects and deriving the true mechanical properties of the film are essential and 
urgent for the development of nanomaterials and nanotechnology, as well as greatly 
improve the comprehension of the elastic and plastic deformation growth in thin 
films.  
 2 
Ever since nanoindentation was applied to thin films, there have been efforts 
to eliminate substrate effects. Many outstanding researchers have used both 
experimental and theoretical methods to study this subject [8,10,13,15,18,70-72]. 
However, due to the complexity from other nanoindentation effects, such as film 
microstructures and pile-up/sink-in situations, the veil on the substrate effect has not 
been fully uncovered. Thus, this research followed the significant work of Nix, 
Doerner, Gao, Oliver, Pharr, and others to investigate and better understand the 
mechanism behind the substrate effect. In this work, 15 thin films (14 amorphous and 
1 nanocrystalline films) were deposited and tested with the Nanoindenter. The 
selection of amorphous structure was aimed at eliminating the microstructure effects 
so that the substrate effect could be decoupled from them. A physical explanation was 
found for an existing empirical function of the substrate effect. By improving this 
function, it was able to derive the true film modulus from the nanoindentation data. 
One unique result in this research was to find that the Poisson?s ratios of the film and 
the substrate played important roles for determining substrate effects and how they 
manifested the composite modulus measured by the nanoindentation. 
 
1.2.  Dissertation Structure 
Chapter 2 gives literature reviews to all the subjects that are related to this 
dissertation work. It includes nanoindentation, continuous stiffness measurement 
(CSM), substrate effects, finite element analysis of nanoindentation, pile-up/sink-in 
phenomenon, film microstructure effects, thin film deposition, and the X-ray 
 3 
diffraction technique. In the nanoindentation part, an overview of instrumented 
indentation theory was given. Two different indentation methods: static and dynamic 
measurements were covered in this section. The relation between the Continuous 
Stiffness Method (dynamic) and the substrate effect was explained in detail. Substrate 
effects were the most important part of this research, and were thus covered in 
significant detail. Finite element analysis played a large role in substrate effects 
indication in this study. Useful simulation results from other researchers were 
illustrated. Pile-up/sink-in and film microstructure, which were two other important 
nanoindentation effects, were also covered in this chapter. Thin film deposition 
techniques and deposition mechanisms were discussed based on how to obtain the 
amorphous structure. X-ray diffraction theory was briefly introduced for the film 
structure determination. 
Chapter 3 covers the experimental setup of this research, which consist of 
three parts: 1. Thin film deposition and characterization; 2. Nanoindentation 
measurements; 3. Finite element analysis setup. 
Chapter 4 deals with the final results and discussions. The experimental data 
from nanoindentation were calibrated for all the amorphous thin films. Young?s 
modulus (E) - Indent depth (h) curves were obtained from the continuous stiffness 
measurement (CSM) so as to investigate the starting point of substrate effects. The 
so-called ?flat region? (with minimum substrate effects) was observed in the early 
stage of nearly every E-h curve for all materials studied. The depth of this region (hcr) 
was found to vary with the material in a non-linear fashion. Based on the 
 4 
experimental data and simulation results, we were able to modify the Doerner & Nix 
empirical function for thin film indentation on a substrate, particularly their parameter 
alpha (?). A mathematical model of the elastic strain energy distribution was 
established to support the new theory. The original alpha was replaced with two new 
parameters that were equal to the film and substrate Poisson?s ratio. The modified 
relation was found to be adept at closely matching all experimental data collected, 
which spanned both soft films on hard substrates and hard films on soft substrates. 
  Chapter 5 provides conclusions and future work.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
CHAPTER 2  
LITERATURE REVIEW 
2.1. Instrumented Indentation Testing 
Instrumented indentation testing (IIT), also known as nanoindentation, is 
achieved using a high-resolution system that can drive an indenter tip made of 
diamond into and withdraw it from a sample material [10,16,29]. During the process, 
the loads and displacements of the tip are recorded by positional sensors, from which 
mechanical properties can be derived. In the past few decades, this technique has been 
developed sufficiently to satisfy the increasing needs of nano-scale materials and 
devices characterizations. Nanoindentation has two major advantages: (1) The 
mechanical properties of nanowires and NEMS are usually very difficult to determine 
by conventional tensile tests. Through nanoindentation, the Young?s modulus and 
hardness can be calculated from the load-displacement curve. Other important 
properties like yield strength, fracture strength, residual stress can be estimated using 
nanoindentation related methods [24,36,62,73-80]. (2) Another key advantage is that 
nanoindentation does not have strict requirements on specimen?s shape, scale, and 
surface conditions (especially for bulk materials). As the indenter tip has a radius less 
than 50 nm, the technique is extremely useful to examine ultra-thin films and 
coatings. 
   
 6 
Figure 2.1a and 2.1b show the schematic and an image of an indentation 
system [29]. There are three basic components of a Nanoindenter: an indenter tip, an 
actuator for applying forces, and a sensor for recording the indent depth. The 
indentation process consists of two stages: loading and unloading. During loading, the 
indenter tip is driven into the material, where both elastic and plastic deformations 
occur. When the tip is unloaded from the sample, only the elastic recovery occurs, 
which effectively separates the elastic properties of the materials from the plastic. A 
typical load-displacement curve is illustrated in figure 2.2 [50].  
 
 
Figure 2.1a Schematic of the basic components of a Nanoindenter [29] 
 
 
 
 7 
 
Figure 2.1b Image of a Nanoindenter XP system 
 
 
 8 
 
 
Figure 2.2 Schematic of the typical load-displacement curve obtained by 
nanoindentation [50] 
 
 
 
 9 
The earliest load-displacement sensing indention experiments can be retraced 
to 1940s. Tabor performed indentations on several metals using a spherical 
indenter[1]. An interesting result was that the hardness impression on samples after 
unloading was indenter shape related. In other words, a round shape impression was 
caused by a spherical tip. Same results were obtained by Stillwell from a conical tip 
[2]. More importantly, they found the existence of elastic recovery during unloading. 
They showed the evidence that the shape of the unloading curve and the elastic 
recovering displacement were related the elastic modulus.  
Their research also led to the understanding that the reduced (measured) 
Young?s modulus (Er) from indentation accounted for the elastic displacement 
occurring in both the indenter and the specimen. The effect can be represented by the 
following equation: 
i
i
r EEE
22 111 ?? ?
+?=                          (1) 
Where E and ? are the Young?s modulus and Poisson?s ratio of the specimen, Ei and ?i 
are Young?s modulus and Poisson?s ratio, respectively of the indenter. For a typical 
diamond tip, Ei =1141 GPa, and ?i=0.07 [1,2]. 
In the 1970s, researchers started to show interest in the roles of the unloading 
curve slope (S) and the projected indentation contact area (A) in the Young?s modulus 
and hardness determination. The representative work in this decade was from 
Bulychev and co-workers [4]. They found a relationship between the reduced 
modulus and the two parameters mentioned above: 
 10 
AEdhdPS rpi2=??????=                         (2) 
Where S is the stiffness of the upper portion of the unloading curve, P is the load, and 
h is the displacement. 
      The term ?nanoindentation? was first used in 1980s. In this decade, 
researchers began to realize the importance of instrumented indentation in thin film 
characterizations. Pethica, Hutchings, and Oliver proposed a simple method for 
modulus calculation [8]. They suggested that the contact area (A) in Eq. 2 can be 
estimated as a function of the final indent depth (hf). 
Loubel and co-workers proposed an idea of deriving the Young?s modulus 
from IIT in 1984 [9]. They suggested that E could be derived from the initial slope of 
the unloading curve, i.e. the indentation load as the function of the indentation depth. 
For a spherical indenter, the slope can be represented as: 
rErdh
dPS
02=??
??
?
?=                           (3) 
where r0 is the radius of the punch. The idea was sensible as the loading portion of the 
test has a complex combination of plastic-elastic deformation. The initial unloading 
then would be only composed of elastic response, and thus contain only the elastic 
information from the specimen to calculate the Young?s modulus. This idea was 
supported empirically by several tested metals that exhibit that the initial part of the 
unloading curve was linear, and the rest of the unloading curve was non-linear, see 
figure 2.3 and 2.4 for the load-displacement curve of <111> silicon and aluminum and 
aluminum [10].  
 11 
 
Figure 2.3 Load-Displacement curve of bulk <111> Silicon [10] 
 
 
Figure 2.4 Load-Displacement curve of bulk Aluminum [10] 
 12 
This theory was improved and expanded by Doerner and Nix in 1986 [10]. 
They proposed a relatively comprehensive method for interpreting the data from 
indentation. They assumed that the Young?s modulus was related to the initial slope of 
the unloading curve and the indentation depth. They also calculated the hardness of 
the film by subtracting the elastic displacement from the total displacement of 
indentation. The hardness equation is shown as:  
A
PH max=                            (4) 
where Pmax is the peak indentation load and A is the projected contact area.  
Moreover, they suggested that there was a ?substrate effect? on the measured 
composite modulus for a film/substrate specimen during indentation [10]. They 
believed that during unloading, the elastic response was not only from the information 
of the film but also from the substrate. They suggested an empirical function of the 
indent depth to describe the film and substrate contributions to the measured young?s 
modulus. Also, King supported Doerner & Nix by performing simulation of thin films 
[13]. Gao built up a mathematical model for the thin film indentation [15]. Their work 
has affected the nanoindentation theory for more than 15 years. As the Doerner & 
Nix/King function and Gao?s model are vital to this study, they will be explained in 
detail in the later section.  
In the 90s, Oliver, Pharr and co-workers were the leaders in the 
nanoindentation field. They proposed the famous Oliver & Pharr method for 
determining bulk materials hardness and elastic modulus from micro/nano indentation 
in 1992 [16]. As mentioned above, their work in investigating the IIT method began in 
 13 
early 1970?s. After years of accumulating indentation data on different materials, they 
questioned Loubel, Doenar, and Nix?s assumption about the linear region at the initial 
unloading curve. They found that for most materials, the unloading curve was 
completely nonlinear (see figure 2.5 for a nonlinear unloading curve). Their solution 
was to fit the unloading curve of the load-displacement data with a power-law relation 
(eq. 5) [16]:  
m
fhhBP )( ?=                          (5) 
where B and m are empirically determined parameters and hf is the final displacement 
after complete unloading of the indenter tip.  
. From the equation above, S can be derived as : 
1
max
max
)( ?
=
?=??????= mf
hh
hhBmdhdPS                 (6) 
The equations of reduced Young?s modulus (Er) and hardness calculation (H) are 
given as:   
A
SE
r ?
pi
2=                              (7) 
 
APH /=                               (8) 
where A is the projected contact area of contact under load, and ? is a constant that 
depends on the geometry of the indenter tip. The projected area is calculated by 
evaluating an empirically determined indenter area function A=f(hc) at the contact 
depth. This area function is also known as the tip shape function. It is related to the 
geometry of the indenter. For a perfect Berkovich tip, A can be represented as: 
 14 
256.24
chA =                            (9) 
It needs to mention that in practical use, the tip geometry will not be perfectly sharp. 
Thus a calibration of the contact area needs to be introduced using: 
????????++++= 3/132/1211256.24 cccc hChChChA             (10) 
where Ci are the fitting constants. 
Oliver and Pharr were the first to suggest (A) as function of hc instead of hf. The 
contact depth (hc) is usually smaller than the displacement (h) for a sink-in situation 
(hc>h, when pile-up). Figure 2.6 shows the schematic of a section through an 
indentation and the relationships between various parameters. In Oliver & Pharr?s 
method, the contact depth is estimated as[16]:   
S
Phh
c ??=                             (11)  
where ? is a constant that depends on the indenter geometry. Nowadays, most of the 
Nanoindenter systems use the Oliver & Pharr method to calculate the materials 
mechanical properties.  
 
