LOW TEMPERATURE MODELING OF I ?V CHARACTERISTICS AND RF SMALL SIGNAL PARAMETERS OF SIGE HBTS Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classified information. Ziyan Xu Certificate of Approval: Fa Foster Dai Professor Electrical and Computer Engineering Guofu Niu, Chair Alumni Professor Electrical and Computer Engineering Vishwani Agrawal James J. Danaher Professor Electrical and Computer Engineering George T. Flowers Dean Graduate School LOW TEMPERATURE MODELING OF I ?V CHARACTERISTICS AND RF SMALL SIGNAL PARAMETERS OF SIGE HBTS Ziyan Xu A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Master of Science Auburn, Alabama December 18, 2009 LOW TEMPERATURE MODELING OF I ?V CHARACTERISTICS AND RF SMALL SIGNAL PARAMETERS OF SIGE HBTS Ziyan Xu Permission is granted to Auburn University to make copies of this thesis at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date of Graduation iii VITA Ziyan Xu, daughter of Wenzhen Li and Jian Xu, was born on November 13, 1984, in Qingdao, P.R. China. She completed her high school degree from No.2 Qingdao High School in 2003. She received her bachelor degree from Hefei University of Technology in 2007, majoring in microelec- tronics. After that she was accepted into the electrical and computer engineering department in Auburn University in the fall of 2007, pursuing her master?s degree. iv THESIS ABSTRACT LOW TEMPERATURE MODELING OF I ?V CHARACTERISTICS AND RF SMALL SIGNAL PARAMETERS OF SIGE HBTS Ziyan Xu Master of Science, December 18, 2009 (B.S., Hefei University of Technology, 2007) 84 Typed Pages Directed by Guofu Niu SiGe HBT has been attached great attention recently to be used for space exploration due to its high-quality performance compared with conventional Si bipolar transistor over an extremely wide temperature range. The currently used compact models fail to correctly function at very low temperature. This work investigates low temperature modeling of I ? V characteristics and RF small signal parameters of SiGe HBTs. Compact model Mextram is used as the starting point. A brief introduction of Mextram model is made. Both main current and base current modeling and their temperature scaling in Mextram model are reviewed. New temperature scalable model of main current and base current is proposed and demon- strated with experimental data from 393 to 43 K. The temperature dependent ideality factor is proved necessary to model the low temperature current-voltage characteristics deviation from Shockley the- ory prediction. The relation triangleV BE = kT/qln(J C1 /J C2 ), which is widely used in bandgap refer- ences (BGR) circuits, is shown no longer valid at low temperature. The e?ect of tunneling on low temperature forward operation current is examined. Trap- assisted tunneling (TAT) current dominates the forward non-ideal base current. The way to dis- tinguish tunneling current and main base current from forward gummel base current measurement is shown. A tunneling current model is developed to fit the lower bias region of forward base current from 110 to 43 K. v Small signal model is used to extract device small signal parameters. A two-step hot-after-cold optimization procedure is successfully used to fit Y-parameters from 1 to 35 GHz. The temperature dependence of important equivalent circuit parameters and implication on cryogenic RF circuit are examined. vi ACKNOWLEDGMENTS My deepest gratitude goes first and foremost to Dr. Guofu Niu, my supervisor, for his constant encouragement and guidance. I would like also to thank the other members of my committee, Dr. Fa Dai, and Dr. Vishwani Agrawal for their assistance. Many thanks also go to Xiaoyun Wei, Lan Luo and Tong Zhang, my groupmates. I feel so lucky to study and work with them. This thesis could not have been possible without the financial assistance of NASA and collab- oration with Prof. John Cressler?s SiGe group at Georgia Tech. At last, I would like to thank my parents and families for their self-less support, understanding and love. vii Style manual or journal used Journal of Approximation Theory (together with the style known as ?aums?). Bibliography follows van Leunen?s A Handbook for Scholars. Computer software used The document preparation package T E X (specifically L A T E X) together with the departmental style-file aums.sty. viii TABLE OF CONTENTS LIST OF FIGURES xi 1INTRODUCTION 1 1.1 Cryogenic Operation of SiGe HBTs ......................... 1 1.2 Thesis Contribution .................................. 4 2INTRODUCTION TO MEXTRAM 6 2.1 Main Current Modeling ................................ 9 2.2 Base Current Modeling ................................ 14 2.2.1 Linear Forward Base Current ......................... 14 2.2.2 Non-Ideal Region ............................... 14 2.3 Temperature Scaling ................................. 14 2.3.1 Saturation Current I S Temperature Scaling................. 15 2.3.2 Base Saturation Current Temperature Scaling ................ 16 2.4 IC-CAP Built-in Mextram Model and Verilog-A Based Mextram .......... 16 3IMPROVED MAIN CURRENT MODELING 19 3.1 I C ?V BE Modeling ................................. 19 3.1.1 Improved N F Temperature Scaling ..................... 21 3.1.2 Improved I S Temperature Scaling ...................... 24 3.1.3 Summary ................................... 27 3.2 N F ?s E?ect in Base Di?usion Charge ........................ 29 4 BGR IMPLICATION 30 4.1 V BE ?T ........................................ 30 4.2 ?V BE ?T ...................................... 30 4.3 V ref ?T ........................................ 31 5BASE CURRENT MODELING 35 5.1 Trap-Assisted Tunneling E?ect............................ 35 5.2 Separation of Main and Tunneling Base Current ................... 37 5.3 Ideality Factor, Saturation Current and Current Gain ................. 38 5.4 Tunneling Base Current Modeling .......................... 41 5.5 Moderate Bias Region ................................ 45 5.6 Summary ....................................... 48 ix 6SMALL SIGNAL MODELING 49 6.1 Equivalent Circuit ................................... 49 6.2 Parameter Extraction ................................. 50 6.2.1 Procedure ................................... 50 6.2.2 Result ..................................... 53 6.3 Substrate Network Implementation .......................... 59 7CONCLUSION 62 BIBLIOGRAPHY 64 APPENDICES 67 AVERILOG-A CODE IMPLEMENTATION WITH KEY IMPROVED MODELS 68 A.1 Improved Ideality Factor Temperature Mapping ................... 68 A.2 Improved Saturation Current Temperature Mapping ................. 69 A.3 Modified Trasfer Current Model ........................... 70 A.4 Modified Base Current Model ............................ 71 x LIST OF FIGURES 1.1 Energy band diagram of a graded-base SiGe HBT. [1] ................ 2 1.2 Calculated base transit time using Philips mobility model and incomplete ionization model. ......................................... 3 2.1 Equivalent Circuit of Mextram Level 504.7. [2] ................... 8 2.2 Comparison collector current between IC-CAP built-in Mextram Model and Verilog-A implementation. .............................. 17 2.3 Comparison base current between IC-CAP built-in Mextram Model and Verilog-A implementation. .................................... 18 3.1 (a) Measured I C versus V BE at temperature from 43 to 393 K. (b) Extracted slope at each temperature compared with ideal 1/V T . (c) Extracted I S at each temperature and I S,T Fitting from Mextram I S,T .......................... 20 3.2 The slope of I C -V BE from device simulation . .................... 21 3.3 N F extracted from measurement versus N F T-scaling models referenced to T nom =300K. ...................................... 22 3.4 Simulated versus measured I C ?V BE . Extracted N F are included. ......... 23 3.5 Simulated versus measured I C ?V BE . Extracted N F and Thurmond?s E g,T included. 26 3.6 I S extracted from measurement versus I S from the three models listed in Table 3.1 over 43-400 K. .................................... 27 3.7 (a) Simulated versus measured I C -V BE at high temperatures. (b) Simulated versus measured I C -V BE at low temperatures. ........................ 28 4.1 (a) Simulated versus measured V BE -T dependence at I 0 =0.1?A, 1?A, and 10?A. (b) Deviation of simulated V BE from measured V BE normalized by thermal voltage. 31 4.2 Wildar bandgap reference circuit. .......................... 32 4.3 (a) Simulated versus measured ?V BE ?T at I 0 =0.1?A, 1?A, and 10?A. (b) Devi- ation of simulated ?V BE from measured ?V BE normalized by thermal voltage. . . 33 xi 4.4 (a) Simulated V REF versus measured V REF for three models. (b) Deviation of simulated V REF from measured V REF with respect to thermal voltage. ....... 34 5.1 Illustration of trap-assisted tunneling in forward biased EB junction. ........ 36 5.2 Forward and reverse gummel measurement data: (a) forward gummel I C -V BE , (b) forward gummel I B -V BE , (c) reverse gummel I E -V BC and (d) reverse gummel I B - V BC . ......................................... 37 5.3 Di?erence between linear fitting with and without iteration at 43 K. ........ 38 5.4 Comparison of ideality factor and saturation current between including and exclud- ing TAT current: (a) I S,BE -T, (b)N EI -T, (c) I S -T, (d)N F -T. ........... 39 5.5 Comparison of ideality factor and saturation current between collector and base: (a) saturation current, (b)ideality factor. ......................... 40 5.6 The slopes of (a) I C ? V BE and (b) extracted from simulation and measurement from 43 to 300 K. ................................... 41 5.7 Comparison between current gain obtained by (a) total I C /I B and (b)excluding tunneling current and current gain obtained at I C = 10 ?5 A (c) by (a) and (b). . . . 