IMPROVED RF NOISE MODELING FOR SILICON-GERMANIUM HETEROJUNCTION BIPOLAR TRANSISTORS Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. Kejun Xia Certificate of Approval: Richard C. Jaeger Distinguished University Professor Electrical and Computer Engineering Guofu Niu, Chair Professor Electrical and Computer Engineering Fa Foster Dai Associate Professor Electrical and Computer Engineering Stuart Wentworth Associate Professor Electrical and Computer Engineering Joe F. Pittman Interim Dean Graduate School IMPROVED RF NOISE MODELING FOR SILICON-GERMANIUM HETEROJUNCTION BIPOLAR TRANSISTORS Kejun Xia 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 15, 2006 IMPROVED RF NOISE MODELING FOR SILICON-GERMANIUM HETEROJUNCTION BIPOLAR TRANSISTORS Kejun Xia Permission is granted to Auburn University to make copies of this dissertation at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date of Graduation iii VITA Kejun Xia, elder son of Shengbin Xia and Meiqin Zhu, was born in Shanxi Town, Wenzhou City, Zhejiang Province, China on June 25, 1978. Upon graduation from Wenzhou No. 1 High School in 1996, he entered Tianjin University, Tianjin, China, from which he received the B.E. and M.S. degrees in electronics in 2000 and 2003, respectively. He entered Electrical and Computer Engineering Department graduate program of Auburn University in 2003 to work towards the Ph.D. degree. iv DISSERTATION ABSTRACT IMPROVED RF NOISE MODELING FOR SILICON-GERMANIUM HETEROJUNCTION BIPOLAR TRANSISTORS Kejun Xia Doctor of Philosophy, December 15, 2006 (M.S., Tianjin University, 2003) (B.E., Tianjin University, 2000) 232 Typed Pages Directed by Guofu Niu Accurate radio frequency (RF) noise models for individual transistors are critical to min- imize noise during mixed-signal analog and RF circuit design. This dissertation proposes two improved RF noise models for SiGe Heterojunction Bipolar Transistors (SiGe HBTs), a semi- empirical model and a physical model. A new parameter extraction method for small signal equiv- alent circuit of SiGe HBT has also been developed. The semi-empirical model extracts intrinsic base and collector current noise from measured device noise parameters using standard noise de-embedding method based on a quasi-static input equivalent circuit. Equations are then developed to model these noise sources by examining the frequency and bias dependences. The model is shown to work at frequencies up to at least half of the peak unit-gain cuto? frequency (f T ), and at biasing currents below high injection f T roll o?. The model is scalable for emitter geometry, and can be easily implemented using currently available CAD tools. For the physical model, improved electron and hole noise models are developed. The impact of the collector-base space charge region (CB SCR) on electron RF noise is examined to determine v its importance for scaled SiGe HBTs. The van Vliet model is then improved to take into account the CB SCR e?ect. The fringe EB junction e?ect is included to improve base hole noise. The base noise resistance is found to be di?erent from the AC intrinsic base resistance, which cannot be explained by the fringe e?ect. Applying a total of four bias-independent model parameters, the combination of new electron and hole noise models based on a non-quasistatic input equivalent circuit provides excellent noise parameter fittings for frequencies up to 26 GHz and all biases before f T roll o?for three generations of SiGe HBTs. The model also has a good emitter geometry scaling ability. The new small signal parameter extraction method developed here is based on a Taylor ex- pansion analysis of transistor Y-parameters. This method is capable of extracting both input non- quasistatic e?ect and output non-quasistatic e?ect, which are not available for any of the existing extraction methods. vi ACKNOWLEDGMENTS The author would like to acknowledge his parents and his wife for their unconditional love and continuous support. He would like to thank his advisor Prof. Guofu Niu for the precious op- portunity to conduct the interesting RF noise research and for the wide-scope instruction on semi- conductor devices. He also thanks his advisory committee, Dr. Richard C. Jaeger, Dr. Fa Foster Dai, and Dr. Stuart Wentworth for their interests and comments. He is grateful to the outside reader Dr. An-Ban Chen for the discussions on Green?s functions. He would like to acknowledge financial support from SRC under grant #2003-NJ-1133 and IBM under an IBM Faculty Partnership Award. The author would like to thank the semiconductor device physics / TCAD research group members, especially Yan Cui and Yun Shi, for their help in microscopic device simulation using software DESSIS. In addition, he is grateful to J. Cressler, D. Sheridan, S. Sweeney and the IBM SiGe team for their data support and contributions. vii Style manual or journal used IEEE Transactions on Electron Devices (together with the style known as ?auphd?). 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 auphd.sty. The plots were generated using Microsoft Viso R? and MATLAB R? . viii TABLE OF CONTENTS LIST OF FIGURES xiii LIST OF TABLES xix 1 INTRODUCTION 1 1.1 Motivation ....................................... 1 1.2 SiGe HBT fundamentals ............................... 2 1.2.1 SiGe as base material ............................. 2 1.2.2 Performance parameters ........................... 5 1.2.3 Improving f T and f max ............................ 6 1.2.4 SiGe BiCOMS technology .......................... 8 1.3 Noise parameters for two-port network ........................ 9 1.4 Frequency and bias dependence of noise parameters for SiGe HBTs ........ 12 1.5 Noise performance trends for SiGe HBTs ...................... 14 1.6 Noise modeling considerations and methodology for SiGe HBTs .......... 15 1.7 Summary ....................................... 16 2 RF NOISE THEORY FOR SIGE HBTS 17 2.1 RF noise sources ................................... 17 2.1.1 Velocity fluctuation noise ........................... 21 2.1.2 GR noise ................................... 29 2.2 Electron noise of base region without distributive e?ect............... 30 2.2.1 1-D solution ................................. 30 2.2.2 General 3-D solution by van Vliet ...................... 45 2.2.3 Evaluation of van Vliet solution for finite exit velocity boundary condition 46 2.3 Extension to including emitter hole noise ...................... 47 2.3.1 3-D van Vliet model ............................. 47 2.3.2 1-D solution ................................. 50 2.3.3 Evaluation of finite surface recombination velocity e?ect.......... 51 2.3.4 Comparison of base electron and emitter hole contributions to S ib ..... 52 2.4 Compact noise model including distributive e?ect.................. 53 2.4.1 Compact noise model assuming uniform f T across EB junction ...... 53 2.4.2 NQS and QS base resistance ......................... 56 2.5 Present noise models and implementation problems ................. 59 2.5.1 SPICE model ................................. 59 2.5.2 Transport noise model ............................ 59 2.5.3 Brutal use of van Vliet model ........................ 60 ix 2.5.4 4kTr bi for r bi noise .............................. 61 2.6 Methodologies to improve noise modeling ...................... 61 2.7 Summary ....................................... 62 3 SMALL SIGNAL PARAMETER EXTRACTION 64 3.1 Necessity of including input NQS e?ect in equivalent circuit ............ 65 3.2 NQS Equivalent circuit ................................ 66 3.2.1 CB SCR e?ect ................................ 67 3.2.2 NQS equivalent circuit ............................ 69 3.3 Parameter extraction ................................. 