Figure 2.5 A load-displacement curve showing a nonlinear unloading portion 
 15 
 
Figure 2.6 Schematic of a nanoindentation impression [16] 
 
Indenters with various shapes are used for different purposes. Pyramidal 
indenters also well known as Berkovich tips are probably the most frequently used 
shape. The three-sided pyramid can ground to a very small point, which will keep its 
geometry to the minimum scales. Besides the Berkovich tip, spherical, cube-corner, 
conical, and Vikers pyramid indenters are also applied. Table 2.1 shows their 
geometries and parameters [50].  
 
Table 2.1 Geometries of different indenter tips [50] 
 
 16 
2.2  Continuous Stiffness Measurement 
Other than the traditional static nanoindentation method, the continuous 
stiffness measurement (CSM), also known as dynamic stiffness measurement, allows 
one to determine the Young?s modulus and hardness along with the displacement 
[29,37]. This alternative of nanoindentation is ideal for very thin film 
characterizations due to the sensitivity of the indent depth. Basically, CSM is no 
different from the static methods. A typical CSM test can be seen as a set of the static 
indentations at the same location but with increasing indent depths. This is 
accomplished by superimposing a small force oscillation on the primary loading 
signal and analyzing the resulting displacement response using a frequency-specific 
amplifier, see figure 2.7 [37]. Thus, it allows continuous measurement of stiffness as 
the tip is driven into the specimen. A typical CSM loading circle is shown in figure 
2.8 [37]. 
 
 17 
 
Figure 2.7 Schematic of continuous stiffness measurement [37] 
 
Figure 2.8 Load-displacement curve from a CSM test [37] 
 18 
CSM has numerous advantages over a single static indentation. First, it can be 
used to evaluate the fatigue in nano-scale thin films and materials. Also the small time 
frame of each indentation in CSM makes it ideal for measuring the mechanical 
properties of polymeric materials. Measurements on multilayer (more than a single 
film) structures and graded materials can also be benefited from the continuous 
method. Furthermore, recent works show that creep measurements of the nanoscale 
materials can be performed by monitoring the change in stress-displacement relation 
in CSM tests [37].  
Applying CSM tests on bulk and thin film materials lead to different results. 
Figure 2.9 and 2.10 show the Young?s modulus-displacement curves of bulk silica and 
SiO2 thin film on a Si substrate, respectively. As one can imagine, the curve for a bulk 
material will be a straight line which shows the bulk elastic modulus. However, for a 
thin film system, the measured modulus represents a composite value containing both 
the film (Ef) and the substrate (Es) information. For SiO2 on Si, Ecomp starts as a 
relatively small value that is closed to Ef. The curve sustains relatively flat for a short 
period and then increases with the displacement towards Es. The CSM curve gives us 
a clear vision that the substrate effects increase as the indenter goes deeper into the 
film. The CSM E-h curves vary with film and substrate properties. For instance, when 
a soft film in on a hard substrate, the curve has a rising trend, while as a hard film on a 
soft substrate, the curve will decrease (see figure 2.11 for a Si3N4 film on the Si 
substrate). The unique relation between the substrate effect and E-h curve makes CSM 
an extreme valuable tool for thin film mechanical characterizations. 
 19 
. 
0
10
20
30
40
50
60
70
80
90
100
0 400 800 1200 1600 2000
Displacement into suface (nm)
Yo
un
g's
 m
od
ulu
s (
GP
a)
 
Figure 2.9 CSM modulus-displacement curve for bulk fused silica 
0
30
60
90
120
150
180
0 50 100 150 200 250 300 350 400 450 500
Displacement into surface (nm)
Yo
un
g's
 M
od
ulu
s (
GP
a)
 
Figure 2.10 CSM modulus-displacement curve for a SiO2 thin film on the Si Substrate 
SiO2 (E=77 GPa) on Si Substrate (E=170 GPa) 
 20 
0
50
100
150
200
250
0 50 100 150 200 250 300 350 400 450 500
Displacement (nm)
Yo
un
g's
 M
od
ulu
s (
GP
a)
 
Figure 2.11 CSM modulus-displacement curve for a Si3N4 thin film on Si substrate 
 
2.3  Thin film substrate effects 
In recent years, as instrumented indentation moved from micro-scale to the 
nano-scale regime, new issues have arose in interpreting nanoindentation data. Among 
them, the substrate effects in thin films, film pile-up/sink-in effects, and the film 
microstructures are the three key issues [50]. 
As mentioned in the last section, while using nanoindentation, the 
measurement of thin film properties is much more difficult than bulk materials 
because the indentation unloading data consists of both film and substrate responses. 
The mechanism of this phenomenon was believed to result from the different 
mechanical properties (Young?s modulus, hardness, Poisson?s ratio, etc) of the film 
and the substrate [10,13,15,71]. As one can imagine, when a hard film is on a soft 
 21 
substrate, the modulus will be reduced from the true film value towards the substrate 
value with increasing indent depth. The opposite trend will occur when a soft film is 
deposited on a hard substrate.  
The substrate effects will largely influence the accuracy of the film mechanical 
properties. An often used guideline recommended by B?ckle is that if the indent depth 
is less than 10% of the film thickness, the substrate can be neglected [3]. Based on 
this, researchers like Doerner, Nix and King suggested there was always a ?flat 
region? in the early stage of the CSM nanoindentation test, where the substrate effect 
could be minimum, and the values indicated the true film Young?s modulus. A typical 
flat region in the Young?s modulus-displacement curve is shown in figure 2.10 and 
2.11. The idea about indenting 10% of the thickness was obtained from the micro or 
even larger scale indentation tests [3]. Based on recent studies, though the guideline 
can still fit most of the cases, it is not always true, especially when the film is much 
softer or harder than the substrate [72]. Also as the film thickness shrinks down to 
nanometer scales, the surface condition effect becomes vital for the first few tens 
nanometer indentation. Thus, simply using 10% rule might lead to an inaccurate 
property value. A more comprehensive, detailed and systematic method needs to be 
found to evaluate substrate effects. 
Doerner and Nix were the first to describe the substrate effect using an 
empirical function. In their work, Doerner and Nix performed nanoindentation on 
tungsten sputtered on Si. They plotted the compliance as a function of the reciprocal 
displacement as shown in figure 2.12 [10], and a unique relation was found. Based on 
 22 
their data, Doerner and Nix suggested a solution for the nanoindentation effective 
modulus-displacement curve, as shown below [10]:  
   
)/()11(11 cht
fsf
eEEEE ?????+?=                    (12) 
where E is the measured Young?s modulus, E?f=Ef /(1-?2f), E?s=Es /(1-?2s), t is the film 
thickness, hc is the indenter tip contact depth, and ? is an empirically determined 
constant for all the materials, which Doenar and Nix suggested a value of 0.25. 
Although, it doesn?t have a physical background, the function works relatively well 
with most of the thin film materials. The Doerner & Nix weight factor exp(-?t/hc) is a 
sigmoid-shaped curve. As shown in figure 2.13 as a function of hc, this factor 
increases from 0 and reaches 1 in infinity. This factor properly describes the influence 
of the substrate along with the indent depth. And figure 2.14 shows how the Doerner 
& Nix function fits the experimental data from CSM nanoindentation. 
                             
 23 
 
Figure 2.12 Compliance vs. reciprocal contact depth for tungsten film on Si [10] 
 
Normalized Displacement (h/t)
0.01
0.25
0
W
eig
ht
 Fa
cto
r
0.1 1 10
0.5
0.75
1
Doerner & Nix 
Factor
Gao's
Factor
W
eig
ht
 Fa
cto
r
 
Figure 2.13 Comparison of Doerner & Nix and Gao?s weighting factors 
 24 
Displacement into Surface (nm)
4000
40
0
Yo
un
g?s
 M
od
ulu
s (
GP
a)
800 1200 1600 2000
80
120
160
Gao Function
Doerner&Nix Function
Yo
un
g?s
 M
od
ulu
s (
GP
a)
 
Figure 2.14 DLC E-h data with theoretical curves from Doerner & Nix and Gao?s 
functions 
 
King supported Doerner & Nix?s work by using finite element analysis [13]. 
In his work, he replaced hc in Doerner & Nix function by the square root of the 
projected contact area a so as to take the indenter geometry into consideration. Three 
major indenter geometries were considered in his study: sphere, square, and triangle. 
The theoretical procedure used by King?s simulation was for analyzing elastic normal 
contact problems for layered media. The simulation results were in certain agreement 
with Doerner & Nix. However, there was a major contribution in King?s paper that 
was against the old work. It was found by simulation that the parameter alpha (?) in 
Doerner & Nix function was not a constant as claimed ?0.25?, but changed with the 
 25 
contact area (depth), see figure 2.15. Doerner & Nix and King?s model often gives 
satisfying results for those films that have a flat region in their early E-h curves. 
However, it is not without problems. The alpha parameter changes with the 
displacement, which might leads to inaccurate data. Also only one variable in the 
function makes it inflexible in fitting experimental data. 
 
Figure 2.15 Alpha as a function of normalized punch size from King?s work [13] 
 26 
Efforts have been made to improve the Doerner & Nix function. Based on 
King?s simulation results [21,26,34,39]. Saha and coworkers modified the old 
function to calculate the thin film Young?s modulus [39]. King assumed a flat 
triangular punch in his analysis, but Doerner & Nix?s data was from a pyramid-shape 
Berkovich tip. Saha connected two efforts by introducing the effective film thickness 
?t-h? to replace t in the old function as shown in figure 2.16. The modified function is: 
a
ht
fsf
c
eEEEE
)(
)11(11
??
???+?=
?
                   (13) 
The modified function fits well as the indent depth is less than 50% of the film 
thickness. However, the predicted reduce modulus greatly deviates from the 
experimental value. This is because changing the Berkovich tip to an ?effective? flat 
punch will greatly overestimate the contribution of the substrate stiffness at large 
indent depth.  
 
 
Figure 2.16 Indentations from a flat punch and the effective flat tip for a Berkovich 
[39] 
 
 27 
Another important study in this field was from Gao and his co-workers [15]. 
Quite different from the theoretical procedure that King employed in his work, Gao 
focused on the total energy transformation from the indenter to a thin film/substrate 
system during nanoindentation. Their idea was simple but intuitive: considering the 
elastic contact problem, the film/substrate system can be seen as a homogeneous body 
made of the substrate material with a phase transformation layer at the surface. Based 
on the energy conservation theory, they derived a mathematical solution from a rigid 
cylindrical stick punching into a semi-infinite elastic body with a different surface 
layer. The function was shown as below: 
??+= )( sfs EEEE                        (14) 
where ? is the weighting factor that depends on the ratio of film thickness and contact 
area radius. The Gao?s factor is shown as: 
 
??
?
??
?
+?
+?
?+=? 22
2
1
1ln)21(
)1(2
1)arctan(2
x
x
x
xxx ?
?pipi          (15) 
 
Where x=t/a. t is the film thickness, a is the radius of the contact area, and ? is the 
Poisson?s ratio. Also, as mentioned above, a is a function of the contact depth hc. 
Like Doerner & Nix factor, Eq. 15 is a sigmoid-shape function that increases 
from 0 and reaches 1 when a or hc is close to infinity. These two factors were 
compared and contrasted in figure 2.13. It is easy to find the major difference is at the 
early stage of indentation. The Doerner & Nix factor has a small flat region, while 
Gao?s factor almost increases from the beginning. This leads to different justifications 
 28 
for the substrate effect during nanoindentation. Gao?s model suggests that the 
substrate effect becomes prominent as soon as the indenter touches the film, while 
Doerner & Nix function assumes there will be a small displacement in the early stage 
that reflects the true film modulus. This is the reason why Gao?s model always fits 
well with the specimens that have the measured modulus changing from the start, i.e 
diamond film on Si substrate. Gao?s mathematical derivation was almost flawless, but 
the model is a little arbitrary. For instance, regardless the trend of the measured 
modulus, as long as the indent depth/film thickness is equal to 0.85, the measured E is 
equal to (Ef+Es)/2.  
In recent years, there were many empirical and analytical solutions for 
modeling the nanoindentation substrate effects [17,18,21,31,32,41,81]. However, 
most of them were not tested with a wide range of materials with different modulus. A 
notable exception was made by Menick and co-workers [21]. They investigated the 
substrate effects by experimentally testing with several groups of thin film materials. 
Based on the data, they compared and contrasted five major expressions for thin film 
indentation including Doerner & Nix and Gao?s functions (figure 2.17a and 2.17b). 
However, one concern was that those materials all had different microstructures. The 
film microstructures are known to greatly influence the nanoindentation data, and thus 
might lead to inaccurate data  
 29 
 
Figure 2.17a Experimental data and the repression values for the TbDyFe/Si film on 
Si. 
 