42 5.8 (a) Base tunneling current at 43, 60 76, 93 and 110 K; (b) The extracted ideality factor is proportional to 1/T; (C) S = N E,TAT ?V T . ................ 43 5.9 The temperature scaling of tunneling saturation current at 43, 60, 76, 93 and 110 K. 46 5.10 Comparison between measurement an modeling results for 43 to 110 K: (a) com- bined with ideal base fitting, (b) TAT current modeling ............... 47 5.11 Measured and modeled I B -V BE from 43-93 K. ................... 48 6.1 Small-signal equivalent circuit used for SiGe HBTs. ................. 50 6.2 Small-signal equivalent circuit used for cold state. .................. 51 6.3 Measured f T -I C as a function of V CB at various temperatures. ........... 51 6.4 Selected I C points for each temperature extraction . ................. 52 6.5 Measured and simulated Y-parameters at 300 K. ................... 54 6.6 Measured and simulated Y-parameters at 223 K. ................... 54 6.7 Measured and simulated Y-parameters at 162 K. ................... 55 xii 6.8 Measured and simulated Y-parameters at 93 K. ................... 55 6.9 Extracted g m -I C results compared with ideal I C /V T with and without self-heating at 300, 223, 162 and 93 K. ................................ 56 6.10 Extracted C ? -I C at 300, 223, 162 and 93 K. ..................... 57 6.11 Extracted C bci -I C and C bcx -I C at 300, 223, 162 and 93 K. .............. 58 6.12 Extracted total C BC (C bci +C bcx )vsV BE from cold and hot extraction. ....... 58 6.13 Extracted R bi at I C = 1mAandR s vs T. ...................... 59 6.14 Small-signal equivalent circuit for substrate network. ................ 60 6.15 Small-signal equivalent circuit for substrate network. ................ 61 7.1 Modified equivalent circuit. ............................. 63 xiii CHAPTER 1 INTRODUCTION After being studied and developed for years, Silicon- Germanium (SiGe) technology has per- vasively used in personal communications devices and military products. The heart of SiGe tech- nology is a SiGe heterojunction bipolar transistor (HBT), which has a high compatibility with BiC- MOS technology, is easy to integrated with existing BiCMOS device [3]. The bandgap engineering by introducing SiGe alloy into the base upgrades both DC and AC performance of conventional Si bipolar transistor (BJT). The Ge-grading induced extra drift field in the neutral bases shown in Fig. 1.1 increases the collector current density (J C ) through increasing electron injection at emitter- base (EB) junction, yielding higher current gain. This induced field also accelerates the minority carrier transportation, shortens the transit time across the base and increases the frequency response [3]. 1.1 Cryogenic Operation of SiGe HBTs The low temperature performance of SiGe HBTs has been investigated for many years [4]. As the bipolar transistor is a minority carrier device, which obeys the Shockley boundary conditions, n 2 i0 is proportional to the exponential of the bandgap. Intuitively, the change in bandgap due to Ge- grading will associate with exponential change in current. Moreover, these change will be naturally exaggerated by thermal voltage V T with cooling. With a glance of the device?s temperature mapping equations, we are able to find that both DC and AC characteristic of SiGe HBTs are favorably a?ected by cooling. The term V T is unavoidably functioning almost everywhere [5]. A good example of SiGe HBTs advantage over Si BJTs is that the induced field o?ers a method to o?set the inherent ? b associated with cooling, yielding an f T improving with cooling [5]. 1 Figure 1.1: Energy band diagram of a graded-base SiGe HBT. [1] In (1.1) and (1.2), the base transit time ? b,Si of Si and ? b,SiGe of SiGe are given respectively [3]. ? b,Si = W 2 b 2D nb ; (1.1) ? b,SiGe = W 2 b D nb kT triangleE g,Ge (grade) ? braceleftbigg 1 ? kT triangleE g,Ge (grade) bracketleftbig 1 ? exp(?triangleE g,Ge (grade) slashbig kT bracketrightbig bracerightbigg . (1.2) Based on a theoretical calculation, the comparison between Si and SiGe base transit time is shown in Fig. 1.2. For simplicity, the base of SiGe HBT is assumed to be 50 nm wide with uniform doping of 10 18 cm ?3 and a total bandgap grading of 100 meV. The Philips unified mobility model [6] is used , while incomplete ionization mobility model reported in [7] is used to model freeze-out by a?ecting di?usivity. Whatever the freeze-out is taken into consideration or not, the transit time of SiGe HBT is much smaller than that of Si BJT. This clearly illustrates the advantages of SiGe HBT in low temperature operation. 2 50 100 150 200 250 300 350 400 0 1 2 3 4 5 6 7 x 10 ?12 Temperature (K) Base Transit Time (s) without freezeout with freezeout Si Base SiGe Base Figure 1.2: Calculated base transit time using Philips mobility model and incomplete ionization model. 3 Due to the excellent analog and RF performance of SiGe HBTs over an extremely wide range of temperatures, together with its built-in total dose radiation tolerance [5], SiGe BiCMOS tech- nology is currently being used to develop electronics for space applications. For instance, SiGe BiCMOS electronic components can operate directly in the extreme ambient environment found on the lunar surface, where temperatures cycle from -180 ? Cto+125 ? C (a 300 ? C swing over 28 days) and radiation exposure exists (both total dose and single event e?ects). Operating electronic systems under ambient conditions, without excessive shielding or by providing heaters inside Warm Electronics Boxes (WEBs) to protect electronics from their surroundings (current practice), can save substantial size, weight, and power, increasing the reliability and decreasing the cost for these missions [8]. The device modeling of high accuracy and convergence is the prerequisite. This is the basic motivation for cryogenic temperature modeling. 1.2 Thesis Contribution In this work, Mextram model is used as a starting point and introduced in Chapter 2. The exquisite model is advanced in modeling quasi-saturation, Kirk e?ect, impact ionization and so on. It is limited, however, in describing the cryogenic behavior of transistors. Next in Chapter 3, an improved model of DC main current is proposed after examination of existing models. It is found that q B factor is far from enough to produce the slope in medium I C ?V BE region [9]. The existing I S T-scaling model does not work below 200 K. The T dependent ideality factor must be included into both I C ?V BE equation and I S T-scaling equation. The same strategy is used to model ideal base current. At cryogenic temperatures, the trap- assisted tunneling dominates non-ideal base current at forward operation. However, the compact models, which may have described the tunneling current at reverse operation, never include the trap-assisted tunneling current. In this work, the trap-assisted tunneling e?ect is examined and is added to model base current. The iteration method is used to distinguish ideal base current and the trap-assisted tunneling current. T-scaling of the tunneling saturation current is also proposed. 4 The temperature dependence of I C ? V BE is very important for designing bandgap refer- ences where the negative temperature coe?cient of V BE is neutralized by the positive tempera- ture coe?cient of ?V BE of two transistors operating at di?erent current densities [10]. ?V BE = kT/qln(J C1 /J C2 ) in existing models and is linearly proportional to absolute temperature T. In Chapter 5, this relation is shown invalid at low temperature. Significantly better fitting of V BE ?T, and ?V BE ?T characteristics are obtained by including T-scaling of N F into this relation. In Chapter 6, the small signal modeling is examined. Parameter extraction strategy and fitting results of Y-parameter over temperature are presented and discussed. Simulation results from 1 to 35 GHz of a first-generation SiGe HBT with a 50 GHz peak f T at 300 K are presented in the temperature range of 393 K to 93 K. All modification is made in Mextram Verilog-A code. However, the I ?V model can be used in other compact models too, as all of the models have similar main current base current equations. 5 CHAPTER 2 INTRODUCTION TO MEXTRAM Mextram model is a widely used vertical bipolar transistor model. Mextram is the acronym of the "most exquisite transistor model". The first Mextram release was introduced as Level 501 in 1985 [11]. Later Level 502, 503 and 504 were respectively released in 1987 [12], 1994 and 2000 [13]. And development was never stopped following the requirement of updated technology. The latest accessible version is Level 504.7 [2]. Following e?ects descriptions are contained in Mextram according to its latest release Level 504.7 [2]: 1. Bias-dependent Early e?ect 2. Low-level non-ideal base currents 3. High-injection e?ects 4. Ohmic resistance of the epilayer 5. Velocity saturation e?ects on the resistance of the epilayer 6. Hard and quasi-saturation (including Kirk e?ect) 7. Weak avalanche (optionally including snap-back behaviour) 8. Charge storage e?ects 9. Split base-collector and base-emitter depletion capacitance 10. Substrate e?ects and parasitic PNP 11. Explicit modeling of inactive regions 12. Current crowding and conductivity modulation of the base resistance 13. First order approximation of distributed high frequency e?ects in the intrinsic base (high- frequency current crowding and excess phase-shift) 14. Recombination in the base (meant for SiGe transistors) 15. Early e?ect in the case of a graded bandgap (meant for SiGe transistor) 6 16. Temperature scaling 17. Self-heating 18. Thermal noise, shot noise and 1/f-noise In Mextram model, there are five internal nodes and 