71 3.3.1 C cs , r cs , ? and r e extraction .......................... 73 3.3.2 C bct , C bcx , C bci , r c and g m extraction ..................... 74 3.3.3 r bx and r bi extraction ............................. 75 3.3.4 C bex and C bej extraction ........................... 77 3.3.5 C bed , r d , g be and ? d extraction ........................ 79 3.4 Results and discussions ................................ 81 3.4.1 Extraction and modeling results ....................... 81 3.4.2 Discussions .................................. 81 3.5 Extraction of bias dependent r bi ........................... 86 3.6 Summary ....................................... 89 4 SEMI-EMPIRICAL NOISE MODEL BASED ON EXTRACTION 91 4.1 Intrinsic noise extraction ............................... 92 4.1.1 Two basic noise de-embedding techniques .................. 92 4.1.2 SiGe HBT noise calculation ......................... 94 4.1.3 Extracted intrinsic noise ........................... 96 4.2 Semi-empirical intrinsic noise model ......................... 99 4.2.1 S ib ...................................... 99 4.2.2 S ic .......................................102 4.2.3 Imaginary part of S icib ? [Ifractur(S icib ?)]......................103 4.2.4 Real Part of S icib ? [Rfractur(S icib ?)] ........................106 4.2.5 Generalized Model Equations ........................108 4.2.6 Noise Parameter Modeling Results ......................111 4.2.7 Sensitivity Analysis .............................112 4.3 Emitter geometry scaling ...............................114 4.3.1 Intrinsic noise scaling ............................115 4.3.2 Extrinsic noise scaling ............................117 4.3.3 Comparison of intrinsic noise with resistance noise .............119 4.4 Implementation in CAD tools .............................119 4.5 Summary .......................................123 5 IMPROVED PHYSICAL NOISE MODEL 125 5.1 CB SCR e?ect on electron noise ...........................125 5.1.1 Model equation derivation ..........................127 5.1.2 Verification and discussion ..........................131 5.2 Fringe BE junction e?ect on base hole noise .....................133 x 5.2.1 Physical considerations ............................135 5.2.2 Model equation derivation ..........................136 5.2.3 R bn , instead of r bi , as base noise resistance .................141 5.3 Improved physical noise model ............................142 5.3.1 Implementation technique ..........................142 5.3.2 Modeling results ...............................144 5.3.3 Geometry scaling ...............................149 5.3.4 Model parameter impacts and extraction guidelines .............150 5.4 Summary .......................................152 BIBLIOGRAPHY 154 APPENDICES 159 AREPRESENTATION TRANSFORMATION FOR TWO-PORT NETWORK 160 A.1 T-matrix for noise representation transformation ...................160 A.2 Derivation of Noise Parameters ............................160 BAPPROXIMATION OF INTRINSIC BASE RESISTANCE NOISE CONSIDERING CURRENT CROWDING EFFECT 162 B.1 General Principles ...................................162 B.2 Circular Emitter BJT .................................163 B.3 Rectangular Emitter BJT ...............................164 CDERIVATION OF NQS DELAY TIME WITH CB SCR 166 DANALYTICAL Y-PARAMETERS 167 D.1 Manual Derivation of Analytical Y-parameters ....................167 D.2 MATLAB Code for Analytical Y-parameters Derivation ...............168 D.3 MATLAB code for Taylor expansion .........................169 E MATLAB CODE FOR SMALL SIGNAL PARAMETER EXTRACTION 171 F MATLAB CODE FOR INTRINSIC NOISE EXTRACTION 177 F.1 MATLAB code ....................................177 F.2 Data of S-parameters and noise ............................179 GVERILOG-A CODE OF VBIC MODEL FOR SEMI-EMPIRICAL NOISE MODEL IMPLE- MENTATION 195 HDERIVATION OF LOW INJECTION VAN VLIET MODEL IN ADMITTANCE REPRESENTA- TION 196 H.1 Fundamentals .....................................196 H.1.1 Operator ...................................196 H.1.2 Green?s theorem for L p ............................197 H.1.3 Dirac delta function ..............................197 H.1.4 ? theorem ...................................198 xi H.2 Problem setup for base low injection noise of PNP transistor ............198 H.3 Green?s function of homogeneous boundary .....................200 H.4 Hole concentration fluctuation and its spectrum ...................202 H.4.1 van Vliet - Fasset form of noise spectrum ..................203 H.4.2 Solution for ? theorem at low injection ...................206 H.5 Terminal noise current spectrum ...........................207 H.5.1 Spectrum due to j o ..............................208 H.5.2 Spectrum due to correlation of j o and ? ...................209 H.5.3 Terminal total noise current density spectrum ................210 H.6 Y-parameters in homogeneous Green?s function ...................210 H.7 Relation between Y-parameter and noise spectrum ..................211 H.7.1 Common-base noise for BJTs ........................212 H.7.2 Common-emitter noise for BJTs .......................212 xii LIST OF FIGURES 1.1 Energy band diagram of a graded-base SiGe HBT. .................. 3 1.2 Cross section of a raised-base SiGe HBT. ...................... 4 1.3 Vertical scaling strategy for SiGe HBT. ........................ 6 1.4 Two-port noise representations. (a) Admittance (Y-) representation, (b) Impedance (Z-) representation, (c) Chain (ABCD-) presentation, and (d) hybrid (H-) repre- sentation. ....................................... 10 1.5 Noise parameters versus I c for a 50 GHz SiGe HBT with A E = 0.24?20?2 ?m 2 . Six frequency points (2 GHz, 5 GHz, 10 GHz, 15 GHz, 20 GHz and 25 GHz) are measured. ....................................... 12 1.6 Simplified common-emitter small signal equivalent circuit with noise sources for SiGe HBTs. ...................................... 13 1.7 HBT optimized noise figure Opt. F min versus frequency for four SiGe HBT BiC- MOS technologies, including three high performance variants at the 0.5-, 0.18-, and 0.13- ?m nodes as well as a cost-reduced (and slightly higher breakdown volt- age) variant at the 0.18- ?m node. .......................... 15 2.1 Illustration of vector dipole current. ......................... 25 2.2 Illustration of base region with built-in electric field. ................ 30 2.3 Equivalent circuit for intrinsic base of bipolar transistor without r bi . (a) With NQS input. (b) With QS input. .............................. 35 2.4 Y-parameter modeling result using equivalent circuit with NQS input. ? 0 =270 cm/vs 2 . ? n = 1.54 ? 10 ? 7s. n 00 = 50/cm 3 . T = 300K. d B =45 nm. V BE =0.8 V. v exit = 1 ? 10 7 cm/s. f T = g m /C b bed =184 GHz for ? = 5. f T = g m /C b bed =83 GHz for ? = 0. A E =1cm 2 .................................. 36 2.5 Delay times of base region. ? 0 =270 cm/vs 2 . ? n = 1.54 ? 10 ? 7s. n 00 = 50/cm 3 . T = 300K. d B =45 nm. A E =1cm 2 . ......................... 37 xiii 2.6 Input NQS delay resistancer b d versus ? atV BE =0.8 V.? 0 =270 cm/vs 2 . ? n = 1.54? 10 ? 7s. n 00 = 50 /cm 3 (N A = 2 ? 10 18 /cm 3 ). T = 300K. d B =45 nm. W E =0.24 ?m. L E =20 ?m. A E =1cm 2 .............................. 38 2.7 Setup for solving Langevin equation. ......................... 41 2.8 Scalar Green functions. ................................ 42 2.9 Vector Green functions. ................................ 43 2.10 Evaluation of van Vliet model for base region noise under d B = 100nm, ? = 5.4 (|E|=70.2 kV/cm), V BE =0.8V, where ?f T ?? g m /C be =698 GHz. A E =1cm 2 . ... 47 2.11 Evaluation of van Vliet model for base region noise under d B = 20nm, ? = 5.4 (|E|=0 kV/cm), V BE =0.8V, where ?f T ?? g m /C be =698 GHz. A E =1cm 2 ...... 