Figure 2.17b Experimental data and the repression values for the Pt film on glass. 
 30 
2.4  Finite element analysis of thin film nanoindentation 
Finite element analysis is another promising technique for investigating thin 
film nanoindentation [44,46,52,70,72,77,82-89]. On the one hand, it can help 
understand the indentation process and elastic and plastic deformation growth in the 
specimen. On the other hand, it can be used to improve the analytical method for 
extracting mechanical properties from the load-displacement data. There are many 
types of FEA software that help simplify thin film indentation simulation, which 
include: ANSYS, ABAQUS, COSMOS, and COVENTOR, etc. The first step of 
simulating the indentation contact problem is to define a mesh and the element type 
that can fit the real thin film/substrate and indenter well. It is also important to know 
the mechanical properties of the indenter, indenter geometry, film thickness, surface 
contact problem, and tribological properties of the film before any simulation process.  
Due to different elements, meshes, dimensions and contact settings, there were 
numerous successful models built for thin film nanoindentation. Among them, a 
simple but efficient 2D model has been accepted and used by Bolshakov, Knapp, Gan 
and many other researchers [46,72,82,84,89]. In this model, the specimen and the 
indenter are described by a two-dimensional axisymmetric mesh. As shown in figure 
2.18 from Knapp?s work [72], the element size is not evenly distributed. Note that the 
element size of the film underneath the indenter is the smallest. Since large 
deformation is expected in this local area, a small size of 10-20 nm will be sensitive 
enough for the accurate calculation. The less interest area usually has a relatively 
large element size to save total computation time. For special interests like the 
 31 
substrate effect investigation, local substrate area (right below the indenter) will also 
be refined to have a very small element size. There are two boundary lines for this 
model. The bottom line is fixed on both lateral and vertical directions, while the left 
line was only prohibited for lateral movement. The target contact pair consists of the 
film surface and one edge of the indenter. A detailed setup will be covered in the next 
chapter. This model is simple but still can provide reasonable results. Besides, due to 
the small mesh size, it provides a relative short run time, which makes it a useful tool 
for a large amount simulation work. FEA of thin film indentation has many 
advantages that the experiments can not provide. It can give us a physical idea about 
how the film and substrate deform and the elastic and plastic strain growth in the 
specimen. It can also calculate some properties which are difficult to be obtained from 
experimental methods, such as residual stress and yield strength. FEA also help to 
understand the nanoindentation process on very hard thin films. It was found in 
certain film materials, such as hard DLC or diamond on a soft substrate, the modulus 
could never reach the suggested values of film modulus (assuming the film modulus 
is comparable with the bulk value), as shown in figure 2.19. Above all, the 
encouraging results confirm that FEA is a promising and really helpful tool for the 
study of thin film nanoindentation.   
 
 
 32 
 
Figure 2.18 Center of a typical 2D axisymmetric mesh for thin film indentation [72] 
 33 
 
Figure 2.19 (a) Experimental and FEA Hardness vs. displacement curves of a-tC film 
on Si; (b) Experimental and FEA elastic Modulus vs. displacement curves of a-tC film 
on Si; 
 
 34 
2.5  Pile-up and Sink-in effects 
Indentation contact area is a key parameter for thin film modulus and hardness 
determination. In the Oliver & Pharr method (first suggested in 1992), the contact 
area was suggested to be a function of the contact depth (hc), which is related to the 
indenter geometry. Also in their method, ?sink-in? was assumed for all kinds of 
material (figure 2.6). This assumption seems reasonable as the indenter is pushed into 
the film, the local area around the tip should also go down to a certain degree. 
However, when a sharp tip like the Berkovich or Cone is used, the indentation process 
is both elastic and plastic. In certain materials, the film material around the tip is 
uplifting plastically instead of normal sink-in, and it can not be explained by the 
elastic contact theory alone. This plastic phenomenon is called Pile-up. Images of 
pile-up and sink-in are shown in figure 2.20a [29]. 
Pile-up doesn?t happen in all materials, however, when it does, it will greatly 
affect determination of film mechanical properties. As shown figure 2.20b [67], the 
contact depth for pile-up materials is larger than those with sink-in. Thus, if 
employing the Oliver & Pharr method for sink-in, the contact area of piling up 
materials will be underestimated, which cause overestimation of the Young?s modulus 
and hardness. Investigations of the pile up mechanism have been performed by many 
researchers [51,60,67,68,85,90-93]. Bolshakov and Pharr examined pile-up/sink-in in 
finite element analysis [67]. Based on their results, the hardness can be overestimated 
up to 60% when pile-up degree is large, while Young?s modulus is going to be 30% 
large. They also examined materials that intended to have pile-up by FEA. It was 
 35 
found that sink-in usually occurred in materials with low yield stress/Young?s 
modulus ratio and little work-hardening ability, i.e, hard ceramics. On the contrary, 
pile-up prevails in soft metals. It was also found that the film/substrate modulus ratio 
played a role for the occurrence of sink-in and pile-up. Normally, pile-up can be found 
as a very soft film is deposited on a hard substrate, while sink-in happens when a very 
hard film on a soft substrate.   
 
 
Figure 2.20a Sink-in and Pile-up profiles during nanoindentation [29] 
 
 
Figure 2.20b  Comparison of the contact depths for sink-in and pile-up [67] 
 
 
Pile-up 
Sink-in 
 36 
The Nix group also put effort in investigating the pile-up mechanism 
[34,39,68]. In McElhaney and Nix?s paper, they presented that the hardness 
impression on the film after indentation might give the information of how to 
calculate the pile-up and sink-in contact areas. As shown in figure 2.21 from Scanning 
Electron Microscopy (SEM) [29], the impression from a pile-up has a convex shape, 
and sink-in usually has a concave impression. A clear schematic is illustrated in figure 
2.22 [68]. Due to the unique relation between the impression and the cross-section 
area of the indenter, the contact area can be estimated if the pile-up degree is known. 
In recent years, as the Atomic Force Microscope (AFM) has been introduced for thin 
film surface characterizations, the pile up degree can be easily determined through an 
AFM scan across the indentation impression as shown in figure 2.23 and 2.24.  
 
 
 37 
 
 
Figure 2.21 (a) SEM image of a pileup indentation impression on Al/Glass; (b) SEM 
image of a sink-in indentation impression on NiP/Cu [29] 
 38 
 
Figure 2.22 Schematic of Pile-up and sink-in contact areas during indentation [68] 
 
 
 
Figure 2.23 AFM (a) image and (b) profile of a 750 indentation on Al/Glass [39] 
 
 39 
 
Figure 2.24 Schematic representation of (a) the projected contact area due to pile-up. 
(b) Cross-section of the contact area again showing pile-up [34] 
 
However, it is still difficult to build up a relation between the pile-up/sink-in 
degrees with film/substrate mechanical properties. The Nix group developed a method 
to eliminate. Joslin and Oliver first suggested to use load over stiffness square (P/S2) 
parameter to eliminate the pile-up and sink-in effects [94]. As shown in Eq. 16 and 
figure 2.25, P/S2 is independent of the contact area, and thus is not corrupted by 
pile-up/sink-in. Later on, the Nix group used this parameter to investigate the 
indentation size effects on hardness [34]. They chose those films and substrates with 
similar Young?s modulus (i.e. Al on glass) to eliminate the substrate effects. However, 
for the film and substrate with different values, Er in Eq. 16 is affected by the 
substrate effects.  
22 4
1
rE
H
S
P pi
?=                          (16) 
 
 40 
 
Figure 2.25 Load over stiffness square (P/S2) vs. indentation depth for fused silica 
film on Si substrate [34] 
 
Compared to the plastic pile-up, sink-in degree is more difficult to determine 
accurately because it involves elastic contact problem. Some serious sink-in situation 
(larger than that the Oliver & Pharr method expects) might occur when indenting on 
very hard materials like diamond. Serious sink-in will usually cause overestimation of 
the contact and resulting to a smaller modulus and hard value. Finite element analysis 
is often used to examine this situation, as shown in figure 2.19 [72]. 
 
 
 
 
 41 
2.6  Thin film microstructure effects (size effects) 
As thin films dimensions begin to approach that of the film microstructural 
features, the materials mechanical properties begin to exhibit a dependence on the 
specimen size [32,45,60,61,69,95,96]. In other words, as film thickness reduces to the 
micro scale, those data obtained from bulk materials are no longer suitable for thin 
film mechanical properties. Usually, plastic behavior, fracture strength and fatigue 
resistance will have the size effects when film thickness reduces to micron scale or 
smaller, while the elastic properties, which depend on the material bonding structure, 
will not exhibit the size effects until the nanoscale. Figure 2.26 [50] shows the 
schematic of length-scale effects on the mechanical properties. Moreover, as the film 
sizes reduce, the micro/nano-structures like grain or cluster size will change, which 
has an effect on both the elastic plastic behavior.  
The thin film microstructures largely complicate the analysis of 
nanoindentation data as they are often textured and do not represent a continuum. For 
instance, the anisotropic elastic properties due to the different crystal orientations, the 
strain gradient due to the crystalline structure, and the grain sizes and boundaries will 
cause the nanoindentation load-displacement to be shifted from the theoretical 
prediction [60,95]. To better understand the substrate effects and the basic thin film 
elastic strain growth mechanism, one should eliminate the microstructures factors to a 
minimum degree. The amorphous materials are qualified for this study, since they 
have no grain structures and are typically elastically isotropic.  
 