79 parameters, including parameters of model flag, parameters of noise and the reference temperature, parameters of temperature scaling, parameters of individual transistor design and parameters to be determined by the fitting the model to the transistor characteristics of a specific device and at a specific temperature. Some parts of the model are optional and can be switched on or o? by setting flags. These are the extended modeling of reverse behaviour, the distributed high-frequency e?ects, and the increase of the avalanche current when the current density in the epilayer exceeds the doping level. The governing Mextram equations are formulated having in mind NPN transistors, but the model can be equally well used for PNP transistors by simple change of the current and charge polarity. Besides, both three-terminal devices (discrete transistor) and four-terminal devices (IC- processes which also have a substrate) can be described. Fig. 2.1 shows the equivalent circuit of Mextram model as it is specified in its latest release Level 504.7 [2]. The branches representing model currents and charges are schematically associated with di?erent physical regions of a bipolar transistor separated by the base-emitter, base-collector, and substrate-collector junctions. All current and charge branches in Mextram are given as explicit functions of external and internal nodal potentials and there are no implicit modeling variables that require internal iterations [11]. 7 Figure 2.1: Equivalent Circuit of Mextram Level 504.7. [2] 8 2.1 Main Current Modeling In Mextram, the integral charge control relation (ICCR) is used for the description of the main current, which is I N in Fig. 2.1. Following repeats the brief derivation of this relation. Most of the derivations are on the reference of [2]. The electron density in the base region can be written as J n = n? n dE fn dx , (2.1) in which n is electron concentration, ? n is the electron minority carrier mobility. The di?erence between quasi-Fermi level is E fn ?E fp = kT ln parenleftBigg np n 2 i parenrightBigg , (2.2) where k is Boltzmann?s constant and n i is the e?ective intrinsic concentration. As the hole is majority in the base, dE fp dx is assumed to be zero. Thus. J n = n? n d parenleftbig E fn ?E fp parenrightbig dx = qD n n 2 i p d dx parenleftBigg np n 2 i parenrightBigg , (2.3) where D n is the electron di?usivity, equal to ? n kT/q. Moving qD n n 2 i p to the left side gives J n p(x) qD n n 2 i dx = d parenleftBigg np n 2 i parenrightBigg . (2.4) Integrating on both sides gives J n x C2 integraldisplay x E1 p(x) qD n n 2 i dx = exp parenleftbigg V B 2 C 2 V T parenrightbigg ? exp parenleftbigg V B 2 E 1 V T parenrightbigg , (2.5) J n = ? q 2 D n n 2 i Q B bracketleftbigg exp parenleftbigg V B 2 E 1 V T parenrightbigg ? exp parenleftbigg V B 2 C 2 V T parenrightbiggbracketrightbigg , (2.6) 9 and Q B = qA em x C 2 integraldisplay X E 1 p(x)dx. (2.7) X E 1 and X C 2 are the positions of the internal emitter node and internal collector node defined in Mextram. Both of them can be considered at the actual junctions. Since the direction from emitter to collector is taken as positive x? direction, and the current density is negative for the forward mode of operation, I N = ?A em J n , (2.8) where A e m is e?ective emitter area. And the main current becomes I N = q 2 D n n 2 i A 2 em Q B bracketleftbigg exp parenleftbigg V B 2 E 1 V T parenrightbigg ? exp parenleftbigg V B 2 C 2 V T parenrightbiggbracketrightbigg . (2.9) In Mextram, the main current based on the ICCR relation is given by I f = I S exp parenleftbigg V B 2 E 1 V T parenrightbigg , (2.10) I r = I S exp parenleftbigg V B 2 C 2 V T parenrightbigg , (2.11) I N = I f ?I r q B . (2.12) Here, the base charge Q B normalized to the base charge at zero bias Q B0 is denoted by q B . Hence, I S = q 2 D n A 2 em n 2 i Q B0 , (2.13) and 10 Q B = Q B0 +Q tE +Q tC +Q BE +Q BC . (2.14) For low injection, Q B is the sum of the base charge at zero bias Q B0 , and the extra charge Q tE and Q tC due to the change in emitter side and collector side depletion region width. Q BE is di?usion charge related to forward operation and depends on the base-emitter bias; Q BC is di?usion charge related to reverse operation and depends on the base-collector bias. Firstly the di?usion charge is neglected for the moment, so the normalized base charge be- comes q 0 = Q B Q B0 = 1 + Q tE Q B0 + Q tC Q B0 . (2.15) Define that V tE = Q tE parenleftbig 1 ?XC jE parenrightbig C jE , (2.16) and V tC = Q tC XC jC C jC , (2.17) so q 0 = 1 + V tE V er + V tc V ef . (2.18) V er and V ef are reverse and forward Early voltage. XC jE is defined in Mextram the fraction of the emitter-base depletion capacitance that belongs to the sidewall. XC jC is the fraction of the collector- base depletion capacitance under the emitter. Taking punch-through into the account, Mextram uses q 1 directly instead of q 0 . q 1 = 1 2 parenleftbigg q 0 + radicalBig q 2 0 + 0.01 parenrightbigg . (2.19) Next consider base di?usion charge only and neglect Early e?ect for the moment. The base doping profile in the electron density is assumed to be linear: n(x) = n(0) parenleftbigg 1 ? x W B parenrightbigg +n(W B ) parenleftbigg x W B parenrightbigg . (2.20) 11 The total electron charge is Q B,elec = 1 2 qA em W B n(0) + 1 2 qA em W B n(W B ). (2.21) Define Q BE = 1 2 Q B0 n 0 , (2.22) and Q BC = 1 2 Q B0 n B . (2.23) They are respectively the charge contributed from the electron density at the base-emitter edge and at the base-collector edge. The base transit time is approximately constant, so Q BE = ? B I. (2.24) Combining with (2.9), the following equation can be got I = I S exp parenleftBig V B 2 E 1 V T parenrightBig 1 + ? BI Q B0 . (2.25) Solving for I get I = 2I S exp V B 2 E 1 V T 1 + radicalbigg 1 + 4I S I K exp parenleftBig V B 2 E 1 V T parenrightBig . (2.26) Define low current and high current I low = I S exp parenleftbigg V B 2 E 1 V T parenrightbigg , (2.27) I high = radicalbig I S I K exp parenleftbigg V B 2 E 1 2V T parenrightbigg . (2.28) Calculating the point where both asymptotes cross get 12 I low = I high =? I = I K . (2.29) I K is the so-called knee current. n 0 = 2Q BE Q B0 = 2? B I Q B0 = 2I I K , (2.30) so f 1 = 4I S I K exp parenleftbigg V B 2 E 1 V T parenrightbigg . (2.31) n 0 = f 1 1 + radicalbig 1 +f 1 (2.32) For reverse, f 2 = 4I S I K exp parenleftbigg V B 2 C 2 V T parenrightbigg . (2.33) n B = f 2 1 + radicalbig 1 +f 2 . (2.34) Then combine early e?ect with base charge di?usion, we can get Q BE = 1 2 q 1 Q B0 n 0 , (2.35) Q BC = 1 2 q 1 Q B0 n B , (2.36) q B = q 1 parenleftbigg 1 + 1 2 n 0 + 1 2 n B parenrightbigg . (2.37) 13 2.2 Base Current Modeling 2.2.1 Linear Forward Base Current In Mextram, the total ideal base current is separated into a bulk and a side-wall current. Both depend on separate voltages. ? f as a parameter that gives the ratio between the main saturation current and base saturation current. The equations are given as: I B 1 = parenleftbig 1 ?XI B 1 parenrightbig I S ? f parenleftbigg exp parenleftbigg V B 2 E 1 V T parenrightbigg ? 1 parenrightbigg , (2.38) I S B 1 = XI B 1 I S ? f parenleftbigg exp parenleftbigg V B 1 E 1 V T parenrightbigg ? 1 parenrightbigg . (2.39) XI B1 is the parameter defined in Mextram to express the part of ideal base current that belongs to sidewall. 2.2.2 Non-Ideal Region The non-ideal forward base current is given by I B 2 I B 2 = I B f parenleftbigg exp parenleftbigg V B 2 E 1 m LF V T parenrightbigg ? 1 parenrightbigg , (2.40) and is simply a diode current with a non-ideality factor m Lf . 2.3 Temperature Scaling Following are the T-scaling equations defined in Mextram. T RK is degree Kelvin at which the parameters are determined. t N = T K T RK , (2.41) V T = parenleftbigg k q parenrightbigg T K , (2.42) 14 V T R = parenleftbigg k q parenrightbigg T RK , (2.43) 1 V triangleT = 1 V T ? 1 V T R . (2.44) 2.3.1 Saturation Current I S Temperature Scaling The T-scaling of saturation current is given as I ST = I S t 4?A B ?A QB0 +dA I S N exp bracketleftbigg ?V gB V triangleT bracketrightbigg . (2.45) Below is the derivation. I S,T = A e m q 2 D n,T n 2 i,T Q B0 , (2.46) I S,T rK = A e m q 2 D n,T rK n 2 i,T rK Q B0 . (2.47) As D n = ? n kT q , (2.48) I S,T = I S,T rK T T rK ? n,T ? n,Tom n 2 i,T n 2 i,T rK . (2.49) Since n 2 i ?T 3 exp parenleftbigg ? E g,T kT parenrightbigg , (2.50) ? n ?T ?m , (2.51) 15 I S,T = I S,T rk t (3?m) N exp parenleftbigg ? V g,T V T + V g,T rK V T rK parenrightbigg , (2.52) ? V g,T V T + V g,T rK V T rK = ? 1 V T bracketleftbig V gB (1 ?t N ) ?BT lnt N bracketrightbig . (2.53) Hence, I ST = I S t 4?m+B T rK V T rK N exp bracketleftbigg ?V gB V T rK parenleftbigg 1 ? 1 t N parenrightbiggbracketrightbigg . (2.54) which is equivalent to (2.45). 2.3.2 Base Saturation Current Temperature Scaling For base current, the T-scaling of saturation current is modeled as I BfT = I Bf t 6?2m Lf N exp bracketleftbigg ?V gj m Lf V triangleT bracketrightbigg , (2.55) and the current gain: ? fT = ? f t (AE?AB?AQ B0 ) N exp bracketleftbigg ?dV g?f V triangle T bracketrightbigg . (2.56) 2.4 IC-CAP Built-in Mextram Model and Verilog-A Based Mextram The analog hardware description language (AHDL) Verilog-A is high-level language devel- oped to describe the structure and behavior of analog system and their components [14]. The capa- bility of Verilog-A to handle state-of-the-art compact bipolar transistor modeling mixed with extra modeling has been demonstrated [15]. The Mextram Level 503 and 504 has been implemented by IC-CAP through work jointly car- ried out by Philips Research Labs, TU Delft, and Agilent EEsof EDA [16]. The IC-CAP built-in C-functions can be used for model parameter extraction [17]. 