48 2.12 Evaluation of van Vliet model for base region noise under d B = 20nm, ? = 5.4 (|E|=70.2 kV/cm), V BE =0.8V. A E =1cm 2 . ..................... 49 2.13 Evaluation of emitter hole noise model. ? p = 1.54?10 ? 7s. ? E =0. p 00 = 6.66/cm 3 . T = 300K. d E =120 nm. V BE =0.8 V. ? p =220 cm/cs 2 . A E =1cm 2 .......... 51 2.14 Comparison of base electron and emitter hole contributions to S ib . ? p = 1.54 ? 10 ? 7s. ? E =0. p 00 = 6.66/cm 3 . T = 300K. d E =120 nm. V BE =0.8 V. ? p =220 cm/cs 2 . ? n = 1.54 ? 10 ? 7s. ? B =5.4. n 00 = 333/cm 3 . d E =20 nm. ? n =450 cm/cs 2 . V exit = 1 ? 10 7 cm/s. ? c = 0.57ps. A E =1cm 2 . ................... 52 2.15 Small signal equivalent circuit for a transistor divided into 1-D sub-transistors. . . . 53 2.16 Compact noise model assuming uniform f T for whole EB junction. ........ 54 2.17 Equivalent circuit for intrinsic base of bipolar transistor with r bi : (a) With NQS input; (b) With QS input. ............................... 57 2.18 Comparison between r bi , r bi,QS and r b d . ? p = 1.54 ? 10 ? 7s. ? E =0. p 00 = 0.466 /cm 3 . T = 300K. d E =120 nm. ? p0 =225 cm/cs 2 . ? n = 1.54 ? 10 ? 7s. ? B =5.4. n 00 = 23.3 /cm 3 (N A = 4.3 ? 10 1 8 /cm 3 ). d E =20 nm. ? n0 =450 cm/cs 2 . V exit = 1?10 7 cm/s. ? c = 0.57ps. C bej =38 fF. W E =0.12 ?m. L E =18 ?m. A E =1 cm 2 ........................................... 58 3.1 Intrinsic NQS small signal equivalent circuit of SiGe HBTs: (a) without ? c ; (b) with ? c . ........................................ 68 3.2 CB SCR e?ect on ? in and ? out . For the 1-D base region, V bs sat = 1 ? 10 7 cm/s, ? n0 = 270 cm/Vs 2 , ? n =0.154 ?s, T=300K. ..................... 68 xiv 3.3 CB SCR and Ge gradient impacts on the importance of input NQS e?ect. For the 1-D base region, d B =20 nm, V bs sat = 1?10 7 cm/s, ? n0 = 270 cm/Vs 2 , ? n =0.154 ?s, T=300 K. V BE =0.8V................................. 69 3.4 Small signal equivalent circuit of SiGe HBTs with substrate tied to emitter. .... 70 3.5 Illustration of Taylor expansions coe?cient extraction for (a) Rfractur(Y BM 11 + Y BM 12 ) 2 , and (b) Ifractur(Y BM 22 ) 1 . .................................. 73 3.6 Small signal equivalent circuit of SiGe HBTs for block B M . ............ 75 3.7 Ifractur(Y BM 21 ?Y BM 12 ) 1 /g m versusIfractur(Y BM 11 +Y BM 12 ) 1 . The slope of fitting line givesr bx +r bi .76 3.8 Ifractur(Y BM 22 ) 1 versus g m C bct . The slope of fitting line gives r bx +?r bi . ......... 77 3.9 Illustration of low current limit extraction for (a) Rfractur(Y BM 11 + Y BM 12 ) 2 , and (b) Ifractur(Y BM 11 +Y BM 12 ) 1 . .................................. 78 3.10 Illustration ofC bet splitting: (a) Linear fitting forC bet versusg m ; (b) ExtractedC bed and C bej versus g m . .................................. 80 3.11 Extracted r d versus 1/I C . r bx and r bi are shown for reference. ........... 81 3.12 Comparison of Y-parameters for experimental data and modeling results at high bias. .......................................... 83 3.13 Intrinsic Rfractur(Y BI 11 ) extraction and modeling results for three biases. ......... 84 3.14 Comparison of S ib obtained from noise de-embedding of experimental noise data and van Vliet model S ib = 4kTRfractur(Y BI 11 ) ?2qI B for a 50 GHz A E = 0.24?20?2 ?m 2 SiGe HBT at three biases. ........................... 85 3.15 Comparison of r bi extracted using equivalent circuit with and without including input NQS e?ect. ................................... 86 3.16 Extracted delay times and modeling results for three biases. ............ 87 3.17 NQS delay time (? in and ? d or ? out ) extraction. ................... 88 3.18 Bias dependent r bi compared with r bi for (a) 50 GHz SiGe HBT and (b) 180 GHz SiGe HBTs. ...................................... 90 4.1 Series block de-embedding using impedance representation. ............ 93 4.2 Parallel block de-embedding using admittance representation. ........... 93 4.3 Small signal equivalent circuit of SiGe HBT used for Y-parameter and noise pa- rameters de-embedding. ................................ 95 xv 4.4 Extracted intrinsic noise sources as a function of frequency. ............ 99 4.5 (a) (S ib ? 2qI B ) versus ? 2 at I c =4.9 mA and 17.9 mA. (b) C ib (denoted as (S ib ? 2qI B )/? 2 ) versus g m . ................................101 4.6 (a) Measured S ic versus frequency at I c =1.40 mA, 10.6 mA and 19.4 mA. (b) C ic (denoted as S ic /Rfractur(Y 21 )) versus g m . .........................103 4.7 Simulated S ic versus frequency at di?erent I c level. ................104 4.8 (a) Ifractur(S icib ?) versus ? at I C =4.9 mA and 17.9 mA. (b) C i icib ? (denoted as Ifractur(S icib ?/?)) versus g 1.8 m . ...............................105 4.9 Rfractur(S icib ?) versus ? 2 at I c =4.9 mA and 17.9 mA. ...................107 4.10 (a) C r2 icib ? (denoted as Rfractur(S icib ?)/? 2 ) versus g 2 m . (b) C r1 icib ? (denoted as Rfractur(S icib ?)[? = 0]) versus g m ..............................108 4.11 Normalized correlation c of the extracted intrinsic noise for 0.24 ? 20 ? 2?m 2 device: (a) Rfractur(c) and Ifractur(c) versus frequency at I c =17.9 mA; (b) Magnitude of c versus frequency at I c =17.9 mA; (c) Rfractur(c) and Ifractur(c) versus I c at f=25 GHz; (d) Magnitude of c versus I c at f=25 GHz. .......................110 4.12 Noise parameters versus frequency for the measured noise data. I c =17.9 mA. A E =0.24 ? 20 ? 2?m 2 .................................112 4.13 Noise parameters versus collector current for the measured noise data. f=25 GHz. A E =0.24 ? 20 ? 2?m 2 .................................113 4.14 Extracted intrinsic noise divided by M vs I c /M at f=15 GHz, where M is the emitter geometry scaling factor. ...........................116 4.15 Noise parameters versus frequency for 0.24?10?2?m 2 SiGe HBT at I c =1.6 mA and 8.0 mA. V CE =2.0V. ...............................117 4.16 Noise parameters versus frequency for 0.24?20?1?m 2 SiGe HBT at I c =1.6 mA and 8.0 mA. V CE =3.0V. ...............................118 4.17 Noise parameters versus frequency for 0.48?10?1?m 2 SiGe HBT at I c =1.6 mA and 7.8 mA. V CE =2.0V. ...............................119 4.18 Normalized noise parameters versus I c /M at f=15 GHz. ..............120 4.19 NF min versus I c , determined by intrinsic noise only, resistance noise only and both of intrinsic and resistance noise for di?erent geometry SiGe HBTs: (a) 0.24?20? 2?m 2 ; (b)0.24 ? 10 ? 2?m 2 ; (c) 0.24 ? 20 ? 1?m 2 ; (d) 0.48 ? 10 ? 1?m 2 . ....121 xvi 4.20 Technique of insertion of correlated noise sources into the intrinsic transistor of VBIC model. .....................................122 4.21 Noise parameters versus frequency simulated by ADS using semi-empirical noise model at I C =15.1 mA. ................................123 4.22 Noise parameters versus frequency simulated by ADS using SPICE noise model at I C =15.1 mA. .....................................124 5.1 Illustration of AC or noise current flows in ideal 1-D intrinsic SiGe HBT. .....126 5.2 Comparison of the intrinsic noise with ? c =0 and ? c =0.75? tr .For? c =0.75? tr , f T =174 GHz. .....................................128 5.3 Comparison between the brutal used van Vliet model and the improved model un- der ? c =0.75? tr . f T =174 GHz. ............................130 5.4 Comparison between van Vliet model, new model and the extracted intrinsic noise from DESIS simulation results. ? c =0.75(? b +? c ) is used in the new model. E?ective d B =20nm, ?