 42 
 
Figure 2.25 Illustration of length-scale effects on the mechanical properties of 
materials [50]  
 
2.7  Thin film deposition mechanism 
There are numerous ways to deposit or grow a thin film on a substrate. It 
includes: sputtering, e-beam evaporation, plasma enhanced chemical vapor deposition 
(PECVD), low pressure chemical vapor deposition (LPCVD), microwave plasma 
chemical vapor deposition (MPCVD), Oxide growth, layer by layer (LBL), Ion beam, 
mass selected ion beam (MSIB), cathodic arc, pulsed laser deposition (PLD). Figure 
2.26 shows the schematics of selected deposition systems for thin films [97]. Among 
those methods, sputtering and CVD methods are most frequently used for industrial 
process and laboratory studies, respectively. Sputtering is a physical process (except 
reactive sputtering), where ion bombardment is the key mechanism for thin film 
 43 
growth. Either DC or RF power could be used to generate argon plasma sputtering the 
graphite electrode. Magnetic fields are often applied in the sputtering chamber to 
increase the deposition rate. The magnetic fields make the electrons spiral and 
increase their mean free path. In other words, it increases the ionization degree of the 
plasma. A bias voltage is often applied to the substrate to increase the ion 
bombardment. Sputtering has several advantages, such as versatility of sputtering 
methods and deposition conditions can be easily controlled by the plasma power and 
gas pressure. For sputtering amorphous films, a high chamber pressure (achieved by 
large gas flow) is needed to reduce the ion mean free path so that the bombard energy 
is not great enough to create crystal structure. A relatively low sputtering power can 
also help create amorphous structure.    
Compared to sputtering, CVD methods involve not only the physical process 
of bombardment but also the chemical reactions on and inside the films. Among them, 
PECVD is a decent process that is often used in semiconductor manufacturing to 
deposit films onto wafers containing metal layers or other temperature-sensitive 
structures. A simple schematic of RF PECVD is shown in figure 2.27.Using LPCVD 
can increase the uniformity of the thin films, and MPCVD is often employed to grow 
very hard materials. During a CVD process, as the source gases are induced into the 
main chamber, the plasma is created between electrodes by the RF or DC power. One 
important feature, the ?plasma sheath? [98] is formed near the electrode surface due to 
the different mobility between ions and electrons. Since electrons have much higher 
thermal velocities than ions, they are lost faster at the electrodes and leave the plasma 
 44 
slightly positive charged. Thus an electric field is generated pointed from plasma to 
the solid surface and accelerates the ions in the plasma to the wall. This confines the 
ions near the surface and creates a sheath with a net positive charge. The sheath has an 
order of Debye length in thickness and usually is darker due to the low electron 
density in that region. In low pressure, the ion sheaths formed are collisionless and 
have low conductivity thus serving as diodes between plasma and electrodes. The 
electrodes then obtain DC self-bias voltages equal to their peak RF voltage. Thus, the 
inverse capacitances of these two diodes determine the DC bias direction. The shorter 
electrode has a smaller capacitance, which brings a larger (negative) bias voltage. 
Thus the DC bias is point from the larger electrode to the shorter one. The field 
accelerates the positive ions to the shorter electrode. This is the reason that substrate 
is mounted to the shorter electrode.  
 
 45 
 
Figure 2.26 Various deposition systems for thin films [97]  
 
 46 
 
Figure 2.27 Schematic of RF PECVD and plasma sheath 
 
2.8  X-ray diffraction technique 
X-ray diffraction is a very useful experimental technique for structural analysis 
of crystals. It can also be used to determine weather the material is amorphous or not. 
X-ray is one kind of electromagnetic wave, whose wavelength (0.01~10nm) is very 
close to the period of crystal lattices and the spacing of atoms in crystals. When x-rays 
pass through a crystal, they will be diffracted by the crystal planes. By scanning the 
X-ray through the specimen, peaks represent the crystal planes will show up in the 
X-ray spectrum, so that the crystal structure information can be obtained. Applying 
XRD on thin film system is different from testing the bulk materials. As the x-ray will 
penetrate a few microns into the material, the X-ray spectrum usually contains the 
crystal structures information for both the film and the substrate. To obtain the film 
x-ray spectrum, another scan is be performed on bulk substrate materials. Subtracting 
the latter from the composite X-ray spectrum leads to data of the film crystal structure. 
CH4+H2 
Si substrate with 
Si3N4 
To pump RF 
Plasma 
Plasma Sheath 
 47 
Amorphous film structure determination is relatively easy to achieve through XRD. 
The amorphous X-ray spectrum should not contain any peaks of the crystal planes. As 
long as the composite and bulk substrate data overlap each other, the film is 
amorphous. Refer to Cullity?s book ?Elements of X-ray Diffraction? for detailed 
information [99]. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 48 
 
 
 
 
 
CHAPTER 3  
EXPERIMENTAL SETUP 
3.1  Thin film selection and deposition 
The materials selected for this research covered a wide range of the Young?s 
modulus values. From soft to hard, the amorphous films were made of polymer, 
near-frictionless carbon (NFC), chromium oxide (CrOx), silicon dioxide (SiO2), iron 
oxide (FeOx), titanium oxide (TiOx), amorphous Si (a-Si), aluminum oxide (AlOx), 
aluminum nitride (AlN), iron boron (FeB), nickel titanium (NiTi), METGLAS, silicon 
nitride (Si3N4), silicon carbide (SiC), and tetrahedral hydrogenated carbon (ta-C:H). 
As it was difficult to get a very hard amorphous diamond-like film, ultra 
nano-crystalline diamond was employed in this study. Bulk silicon was used as the 
substrate materials, all of which were p-type <100> direction wafers. The substrate 
thickness was 500?m ?15?m.  
Thin films were deposited using various techniques. The film thickness, 
deposition methods and target materials/source gases are listed in table 3.1. Metals 
oxides like CrOx, FeOx, TiOx, and AlOx were deposited using the sputtering system. 
Important sputtering parameters are listed in table 3.2. Note that the sputtering power 
and gas flows vary with materials. The best parameters for amorphous films growth 
were experimentally determined by depositing films with different combinations of 
parameters (i.e. fix the power value then change the gas flow ratio, and fix gas ratio to 
 49 
change the power). Based on the characterization results, the parameters that 
generated the ?most? amorphous (no peaks in XRD, small cluster size, and smooth 
surface, etc) film were selected. The image of a sputtering system is shown in figure 
3.1. 
The NFC and ta-C:H films were grown using the RF PECVD method at Argonne 
National Lab [45]. The near-frictionless carbon films possess unique mechanical and 
tribological properties that have exhibited the lowest ever recorded friction coefficient, 
0.001, and ultra-low wear rates of 10?11?10?10 mm3 N?1 m?1, even under dry sliding 
conditions and at very high contact pressures [57,62,73,79,80,100-106]. Tetrahedral 
hydrogenated carbon (ta-C:H) is one kind diamond-like carbon materials that possess 
relatively high Young?s modulus and hardness. Figure 3.2a and 3.2b show a schematic 
of a PECVD system and an image of the CVD chamber, respectively. In PECVD 
mode, the flow rate of the precursor gases was limited to a maximum of 60 sccm. All 
substrates were pre-sputtered with Argon plasma using RF powers between 
900-1100W for cleaning purpose. The pre-sputtering time ranged from 5 to 15 
minutes depending on the process requirement. The base pressure prior to deposition 
was 3x10-7 torr, while the deposition pressure was 20-24 mtorr. The self bias voltage 
applied was -375V~-425V. This bias was higher than the voltage (-100V) mentioned 
in literature review that gives the highest Young?s modulus and density. Actually, 
different experimental conditions like gas flow rates, sources gas ratio, RF field-plate 
(FP) power and the chamber volume will cause this bias voltage changing.  
 
 50 
The source gases for NFC and ta-C:H deposition were methane (CH4) and 
hydrogen (H2). The different ratio between these two gases brings different properties 
(structure) of the films. For NFC films, the ratio was 75% hydrogen (42 sccm flow 
rate) and 25% methane (14 sccm flow rate). A much higher methane ratio was used 
for ta-C:H deposition to increase the film strength [107]. The total flow rate should 
not be over 60 sccm during the whole deposition process. The process was running at 
Room Temperature. The estimated deposition rate was around 1.85nm/min, so it took 
about 9 hours to grow a 1?m NFC or ta-C:H film. The NFC films with different 
thicknesses were deposited for the experiments.  
Dielectric films like SiO2 and Si3N4 were deposited using LPCVD method. A low 
pressure CVD can help grow smooth and uniform amorphous films For Si3N4, the 
source gases were dichloro-Silane (SiH2Cl2, 140sccm) and Ammonia (NH3, 30sccm). 
The deposition temperature was 850 ?C and the pressure was around 420 mtorr. It 
took 90 min to deposit 300nm silicon nitride on both sides of the wafer. SiO2 were 
also deposited using this method with the source gases of silane and oxygen. Detailed 
information can be found in Jaeger?s book ?Introduction to microelectronic 
fabrication? [108]. The UNCD film was deposited through MPCVD [75], and the 
polymer film was grown using a typical layer-by-layer method [109].  
 
 
 
 
 51 
Table 3.1 Deposition information for all the specimens 
Thin Films Deposition Method Target materials & Source Gases Film Thickness (nm) 
Polymer LBL Polymer 225 
NFC PECVD CH4+H2 150, 380, 1125 
CrOx Sputtering Cr+O2 125, 2000 
SiO2 LPCVD SiH4+O2 395 
FeOx Sputtering Fe+O2 125 
TiOx Sputtering Ti+O2 120 
a-Si Sputtering Si 450 
AlOx Sputtering Al+O2 120 
AlN Sputtering Al+N2 1100 
FeB Sputtering Fe+B 1010 
Metglas Sputtering METGLAS 278 
Si3N4 LPCVD Si+SiH2Cl2+NH3 300 
SiC Sputtering SiC 500 
a-C:H PECVD C2H2+H2 868 
UNCD MPCVD CH4+H2 560 
 
 
 
 
 
 52 
Table 3.2 Selected sputtering parameters for amorphous CrOx 
Sputtering Power DC 
Target Chromium 
Substrate Silicon 
Base Pressure  6x10-6 torr 
Pre-sputtering Time 120 sec 
Pre-sputtering power 150 watts 
Sputtering Time 600 sec 
Sputtering Power 150 w 
Gas 1 flow (Ar) 27 sccm 
Gas 1 flow (O2) 13 sccm 
Deposition Temperature Room Temperature 
Substrate Holder Rotate 50% 
Ignition Pressure 70 mtorr 
Deposition Pressure 7.8~8.0 mtorr 
 
 53 
 
Figure 3.1 Image of a Denton sputtering system 
 
 
 
 
 
Courtesy of Materials Engineering 
at Auburn University 
 54 
 
Figure 3.2a Schematic of the RF PECVD system at Argonne National Lab 
Courtesy of Argonne National Lab 
 
 
Figure 3.2b Image of the PECVD chamber at Argonne National Lab 
Courtesy of Argonne National Lab 
 
 
 55 
3.2  Thin film Characterizations 
The deposited films were tested with XRD for confirming the amorphous 
structure. SEM and AFM characterizations were employed to observe the surface 
conditions of the films.  
 
 
Figure 3.3 Image of a Jeol 5000 atomic force microscope 
Courtesy of Materials Engineering 
at Auburn University 
 56 
 
Figure 3.4 Image of a Jeol 7000F filed emission scanning electron microscope 
 
3.3  Nanoindentation Setup 
The indentation tests were performed with an MTS Nanoindenter XP system 
at ambient pressure and temperature (Figure 2.1b). The Nanoindenter tip was a 
Berkovich type diamond pyramid. For the CSM (dynamic) method, the allowable 
thermal drift rate was limited as 0.05 nm/s. The Poisson's ratio of the film was 
assumed as equal to that of the same bulk material. The Poisson?s ratio of the 
substrate was set as 0.28 for <100> silicon [110]. The continuous stiffness 
measurement was depth-controlled and the harmonic displacement target was set to 2 
nm. For each thin film, 20 or more CSM tests were run with same final indent depth. 
Film Young?s modulus was determined from the load-displacement curve and the 
Courtesy of Materials Engineering 
at Auburn University 
 57 
CSM modulus-displacement curve. The contact depth (hc) and the projected contact 
area (A) were calculated independently using Eq. (10) and (11) in Chapter 2.  
 