16 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 ?8 10 ?6 10 ?4 10 ?2 V be (V) I C (A) ICCAP Built?in Model Verilog?A 393K 162K Figure 2.2: Comparison collector current between IC-CAP built-in Mextram Model and Verilog-A implementation. Verilog-A code 504.7 is downloaded from Mextram website.We first need to make sure the Verilog-A code is "functional correct" by comparing with built-in model in IC-CAP. Forward Gum- mel simulation at 393, 300, 223 and 162 K are performed in both ways. The simulation results shown in Fig. 2.2 and Fig. 2.3 are highly consistent, which proves that Verilog-A code used is cor- rect. Convergence problem at lower temperature is encountered using both Verilog-A code and the built-in model. 17 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 ?10 10 ?8 10 ?6 10 ?4 10 ?2 I B (A) V BE (V) ICCAP Built?in Model Verilog?A 393K 162K Figure 2.3: Comparison base current between IC-CAP built-in Mextram Model and Verilog-A im- plementation. 18 CHAPTER 3 IMPROVED MAIN CURRENT MODELING Shockley theory predicts thatI C ?V BE has an exponential relation in moderate injection region, so on a semilog scale the I C ? V BE slope is 1/V T . Below 100 K, however, we find that the T- dependence of I C ? V BE becomes increasingly weaker. Whereas in other compact models like Gummel-Poon and VBIC, ideality factor N F is included [18] [19], though its value is almost unity constant, in Mextram the deviation of I C ? V BE from 1/V T is modeled with q B [17] as talked in Chapter 2, and it was believed that q B is su?cient in modeling the slope of I C ? V BE , and using ideality factor could complicate parameter extraction. In this chapter, only q B factor is shown far from enough to reproduce the deviation and a method including T dependent ideality factor in both I C ?V BE equation and I S ?T equation for modeling will be proposed compared with experimental results. 3.1 I C ?V BE Modeling For clarity of discussion, we consider forward mode only where I C relates to V BE by [17] [18] [19] I C = 1 q B I S parenleftbigg exp parenleftbigg V BE N F V T parenrightbigg ? 1 parenrightbigg . (3.1) Here q B accounts for modulation of base charge (or e?ective charge for HBTs), I S is the saturation current, and N F is the ideality factor. V T = kT/q. In Shockley?s junction theory, N F =1, and any deviation of I C -V BE in medium injection can only be modeled through q B . This approach is taken by Mextram as it was believed to be more physical. In the SPICE Gummel-Poon (SGP) and VBIC models, N F is used as a fitting parameter, and does not have a clear physical meaning. As both q B and N F a?ect the slope of I C ? V BE , as long as I C ? V BE is well fitted, both approaches can be used. It is found that, at low temperatures, the slope of measured I C ? V BE , 19 0.2 0.4 0.6 0.8 1 10 ?10 10 ?5 10 0 V BE (V) I C (mA) 0 200 400 0 100 200 300 temperature (K) 1/I C *dI C /dV BE (1/V) 0 200 400 10 ?100 10 ?50 10 0 temperature (K) I S,t (A) Extracted Ideal 1/V T Extracted I S Mextram (a) (b) (c) temperature increase 162K 192K 300K 43K 393K Figure 3.1: (a) Measured I C versus V BE at temperature from 43 to 393 K. (b) Extracted slope at each temperature compared with ideal 1/V T . (c) Extracted I S at each temperature and I S,T Fitting from Mextram I S,T . shown in Fig. 3.1 (a), significantly deviates from the ideal 1/V T value, which cannot possibly be reproduced using q B , which is primarily due to reverse Early e?ect at medium injections where the I C ?V BE relation is linear on a semilog scale. To identify the physical reasons, we performed both drift-di?usion and hydrodynamic device simulation of the SiGe HBT used. The simulated I C ? V BE slope, however, shows a much less deviation from ideal value than what we observed in measurement. The results are shown in Fig. 3.2. As all of the higher order e?ects in compact models are naturally included in device simulation, we conclude that such a deviation is likely not caused by e?ects modeled by q B , and is due to unknown physics to the best of our knowledge. Given that existing q B models fail to model I C ?V BE slope, the use of N F becomes necessary. Our strategy is to use N F as the main parameter for fitting the slope of I C ?V BE in the medium I C range, where I C ?V BE is virtually linear on semilog scale, and use q B for fine tuning. Furthermore, Fig. 3.1 (c) show I S extracted versus temperature and Mextram modeling. The di?erence at low temperature the I S T-scaling should be revaluated too. As convergence problem is encountered when simulation runs at low temperature in Mex- tram and the high consistency in describing moderate I C ?V BE characteristics between VBIC and 20 0 50 100 150 200 250 300 0 50 100 150 200 250 300 1/ I C *dI C /dV BE (1/V) temperature(K) Drift?diffusion Hydrodynamic Measurement Ideal 1/V T Figure 3.2: The slope of I C -V BE from device simulation . Mextram, the evaluation below will be first preformed by VBIC using Verilog-A code, and then transferred to Mextram. 3.1.1 Improved N F Temperature Scaling Here, we propose a new nonlinear N F,T equation N F,T = N F,nom parenleftBigg 1 ? T ?T nom T nom parenleftbigg A NF T nom T parenrightbigg X NF parenrightBigg . (3.2) The term nom means the reference temperature at which all parameters are determined. Fig. 3.3 plots the extracted N F , a constant N F,T , a linear N F,T and the N F,T from (3.2). (3.2) produces the best fitting and is used below. By including extracted N F into Verilog-A, however, the simulated I C ? V BE still cannot fit data below 192 K, as shown in Fig. 3.4. I C ? V BE slope is now correct though. The data clearly show that the main problem is that I S (intercept with Y-axis) is underestimated by the I S ?T model 21 0 50 100 150 200 250 300 350 400 0.9 1 1.1 1.2 1.3 1.4 temperature (K) NF(T) Constant NF Linear Model New Model T nom = 300K (27?C) Symbol : N F Extracted from Measurement Figure 3.3: N F extracted from measurement versusN F T-scaling models referenced toT nom =300K. at lower temperatures. Therefore we need to develop a better I S ?T model that gives higher I S at lower T but does not change I S at higher T. 22 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 10 ?10 10 ?8 10 ?6 10 ?4 10 ?2 10 0 10 2 V BE (V) I C (mA) Measurement Simulation 393K 192K 162K 76K Simulation Includes Extracted N F and Linear E g,t Figure 3.4: Simulated versus measured I C ?V BE . Extracted N F are included. 23 3.1.2 Improved I S Temperature Scaling In existing compact models, the I S temperature dependence is described by I S,T = I S,nom parenleftbigg T T nom parenrightbigg X IS exp ? ? ? ?E a,nom parenleftBig 1 ? T T nom parenrightBig V T ? ? ? , (3.3) which has been derived in Chapter 2. To increase I S in lower temperatures, first comes the thought of bandgap. If a linear bandgap temperature scaling is used, E g,T = E g,0 ??T, (3.4) E a,nom = E g,0 , (3.5) X IS = ? + 1 ?m, (3.6) in which the value of ? is 3. If Lin and Salama?s bandgap temperature scaling is used, E g,T = E g,0 +AT ?BT lnT, (3.7) E a,nom = E g,0 , (3.8) X IS = ? + 1 ?m+B T nom V T nom . (3.9) In fact, for Lin and Salama?s method, part of nonlinear coe?cient of Eg,t is lumped into the X IS term, as proved in [20] . The value of X IS needs to be adjusted. 24 But during parameter extraction, both X IS and E a are fitting parameter, inevitably, we can only get their value to achieve the optimized fitting result, and are not able to tell which bandgap t-scaling is used. That is, the failure of fitting shown in Fig. 3.4 at low temperature is not caused by linear bandgap t-scaling. Next we extend the above T-scaling equation for the classical Thurmond bandgap T-scaling model, E g,T = E g,0 ??T 2 /(T +?) [21]. ? and ? are fitting parameters. A lengthy but straightfor- ward derivation leads to a new I S,T T-scaling equation that can be written in the same form as (3.3), but with a T-dependent E a [22]. E a,T = E g,0 + ??TT nom (T +?)(T nom +?) . (3.10) The activation energy at T = T nom , denoted as E a,nom ,is E a,nom = E g,0 + ??T 2 nom (T nom +?) 2 . (3.11) This leads to a T nom referenced E a,T E a,T = E a,nom ? ??T nom 2 (T nom +?) 2 + ??TT nom (T +?)(T nom +?) . (3.12) Including nonlinear E g,T , the simulated I C -V BE at low temperature are is still far away from mea- surement as shown in Fig. 3.5. Clearly a new approach needs to be developed to improve T-scaling in current models. [22] and [23] proposed to include a T-dependent N F into I S,T . Intuitively, the increase of N F with temperature will gear down the decrease of I S . Below we will show that using a T-dependent N F in I S,T can significantly improve I C ?V BE fitting below 200K. The including of N F,T leads to a I S,T function as I S,T = I S,nom parenleftbigg T T nom parenrightbigg X IS N F,T exp ? ? ? ?E a,T parenleftBig 1 ? T T nom parenrightBig N F,T V T ? ? ? . (3.13) 25 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 10 ?10 10 ?8 10 ?6 10 ?4 10 ?2 10 0 10 2 V BE (V) I C (mA) Measurement Simulation 393K 192K 162K 76K Simulation Includes Extracted N F and Nonlinear E g,t Figure 3.5: Simulated versus measured I C ?V BE . Extracted N F and Thurmond?s E g,T included. Table 3.1: Temperature scaling models examined in this work. Name Model 1 Model 2 Model 3 N F,T 1.0 (3.2) (3.2) E a,T E a,nom E g,0 (3.12) Now we examine three models of di?erent complexity shown in Table 3.1. Model 1 is the SGP model with N F =1. Model 2 uses nonlinear N F,T , but with E a =E g,0 . This has been shown to be equivalent to using Lin and Salama?s nonlinear bandgap model (also used in HICUM). Model 3 implements N F,T in (3.1) and (3.3). Fig. 3.6 compares I S modeling results for all three models. Model 1 gives several decades lower I S below l00 K. Model 2 and 3 are both close to the extracted I S and give comparable I S,T fitting. On closer examination, model 3 is better. If ? = 0 in (3.11), the nonlinear E g,T will be simplified as a linear equation. E a,nom in Model 3 will be equal to E g,0 . Model 3 with ? = 0 can provide better fitting than Model 1, but worse fitting than Model 3 with ? = 686. 