=5.4, |E|=70.2 kV/cm. .........................131 5.5 Illustration of base distribution e?ect by dividing the base resistances into five seg- ments of three types. .................................134 5.6 Small signal equivalent circuit of five segments model. ...............135 5.7 Comparison of simulation and new model for base hole noise in hybrid represen- tation at one bias V BE =0.90V. ............................140 5.8 Comparison of simulation and new model for base hole noise in chain representa- tion at one bias V BE =0.90V. .............................141 5.9 (a) K 1 extracted for simulated 85 GHz and 183 GHz peak f T SiGe HBTs. (b) K 2 extracted for simulated 85 GHz and 183 GHz peak f T SiGe HBTs .........142 5.10 Comparison between thermal resistances R bn and small signal resistance r bi .....143 5.11 Noise parameters versus frequency for A E = 0.24?20?2?m 2 50 GHz SiGe HBT at I c =19.4 mA. ....................................144 5.12 Noise parameters versus I C for A E = 0.24 ? 20 ? 2 ?m 2 50 GHz SiGe HBT at f=15 GHz. .......................................145 5.13 Comparison between thermal resistances R bn and small signal resistance r bi .....146 5.14 Noise parameters versus frequency for A E = 0.12 ? 18 ?m 2 160 GHz SiGe HBT at I c =11.7 mA. ....................................147 xvii 5.15 Noise parameters versus I c for A E = 0.12 ? 18 ?m 2 160 GHz SiGe HBT at f=26 GHz. ..........................................148 5.16 Noise parameters versus frequency for 0.12?20?4 90 GHz SiGe HBT atI C =34.8 mA. ..........................................149 5.17 Noise parameters versus I C for 0.12 ? 20 ? 4 90 GHz SiGe HBT at f=20 GHz. . . 150 5.18 Noise parameters versus frequency for scaled 50 GHz SiGe HBTs (A E = 0.24 ? 10 ? 2?m 2 ).......................................151 5.19 Noise parameters versus frequency for scaled 90 GHz SiGe HBTs (A E =0.12?8? 4?m 2 )..........................................152 5.20 Noise parameters versus frequency for scaled 160 GHz SiGe HBTs (A E =0.12 ? 12?m 2 ).........................................153 A.1 Noise Figure ......................................161 B.1 Approximation induced error versus V B x B i for rectangular emitter BJT. .......165 B.2 Comparison between approximation method and the traditional 4kT/r bi method for rectangular emitter BJT. ..............................165 H.1 Schematic geometry of a PNP transistor. .......................199 H.2 Illustration of surface integral. ............................202 H.3 Admittance representation for BJT noise: (a) Common-base; (b) Common-emitter. 212 xviii LIST OF TABLES 1.1 Comparison of key performance parameters for di?erent SiGe HBT generations . . 8 3.1 Extracted small signal parameter values of A E = 0.12 ? 6 ? 1 ?m 2 SiGe HBT . . . 82 4.1 Extracted small signal parameter values of 0.24 ? 20 ? 2?m 2 SiGe HBT ...... 97 4.2 Parameter values of the simplified noise model for Experiment 0.24 ? 20 ? 2 ?m 2 50 GHz SiGe HBT . ..................................111 4.3 Parameter sensitivity at I C =17.9mA, f=25GHz. A E =0.24 ? 20 ? 2?m 2 . .....114 4.4 Extracted r bx , r bi for 50 GHz SiGe HBTs with di?erent emitter geometries .....117 5.1 Extracted delay time from DESSIS simulation data .................130 5.2 Model parameters, r bi * and r bx * for reference ....................151 A.1 Transformation Matrices to Calculate Noise Matrices ................160 xix CHAPTER 1 INTRODUCTION This chapter opens with a discussion of the motivation for this research on improving RF noise modeling for SiGe Heterojunction Bipolar Transistors (SiGe HBTs). The fundamentals of SiGe HBT physics and the two-port noise representation theory are then introduced, followed by a description of the basic characteristics of noise parameters for SiGe HBTs and the noise perfor- mance scaling trend. Finally the chapter is summarized and the organization of this dissertation is provided. 1.1 Motivation The rapidly developing wireless communication systems have given the human race an infor- mation net composed of thousands of communication satellites in space, millions of base-stations on the ground and billions of personal communicators in people?s hands. Detailed studies on reduc- ing the noise in the mixed-signal analog and RF circuits used in wireless systems are therefore vital to improve the sensitivity of transceivers, and thus save base-station density and enhance the flexi- bility of handsets. One of the key concerns is the minimization of RF noise in transistor amplifiers through device level design and circuit level design. By introducing a graded germanium profile in the base and a higher level of base doping, SiGe HBT enjoys a higher unit-gain cuto? frequency and a smaller base resistance than traditional Sili- con Bipolar Transistors (Si BJTs) and maintains a comparable current gain [1]. All these features contribute to the lower noise level of SiGe HBTs compared to Si BJTs. 1 For RF circuits based on SiGe HBTs, optimizing the design is very important to reduce noise. This clearly requires accurate SiGe HBT noise models and e?cient parameter extraction tech- niques, particularly at the increasingly higher frequencies. The noise modeling approaches cur- rently used for the compact bipolar models are not su?ciently accurate for robust circuit simula- tion [1], and must be refined to make possible predictive low-noise RF circuit design. The purpose of this study is to improve RF noise modeling for SiGe HBTs by developing more accurate compact models for intrinsic transistor noise sources. A semi-empirical noise model and a physical noise model are presented in this dissertation. A novel small signal parameter extraction method is also presented. These results were presented in the 2006 IEEE Transactions on Electron Devices [2], the 2004, 2005 and 2006 IEEE BCTM Conference Proceedings [3?6], and the 2006 IEEE SiRF Conference Proceedings [7], while others are forth coming [8,9]. 1.2 SiGe HBT fundamentals 1.2.1 SiGe as base material The key feature of SiGe HBT is the use of SiGe alloy as the base. Since the energy bandgap of Ge (0.66 eV) is smaller than that of Si (1.12 eV), the bandgap of SiGe is smaller than that of silicon and depends on the Ge mole composition x (?E g,SiGe = 0.74x). The Ge-induced band o?set occurs predominantly in the valence band. A properly defined base Ge profile determines the DC, AC and noise characteristics of SiGe HBTs, and gives SiGe HBTs performance advantages over silicon BJTs [1]. Fig. 1.1 shows a typically graded Ge profile and the resulting energy band diagram for a SiGe HBT. The band diagram shows a finite band o?set at the EB junction, denoted as ?E g0 , along with a larger band o?set at the CB junction, leading to a built-in electric field in the neutral base region that facilitates electron transport from emitter to collector and hence reduces base transit time and improves AC frequency response. If the profile is linear and the base doping 2 EB C Ge C E V E f E E arrowrightnosp Figure 1.1: Energy band diagram of a graded-base SiGe HBT. is uniform, the built-in field is homogeneous within the base region, that is E = ? ?V T d B , (1.1) where d B is the base width, ? denotes the di?erence between the bandgaps at the two base ends in unit of thermal voltage, i.e. ? =?E g,Ge (grade)/V T . Another important consequence of a graded Ge profile is the exponentially decreasing output conductance g o , which is reflected by the Early voltage, V A . g o is negligible for SiGe HBTs. The concept of adding a drift field in the base is surprisingly old, and was pioneered by Kroe- mer [10,11]. However, it took 30 years to realize due to material growth limitations. Nowadays, SiGe alloy can be grown epitaxially on silicon using the ultrahigh vacuum / chemical vapor depo- sition (UHV/CVD) technique. For SiGe HBT, the addition of Ge