3.4  Finite Element Analysis Setup 
The FEM was performed using ANSYS 11.0. The model was built-up using 
2-D elements. To simplify the simulation, two assumptions have been made 
beforehand: the film surface roughness and residual stress were ignored during 
simulations. Following are the detailed procedures of setting up the parameters in 
ANSYS: 
1. The Preferences setup was Structural. 
2. Element type for film, substrate, and tip was PLANE183 (8 node). The target and 
contact surface elements would be setup later. 
3. The real constant for PLANE183 was using default/none. The real constant for the 
target-contact pair would be setup later. 
4. The materials properties were set as linear and elastic isotropic for both the 
film/substrate and the tip. The tip was designed to only have elastic deformation. 
The film Young?s modulus was defined using the experimental data obtained from 
nanoindentation. The Si substrate modulus was defined as 170 GPa and the 
Poisson?s ratio was set to ?=0.28 for <100> wafers. The indenter tip?s modulus 
and Poisson?s ratio were set to E=1141 GPa, ?=0.07. 
5. Modeling was used to define the shape of the film, substrate and tip. The thickness 
of film was designed to match the true value of each thin film. The width was 
 58 
defined as 10 ?m for all the cases. The substrate had the same width with the film, 
and the thickness was designed as 20 ?m. The film and substrate were designed to 
be perfectly attached to each other. Usually, the film and substrate thickness ratio 
would be close to 1/20, which was good enough for analyzing the elastic strain 
grown in the specimen. To simplify the simulation, the tip was designed as a half 
shape of a Berkovich tip. The right triangle has an angle of 65.3?, which is half of 
the Berkovich total angle (from one edge to the opposite face). The length of the 
tip was designed as 1 ?m to match the film thickness. 
6. The mesh needs to be first defined on the film, substrate, and tip, before defining 
the target-contact surface. The element size of the film in the local contact area is 
about 10 nm, where a high stress/strain concentration is expected. The size 
increases along with the width. But along the thickness, the element size doesn?t 
change as the film is very thin. The substrate and tip meshes are defined with the 
same manner. The total element number is around 20000~30000. The estimated 
calculating time is around 5-10 min.  
7. The target-contact pair is designed for two surfaces (lines) touching each other. In 
the pair, element type TARGE169 is designed for the indenter tip edge, and the 
CONTA172 is designed for the film surface. The real constant for the pair used in 
this study is the system defaults. Changing the real constant may improve the 
model in future study.  
8. The simulation is quasi-static, and the tip movement is controlled by 
displacement. . The displacement value for each film matches the experimentally 
 59 
obtained critical indent depth. 
A schematic of the central portion of the mesh design for this FEA study is shown 
in figure 3.5. 
 
 
Figure 3.5 Schematic of the finite element analysis of the nanoindentation. 
 
 
 
 
 
 
 
 
 
Load 
Film 
Indenter Tip 
  
  
  
  
  
Fix
ed
 B
ou
nd
ary
 
Substrate 
 60 
 
 
 
 
 
CHAPTER 4  
RESULTS AND DISCUSSIONS 
4.1  Film thickness measurement & XRD results  
Accurate measurement of the thin film thickness was extremely important to 
this work. As many specimens had thickness less than 200 nm, normal profilometers 
that were used for micro-scale devices no longer met the requirement. Thus, SEM 
characterization was employed to determine the thickness. The specimen was cut into 
pieces after deposition on a silicon wafer. One of them was selected for SEM 
observation, which should have a flat and clean cross-section. Sometimes, several 
chips of the specimen from different locations on the wafer needed to be characterized 
to gain an average value of the film thickness. During SEM tests, the chips were 
vertically attached to the sample holder so that the cross-section can be focused 
directly. Figure 4.1a shows a SEM image for the ta-C:H film cross-section. A uniform 
885 nm thick layer was observed on the bulk silicon. However, clear boundary 
between the film and the substrate could not always be found in normal SEM 
observation. In this case, the SEM composition mode needed to be employed. As 
shown in figure 4.1b, different materials would have various gray-scale colors in the 
image, which made it easy to define the film layer. Note that the ?compo? sign at the 
lower left corner of the picture distinguishes it from the regular SEM images. XRD 
results were shown in figure 4.2. Most of the spectra consist of two major peaks, 
 61 
which belong to the silicon substrate (002) and (004) planes. There are no obvious 
peaks that belong to the target films shown in the spectra. The films are amorphous.  
 
 
Figure 4.1a Cross-sectional SEM image of ta-C:H film on Si substrate  
 62 
 
Figure 4.1b Composition SEM image of the cross-section of NFC/Si 
 
 
 63 
0
1000
2000
3000
4000
5000
6000
7000
8000
20 30 40 50 60 70 80
2theta
Int
en
sit
y
NFC
SiC
Am-Si
Metglas
CrOx
NiTi
SiNx
SiO2
Si (002)
Peak
Si (004)
Peak
 
Figure 4.2 XRD spectra for tested materials 
 
4.2  Indentation Impression (Sink-in & Pile-up) Characterizations 
The indentation impressions were characterized with a scanning electron 
microscope (SEM) and an atomic force microscope (AFM). Figure 4.3 shows a SEM 
image of an impression on NFC/Si after unloading. It supplies general information 
about the nanoindentation process on the film. For instance, the film surface 
conditions can be roughly estimated as well as the size and shape of the impression 
area. It helps to determine whether sink-in (elastic contact) or pile-up (elastic-plastic 
contact) happened based on McElhaney and Nix?s theory introduced in Chapter 2. 
However, SEM can not tell the vertical degree of the pile-up or sink-in impression. 
This was why atomic force microscopy was employed.  
 
 64 
As a powerful tool, AFM can provide 2-dimensional, 3 dimensional and 
profile images of an indentation impression. In this study, both sink-in and pile-up 
phenomena were observed on different tested thin films. Figure 4.4 and 4.5 show the 
sink-in impressions on NFC/Si (soft film on hard substrate) and ta-C:H (hard film on 
soft substrate). The triangle shape profile of sink-in resembles the indenter geometry. 
Unlike the sink-in elastic contact, the plastically deformed pile-up greatly affects the 
indenter contact area and the accurate determination of film Young?s modulus and 
hardness. Thus, extra caution is needed when dealing with pile-up thin films. As 
mention above, pile-up usually happens in soft metals on hard substrate. Shown in 
figure 4.6a, the a-Si film exhibits minor pile-up on silicon substrate as the indentation 
depth was within the film thickness (The ratio of the pile-up level and indentation 
depth is less than 1/50). Similar pile-up degree was found in SiO2 and other soft metal 
oxide films. An overestimation of the Young?s modulus at this level of pile-up is less 
than 2.5% based on Bolshakov and Pharr?s simulation results when hf/hmax <0.7 ( the 
ratio of final depth after unloading to the depth at maximum load) [67]. However, if 
the indentation depth is greater than the film thickness, the indenter will penetrate into 
the substrate. It is known that the pile-up from the Si substrate will greatly affect the 
total pile-up degree on the film [111]. As shown in figure 4. 6b, a 500nm indentation 
on a-Si film (450 nm thick) has a much greater pile-up (The ratio of the pile-up level 
and indentation depth is greater than 1/6!) than that from the 300 nm indentation in 
figure 4.6a. The effect from the substrate indentation part causes overestimation of the 
Young?s modulus up to 30%. Thus, in this research, the data acquisition from 
 65 
nanoindentation was strictly limited to the film thickness region so as to minimize the 
pile-up effect from the Si substrate.       
 
 
Figure 4.3 SEM image of an indentation impression on NFC/Si. 
 
 
 
 
 
 
 
 
 
 66 
   
 
Figure 4.4 AFM images of a sink-in impression of 300 nm indentation on NFC/Si 
 
 
 
 
 
Figure 4.5 AFM images of a sink-in impression of 500 nm indentation on ta-C:H/Si 
 
 67 
 
 
Figure 4.6a AFM images of a pile-up impression of 300 nm indentation on a-Si film 
(450 nm) on silicon substrate. (The indenter remained in the film and didn?t go into 
the substrate) 
 
 
 
 
 
 
 
 
 
 
 
 
 68 
 
 
Figure 4.6b AFM images of a pile-up impression of 500 nm indentation on a-Si film 
(450 nm) on silicon substrate (The indenter penetrated the film and went into the Si 
substrate) 
 
4.3  Film Young?s modulus and Critical indent depth determination 
Raw CSM nanoindentation data of the thin films were input into Analyst (data 
processing software for the MTS-Nano Nanoindenter XP system) for calibration. 
Figure 4.7a and 4.7b show a typical calibrated CSM modulus-displacement curve for 
the SiO2 and SiNx film on silicon a substrate, respectively. ?Flat regions? were 
observed in the early stage of E-h curves for both films softer the substrate and films 
harder than the substrate. The value in this region strongly indicated the true film 
modulus with little affection from the substrate. In this study, an empirical guideline 
 69 
was suggested to roughly estimate the region length: The end of the flat region or the 
critical indentation depth (hcr) was defined when the measured modulus value 
changed more than 1% from the identified average modulus (Ef) calculated from the 
flat region. For instance in figure 4.7a, hc was first roughly estimated as 30 nm. The 
film Young?s modulus was then calculated as an average value from this estimated flat 
region (77 GPa for SiO2). Finally, hcr was defined as the measured value increased (or 
decreased for a harder film on soft substrate) 1% from 77 GPa (70 nm for SiO2). 
There was one exception from the nanocrystalline diamond (UNCD) film. Due to the 
large modulus value, its E-h curve has a very short flat region, and the modulus 
almost started to decrease from the beginning. The hcr, Ef, and their ratio to the film 
thickness and substrate modulus are listed in table 4.1. It is necessary to mention that 
all the films had <100> Si wafers as the substrate. The average Young?s modulus of 
the Si substrates was determined as 170 ? 5 GPa from at least 20 indentations on each 
wafer. The Poisson?s ratio was suggested as 0.28 for <100> direction [110].  
           
 70 
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300 350 400 450 500
 Displacement Into Surface (nm)
Yo
un
g's
 M
od
ulu
s (
GP
a)
36 nm
77 GPa
 
Figure 4.7a SiO2 film CSM modulus-displacement curve 
0
40
80
120
160
200
240
0 50 100 150 200 250 300 350 400 450 500
Displacement into Surface (nm)
Mo
du
lus
 (G
Pa
)
 67 nm
208 GPa
 
Figure 4.7b Si3N4 film CSM modulus-displacement curve 
 
 71 
Table 4.1 hcr, Ef, Ef, Ef/Es, and hcr/t for all tested specimens 
Thin Films Ef (GPa) t (nm) hcr (nm) Ef/Es hcr/t 
Polymer 12 225 20 0.07 0.088 
NFC 52 1125 102 0.31 0.090 
CrOx 60 130 12 0.35 0.092 
SiO2 77 395 36 0.45 0.091 
FeOx 90 105 11 0.53 0.107 
TiOx 103 120 13 0.61 0.108 
a-Si 108 450 48 0.64 0.107 
AlOx 115 120 15 0.68 0.125 
AlN 133 1000 158 0.78 0.158 
FeB 143 1000 188 0.85 0.188 
Metglas 180 278 200 1.06 0.719 
SiNx 208 300 67 1.22 0.223 
SiC 212 500 93 1.25 0.186 
ta-C:H 230 868 87 1.35 0.100 
UNCD 573 560 20 3.35 0.036 
 
The relation between the ratio of the critical indent depth to the film thickness 
(hcr/t) and the ratio of film/substrate modulus (Ef/Es) is described in figure 4.8. 
Different from the traditional way of using displacement as a variable, figure 4.8 has 
Ef/Es as the x coordinate, because the critical indent depth was influenced by Ef/Es, not 
 72 
vice versa. According to the figure, many tested materials still follow B?ckle?s 10% 
indent depth rule to avoid substrate effects. We should give more credibility and 
confidence in thin film nanoindentation technique. However, it is important to notice 
that as the films elastic modulus approached that of the substrate the hcr percentage 
started to increase in a unique nonlinear fashion. Moreover, as the film was much 
harder or softer than the substrate, this percentage drops lower than the 10% line. The 
solid curves in the plot are drawn to suggest a trend and are not generated from curve 
fitting or an analytical solution. The existence of the ?flat region? and its relation to 
the film/substrate materials led us to further investigate the Doerner & Nix equation. . 
Also, as the flat region of E-h in early indentation indicates the true film modulus, 
accurate determination of the trend of hcr/t vs. Ef/Es might help us understand the 
mechanism  
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.5 1 1.5 2 2.5 3 3.5 4
Film Modulus/Substrate modulus (Ef/Es)
Cr
itic
al 
ind
en
t d
ep
th/
Fil
m 
thi
ck
ne
ss 
(hc
r/t)
TiOx AlOxPolymer
AlN
FeB
  a-SiCrOx
   NFC  SiO2
METGLAS
SiN
ta-C:H UNCD
SiC
 FeOx
 
Figure 4.8 hcr/t vs. Ef/Es for all tested specimens 
 73 
From now on, softer films on Si substrate (Ef<Es) and harder films on Si 
substrate (Ef>Es) will be dealt with respectively. Before defining the theoretical 
solution for the hcr/t curves, there is one more important aspect required to be 
analyzed: the film thickness effect to hcr. In other words, for each film material, does 
hcr/t change with film thickness? Nanoindentation tests were performed on NFC film 
with different thickness, as shown in table 4.2 and figure 4.9. It is found that the all 
the NFC films had similar positions in the plot, which indicated the films thickness 
should have no effect on the ratio as long as the Ef/Es is the same.   
 