26 0 50 100 150 200 250 300 350 400 10 ?100 10 ?80 10 ?60 10 ?40 10 ?20 temperature (K) I S (T) (A) Model 1 Model 2 Model 3 Symbol: Measurement Model 1 Model 2&3 Figure 3.6: I S extracted from measurement versus I S from the three models listed in Table 3.1 over 43-400 K. 3.1.3 Summary Fig. 3.7 compares the I C -V BE simulated with measurement. Above 110 K, even the sim- plest model, Model 1, can produce reasonably good I C ? V BE at moderate injection as shown in Fig. 3.7 (a). From 43-93 K, however, model 1 fails, as shown in Fig. 3.7 (b). Model 3 shows the best result at 43 K. In model 3, if linear E g,T is used, i.e. ? = 0, the resulting E a,nom is 0.028 eV lower than the E a,nom in model 3 from (3.11). Typical value ? = 4.45e ? 4 and ? = 686 is used in (3.11). Although it looks like that model 2 is based on a linear E g,T model, with the choice of E g,T = E g0 , it could be essentially applying the nonlinearE g,T model of Lin and Salama?s [24]. Part of nonlinear coe?cient of E g,T also appears as part of the X IS term. In other words, both E a,nom and X IS need to be adjusted for model 2 to work. This is also confirmed by simulation. 27 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 10 ?10 10 ?8 10 ?6 10 ?4 10 ?2 10 0 I C (mA) Measurement Model 1 Model 2 Model 3 T=110K, 136K,162K, 192K, 223K, 262K, 300K, 393K (a) temperature increase 110K 393K 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 10 ?10 10 ?8 10 ?6 10 ?4 10 ?2 10 0 I C (mA) V BE (V) (b) temperature increase 43K 93K T=43K, 60K,76K, 93K Figure 3.7: (a) Simulated versus measured I C -V BE at high temperatures. (b) Simulated versus measured I C -V BE at low temperatures. 28 3.2 N F ?s E?ect in Base Di?usion Charge As including non-linear temperature dependent N F into I C ? V BE is necessary, accordingly, to correctly model base di?usion charge, (2.25),(2.26), (2.27), (2.28), (2.31) and(2.33) should be adjusted to I = 2I S exp parenleftBig V B 2 E 1 N F,T V T parenrightBig 1 + ? BI Q B0 , (3.14) I = 2I S exp parenleftBig V B 2 E 1 N F,T V T parenrightBig 1 + radicalbigg 1 + 4I S I K exp parenleftBig V B 2 E 1 N F,T V T parenrightBig , (3.15) I low = I S exp parenleftbigg V B 2 E 1 N F,T V T parenrightbigg , (3.16) I high = radicalbig I S I K exp parenleftbigg V B 2 E 1 2N F,T V T parenrightbigg , (3.17) f 1 = 4I S I K exp parenleftbigg V B 2 E 1 N F,T V T parenrightbigg , (3.18) and f 2 = 4I S I K exp parenleftbigg V B 2 C 2 N F,T V T parenrightbigg . (3.19) 29 CHAPTER 4 BGR IMPLICATION The temperature dependence of I C ?V BE is very important for bandgap reference (BGR) de- sign. The main concept of BGR is to use the positive temperature coe?cient of triangleV BE generated by two transistors operating at di?erent current densities to compensate the negative temperature coef- ficient ofV BE , to make a zero temperature coe?cient reference voltage output [10]. As a strongly T- dependent ideality factor N F has been included into I C ?V BE relation. ?V BE =N F V T ln(J C1 /J C2 ) is no longer linearly proportional to T. In this Chapter, the three models in Table 3.1 in Chapter 3 are used to examine V BE ? T and ?V BE ?T. Implication to BGR output is simulated with a Widlar BGR circuit. 4.1 V BE ?T The V BE versus T characteristics at three fixed I C , I C =0.1?A, 1?A, and 10?A, are shown in Fig. 4.1 (a). The data are interpolated from forward gummel measurement and simulation. Above 200K, all three models can correctly model the nonlinear V BE ?T dependence. Below 200K, although the deviation of simulated V BE from measured V BE is small, it is significant when normalized by thermal voltage, as shown in Fig. 4.1 (b). 4.2 ?V BE ?T A Widlar BGR as Fig. 4.2 is built in ADS simulator using three models from Table 3.1 to evaluate the improvement of the new temperature dependence model in more realistic environment. All of the elements in BGR except the transistors are ideal components including the ideal current source I 0 . 30 0.4 0.6 0.8 1 1.2 V BE (V) 0 50 100 150 200 250 300 350 400 ?5 ?4 ?3 ?2 ?1 0 1 temperature (K) (V BE ?V BE,meas )/V T Model 1 Model 2 Model 3 Symbol: Measurement I 0 increase I 0 = 1e?7A, 1e?6A, 1e?5A Model 1 Model 2 Model 3 (a) (b) Figure 4.1: (a) Simulated versus measured V BE -T dependence at I 0 =0.1?A, 1?A, and 10?A. (b) Deviation of simulated V BE from measured V BE normalized by thermal voltage. As for simulation, the V BE di?erence, ?V BE =V BE,1st -V BE,2nd , is generated by two transistors having 8 times di?erent current densities, whereas the measured triangleV BE is actually interpolated from forward gummel data between two points which has 8 time current di?erence, as testing is not available. Comparison between measurement and three models is plotted in Fig. 4.3 (a). Fig. 4.3 (b) shows ?V BE /V T . Model 1 gives a linear ?V BE ? T dependence as expected. Model 2 and 3 can both reasonably reproduce the nonlinear temperature dependence of ?V BE . ?V BE /V T is a constant above 200 K, and this constancy is the basis for producing a PTAT voltage in BGR design. 4.3 V ref ?T Fig. 4.4 compares V REF , the simulated output voltage of BGR, with measurement results. The measurement results in Fig. 4.4 (a), not from a real BGR circuit measured, is again the estimated V REF using measured I C -V BE data, which provides a reference value for the comparison. V REF using Model 1 is several V T lower than measurement. Model 2 and 3 can dramatically improve the 31 Figure 4.2: Wildar bandgap reference circuit. 32 0 0.02 0.04 0.06 0.08 ? V BE (V) 0 50 100 150 200 250 300 350 400 2 2.5 3 temperature (K) ? V BE /V T Model 1 Model 2 Model 3 Model 1 Model 2&3 Symbol: Measurement I 0 = 1e?7A, 1e?6A, 1e?5A I 0 increase Symbol: Measurement (a) (b) Figure 4.3: (a) Simulated versus measured ?V BE ?T at I 0 =0.1?A, 1?A, and 10?A. (b) Deviation of simulated ?V BE from measured ?V BE normalized by thermal voltage. simulation results, and further benefit the simulation of large ICs. The increase of ?V BE /V T and decrease of V ref with cooling below 200 K may need to be considered and exploited for better BGR design at cryogenic temperatures. 33 1.1 1.15 1.2 1.25 V REF (V) 0 50 100 150 200 250 300 350 400 ?6 ?4 ?2 0 2 temperature (K) (V REF ?V REF,meas )/V T Model 1 Model 2 Model 3 Symbol: Measurement (a) (b) 1.222V Figure 4.4: (a) SimulatedV REF versus measuredV REF for three models. (b) Deviation of simulated V REF from measured V REF with respect to thermal voltage. 34 CHAPTER 5 BASE CURRENT MODELING This chapter addressesI B ?V BE characteristics. In Mextram, theI B ?V BE modeling is achieved in the use of current gain factor ? f and related to I S . The current gain, however, as shown in Fig. 5.7 (a), varies strongly with bias at low temperature and becomes more I C dependent. This will couple the inaccuracy of I C modeling into I B and make the extraction more complex. Here, the same strategy used in I C modeling is used for "ideal" I B . Saturation current and ideality factor of I B itself will be used in the model, which will facilitate the modeling by avoiding unnecessary entanglement. At low temperature, the base current has a obvious increase in low bias range, which makes the curve deviate from linearity. The excess current is contributed by forward-bias tunneling, i.e., trap- assisted tunneling (TAT). We find that ideality factor extracted from I B ? V BE moderate region is exaggerated to some extent at low temperature because of tunneling current, while it only happens at 43KinI C ?V BE . Using a method of iteration, we can separate the main base current and tunneling current and quantify the e?ect of tunneling on the total current gain fallo? at low temperature. A new temperature scaling of tunneling saturation current equation is proposed by including ideality factor N E,TAT . 5.1 Trap-Assisted Tunneling E?ect It has already been found that tunneling generation is an important source of leakage in ad- vanced silicon device [25] [26] [27] [28]. The trend of down scaling and high doping in device technology leads to a strong electric field around the p-n junction, making the e?ect of tunneling significant [29]. 35 Figure 5.1: Illustration of trap-assisted tunneling in forward biased EB junction. Generally, when the E-B junction is reversed biased, band-to-band tunneling (BBT) dominates due to the overlap between valance band in the base region and conduction band in the emitter region [30]. When the E-B junction is forward biased, the base current increase is mainly due to TAT, which is the case of this work. Under the later circumstances, a defect with an energy state deep in the band gap, called a "trap", assists the tunneling process shown in Fig. 5.1. An electron located at x 1 can tunnel to a trap at x later recombines with a hole tunneling to x. The higher doping level is, the easier it is to observe this phenomenon [31]. For collector current the tunneling process is more complicated [3]. Fig. 5.2 (a) and (b) show the gummel measurement data. For our device, the TAT current in I C can be only observed at 43K while can be observed in I B below 110K. This is the reason for current gain fallo? at low temperature [32] [31], which will also be proved later. The tunneling current can be mainly observed at high doped E-B junction and hardly seen at low doped C-B junction. This is consistent with reverse gummel I E and I B shown in Fig. 5.2 (c) and (d). 