in the base increases the collector current density, J C . This is made possible by the increased electron injection at the EB junction, which yields more 3 emitter-to-collector charge transport for a given BE voltage. Such an increase in J C also results in an increase in the DC current gain, ?. Consequently the base doping can be increased if the DC current gain is maintained at the same level as for Si BJTs. This reduces the base resistance, leading to further improved AC performance and reduced RF noise. Fig. 1.2 shows the cross-sectional structure of a raised-base SiGe HBT [12]. Carbon is doped during SiGe epitaxy to prevent boron backward di?usion into collector. Selectively Implanted Collector (SIC) [13] and Shallow Trench Isolation (STI) [14] are used to improve transistor per- formance. These techniques will be described in detail below. The most important parasitics are labeled in Fig. 1.2 and consist of the emitter resistance r e , extrinsic base resistance r bx , extrinsic collector resistance r cx , substrate resistance r cs , extrinsic EB capacitance C bex , extrinsic CB ca- pacitance C bcx , and collector-substrate junction capacitance C cs . The intrinsic base resistance r bi , intrinsic CB capacitance C bci and intrinsic collector resistance r ci are also shown for reference. E B B C e r bx r bi r cx r ci r bci C bcx C SIC cs r cs C bex C STI STI DT Figure 1.2: Cross section of a raised-base SiGe HBT. Main parasitics are labeled. 4 1.2.2 Performance parameters For low injection, a key SiGe HBT AC figure-of-merit, the unity-gain cuto? frequency (f T ), can be written generally as [1] f T = 1 2?? ec ? 1 2? bracketleftbigg C bej +C bcx +C bci g m +? b tr +? e +? c + (r e +r cx +r ci )(C bcx +C bci ) bracketrightbigg ?1 , (1.2) where ? ec is the total emitter-to-collector delay time, g m (? qI c /kT) is the intrinsic transconduc- tanceatlowinjection,C bej istheEBdepletioncapacitance,? b tr isthebasetransittime,? e istheemit- ter charge storage delay time, and ? c is the collector transit time due to the CB space charge region (CB SCR). Physically, f T is the common-emitter, unity current gain cuto? frequency (H 21 = 1), and can be conveniently measured using S-parameter techniques. f T can be improved by reducing transit times and using a smaller resistive collector. For an ideal HBT,f T increases versus collector current I c and finally saturates, a direct result of (1.2). However in reality, f T will roll-o? when I c exceeds some threshold value due to the Kirk e?ect or base push-out [15]. That is, f T has a peak value at certain current density J C,peak . Another figure-of-merit that is often used to describe device AC performance is the maximum oscillation frequency f max , reflecting the power gain of a transistor. f max is the common-emitter, unity power gain frequency, and can be related to f T by a first order equation [1] f max ? radicalBigg f T 8?(C bci +C bcx )(r bx +r bi ) . (1.3) There are various definitions of power gain (e.g. U, MAG, MSG), all of which can be measured from the S-parameters [1]. Clearly f max depends not only on the intrinsic transistor performance (f T ), but also on the device parasitics associated with the process technology and its structural implementation. Reducing the base resistance and CB capacitance is decisive for improving f max . 5 For general applications, the CB junction is reversely biased. IfV CB is high enough, ionization occurs within the CB SCR. I c increases dramatically due to carrier multiplication, resulting in device breakdown. BV CBO is the CB breakdown voltage when the emitter is floated. BV CEO is the CB breakdown voltage when the base floats. As shown below, increasing BV CEO will decrease f T . Product BV CEO ?f T , the so-called Johnson limit, is a physical constraint on device optimization. 1.2.3 Improving f T and f max Common sense dictates that for transistors, the smaller they are, the faster they will perform. Indeed, the performance of SiGe HBTs has been greatly enhanced by scaling down accompanied with innovative structure designs, both in vertical and lateral dimensions. The f T of the first func- tional SiGe HBT demonstrated in 1987 [16] is about 50 GHz. Nowadays, SiGe HBTs with both f T and f max greater than 300 GHz have been achieved [17], and this trend continues. Vertical scaling As Ge B P P As B d C w Figure 1.3: Vertical scaling strategy for SiGe HBT. Fig. 1.3 illustrates the vertical scaling strategy. 6 ? Base Here the base width d B is reduced, and a higher Ge ramp is applied, both of which help to reduce base transit time ? b tr . For advanced devices, e.g. 200 GHz SiGe HBTs, ? b tr is less than the total of other transit times. Base doping is also increased to reduce base resistance. ? Collector A higher collector doping N C and a narrower lightly doped collector thickness w C are used to reduce collector transit time ? c . For aggressively scaled devices, ? c dominates the total transit time. A higher level of doping also helps to defer the Kirk e?ect. However, the breakdown voltage is reduced due to the higher CB SCR electric field. Additionally, higher collector doping leads to larger CB capacitance, which reduces f max . Therefore, there is a trade o? between f T , f max and breakdown voltage for N C . ? Emitter The doping is increased to reduce r e and ? e , and the arsenic dopant can be replaced with phosphorus to obtain higher doping concentrations. Generally speaking, ? e is negligible due to HBT?s high DC current gain ?. Lateral scaling The emitter width W E is the key factor for lateral scaling, and generally serves as an indicator of the technology generation. When W E is narrowed, both the intrinsic base resistance r bi and the intrinsic CB capacitance C bci are reduced, and hence f max is improved. f T , however, cannot be improved by this approach. With W E scaled down, extrinsic base and collector parasitics become significant for f max , and must be reduced by scaling and ad hoc techniques . ? R bx Increasing base doping will reduce R bx , but at the price of increasing the CB capac- itance. The solution to this dilemma is to use the so called raised base technique, as shown in Fig. 1.2 [12], where highly doped polysilicon is deposited on top of the SiGe:C layer. Self-aligned low resistive silicide is generally used for such a raised extrinsic base, and a double base contact can be used to reduce the base resistance further. 7 ? C bcx Using implantation through the emitter window, only the collector of the intrinsic device is highly doped to obtained highf T . The remainder of the collector, which is on top of the highly doped collector buried layer, is lightly doped to obtain small C bcx . This is known as the Selectively Implanted Collector (SIC) technique [13]. Shallow trench isolation [14], as shown in Fig. 1.2, can be used to reduceC bcx further by reducing the extrinsic CB junction area. Table 1.1 summarizes the key performance parameters for the five generations of SiGe HBTs readily fabricated in industry [18]. Table 1.1: Comparison of key performance parameters for di?erent SiGe HBT generations [18] Generation I II III IV V W E (?m) 0.5 0.25 0.18 0.13 0.12 f T (GHz) 47 47 120 210 375 f max (GHz) 65 65 100 285 210 ? 100 100 350 300 3500 BV CEO (V) 3.4 3.4 1.8 1.7 1.4 BV CBO (V) 10.5 10.5 6.5 5.5 5.0 J C,peak (mA/?m 2 ) 1.5 1.5 8 12 23 1.2.4 SiGe BiCOMS technology Today?s SiGe HBT technology combines the high speed, low noise SiGe HBTs, aggressively scaled Si CMOS, and a full-suite of on-chip passives together, to create the so-called SiGe BiCMOS technology. SiGe technology has thus emerged as a serious contender for many high-speed digital, RF, analog and microwave applications [1]. At present, there are more than 25 SiGe HBT industrial fabrication facilities on line, and their numbers are growing steadily. Design kits for first four generations of SiGe BiCMOS systems have already been released by IBM. More details of the industrial ?state-of-the-art? for SiGe HBT BiCMOS technology can be found in [19]. 