Table 4.2 hcr/t and Ef/Es data NFC films with different thickness 
Thin Films Ef (GPa) Ef/Es  hcr (nm) t (nm) Hcr/t 
NFC 52 0.31 102 1125 0.090 
NFC 48 0.28 34 380 0.088 
NFC 50 0.29 14 150 0.093 
 
 74 
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Film Modulus/Substrate modulus (Ef/Es)
Cr
itic
al 
ind
en
t d
ep
th/
Fil
m 
thi
ck
ne
ss 
(hc
r/t)
TiOx AlOx
Polymer
AlN
FeB
  a-Si
CrOx
 NFC with
different thickness
 SiO2
 FeOx
 
Figure 4.9 Positions of the NFC films with different thickness in hcr/t vs. Ef/Es (Ef<Es) 
 
4.4  Doerner & Nix function investigation 
As mentioned in chapter 2, there are two major functions for thin film 
nanoindentation: Doerner & Nix, and Gao. The Doerner & Nix function suggested 
that within a small indentation depth, the substrate effects can be neglected. While in 
Gao?s model, the measured modulus increases as long as the indenter touches the film. 
Since our experimental data matches Doerner & Nix?s description, investigation will 
be mainly focused on their model. As shown Eq. (2), Doerner & Nix equation was 
developed to describe the indentation E-h curve using an empirically determined 
constant ? as 0.25. Using the same criterion of 1% change of the modulus value 
(E=(1+1%)Ef) to determine hcr, one can change Eq, (12) to the function below: 
 75 
)1001ln()1ln( ??= sf
cr
EEt
h ?                       (17) 
 
Re-plotting Figure 4.8 with the theoretical curves obtained from Eq (17) in 
Figure 4.10a and 4.10b (for Ef<Es and Es<Ef, respectively), one can find the 
experimental and Doerner & Nix curves have the similar trend. However, for films 
softer than the substrate, the experimental points were obviously above the theoretical 
line. In other words, for each film/substrate system, the flat region was longer then 
that of the Doerner & Nix function expected. When Ef approached Es, this deviation 
became greater. From the figure, we directly got an idea that ? at least was not a 
universal constant for all the materials. And mort importantly, the hcr was not solely 
determined by the modulus difference between film and substrate, but also was 
affected by ?. For films harder than the substrate, Figure 4.10b indicates that the 
Doerner & Nix function does not always underestimate the flat region length. The 
diamond film had serious substrates and sink-in effects even from the beginning of 
the indentation, for which the Doerner & Nix function can not extrapolate. 
 76 
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1
Film Modulus/Substrate modulus (Ef/Es)
Ind
en
t d
ep
th/
film
 th
ick
ne
ss 
(hc
r/t)
Theoretical
Experimental
 
Figure 4.10a The experimental and theoretical curves of hcr/t-Ef/Es for Ef<Es 
0
0.2
0.4
0.6
0.8
1 1.5 2 2.5 3 3.5 4
Film Modulus/Substrate modulus (Ef/Es)
Cr
itic
al 
Ind
en
t d
ep
th/
film
 th
ick
ne
ss 
(hc
r/t)
Theoretical
Experimental
 
Figure 4.10b The experimental and theoretical curves of hcr/t-Ef/Es for Ef>Es 
 
 77 
To better understand how alpha (?) affects the Doerner & Nix function, the 
experimental CSM curve of SiO2 was compared with theoretical curves having ? of 
0.1, 0.25, and 0.5 in figure 4.11. It was obvious that a smaller ? indicated a faster 
(more abrupt) changing of the measured modulus from the film value to the substrate 
value. In other words, smaller ? yields shorter flat region (hcr) and greater substrate 
effects.  On the contrary, a relatively large ? slows down the changing trend of the 
measured modulus and has a greater hcr.  
Based on Eq. 17 and the experimentally determined hcr, alpha (?) at hcr for 
each material can be calculated and plotted versus Ef/Es, as shown in figure 4.12a and 
4.12b. It is found ? (at hcr) increases as Ef<Es, and decreases when film is hard than 
the substrate. .  
0
20
40
60
80
100
120
140
160
180
200
0 50 100 150 200 250 300 350 400 450 500
Displacement (nm)
Yo
un
g's
 M
od
ulu
s (
GP
a)
?=0.1
?=0.5
?=0.25
 
 Figure 4.11 SiO2 modulus-displacement curve with theoretical curves having alpha 
(?) of 0.1, 0.25, and 0.5 
 78 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Film/substrate Young's modulus ratio (Ef/Es)
alp
ha
 at
 th
e c
riti
cal
 in
de
nta
tio
n d
ep
th 
(hc
r)
 
Figure 4.12a Plot of ? value at critical indentation depth vs. Ef/Es for tested materials 
with Ef<Es 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 1.5 2 2.5 3 3.5 4
Film/substrate Young's modulus ratio (Ef/Es)
alp
ha
 at
 th
e c
riti
cal
 in
de
nta
tio
n d
ep
th 
(hc
r)
 
Figure 4.12b ? value at hcr vs. Ef/Es for tested materials with Ef>Es 
 79 
Now, an important question arises: Is ? a constant for each film/substrate 
combination? In applying the experimental data to Eq. (12) we were able to plot the 
parameter ? in the Doerner & Nix function versus indentation depth. It was interesting 
to find that ? was not only different for each film material, but was not constant with 
displacement either. A typical ?-h curve from a single CSM indentation test on the 
NFC film is shown in figure 4.13. A similar trend was found in all other materials 
tested for both soft on hard (SiO2/Si, figure 4.14) and hard on soft (ta-C:H/Si, figure 
4.15) situations. It is interesting to note that the ?-h curve peaks are at the critical 
indentation depth (hcr) when contrasting with the film E-h curve. As the substrate 
effect is negligible in the flat region, ? appeared to have no physical meaning here, or 
more precisely is close to infinity.  
 80 
0
50
100
150
Yo
un
g's
 M
od
ulu
s (
GP
a)
0
0.1
0.2
0.3
0.4
0.5
0 500 1000 1500 2000
Displacement (nm)
Al
ph
a
hcr
Ef
Yo
un
g's
 M
od
ulu
s (
GP
a)
Al
ph
a
Yo
un
g's
 M
od
ulu
s (
GP
a)
Al
ph
a
Yo
un
g's
 M
od
ulu
s (
GP
a)
Al
ph
a
 
Figure 4.13 Plot of ? vs. h for the NFC film contrasted with the E-h curve 
 81 
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500
Displacement (nm)
Yo
un
g's
 M
od
ulu
s (
GP
a)
0.1
0.2
0.3
0.4
0.5
0 100 200 300 400 500
Displacement (nm)
Al
ph
a
hcr
Yo
un
g's
 M
od
ulu
s (
GP
a)
Al
ph
a
Yo
un
g's
 M
od
ulu
s (
GP
a)
Al
ph
a
 
Figure 4.14 Plot of ? vs. h for the SiO2 film contrasted with the E-h curve  
 82 
0
50
100
150
200
250
300
0 100 200 300 400 500
M
od
ulu
s (
GP
a)
0
0.1
0.2
0.3
0.4
0.5
0 100 200 300 400 500
Displacment (nm)
Al
ph
a
hcr
M
od
ulu
s (
GP
a)
Al
ph
a
M
od
ulu
s (
GP
a)
Al
ph
a
 
Figure 4.15 Plot of ? vs. h for the ta-C:H film contrasted with the E-h curve  
 
4.5  Alpha (?) parameter determination 
Based on the results in last section, it is reasonable to suggest alpha (?) as a 
power function of h shown in figure 4.16. Numerous empirical solutions of ? were 
proposed in the previous preliminary work. Although a power function like Eq. (18) 
 83 
fits better with the experimental data than the Doerner & Nix function, it lacks a 
physical explanation to support the assumption. Moreover, a changing alpha 
parameter will only bring more complexity to the empirical Doerner & Nix function. 
It is against the original idea of building up a simple and universal solution to 
extrapolate the thin film indentation mechanism. To find the true physical meaning 
behind alpha, finite element analysis is needed.  
( )kth /=?                               (18) 
 
0
0.1
0.2
0.3
0.4
0.5
0 500 1000 1500 2000
Displacement (nm)
Al
ph
a
a suggested power 
function for alpha
Al
ph
a
 
Figure 4.16 A suggested power function of h for alpha. 
 
 
 
 
 84 
Before introducing the FEA results, let us rearrange Eq. (12) to get a better 
picture of the role of alpha plays in the Doerner & Nix function:  
)/()/( 1]1[11 cc ht
s
ht
f
eEeEE ?? ?? ?+??=                   (19) 
From Eq. (19), the Doerner & Nix function can be suggested as contributions 
from the film and substrate compliance weighted by the exponential fraction 
coefficient. As the sum of the fractions equals to 1 and there is only one parameter (?) 
controlling the factor, the Doerner & Nix function describes a consistent change of the 
elastic strain intensity distribution in film and substrate along with the displacement. 
The assumption is reasonable in a mathematical (theoretical) concern. However, in 
reality, the elastic strain growth in the specimens will be affected by the film/substrate 
mechanical properties. Thus some degree of the discontinuity of the elastic strain 
distribution can be expected.    
A schematic of the estimated elastic strain growth in the specimen is shown in 
figure 4.17. Let us imagine the strain change in the specimen at different indentation 
depth. At the beginning of the indentation, elastic deformation starts to develop in the 
specimen and most of the strain energy is confined in the film. The strain intensity in 
the film is much larger than that in the substrate, which is why there could be a flat 
region that reflects the true film modulus. As the tip penetrates deeper into the film, 
the elastic strain intensity begins to grow in both the film and the substrate but in a 
different manner. Taking NFC/Si as an example, the NFC atoms are easier to move 
from their original positions than silicon atoms due to the smaller shear modulus. 
Thus, the strain development in the NFC film should not be uniform in any direction 
 85 
but follow a preferred manner of growing laterally. Thus for each small indent depth 
change (?h), the elastic strain growth in the film (??f) is larger than that in the 
substrate (??s).  
 