36 0.6 0.8 1 1.2 10 ?10 10 ?5 V BE (V) I C (A) 0.6 0.8 1 1.2 10 ?10 10 ?5 V BE (V) I B (A) 0.6 0.8 1 1.2 10 ?10 10 ?5 I E (A) V BC (V) 0.6 0.8 1 1.2 10 ?10 10 ?5 I B (A) V BC (V) (a) (b) (c) (d) TAT TAT 300K 43K 300K 300K 300K 43K 43K 43K Figure 5.2: Forward and reverse gummel measurement data: (a) forward gummel I C -V BE , (b) forward gummel I B -V BE , (c) reverse gummel I E -V BC and (d) reverse gummel I B -V BC . 5.2 Separation of Main and Tunneling Base Current As the base tunneling current is not negligible below 110K, we definitely need to accurately estimate the magnitude of tunneling current and quantify its e?ect on the main base current. In other words, the tunneling current has to be taken o? from total base current, avoiding overvaluing main current. Below, we use a iteration method to separate the main and tunneling current: 1) Select a linear region on I B ?V BE and perform linear fitting. Extract slope and intercept of this part. 2) Subtract the linear fitting from total base current and get a nearly linear line at low V B E region. Linear fitting is again performed for this low V BE . 3) Subtract fitting result obtained by Step 2 from total I B . 4) Perform linear fitting on the result obtained by Step 3. Extract slope and intercept again. 37 0.95 1 1.05 1.1 1.15 1.2 10 ?10 10 ?9 10 ?8 10 ?7 10 ?6 10 ?5 10 ?4 10 ?3 V BE (V) I B (A) Measurement with iteration without iteration 43K Figure 5.3: Di?erence between linear fitting with and without iteration at 43 K. 5) Repeat the Step 2, 3 and 4 until the di?erence of slope extracted by two successive times is smaller than a set limit. Fig. 5.3 shows the di?erence between linear fittings of base current with and without iteration at 43 K. Without iteration means that main current is directly fitted only by Step 1 from the a certain region without excluding the tunneling current. The slope including tunneling current is smaller than that excluding tunneling current, increasing ideality factor from 1.290 to 1.522, a significant number. The saturation current is correspondingly downgraded. A zoom-in plot shows the with iteration the summation of two linear fitting can better fit the measurement. The same method is applied on I C ?V BE only at 43 K. 5.3 Ideality Factor, Saturation Current and Current Gain Using the method proposed above, we are able to accurately extract the ideality factors and saturation currents of main base and collector currents. Fig. 5.4 shows the extracted saturation 38 0 100 200 300 10 ?100 10 ?50 Temperature (K) I S,BE (A) 0 100 200 300 1 1.2 1.4 1.6 1.8 Temperature (K) N EI including TAT excluding TAT 0 100 200 300 10 ?100 10 ?50 Temperature (K) I S (A) 0 100 200 300 1 1.1 1.2 1.3 1.4 Temperature (K) N F Figure 5.4: Comparison of ideality factor and saturation current between including and excluding TAT current: (a) I S,BE -T, (b)N EI -T, (c) I S -T, (d)N F -T. currents and ideality factors. "Excluding TAT" means the results are obtained from the ideal base current, i.e., subtracting tunneling current from total base current. It is clear that excluding the TAT current will result in smaller ideality factor and larger saturation current especially for low temperature. Comparing the base saturation current with collector saturation current, we find that the I BEI has weaker temperature dependence than I S as shown in Fig. 5.5. The slopes of I C ? V BE and I B ? V BE extracted from Hydrodynamic simulation, measurement are overlaid in Fig. 5.6. The slope of simulated I C ? V BE is smaller than ideal 1/V T at temperature below 100 K, while that of I B ? V BE is almost same as ideal 1/V T . The di?erence is mainly caused by early e?ect and Ge-ramp e?ect. The slopes of both I C ?V BE and I B ?V BE extracted from measurement excluding tunneling current are larger than those of including tunneling current, but still smaller than those of simulation. This verifies that the ideality factor larger than 1 at low temperature [9] is substantial and at least not due to tunneling current. The freezeout model has not been applied in simulation, so freezeout e?ect may be responsible for that. However the freezeout should not a?ect highly doped 39 0 100 200 300 10 ?100 10 ?80 10 ?60 10 ?40 10 ?20 10 0 Temperature (K) I S /I S,300K and I S,BE /I S,BE,300K 0 100 200 300 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 Temperature (K) ideality factor N F N EI I S I S,BE (a) (b) Figure 5.5: Comparison of ideality factor and saturation current between collector and base: (a) saturation current, (b)ideality factor. 40 0 100 200 300 0 50 100 150 200 250 300 Temperature (K) 0 100 200 300 0 50 100 150 200 250 300 Temperature (K) ideal 1/V T HD simulation Excluding TAT Including TAT 1/I C *dI C /dV BE (1/V) 1/I B *dI B /dV BE (1/V) Figure 5.6: The slopes of (a) I C ? V BE and (b) extracted from simulation and measurement from 43 to 300 K. emitter much and base ideality factor is also larger than 1, therefore the physics underneath needs further exploration. Next we exam the e?ect of tunneling on current gain. Fig. 5.7 (a) is obtained by directly dividing measured I C by I B . Fig. 5.7 (b) only by dividing main I C by main I B , which means the TAT e?ect has been excluded. The current gain now does not fall o? but increases with cooling. 5.4 Tunneling Base Current Modeling We have separated tunneling current from main current. We will focus at 43, 60, 76, 93 and 110 K to develop the tunneling current model. The trap-assisted tunneling e?ect has been described by an expression that for weak electric fields reduces to the conventional Shockley-Read-Hall expression for recombination via traps and the model has one extra physical parameter, the e?ective mass m ? [29]. This method, however, is more preferably used in device simulations. 41 10 ?6 10 ?4 10 ?2 0 100 200 300 400 500 I C (A) ? 10 ?6 10 ?4 10 ?2 0 50 100 150 200 250 I C (A) ? 0 50 100 150 200 250 300 100 200 300 Temperature (K) ? 43K 60K 76K 93K 110K 136K 162K 192K 223K 262K 300K Total I C / I B Excluding TAT (a) (b) (c) Figure 5.7: Comparison between current gain obtained by (a) total I C /I B and (b)excluding tunnel- ing current and current gain obtained at I C = 10 ?5 A (c) by (a) and (b). Fig. 5.8 (a) shows the tunneling current at these five temperatures. In the semilog scale, the current is not strictly linear line. However, we have found that two linear fittings in semilog scale is su?cient to model the total base current before high injection. So an exponential I ?V relation is su?cient and we need to carefully select the fitting range to get the ideality factor N E,TAT and saturation current I S,BE,TAT . Fig. 5.8 (b) shows N E,TAT at five temperatures and it is proportional to 1/T. In most of the widely used compact models, such as Spice Gummel Poon, VBIC and Mex- tram, the non-ideal base current, i.e., base current at low bias, is modeled by Shockley-Read-Hall recombination [33] in an expression like I BEN = I S,BEN exp parenleftbigg V BE N EN V T parenrightbigg , (5.1) in which N EN is constant equal to 2. 42 0.005 0.01 0.015 0.02 0.025 1 2 3 4 5 1/Temperature (1/K) N E,TAT 40 60 80 100 120 0.016 0.017 0.018 0.019 0.02 Temperature (K) S (V) 0.85 0.9 0.95 1 10 ?11 10 ?10 10 ?9 V BE (V) I B,TAT (A) (a) (b) (c) 110K 43K Figure 5.8: (a) Base tunneling current at 43, 60 76, 93 and 110 K; (b) The extracted ideality factor is proportional to 1/T; (C) S = N E,TAT ?V T . 43 In our case, the N E,TAT increases from 1.9 at 110 K to 4.8 at 43 K, indicating that TAT current is dominant in low bias region at very low temperature. Besides, we found that S = N E,TAT ?V T is independent of temperature. So here I BE,TAT = I S,BE,TAT exp parenleftbigg V BE S parenrightbigg , (5.2) which is similar to [25], is added in addition to (5.1). Temperature dependence of tunneling saturation current I S,BE,TAT in [25] comes from built-in potential ? bi . As E-B junction doping is very high in our device, then q? bi = Eg. So firstly, we model the temperature dependence of I BE,TAT,T as I S,BE,TAT,T = I S,BE,TAT,0 exp parenleftbig ?k 1 E g,T parenrightbig . (5.3) Both linear E g,T = E g,0 ??T (5.4) and nonlinear E g,T = E g,0 ? ?T 2 ? +T (5.5) are used and model results compared with extraction from measurement are shown in Fig. 5.9. The more accurate non-linear bandgap T scaling gives worse results than linear bandgap T scaling. so we propose to include N E,TAT which has a negative temperature coe?cient into the modeling and get I S,BE,TAT,T = I S,BE,TAT,0 exp parenleftbigg ?k 1 E g,T N E,TAT,T parenrightbigg . (5.6) Making it related to the reference temperature, I S,BE,TAT,T = I S,BE,TAT exp parenleftbigg k 1 E g,ref N E,TAT ? k 1 E g,T N E,TAT,T parenrightbigg . (5.7) 44 I S,BE,TAT ,E g andN E,TAT are therefore the tunneling saturation current, bandgap and ideality factor at referenced temperature, which is normally 300 K. The modeling result is also shown in Fig. 5.9 and gives a equally good fitting using only linear E g,T . Replacing N E,TAT with S/V T can reduce the number of parameters. So I S,BE,TAT,T = I S,BE,TAT exp parenleftbigg k 1 S parenleftbigg E g,ref V T,ref ? E g,T V T parenrightbiggparenrightbigg , (5.8) and k 1 /S can be replaced by only one parameter, and I S,BE,TAT is tunneling saturation current at reference temperature. Hence, I S,BE,TAT,T = I S,BE,TAT exp parenleftbigg k 1 parenleftbigg E g,ref V T,ref ? E g,T V T parenrightbiggparenrightbigg . (5.9) The modeling results for 43 to 110 K are shown in Fig. 5.10 (b). Good fitting results of tunneling current are obtained. 5.5 Moderate Bias Region Similar to I C , the equation (2.38) and (2.39) of I B 1 , the "ideal" base current, are now adjusted to I B 1 = parenleftbig 1 ?XI B 1 parenrightbig I BEI,T parenleftbigg exp parenleftbigg V B 2 E 1 N EI,T V T parenrightbigg ? 