8 1.3 Noise parameters for two-port network In this study, the substrate of the SiGe HBT is always tied to its collector to facilitate S- parameter measurements using a GSG probing system. The resulting SiGe HBT is a two-port network. The noise level of such two-port networks can be measured in terms of Noise Factor, F, which is defined as F = (SNR) signal source (SNR) output . (1.4) Here SNR is the signal-to-noise power ratio. F is usually measured in dB and its value is referred to as the Noise Figure NF , i.e. NF = 10Log10(F). For a two-port network connected to a signal source, F is determined by both the noise parameters (F min or NF min , R n and Y opt ) of the two-port network and the signal source admittance Y S as [20] NF = NF min + R n G S |Y S ?Y opt | 2 , (1.5) where G S is the real part of Y S . The noise parameters can be measured using noise measurement facilities, and their meanings can be explained as follows: ? F min , the minimum noise factor. Its value in dB is the so called minimum noise figureNF min , i.e. 10Log10(F min ). ? R n , the noise resistance, is commonly normalized by the intrinsic impedance Z 0 , and thus is unitless. ? Y opt , the optimum noise matching admittance, is a complex number with a real part G opt and an imaginary part B opt . Its inverse value is denoted as Z opt . Experimentally, the reflection coe?cient ? opt is measured instead of Y opt . Note that ? opt = Mag ?e (j?Angle/180??) . Y opt can 9 be obtained from ? opt as Y opt = 1 Z 0 ? 1 ?? opt 1 +? opt . (1.5) implies that if a two-port network is noise matched (Y S = Y opt ), the noise figure is minimized. The available power gain under noise matching conditions is known as the associated power gain, G ass A . It can be calculated by G ass A = vextendsingle vextendsingle vextendsingle vextendsingle Y 21 Y 11 +Y opt vextendsingle vextendsingle vextendsingle vextendsingle 2 G opt Rfractur[Y 22 ? (Y 12 Y 21 )/(Y 11 +Y opt )] . The noise parameters of a two-port network are fully determined by the noise sources that are distributed within the network. All of the distributive noises can be lumped into two equivalent noise sources located at the port terminals, and they are generally correlated [20]. Fig.1.4 shows four commonly used representations for lumped noise sources, (a) admittance or Y- representation, (b) impedance or Z- representation, (c) chain or ABCD- or A- presentation, and (d) hybrid or H- representation. Note the source polarities in (c) and (d). For each representation, the noise Power 1 i 2 i Noiseless Two port Network Noiseless Two port Network Noiseless Two port Network Noiseless Two port Network a i a v 1 v 2 v h v h i + + + + ()a ()b ()c ()d Figure 1.4: Two-port noise representations. (a) Admittance (Y-) representation, (b) Impedance (Z-) representation, (c) Chain (ABCD-) presentation, and (d) hybrid (H-) representation. 10 Spectral Density (PSD) of the two noise sources, as well as their correlation, can be described by a noise correlation matrix at each frequency point (?). PSD matrices for the four representations are defined as S Y (?) = ? ? ? S i 1 i ? 1 (?) S i 1 i ? 2 (?) S i 1 i ? 1 (?) S i 2 i ? 2 (?) ? ? ? ,S Z (?) = ? ? ? S v 1 v ? 1 (?) S v 1 v ? 2 (?) S v 2 v ? 1 (?) S v 2 v ? 2 (?) ? ? ? , S A (?) = ? ? ? S v a v ? a (?) S v a i ? a (?) S i a v ? a (?) S i a i ? a (?) ? ? ? ,S H (?) = ? ? ? S v h v ? h (?) S v h i ? h (?) S i h v ? h (?) S i h i ? h (?) ? ? ? . (1.6) Each of these matrices, denoted as S origin , can be transformed into another, denoted as S destination , by S destination = TS origin T ? . (1.7) Here the superscript ? represents the transpose conjugate operator. The T-matrices are summarized in Appendix A. Noise parameters, determined by lumped noise sources, can be directly calculated from the chain representation noise matrix elements, i.e. S v , S i and S iv ? as [20] R n = S v 4kT , G opt = radicalBigg S i S v ? bracketleftbigg Ifractur(S iv ?) S v bracketrightbigg 2 , B opt = ? Ifractur(S iv ?) S v , NF min = 1 + 2R n bracketleftbigg G opt + Rfractur(S iv ?) S v bracketrightbigg . (1.8) 11 The derivation is given in Appendix A. Inversely, the chain representation noise matrix can be calculated from noise parameters using (1.8) as S A = ? ? ? S v S vi ? S iv ? S i ? ? ? = 4kT ? ? ? R n NF min ?1 2 ?R n Y ? opt NF min ?1 2 ?R n Y opt R n |Y opt | 2 ? ? ? . (1.9) (1.9) will be used in the noise de-embedding procedure described in Chapter 4. 1.4 Frequency and bias dependence of noise parameters for SiGe HBTs Generally speaking, all the noise parameters are both frequency and bias dependent. Fig. 1.5 shows the measured noise parameters versus collector current I c for a 50 GHz SiGe HBT with emitter area A E = 0.24 ? 20 ? 2 ?m 2 . Six frequency points (2 GHz, 5 GHz, 10 GHz, 15 GHz, 20 GHz and 25 GHz) are shown. 0 5 10 15 20 0 1 2 3 4 5 I C (mA) NF min (dB) 0 5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 I C (mA) Rn (/Z0) 0 5 10 15 20 0 0.01 0.02 0.03 0.04 0.05 0.06 I C (mA) Gopt (S) 0 5 10 15 20 ?0.04 ?0.03 ?0.02 ?0.01 0 0.01 I C (mA) Bopt (S) 2 GHz 5 GHz 10 GHz 15 GHz 20 GHz 25 GHz 50 GHz SiGe HBT 0.24?20?2 ?m 2 V CE =1.5 V (a) (b) (c) (d) Figure 1.5: Noise parameters versus I c for a 50 GHz SiGe HBT with A E = 0.24?20?2 ?m 2 . Six frequency points (2 GHz, 5 GHz, 10 GHz, 15 GHz, 20 GHz and 25 GHz) are measured. 12 To qualitatively understand these frequency and bias dependence, it is necessary to derive the noise parameters analytically. Fig. 1.6 shows a simplified common-emitter small signal equivalent circuit with noise sources for SiGe HBTs. C bct is the total CB capacitance (= C bci +C bcx ) and r bt is the total base resistance. Since r e is not included in Fig. 1.6, r bt ? r bx +r bi +r e (1+?). C bet is the total EB capacitance, and is the sum of the EB di?usion capacitance C bed and the EB junction depletion capacitance C bej . g be is the EB low frequency conductance. r bt is assumed to have 4kTR noise PSD, i.e. S vrbt = 4kTr bt . Uncorrelated 2qI shot noise PSDs are assumed for the base and collector current noises, i.e. S ib = 2qI b ,S ic = 2qI c andS icib ? = 0. CB ionization noise is not taken into account here and throughout this work, since only low V CB operation is concerned. Using the m gv C E v bet C bt r rbt v b i c i be g bct C B Figure 1.6: Simplified common-emitter small signal equivalent circuit with noise sources for SiGe HBTs. two-port network noise theory in Section 1.3, the noise parameters can be derived as [19] F min ? 1 + N ? + radicaltp radicalvertex radicalvertex radicalbt 2I c V T r bt parenleftBigg f 2 f 2 T + 1 ? + N 2 ? parenrightBigg ? 1 + radicaltp radicalvertex radicalvertex radicalbt 2I c V T r bt parenleftBigg f 2 f 2 T + 1 ? parenrightBigg , (1.10) R n ? V T 2I c +r bt , (1.11) Y opt ? F min ? 1 2R n , (1.12) where N is the EB junction ideality factor and N ? 1. 13 ? NF min orF min (1.10) means that at a fixed bias, NF min increases versus frequency as shown in Fig. 1.5 (a). At low biases,f T ? 1/I c according to (1.2), consequentlyF min ?1 ? 1/ radicalbig I c . Therefore NF min increases when I c decreases at very low I c levels. For high biases before f T roll-o?, f T is nearly constant, hence F min ? 1 ? radicalbig I c , meaning that NF min increases versus I c . These trends are indeed shown in Fig. 1.5 (a). Analysis shows that F min has a minimum value, the so-called optimized noise figure (Opt. F min ), which is approximately radicalbig f radicalBig 8? 