Figure 4.17 Schematic of certain degree of the elastic strain growth in the specimen 
with increasing indent depth: (a) 20 nm; (b) 50nm; (c) 130nm; (d) 500nm; (e) 1000nm; 
(f) 2000nm 
Film 
Substrate 
Estimated elastic strain 
region 
indenter 
Film 
Substrate 
Film 
Substrate 
Film 
Substrate 
(a)  (b) 
(c) (d) 
Film 
Substrate 
(e) 
Film 
Substrate 
 (f) 
 86 
Finite element analysis (FEA) was employed in this study to observe the 
elastic strain growth manner in the film and substrate during nanoindentation. Figure 
4.18a to 4.18e show the FEA results of indentation at 8, 30, 50, 100, and 250nm into 
the NFC/Si specimen. The simulation results strongly support the suggested 
schematic of the elastic strain growth in figure 4.17. At 8 nm indentation, most of the 
elastic strain is confined in the film. From 8nm towards 100 nm, though the substrate 
starts to deform elastically, the major part of the elastic strain still prefers to grow 
latterly in the film. After 100nm indentation for NFC, the elastic strain intensity in the 
film is getting saturated and the film can no longer sustain much more strain 
accumulation, and should pass most of the deformation energy into the substrate. 
From this point, the strain growth in the substrate becomes faster than that in the film. 
Thus, the measure modulus starts to increase quickly toward the substrate value. 
Moreover, it is found that the radius of the elastic strain contact area at the interface of 
film and substrate is another important parameter for characterizing substrate effects. 
Based on the simulation results, we found that in the early stage of the indentation, the 
strain contact area grew faster and reached a certain size. When substrate effects start 
to prevail, the contact area grew much slower, which strongly indicated the elastic 
strain saturation in the film. The elastic contact area information could help us better 
understand the strain development in the early stage of indentation and will be 
carefully investigated in later study. 
     Moreover, a gap was found between the same scale of the elastic strain (i.e. 
0.1% strain) at the interface of film and substrate. This illustrates the inconsistent 
 87 
elastic strain flow from film to substrate. Similar discontinuity was observed when 
indenting on hard film/soft substrate combination as shown in figure 4.19 for 
ta-C:H/Si. The gap degree is influenced by the different mechanic properties of the 
film and substrate (Young?s modulus and Poisson?s ratio). It is also interesting to see 
that even for the same materials combination (i.e, Silicon film on diamond substrate, 
and diamond film on Si substrate), the manner of the strain discontinuity is different, 
see figure 4.20a and 4.20b.    
 
Figure 4.18a FEA image of the elastic strain distribution in NFC/Si specimen at 8nm 
indentation depth. 
 88 
  
Figure 4.18b FEA image of the elastic strain distribution in NFC/Si specimen at 30nm 
indentation depth. 
 
Figure 4.18c FEA image of the elastic strain distribution in NFC/Si specimen at 50nm 
indentation depth. 
 89 
 
Figure 4.18d FEA image of the elastic strain distribution in NFC/Si specimen at 
100nm indentation depth. 
 
Figure 4.18e FEA image of the elastic strain distribution in NFC/Si specimen at 
250nm indentation depth. 
 90 
 
 
Figure 4.19 FEA image of the elastic strain distribution in ta-C:H/Si at 100nm 
indentation depth. 
 
 91 
 
Figure 4.20a FEA result of indenting 20 nm into diamond film on Si substrate 
 
 
Figure 4.20b FEA result of indenting 20 nm into Si film on diamond substrate 
 92 
Based on the simulation results of the inconsistent elastic strain growth in the 
film and substrate, the single alpha parameter in the Doerner & Nix function might 
not be sufficient enough to describe the whole situation. Thus, we suggested that there 
might be two parameters that control the weighting factors of the film and substrate 
separately. Eq. (19) is based on the compliance (form of reciprocal of the Young?s 
modulus) of the specimen. Let us rearrange the equation again so that E is directly 
presented, and assume there are two parameters ?1 and ?2 in the function: 
]1[ )/()/( 21 cc htshtf
sf
eEeE
EEE
?? ?? ??+?
??=
                 (20) 
According to the equation above, exp(-?1*t/hc) is the weighting factor that determines 
the contribution of the film part to the composite modulus, while [1- exp(-?2*t/hc)] 
represents the substrate?s fraction. In applying the experimental data to Eq. (20), the 
results were very satisfying. It was found that, if fixing ?2 as a constant of 0.27~0.29, 
?1 would also result in a constant value. Figure 4.21 shows the curve of ?1 vs. 
displacement (h) for SiO2 (when ?2=0.27) contrasted with the old alpha from the 
Doerner & Nix function. A list of the ?1 and ?2 values for each thin film system is 
shown in table 4.3. Using the values from table 4.3, the new function (Eq. 20) was 
found to be adept at closely matching most experimental data collected, which 
spanned both soft films on hard substrates and hard films on soft substrates. The 
experimental E-h data were compared with the repression curves from the Doerner & 
Nix function, Gao?s function, and Eq. (20) in Figure 4.22 through 4.34.  
 
 93 
It is very interesting to see that the constant ?2, which controls the substrate 
weighting factor, is equal to the Poisson?s ratio of <100> Si (0.28). As the Si wafers 
are not always perfectly oriented to (100), it is reasonable to have a variation between 
0.27~0.29 for the Si substrates used in this research. According to the data in table 4.3, 
the constant ?1, on the other hand, is very close to the Poisson?s ratio of the film. Is it 
true that ?1 and ?2 are actually the film and substrate Poisson?s ratio? An analytical 
approach was introduced in the next section to relate these parameters. 
0
0.2
0.4
0.6
0.8
0 150 300 450
Displacement (nm)
Al
ph
a v
alu
es
Alpha#2 from new function
when alpha#1=0.28
Single Alpha from Doerner
& Nix Function
 
Figure 4.21 Comparison of ?1 -h data for SiO2 with the Doerner & Nix function 
 
 
 
 
 
 94 
Table 4.3 List of the ?1 and ?2 values for each thin film with the film Poisson?s ratio 
Thin films Alpha#2 (?2) Alpha#1 (?1) Film Poisson?s ratio 
Polymer 0.28 0.45 0.45 [112] 
NFC 0.27 0.33 0.33 [113] 
CrOx 0.28 0.27 0.25 [114] 
SiO2 0.27 0.2 0.17 (quartz) [115]  
FeOx 0.28 0.35 N/A  
TiOx 0.27 0.28 0.28 [116] 
a-Si 0.28 0.25 0.25 [117] 
AlOx 0.29 0.27 0.27 [116] 
AlN 0.28 0.27 0.25 (crystal) [118] 
FeB 0.27 0.25 0.25 [119] 
Metglas 0.28 0.31 0.33 [120] 
SiNx 0.28 0.3 0.28 (crystal) [116] 
SiC 0.28 0.3 0.22 ? 0.27 [121] 
ta-C:H 0.28 0.28 0.28 [122] 
UNCD 0.28 0.1 0.07 [29] 
 
 
 
 95 
0
40
80
120
160
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lus
 (G
Pa
)
SiO2 experimental
New function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.22 E-h curve of SiO2 film with the new, Doerner & Nix, and Gao models.  
0
40
80
120
160
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lus
 (G
Pa
)
a-Si experimental
New function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.23 E-h curve of a-Si film with the new, Doerner & Nix, and Gao models. 
 96 
0
40
80
120
160
200
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Modul
us
 (G
Pa
)
FeB experimental
New function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.24 E-h curve of FeB film with the new, Doerner & Nix, and Gao models.  
0
40
80
120
160
200
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Modu
lus
 (G
Pa
)
TiOx experimental
New function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.25 E-h curve of TiOx film with the new, Doerner & Nix, and Gao models.  
 97 
0
40
80
120
160
200
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lus
 (G
Pa
)
AlN experimental
New functoin
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.26 E-h curve of AlN film with the new, Doerner & Nix, and Gao models. 
0
20
40
60
80
100
120
140
160
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lu
s (
GP
a)
NFC experimental
New function
Doerner & Nix
Gao Eq. 
 
Figure 4.27 E-h curve of NFC film with the new, Doerner & Nix, and Gao models. 
 98 
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lus
 (G
Pa
)
FeOx experimental
New function
Doerner & Nix Eq.
Gao Eq.
Figure 4.28 E-h curve of FeOx film with the new, Doerner & Nix, and Gao models. 
0
40
80
120
160
200
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lus
 (G
Pa
)
AlOx experimental
New Function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.29 E-h curve of AlOx film with the new, Doerner & Nix, and Gao models. 
 99 
0
40
80
120
160
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
dul
us
 (G
Pa
)
CrOx experimental
New function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.30 E-h curve of CrOx film with the new, Doerner & Nix, and Gao models. 
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1
Normalized Displacement  (h/t)
Mo
du
lus
 (G
Pa
)
SiC experimental
New function
Doerner & Nix
Gao Eq.
Figure 4.31 E-h curve of SiC film with the new, Doerner & Nix, and Gao models. 
 100 
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lu
s (
GP
a)
SiN experimental
New function
Doerner & Nix
Gao Eq.
 
Figure 4.32 E-h curve of Si3N4 film with the new, Doerner & Nix, and Gao models. 
0
60
120
180
240
300
0 0.1 0.2 0.3 0.4 0.5
Normalized Displacement (h/t)
Modul
us
 (G
Pa
)
ta-C:H experimental
New function
Doerner & Nix Eq.
Gao Eq.
 
Figure 4.33 E-h curve of ta-C:H film with the new, Doerner & Nix, and Gao models. 
 101 
0
50
100
150
200
0 0.2 0.4 0.6 0.8 1
Normalized Displacement(h/t)
Mo
du
lus
 (G
Pa
)
Metglas experimental
New function
Doerner & Nix 
Gao Eq.
Figure 4.34 E-h curve of Metglas film with the new, Doerner & Nix, and Gao models. 
 
There are two films that didn?t match well with the two-alpha function: 
Polymer and nanocrystalline diamond (UNCD) as shown in figure 4.35a and 4.35b. 
The deposition method for the polymer was layer-by-layer. Since it is not a 
continuous deposition method, it is possible that the polymer film is not as uniform as 
other amorphous materials. As there is an interface between each layer of the polymer, 
penetration from one layer to the next needs extra load (energy) due to the existence 
of the residual stress [47]. This may be the reason why the E-h curve in figure 4.36a 
increases faster than expected. However, the new function is still doing an excellent 
job for matching the experimental data in the early stage of indentation.  
The error in nanocrystalline diamond data is caused by another mechanism. 
Based on knapp?s simulation results [72], nanoindentation on diamond film results in 
 102 
serious sink-in situation, whose degree is much larger than that the Oliver & Pharr 
method suggests. A unique technique for measuring the UNCD modulus was by 
Espinosa, Prorok and co-workers [62,73-76,78-80]. The so-called membrane 
deflection technique (MDE) employs tests on freestanding structures of the diamond 
film. Without having a substrate (effect), the Young?s modulus of UNCE was 
accurately measured as 960 GPa. This value is almost 40% larger than that obtained 
from the regular nanoindentation test. Due to the serious sink-in degree, the measured 
modulus was greatly underestimated even from the beginning of indentation. This is 
the reason why UNCE E-h curve never reaches the true film value in figure 4.36b. 
Besides, the diamond film has nanocrystalline structures, which might also affect the 
nanoindentation data due to the existence of small grain structures. Both of these 
mechanisms limit the use of the new function in UNCD specimens. 
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1
Normalized Displacement (h/t)
Mo
du
lu
s (
GP
a)
Polymer experimental
New function
 
Figure 4.35a E-h curve of polymer film with the new two-alpha model. 
 103 
0
200
400
600
800
1000
0 0.2 0.4 0.6 0.8 1
Displacement Into Surface (nm)
Mo
du
lus
 (G
Pa
)
UNCD experimental
New functionUNCD E=960 GPa, measured from MDEtests
 