1 parenrightbigg (5.10) and I S B 1 = XI B 1 I BEI,T parenleftbigg exp parenleftbigg V B 1 E 1 N EI,T V T parenrightbigg ? 1 parenrightbigg . (5.11) I BEI,T andN EI,T are temperature scalable forward base saturation current and forward base ideality factor. Good fitting results combined with tunneling current results against measurement are shown in Fig. 5.10 (a) for 43 to 100 K. The same has been implemented in I EX to model reverse base current. 45 40 50 60 70 80 90 100 110 10 ?34 10 ?33 10 ?32 Temperature (K) I S,BE,TAT (A) Measurement Extraction Linear E gT Nonlinear E gT Nonlinear E gT with N tn Figure 5.9: The temperature scaling of tunneling saturation current at 43, 60, 76, 93 and 110 K. 46 0.9 1 1.1 10 ?10 10 ?9 10 ?8 10 ?7 10 ?6 10 ?5 V BE (V) I B (A) 0.85 0.9 0.95 1 10 ?11 10 ?10 10 ?9 V BE (V) I B,TAT (A) measurement modeling 110K 43K 110K 43K Figure 5.10: Comparison between measurement an modeling results for 43 to 110 K: (a) combined with ideal base fitting, (b) TAT current modeling . 47 0.9 0.95 1 1.05 1.1 1.15 10 ?5 10 0 I B (mA) V BE (V) 0.2 0.4 0.6 0.8 1 10 ?5 10 0 I B (mA) Measured data Simulated data 393K 136K 93K 43K T = 43K, 60K, 76K, 93K Figure 5.11: Measured and modeled I B -V BE from 43-93 K. 5.6 Summary Fig. 5.11 show the I B ?V BE results from 43-393 K. New I B model for moderate and low bias region work together to give a good simulation result over the whole temperature range. 48 CHAPTER 6 SMALL SIGNAL MODELING Small signal modeling is an important tool used for parameter extraction and linear RF circuit design. As SiGe HBTs are widely used in RF circuit, here the small signal equivalent circuit will be used to discuss the device RF performance. Most of the work has been published in [8]. 6.1 Equivalent Circuit Fig. 6.1 and Fig. 6.2 shows the small-signal equivalent circuit used in this work. This is a typical equivalent circuit for SiGe HBTs, with a topology similar to the large-signal equivalent circuit in compact models such as VBIC and Mextram. An exception is the addition of the C cso capacitance, which in addition to C cs and R s is necessary to fit the imaginary part of Y 22 . This capacitance was proposed in [34], but was attributed to the overlapping of the emitter and collector interconnection metals. We believe, however, that this C cso is physically the peripheral deep trench coupling capacitance between the N + buried layer and p-substrate. Note that such a capacitance has not been used in other investigations (e.g., [35]). 49 B E C C C C C R R + V - C r S g V R R C C R r Figure 6.1: Small-signal equivalent circuit used for SiGe HBTs. 6.2 Parameter Extraction Fig. 6.3 shows the measured f T -I C as a function of V CB at 300, 223, 162 and 93 K. With cooling, peak f T increases, and the f T roll-o? current increases as well. To make the parameter extracted at each temperature comparable, similar I C points are chosen for each T as shown in Fig. 6.4. The higher V BE range is chosen to cover the rise and fall portions of the f T -I C curves. At each temperature, hot S-parameter measurements were made on-wafer from 1 to 35 GHz by sweeping V BE for V CB = ?0.5, 0, 1, and 2 V. 6.2.1 Procedure There have been several direct extraction methods reported for SiGe HBTs [34] [35]. However, these methods do not yield good results when applied to the present data. Here a combination of direct extraction and a two-step optimization procedure is chosen, as detailed below. 50 B E C S C +C C +C C C R R Figure 6.2: Small-signal equivalent circuit used for cold state. 10 ?2 10 0 0 20 40 60 I C (mA) f T (GHz) 10 ?2 10 0 0 20 40 60 I C (mA) f T (GHz) 10 ?2 10 0 0 20 40 60 I C (mA) f T (GHz) 10 ?2 10 0 0 20 40 60 I C (mA) f T (GHz) V CB = ?0.50V V CB = 0.00V V CB = 1.00V V CB = 2.00V 300K 223K 93K 162K Figure 6.3: Measured f T -I C as a function of V CB at various temperatures. 51 10 ?2 10 ?1 10 0 10 1 0 10 20 30 40 50 60 70 I C (mA) f T (GHz) 93K 162K 223K 300K Symbol: Seleted I C points for parameter extraction Figure 6.4: Selected I C points for each temperature extraction . A common first step in [34] [35] is extracting C beo and C bco from cold Y-parameter data. The total C be is obtained from imaginary part of Y 11 +Y 12 for each V BE , and then fitted into C be =C beo + C je (1+V be /V de ) ?M je . Our experience shows that this approach is very unreliable, as many solutions exist for C beo , C je , V de and M je . Furthermore, it is found that in order to fit the imaginary part of Y 11 above 20 GHz, C beo must be allowed to increase with I c . The exact physical origin of this increase of C beo with I c is not yet well understood. It could, for instance, be a manifestation of sidewall injection which is not explicitly accounted for in this equivalent circuit. The most reliable way to extract C beo , is to use optimization of the imaginary part of Y 11 for hot data (Y-parameters measured at higher V BE when device is turned-on), as it contributes to Ifractur(Y 11 ). During optimization, the value of C bco constantly approaches zero, at which we fixed C bco accordingly. One main disadvantage of optimization is the multiple and sometimes unphysical solutions that result. When optimization is directly used on hot Y-parameters without constraints, the resulting C bci tends to be unrealistically small, and the V cb and I c dependence of C bci , C bcx , C cs and R s are 52 not physical. This problem is overcome by fitting the cold Y-parameters first to determine C cso , C cs , and R s first. These parameters are then fixed for the required V CS . Two sets of cold measurements were taken, one V CB sweep at V BE =0 V, and one V BE sweep at V CB =0 V. In the cold state, it is not possible to distinguish C bci from C bcx , or to distinguish C beo from C ? , and thus a simplified circuit, as shown in Fig. 6.2, is used. C cso is found to be independent of V BE or V SC . C cso is3fFat300K, and weakly temperature dependent. We believe C cso is the peripheral coupling capacitance between buried layer and p-substrate through the deep trench oxide. C cs decreases slightly with V CS ,as expected. Next, R e is extracted from a standard R e flyback measurement. R e is very di?cult to uniquely determine from optimization. The R e from optimization is much larger than the R e from flyback, and can be bias dependent, which is unphysical. Another parameter that is equally di?cult to uniquely determine from optimization is R bx . Here it is determined from the high I B limit of an overdriven measurement. R e and R bx are fixed over bias, and found to be temperature independent as well. R e =6 ? and R bx =25 ? for the device are used. All other parameters are then determined by fitting the hot Y-parameter over frequency data. This process is then repeated for each bias step. 6.2.2 Result Fig. 6.5, 6.6, 6.7, 6.8 show the Y-parameter modeling results at 300, 233, 162 and 93 K. V CB =0V was used here. Equally good fitting is achieved at other V BE ?s, except at high injection biases well after the f T roll-o?, where the equivalent circuit begins to fail. The real part of Y 11 above 30 GHz at higher I C is not yet well-fitted with the present circuit. The attempt to include distributive e?ects (by adding a parallel capacitance toR bi ) and input non-quasi-static (NQS) e?ects (by adding a delay resistance to the di?usion component of C ? ) did not further improve Y 11 fitting. The measured imaginary part of Y 11 shows a drop above 25 GHz, which cannot possibly be fitted with the equivalent circuit. As this occurs only at 93 K, The conclusion is likely due to measurement error, but this is still being explored. In general, both measurements and modeling become more di?cult at 93 K. 53 0 5 x 10 ?3 ? (Y 11 ) 0 2 4 x 10 ?3 ? (Y 11 ) 0 0.05 ? (Y 21 ) ?0.02 ?0.01 0 ? (Y 21 ) ?4 ?2 0 x 10 ?4 ? (Y 12 ) ?1 ?0.5 0 x 10 ?3 ? (Y 12 ) 0 10 20 30 40 0 1 2 x 10 ?3 ? (Y 22 ) frequency (GHz) 0 10 20 30 40 0 1 2 3 x 10 ?3 ? (Y 22 ) frequency (GHz) Measurement Simulation I C = 0.5210mA I C = 1.128mA I C = 1.409mA I C = 2.903mA Figure 6.5: Measured and simulated Y-parameters at 300 K. 0 5 x 10 ?3 ? (Y 11 ) 0 2 4 x 10 ?3 ? (Y 11 ) 0 0.05 ? (Y 21 ) ?0.03 ?0.02 ?0.01 ? (Y 21 ) ?2 ?1 0 x 10 ?4 ? (Y 12 ) ?1 ?0.5 0 x 10 ?3 ? (Y 12 ) 0 10 20 30 40 0 0.5 1 1.5 x 10 ?3 ? (Y 22 ) frequency (GHz) 0 10 20 30 40 0 1 2 x 10 ?3 ? (Y 22 ) frequency (GHz) Measurement Simulation I C = 0.5538mA I C = 1.017mA I C = 1.658mA I C = 2.902mA Figure 6.6: Measured and simulated Y-parameters at 223 K. 54 0 5 x 10 ?3 ? (Y 11 ) 0 5 x 10 ?3 ? (Y 11 ) 0 0.05 ? (Y 21 ) ?0.03 ?0.02 ?0.01 ? (Y 21 ) ?2 ?1 0 x 10 ?4 ? (Y 12 ) ?1 ?0.5 0 x 10 ?3 ? (Y 12 ) 0 10 20 30 40 0 1 2 x 10 ?3 ? (Y 22 ) frequency (GHz) 0 10 20 30 40 0 1 2 x 10 ?3 ? (Y 22 ) frequency (GHz) Measurement Simulation I C = 0.7226mA I C = 1.002mA I C = 1.703mA I C = 3.032mA Figure 6.7: Measured and simulated Y-parameters at 162 K. 0 5 x 10 ?3 ? (Y 11 ) 0 2 4 x 10 ?3 ? (Y 11 ) 0 0.05 ? (Y 21 ) ?0.03 ?0.02 ?0.01 ? (Y 21 ) ?1 0 1 x 10 ?3 ? (Y 12 ) ?1.5 ?1 ?0.5 0 x 10 ?3 ? (Y 12 ) 0 10 20 30 40 0 1 2 x 10 ?3 ? (Y 22 ) frequency (GHz) 0 10 20 30 40 0 2 4 x 10 ?3 ? (Y 22 ) frequency (GHz) Measurement Simulation I C = 0.4330mA I C = 1.099mA I C = 1.924mA I C = 2.994mA Figure 6.8: Measured and simulated Y-parameters at 93 K. 55 0 0.5 1 1.5 2 2.5 0 0.02 0.04 0.06 0.08 0.1 I C (mA) g m (S) 0 2 4 0 0.1 0.2 0.3 0.4 I C (mA) g m (S) 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 I C (mA) g m (S) 0 2 4 6 0 0.05 0.1 0.15 0.2 I C (mA) g m (S) extracted g m I C /V T including selfheating I C /V T0 V CB =?0.50V V CB =0.00V V CB =1.00V V CB =2.00V 300K 223K 93K 162K Figure 6.9: Extracted g m -I C results compared with ideal I C /V T with and without self-heating at 300, 223, 162 and 93 K. Fig. 6.9 shows extracted g m -I C as a function of V CB . The I C /V T both with and without self- heating are also shown as base-line references. R th is extracted at each temperature using the method in [36]. A significant deviation of g m from ideal I C /V T is observed, and the deviation increases with I C . Note that the degradation of g m compared to I C /V T is significant even before peak f T . Interestingly, for di?erent V CB , as long as I C is the same, g m is approximately same. This means despite the complex impact of V CB on I C at high injection, the device intrinsic transconductance remains the same at a given I C . This has direct implications on high frequency analog circuit biasing, as V CB does not a?ect g m as long as the biasing current can be fixed (e.g. with a current source). Fig. 6.10 shows the extraction results ofC ? -I C . Again, littleV CB dependence is observed, even for V CB =-0.5 V, which shows an early turn-on of Kirk e?ect. This early turn-on of Kirk e?ect at V CB =-0.5 V is manifested through a rapid rise of the intrinsic CB capacitance C bci , as shown below. Although g m -I C shows deviations from linearity, C ? -I C is to a large extent linear, even past peak f T . This leads to a rapid rise of the di?usion transit time with increasing I C . 