2 r bt (C bej +C bct )/? 1/2 at low frequencies and f4? radicalbig 2r bt (C bej +C bct )(? b +? e +? c ) at high frequencies. Increasing f T and decreasing r bt can significantly reduce the optimized noise figure (Opt.F min ). ? R n (1.11) shows thatR n drops versusI c and saturates to the value ofr bt , which is consistent with Fig. 1.5 (b). The frequency dependence cannot be explained using this simplified equivalent circuit. ? Y opt (1.12) shows that the imaginary part of Y opt or B opt is negligible. This is qualitatively true, as can be seen by comparing Fig. 1.5 (d) with Fig. 1.5 (c). The real part of Y opt or G opt increases versus frequency, as shown by Fig. 1.5 (c), which is consistent with (1.12). The bias dependence of G opt , however, cannot be easily explained since both F min and R n are bias dependent. As Fig. 1.5 (c) shows, G opt increases versus I c . 1.5 Noise performance trends for SiGe HBTs Advances in scaling technology and a series of innovations in processing and structure have led to a steady increase in the peak f T and a reduction in the base resistance r bt . According to the discussion above, these f T and r bt trends will improve noise performance, driving a reduction in NF min with each generation. SiGe BiCOMS technologies thus enable circuit designers to im- plement noise-sensitive applications at an increasingly broader frequency range based on silicon technology. Fig. 1.7 shows the optimized noise figure Opt. F min versus frequency for four SiGe 14 HBT BiCMOS technology generations, including three high performance variants at the 0.5-, 0.18- , and 0.13- ?m nodes as well as a cost-reduced (and slightly higher breakdown voltage) variant at the 0.18- ?m node. The performance of GaAs PHEMT is also illustrated for reference. The noise figure has been greatly decreased for the 0.13- ?m node. F min remains below 0.4 dB beyond 12 GHz, rising to only 1.3 dB at 26.5 GHz. This level of performance falls within the range estab- lished using the data sheets for GaAs PHEMT currently on the commercial market, placing silicon within one generation of this benchmark [19]. Figure 1.7: HBT optimized noise figure Opt. F min versus frequency for four SiGe HBT BiCMOS technologies, including three high performance variants at the 0.5-, 0.18-, and 0.13- ?m nodes as well as a cost-reduced (and slightly higher breakdown voltage) variant at the 0.18- ?m node. (Original figure was shown by D. Greenberg at IEEE MTT-S International Microwave Symposium, Fort Worth, 2004. The above figure is copied from [19]) . 1.6 Noise modeling considerations and methodology for SiGe HBTs A good noise model should give an excellent fit for all the noise parameters. This calls for an accurate compact noise model for device noise. The noise models used for extrinsic parasitics are 15 well established and are su?ciently accurate for this purpose, so for this study the only concern is the intrinsic device noise modeling. As we are developing models for the wide sense steady (WSS) RF noise, the noise modeling can be completely based on small signal equivalent circuit due to the small signal nature of RF noise. Both semi-empirical and physical noise models are developed here. Emitter geometry scaling, especially emitter length scaling are examined for both models. MATLAB programs are used for small signal parameter extraction and noise calculation. The Verilog-A language is used to implement the new semi-empirical noise model in VBIC, a large signal BJT model applicable for circuit simulators such as Advanced Design System (ADS) supplied by Agilent Technologies. 1.7 Summary This chapter describes the motivation for the research and the theoretical background for SiGe HBTs and RF noise modeling. In this dissertation, Chapter 2 gives the RF noise theory for SiGe HBTs. Chapter 3 explores the small signal extraction method for a small signal equivalent circuit including the input non-quasistatic e?ect. Chapter 4 describes the intrinsic noise source extrac- tion technique and the new semi-empirical noise model developed based on the extraction results. Chapter 5 describes the new physical noise model developed for this study based on the improved electron and base hole noise models. 16 CHAPTER 2 RF NOISE THEORY FOR SIGE HBTS This chapter opens with a description of two noise sources that are important for semiconduc- tors, namely carrier velocity fluctuation and carrier population density fluctuation. Then two sets of base electron noise PSDs are presented without including the distributive e?ect, one of which is the solution for a new 1-D Langevin equation including finite exit velocity boundary condition for CB junction, the other is van Vliet?s general solution for 3-D Langevin equation without including finite exit velocity boundary condition for CB junction. The 1-D solution is used to evaluate van Vliet?s 3-D solution for the finite exit velocity e?ect. Both solutions are extended to include emitter hole noise. The crowding theory that deals with base distributive e?ect using segmentation method is then described. The compact noise model with three noise sources derived from the crowding theory by assuming uniform f T across the whole EB junction is discussed in detail. Finally the disadvantages of previous noise models used in CAD tools and the literatures are reviewed, and new methods are proposed, with which to develop improved models. 2.1 RF noise sources During the motions induced by external forces, carriers in semiconductors inevitably inter- act with lattice perturbations, impurities or other carriers, leading to observable terminal voltage or current variations from their ideal values. The noise measured at device terminals is referred to as macroscopic noise, while the spatially distributive fluctuations, such as carrier velocity, position, population density, are referred to as microscopic noise. To mathematically describe 17 the microscopic noise, two approaches can be followed, the microscopic and mesoscopic meth- ods [21,22]. The quantum mechanical microscopic method describes statistical carrier distribu- tions within whole phase space or states based on Liouville / von Neumann?s equations. Both the fluctuation-transportation equation and characterization of microscopic noise sources can be ob- tained in terms of Markov random processes. For example, a semi-classical k-space Boltzmann transport equation (BTE) with appropriate Langevin source has been derived [23?26], from which it is possible to obtain hydrodynamic or more simplified drift-di?usion models that include the microscopic noise sources. This approach is beyond the scope of this dissertation, and will not be further discussed. The mesoscopic approach deals with carrier fluctuations within coarse-grained time and space limits, e.g. t>>? cs and l>>l 0 , where ? cs is the expected collision time and l 0 is the expected free path distance. By satisfying such limits, the impact of the carrier?s initial state vanishes. Di?erent carriers may have the same statistical characterizations, so for carriers contained within a small mesoscopic volume, it is only necessary to study the statistical character- izations of one electron in order to know the statistical characterizations of the whole volume. Therearetwomainmicroscopicnoisesourcesinsemiconductormaterialsordevices. Theyare generation-recombination (GR) noise, which represents the carrier population density fluctuation due to transitions between bands and localized states (donors, traps, Shockley-Read-Hall centers, etc.), and velocity fluctuation or di?usion noise, which is associated with the Brownian motion of free carriers in the classical treatment or electron-phonon and electron-impurity scattering in the quantum treatment [21]. The underlying microscopic events are interband transitions for GR noise and intraband transitions for velocity fluctuation noise. To describe each of the two noise sources mesoscopically, two methods can be used: the coarse-grained mesoscopic Master equation (ME) approach and the Langevin approach. 