Figure 4.35b E-h curve of UNCD film with the new two-alpha model  
 
4.6  Physical explanation of ?1 and a2 in modified Doerner & Nix function 
The physical meaning behind ?1 and a2 needs to be found to leverage the 
empirical function to theoretical level. In this section, a mathematical approach was 
employed to better understand the physical meaning behind the Doerner & Nix 
function and the parameter ?. As mentioned in Chapter 2, Gao?s analytical function 
for thin film indentation was derived from considering the energy transformation from 
indenter to the specimen. Thus, Gao?s weighting factor gives the ratio of the strain 
energy stored in the film to the total energy stored in the specimen. Doerner & Nix 
weighting factor resembles the shape of Gao?s factor, which suggests that it is also a 
ratio of strain energy distribution. As mentioned above, Doerner & Nix function 
describes a consistent change of the elastic strain intensity distribution in film and 
 104 
substrate. Assuming the total elastic strain energy in the specimen is confined in an 
?effective? volume with uniform energy density, Doerner & Nix weighting factor can 
be seen as a ratio of the segment in the film to the total volume. According to the 
shape of elastic strain region in finite element analysis, also to simplify the 
mathematical model, the volume is assumed to have a spherical shape. As shown in 
figure 4.36, the film segment of the spherical volume always has a vertical length of t, 
which is the film thickness. The radius of the effective energy volume (R) increases 
with the displacement (h) as the indenter goes deeper into the specimen.  
 
t
Indenter
Film
Substrate
Effective volume of elastic 
strain energy
Radius (R) increases with 
indentation depth (h)  
Figure 4.36 Effective elastic strain energy volume developed during 
nanoindentation 
 
 
 105 
The formula for the volume of a spherical segment is shown in Eq. (21) 
according to figure 4.37: 
)3(3 2 xRxVcap ?= pi                       (21) 
As the volume for a sphere is equal to (4/3)?R3, the sphere segment ratio can be 
represented as: 
32
3
2
4
1
4
3
3
4
)3(3
?????????????=
?
= RxRx
R
xRx
V
V
total
cap
pi
pi
               (22) 
For thin film indentation, x=film thickness (t), and R is a function of indentation depth 
(h). ?=[1-Vcap/Vtotal] should resemble the Doerner & Nix factor [exp(-?t/h)], if correct R 
function is suggested.  
x
R
b
R-x
 
Figure 4.37 Important parameters for the spherical segment  
 
Mathematical approach has been performed to obtain the most suitable radius 
function. It was found that a simple linear relation between R and h fitted best with 
the Doerner & Nix factor: 
2*
thKR +=                             (23) 
 
 106 
where K is a constant and the Y-intercept equals half of the film thickness. 
Substituting the above equation into Eq. (22), the energy fraction coefficient is plotted 
in figure 4.38a with the Doerner & Nix factor (?=0.25) for indentation on a 500nm 
thick film. The two factors are in good agreement. Now let?s see how the alpha 
parameter influences R in Eq. (22) and (23). Changing ? from 0.25 to 0.05 (figure 
4.38b) results in a sharp increase of the Doerner & Nix factor. As mentioned in 
section 4.3, a smaller ? yields a greater substrate effect. To match with this trend, R in 
Eq. (23) must be increased to account for more volume in the substrate for the rising 
substrate effect. From figure 4.38a to 4.38b, K in Eq. (23) increased from 1.2 to 5.5. 
In applying a 10nm indentation depth in a 500nm film, R and K for different ? values 
are listed in table 4.4. It was found find that ? has a linear relation with 1/K, see figure 
4.39. Also R decreases non-linearly with increasing ? (figure 4.40). 
 
 
 
 107 
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000
Displacement (nm)
W
eig
ht 
Fa
cto
r
Energy sphere model 
factor (R=1.2h+t/2)
Doerner & Nix 
Factor (?=0.25)
W
eig
ht 
Fa
cto
r
?
 
Figure 4.38a Comparison of the Doerner & Nix factor (?=0.25) and the analytical 
model from Eq. (22). 
 
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000
displacement (nm)
W
eig
hti
ng
 Fa
cto
r
Doerner & Nix 
Factor (?=0.05)
Energy sphere model 
factor (R=5.5h+t/2)
?: 0.25barb2right 0.05
K: 1.2 barb2right 5.5We
igh
tin
g F
ac
tor
?
?
??
?
 
Figure 4.38b Comparison of the Doerner & Nix factor (?=0.05) and the analytical 
model from Eq. (22). 
 
 108 
Table 4.4 ? parameter in Doerner & Nix function and corresponding K and R values 
in the analytical model from Eq. (22) when indenting 10 nm into a 500nm film 
Alpha (?) K 1/K R 
0.05 5.52 0.18 305 
0.1 3.01 0.33 280 
0.15 2.01 0.5 270 
0.2 1.50 0.67 265 
0.25 1.21 0.83 262 
0.3 1.07 0.95 260.5 
0.35 .0.93 1.11 259 
0.4 0.82 1.25 258 
0.45 0.71 1.41 257.2 
0.499 0.66 1.53 256.5 
 
 
 109 
y = 3.0228x + 0.0463
0
0.6
1.2
1.8
0 0.1 0.2 0.3 0.4 0.5 0.6
alpha
1/K
 
Figure 4.39a Plot of 1/K vs. ? for a 10 nm indentation on a 500nm film 
250
260
270
280
290
300
310
0 0.1 0.2 0.3 0.4 0.5 0.6
alpha
Ra
diu
s o
f th
e e
ffe
cti
ve
 el
as
tic
 re
gio
n (
nm
)
 
Figure 4.39b Plot of R in Eq. (23) vs. ? for a 10 nm indentation on a 500nm film 
 
 
 110 
The section above has shown the Doerner & Nix function can be related to the 
strain energy volume model. However, the increasing trend of the volume radius with 
decreasing ? does not match the real situation. The major drawback is from the 
assumption of continuous elastic strain growth through the specimen. Considering 
that there are two parameters that specify the film and substrate elastic strain growth 
respectively, the strain volume radius in the film (Rf) should not match with that in the 
substrate (Rs). Figure 4.41 shows the different changing ways of R between single 
alpha (?) and two parameters situation to achieve the same volume ratio increase. 
 
One ?
?=0.25?=0.45
total
cap
V
V
Two?s
????2=0.25
????1=0.25
????2=0.25
total
cap
V
V
????1=0.45
?
??
?
????
????
????
????
 
Figure 4.40 Comparison between 1-? and 2- ? models 
 
   As the specimens all have Si as substrates, let us fix ?2 and see how ?1 affects the 
radius change in the film. In applying Equation (24), Rf  was calculated and plotted 
with corresponding ?1 in figure 4.41. It is found Rf  increases with ?1 with an opposite 
 111 
trend from figure 4.39b for single alpha factor.  
)]3(3)3(334[
)3(3
223
2
tRttRtR
tRt
V
V
fss
f
total
cap
?+??
?
= pipi
pi
pi
             (24) 
 
????1 (Solid Points) & Poisson?s Ratio (Open Points) 
0.20
225
150R
ad
ius
 of
 th
e e
las
tic
 en
er
gy
 vo
lum
e (
nm
)
0.3 0.4 0.5 0.6
300
375
450
0.1
475
420
530
585
640 R
ad
ius
 fo
r 0
.13
% 
str
ain
 bo
un
da
ry
 (n
m)
????
Ra
diu
s o
f t
he
 el
as
tic
 en
er
gy
 vo
lum
e (
nm
)
????
Ra
diu
s o
f t
he
 el
as
tic
 en
er
gy
 vo
lum
e (
nm
) Ra
diu
s f
or
 0.
13
% 
str
ain
 bo
un
da
ry
 (n
m)
 
Figure 4.41 Radius of the elastic strain boundary vs. alpha1 & Poisson?s Ratio   
 
Now, if the film Poisson?s ratio is equal to ?1, it should manifest Rf in a same 
way as shown in figure 4.41. First, it is necessary to investigate how the Poisson? ratio 
affects the elastic strain energy in the specimen. As the indenter is driven into the film, 
local material is pushed away both vertically and laterally (figure 4.42). The laterally 
displaced film is mainly controlled by the shear mechanisms. The film shear modulus 
can be represented as: 
 112 
)1(2 ??
?
+==
EG
xy
xy                          (25) 
Assuming indentation is performed on two films with similar Young?s modulus but 
different Poisson? ratio (i.e, ?1=0.1, and ?2=0.3). According to Eq. (25), the one 
with smaller Poisson?s ratio should have larger shear modulus and thus be more 
difficult to deform laterally.  
According to that, the elastic strain boundary (volume) should be lesser for a 
material with smaller Poisson?s Ratio. The finite element analysis results proved 
this assumption. Figure 4.43 shows 10 nm indentations on 500 nm NFC film with 
the same Young?s Modulus and same Si substrates but different Poisson?s Ratio. 
From 4.43(a) to 4.43(c), it is clear to see the film elastic strain boundary at the 
interface grows with the change of Poisson?s Ratio. Using the equation below from 
figure 4.37 to calculate the effective radius of the total elastic energy volume [R in 
Eq. (22)]: 
x
xbR
2
22 +
=                           (26) 
 
The Rf-Poisson?s ratio (?) curve is also plotted in figure 4.41 for comparison and 
contrast. Although the absolute values of these two curves are not the same 
(because the elastic strain boundary was randomly picked-up, which may not 
resemble the true radius of the effective elastic energy volume), it was found that 
alpha and Poisson?s ratio manifested R in a same manner. Based on this 
mathematical approach and the matching experimental data, we suggest that the 
 113 
two constants in the modified Doerner & Nix function are film and substrate 
Poisson?s ratios.  
 
Film
Substrate
Indenter
 
Figure 4.42 Schematic of an indentation impression in the film 
 114 
 
Figure 4.43 FEA elastic strain distributions in the films with the same Young?s 
modulus but different Poisson?s Ratios: (a) ?=0.1, (b) ?=0.3, (c) ?=0.49 
 
 
 
 
 
(a) 
(b) 
(c) 
 115 
 
 
 
 
 
CHAPTER 5  
CONCLUSIONS  
 
In this study, 15 specimens were tested with continuous stiffness 
nanoindentation to investigate the origin of the substrate effects and its relation to the 
elastic strain properties in the film and substrate. 14 of them were deposited as 
amorphous films so that the film microstructures effects on nanoindentation could be 
minimized. Another tested material was ultra-nanocrystalline diamond film. It was 
found that as long as the indentation depth was within the film, pile-up degree was 
negligible and only had minor effects on measure Young?s modulus. The films were 
characterized with XRD, SEM, AFM and other systems before indentation data 
collection.  
Two existing models for the thin film nanoindentation mechanism: Doerner & 
Nix and Gao models, were studied and compared with our experimental data. It was 
found that the Doerner & Nix function always yielded more satisfying results for 
those films that had a flat region in the early stage of the E-h curve, while Gao?s 
model always fit well with the material whose Young?s modulus increased (or 
decreased) from the beginning of the indentation. Since most of our experimental 
curves had flat regions, the study was mainly focused on modifying Doerner & Nix 
function.  
 116 
The flat region length or the critical indent depth (hcr) was determined for each 
material. A unique non-linear trend was found between the critical indent depth/film 
thickness ratio and film/substrate modulus ratio. It was also found that the thickness 
would not affect the hcr/ t as long as Ef/Es was similar. The Doenar and Nix equation 
for substrate effects were investigated carefully in this study. It was found parameter ? 
in the function was not a constant as originally suggested. It changed with the 
indentation depth. To improve Doerner & Nix function, two parameters were 
suggested in the equation instead of one. Based on the experimental determination 
and analytical modeling, the two parameters were suggested as the Poisson?s ratio of 
the film and substrate. The modified function was found to better describe most of the 
tested materials.  
 
 
 
 
 
 
 
 
 
 
 
 117 
 
 
 
 
 
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