56 0 1 2 3 0 50 100 150 I C (mA) C ? (fF) 0 2 4 6 0 50 100 150 200 250 I C (mA) C ? (fF) 0 2 4 6 0 50 100 150 200 250 I C (mA) C ? (fF) 0 2 4 6 0 100 200 300 I C (mA) C ? (fF) V CB =?0.5V V CB =0.00V V CB =1.00V V CB =2.00V 300K 223K 93K 162K Figure 6.10: Extracted C ? -I C at 300, 223, 162 and 93 K. Fig. 6.11 shows the extracted C bci -I C and C bcx -I C results. Both C bci and C bcx decrease with increasing V CB . C bci increases with I C , as expected, which is stronger at lower V CB , while C bcx is independent of I C . This is physically meaningful since Kirk e?ect primarily occurs at the intrinsic selectively implanted collector (SIC) region. Kirk e?ect is the worst at V CB =-0.50 V, causing the rapid increase ofC bci withI C atV CB =-0.50 V. Observe that the increase ofC bci withI C corresponds to f T rolling o?. At low injection, both C bci and C bcx show a weak temperature dependence. The total C bc (C bci + C bcx )vsV BE are shown in Fig. 6.12. The extraction results from cold data fitting (only V CB =0 V is taken) are combined with results from hot data fitting. Observe that the cold and hot extraction results are very consistent. The temperature dependence of R bi and R s are shown in Fig. 6.13. The bias dependence of R s is very weak, as expected. R bi shows the usual bias dependence, and here we use the R bi extracted at I C =1 mA. With cooling, R s decreases from 3313 ? at 300 K to 335.3 ? at 93 K, while R bi increases from 239.6 ? at 300 K to 563.3 ? at 93 K. This di?erence is caused by di?erent doping levels in the p-substrate and p-base. The higher base doping level leads to much stronger impurity 57 0 1 2 3 4 5 0 5 I C (mA) C bci (fF) 0 1 2 3 4 5 0 5 10 I C (mA) C bcx (fF) 0 1 2 3 4 5 0 5 C bci (fF) 0 1 2 3 4 5 0 5 10 I C (mA) C bcx (fF) 0 5 C bci (fF) 0 1 2 3 4 5 0 5 10 I C (mA) C bcx (fF) 0 1 2 3 4 5 0 5 I C (mA) C bci (fF) 0 1 2 3 4 5 0 5 10 I C (mA) C bcx (fF) V CB =?0.50V V CB =0.00V V CB =1.00V V CB =2.00V 300K 223K 162K 93K Figure 6.11: Extracted C bci -I C and C bcx -I C at 300, 223, 162 and 93 K. 0 0.5 1 0 5 10 15 20 V BE (V) C bci +C bcx (fF) 0 0.5 1 1.5 0 10 20 30 V BE (V) C bci +C bcx (fF) 0 0.5 1 1.5 0 5 10 15 V BE (V) C bci +C bcx (fF) 0 0.5 1 1.5 5 10 15 20 V BE (V) C bci +C bcx (fF) V CB =?0.50V V CB =0.00V V CB =1.00V V CB =2.00V "cold" extraction VCB = 0.00V 300K 162K 223K 93K Figure 6.12: Extracted total C BC (C bci +C bcx )vsV BE from cold and hot extraction. 58 50 100 150 200 250 300 350 200 300 400 500 600 T (K) R bi ( ? ) 0 1000 2000 3000 R s ( ? ) Figure 6.13: Extracted R bi at I C = 1mAandR s vs T. scattering at lower temperature in the base and hence the mobility decrease and resistance increase. Carrier freezeout could also be responsible, as it is worse for higher doping as well. The increase of R bi with cooling can degrade noise figure; this, however, is o?set by the decrease of thermal voltage and increase of f T and current gain. The significant decrease of R s will compromise the low RF loss advantage of a high resistivity (300 K) substrate, particularly for inductors and transmission lines. This increased substrate loss with cooling should be taken into consideration in cryogenic RF circuit design. 6.3 Substrate Network Implementation The experience with small signal modeling showed that the substrate network is important for modeling the device high frequency behavior. So a substrate network is extended to Mextram Verilog-A code. To include distributive characteristics of low doping substrate, a substrate network is added between branches (C 1 ,S) and (C,S). Fig. 6.14 shows the substrate small signal equiva- lent circuit used in [37]. R sub and C sub are used to model distributive characteristics of substrate. 59 C1 C S C C R R C Figure 6.14: Small-signal equivalent circuit for substrate network. C CS is the CS junction capacitance. C DT is physically the peripheral deep trench coupling capac- itance between the N + buried layer and p-substrate equal to C CSO in the small signal topology. In isothermal fitting of Y 22 +Y 12 from cold measurement, C SUB and C DT show very weak temperature dependence. Hence, C SUB and C DT can be fixed at the values extracted from reference temperature. The extracted R SUB is consistent with measured substrate resistivity as shown in Fig. 6.15. Though it is not very consistent with the extraction from small signal, they share a similar temperature dependence. 60 50 100 150 200 250 300 350 400 0 500 1000 1500 2000 2500 Temperature (K) R Sub ( ? ) Extraction from Y 22 + Y 12 Measurement Figure 6.15: Small-signal equivalent circuit for substrate network. 61 CHAPTER 7 CONCLUSION At low temperatures, the deviation of I C ?V BE slope from ideal 1/V T is much larger than what can be modeled with q B . Furthermore, below 100K, the T-dependence of I C ?V BE and I B ?V BE becomes increasingly weaker than predicted by Shockley theory. A N F factor that increases with cooling has been used to model this deviation, such that the slope 1/(N F V T ) does not increase as much as the ideal 1/V T . Based on a similar consideration, the T-dependence of N F is included in the T-dependence model of I S . The same strategy is both used for I C and I B , so current gain ? f is no longer necessary. In this work, only the forward mode of operation has been discussed. Similar modification, however, has been made to model the reverse mode. The temperature dependence of V BE is also examined. The triangleV BE ? T is not linear at low temperature. This should be taken into account when designing bandgap reference. I B EN is added between B 2 and E 1 to account for forward bias trap-assisted tunneling current, which becomes important below 100K. Substrate network is included as part of the model, as shown in Fig. 7.1. The modifications and extensions based on Mextram is primarily to increase its appli- cable temperature range. Most of the modifications and extensions can be directly applied to other compact models as well. Verilog-A is used for model implementation, and IC-CAP is used for parameter extraction. 62 Figure 7.1: Modified equivalent circuit. 63 BIBLIOGRAPHY [1] J. D. 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Berroth, ?An accurate method to determine the substrate network elements and base resistance,? in Proc. of IEEE BCTM, pp. 93?96, 2003. 66 APPENDICES 67 APPENDIX A VERILOG-A CODE IMPLEMENTATION WITH KEY IMPROVED MODELS A.1 Improved Ideality Factor Temperature Mapping NF T = NF*(1.0+(1-tN)*pow(ANF/tN,XNF)); NR T = NR*(1.0+(1-tN)*pow(ANR/tN,XNR)); NEI T = NEI*(1.0+(1-tN)*pow(ANE/tN,XNE)); NCI T = NCI*(1.0+(1-tN)*pow(ANC/tN,XNC)); 68 A.2 Improved Saturation Current Temperature Mapping tvgb = ?EAA*?EAB*tN*pow(Trk,2)/((?EAB+Trk)*(?EAB+Tk)) -?EAA*?EAB*pow(Trk,2)/pow((Trk+?EAB),2); VGB T = VGBR+tvgb; IS T = IS*pow((pow(tN,XTI)*exp(-VGB T *(1.0-tN)/Vt)),(1.0/NF T )); ISR T = IS*pow((pow(tN,XTI)*exp(-VGB T *(1.0-tN)/Vt)),(1.0/NR T )); VGBE T = VGBE+tvgb; IBEI T = IBEI*pow((pow(tN,XTE)*exp(-VGBE T *(1.0-tN)/Vt)),(1.0/NEI T )); VGBC T = VGBC+tvgb; IBCI T = IBCI*pow((pow(tN,XTC)*exp(-VGBC T *(1.0-tN)/Vt)),(1.0/NCI T )); ISBETAT T = ISBETAT*exp(K1*(-VGB T *(1.0-tN)/Vt)); 69 A.3 Modified Trasfer Current Model If0=4.0*IS TM /IK TM ; f1 = If0 * exp (( Vb2e1*VtINV) *(1/ NF T )); n0 = f1 / (1.0 + sqrt(1.0 + f1)); f2 = 4.0 * ISR TM / IKR TM * exp (ln(eVb2c2star) *(1/ NR T )); Ir = ISR TM * exp (ln(eVb2c2star) *(1/ NR T )); If=IS TM * exp (( Vb2e1*VtINV) *(1/ NF T )); In =( If-Ir) /qBI ; 70 A.4 Modified Base Current Model if (XREC == 0.0) Ib1 = (1.0 - XIBI) * IBEI T * (exp (( Vb2e1*VtINV) *(1/ NEI T )) - 1.0); else Ib1 = (1.0 - XIBI) * (1.0 - XREC) * IBEI T * (exp (( Vb2e1*VtINV) *(1/ NEI T )) - 1.0) + (1.0 - XIBI) * (1.0 + Vtc / VEF T )* XREC * (IBEI T * (exp (( Vb2e1*VtINV) *(1/ NEI T )) - 1.0) + IBEI T * (exp ((ln(eVb2c2star)) *(1/ NCI T )) - 1.0)); Ib1s = XIBI * IBEI T * (exp (( Vb1e1*VtINV) *(1/ NEI T )) - 1.0); ?expLin(tmpExp,Vb2e1 * VtINV / MLF) Ib2 = IBF TM * (tmpExp - 1.0) + GMIN * Vb2e1; Ibetat = ISBETAT TM *exp(Vb2e1/STN); ?expLin(tmpExp,0.5 * Vb1c4 * VtINV) Ib3 = IBR TM * (eVb1c4 - 1.0) / (tmpExp + exp(0.5 * VLR * VtINV)) + GMIN * Vb1c4; g1 = If0 * eVb1c4; g1 = 4.0 * ISR TM / IKEX TM * exp (( Vb1c4*VtINV) *(1/ NCI T )); g2 = 4.0 * eVb1c4VDC; nBex = g1 / (1.0 + sqrt(1.0 + g1)); pWex = g2 / (1.0 + sqrt(1.0 + g2)); Iex = IBCI T * (2*exp (( Vb1c4*VtINV) *(1/ NCI T )) / (1.0 + sqrt(1.0 + g1)) - 1.0); 71