18 The ME describes how the transition probability density evolves versus time for Markov pro- cesses as ?P(a,t|a prime ) ?t = integraldisplay da primeprime [P(a primeprime ,t|a prime )W a primeprime a prime ?P(a,t|a prime )W aa primeprime], (2.1) where P(a,t|a prime ) is the probability density of state a at time t with initial state of a prime at time zero, W aa primeprime is the probability (density) per unit time of an instantaneous transition at time t from a to a primeprime , and similarly for W a primeprime a prime. a and a prime are the state vectors. (2.1) can be transformed into the Kramer- Moyal expansion. Often only the first and the second orders of such expansions are important. Truncating all higher orders (>2) yields the Fokker-Planck equation (FPE) ?P(a,t|a prime ) ?t = ? summationdisplay i ? ?a i [A i (a)P] + 1 2 summationdisplay ij ? ?a i ?a j [B ij (a)P], (2.2) where i and j denote di?erent variables of state a, and A and B are the first and second order Fokker-Planck moments, explicitly A i (a) = integraldisplay (a prime i ?a i )W aa primeda prime ,B ij (a) = integraldisplay (a prime i ?a i )(a prime j ?a j )W aa primeda prime . (2.3) Moments A and B fully describe the stochastic process a, and hence determine the phenomenolog- ical noise source in the Langevin description, as shown below. From (2.2) and applying a Laplace transformation gives [21] ? a prime ?t = ?M a prime,M ij = ? ?A i (a prime ) ?a j . (2.4) This is the phenomenological equation that gives the average behavior of deviation for a(t) from its initial state a prime (0). Such phenomenological equations may involve external or internal driving forces and sometimes friction forces. Theoretically, any higher order moment, such as covariances 19 and correlations, can be obtained in the same way as (2.4). Note that the Fourier transform of an autocorrelation gives the power spectrum density (PSD). In the Langevin approach, (2.4) is written without the conditional averaging bracket by adding Langevin sources as ??a(t) ?t = ?M?a(t) +?(t). (2.5) Clearly ?(t) is used to mimic the random forces that produce the fluctuation of a(t). Two necessary and su?cient requirements for ?(t) to provide the same first order and second order moments of ?a(t), are [21] = 0,= B[a prime (0)]?(t?t prime ). (2.6) From (2.6), the PSD of ?(t) can be obtained by performing a Fourier transformation as S ?,? = 2B[a prime (0)]. (2.7) The coe?cient is 2, since single side band PSD is considered here (for measurement only positive frequencies are allowed). (2.7) explains the meaning of the Langevin force and how to obtain the PSD of such a force from the Master equation. The advantage of the Langevin approach is that the calculation of fluctuation-transportation can then be fully resolved based on familiar partial di?erential equations (PDEs). The next section describes the PSD of the microscopic noise sources using these two methods. The emphasis here is on velocity fluctuation noise since it is the major noise source for modern SiGe HBTs with narrow base widths. 20 2.1.1 Velocity fluctuation noise In the mesoscopic approach, carrier velocity fluctuations are modeled as the Brownian motion of a single particle that takes place in the corresponding band [21], i.e. electrons in the conductance band and holes in the valence band. The ground breaking work on the theory of Brownian motion was conducted by Einstein in 1905 in one of the classic papers he published that year [27] using a Fokker-Planck PDE he had derived, where the relation= 2Dtwas first established. Another breakthrough was made in 1908 by Langevin using what is now called Langevin force method [28]. TheBrownianmotiontheorywasfurtherelucidatedbyUhlenbeckandOrnsteinin1930[29], where the motion was treated using both the Langevin method and the Fokker-Planck method and the motion was proved to be Gaussian. A complete analysis of Brownian motion can be found in the 1943 classic paper by Chandrasekar [30]. The results of the PSD of velocity fluctuations,S ?v,?v (?), are shown below using the ME/Fokker-Planck and Langevin methods respectively, after which the methods to establish the microscopic noise source representations for velocity fluctuations will be discussed. Brownian motion, which is assumed to be a Markov stochastic process in phase space ( ?? r , ?? v ), can be described by the Fokker-Planck equation (2.2). Further assuming that the scattering events are characterized by a collision time ? cs , and that the carrier e?ective mass m ? is isotropic, the Fokker-Planck moment A and B can be derived as [29,30] A = ? ? ? ?? v ? 1 ? cs ?? v ? ? ? ,B= ? ? ? OO O 2 ? cs kT m ? I ? ? ? , (2.8) where O and I are rank-two zero and unitary tensors, respectively. The solution P( ?? r, ?? v,t| ?? r 0 , ?? v 0 ) for (2.2) and (2.8) can be found in [22,30]. Using the obtained solution and the = 2Dt 21 relation, the di?usion coe?cient can be obtained as D = ? cs kT m ? = V T ?, (2.9) where ? is the carrier mobility defined as e? cs /m ? . (2.9) states the Einstein relation. More accurate D and ? expressions can be derived from the Boltzmann transport equation. It shows that (2.9) holds only when carriers satisfy the Boltzmann statistics [31]. Therefore (2.9) is only valid for low field transport. As only the velocity of ( ?? r , ?? v ) is interested here, (2.2) is integrated through the whole ?? r space to obtain the equation for P( ?? v,t|, ?? v 0 ), briefly designated as P v , ?P v ?t = 1 ? cs ? v ? ( ?? vP v ) + 1 ? cs kT m ? ? 2 v P v . (2.10) The solution is [21,30] P( ?? r, ?? v,t| ?? r 0 , ?? v 0 ) = bracketleftbigg m ? 2?kT(1?e ?2t/? cs ) bracketrightbigg 1/2 exp bracketleftBigg ? m ? 2kT vextendsingle vextendsingle ?? v ? ?? v 0 e ?t/? cs vextendsingle vextendsingle 2 1 ?e ?2t/? cs bracketrightBigg . (2.11) The conditional mean velocity and variance can then be obtained as < ?? v (t) > ?? v 0 = ?? v 0 e ?t/? cs , ?? v 0 ?< ( ?? v (t)? < ?? v (t) > ?? v 0 ) 2 > ?? v 0 = kT m ? (1?e ?2t/? cs )I. (2.12) (2.12) approaches Maxwell?s equilibrium distribution when t>>? cs . From (2.11), the autocorre- lation function of velocity can be obtained through integration as [22,32] R ?v,?v (t,t prime ) = bracketleftbigg ? kT m ? e ?(t+t prime )/? cs + kT m ? e ?|t?t prime |/? cs bracketrightbigg I. (2.13) 22 By setting t = t prime , (2.13) is correctly reduced to (2.12). Due to the coarse-grain procedure inherent in mesoscopic method, i.e. t,t prime >> ? cs , the stationary autocorrelation function is obtained from (2.13) R ?v,?v (?) = 1 ? cs De ?|?|/? cs I, (2.14) where ? ? t?t prime and D is given in (2.9). Now the single side band PSD of velocity fluctuation can be readily obtained from a Fourier transformation of (2.14) multiplied by two. The result is S ?v,?v (?) = 4D 1 +? 2 ? 2 cs I, (2.15) Since ? cs is of the order of picoseconds, S ?v,?v (?) ? 4DI is valid up to one hundred GHz. For the Langevin method, two white-noise sources are introduced for ?? r and ?? v. The equations for the fluctuations of ?? r and ?? v are d? ?? r dt = ? ?? v +? r , d? ?? v dt = ? 1 ? cs ?? v +? v . (2.16) ? v has a physical meaning of the stochastic collision force. According to (2.7), the PSD of ? r and ? v must obey S ?,? = 2B = ? ? ? OO O 4 ? cs kT m ? I ? ? ? , (2.17) which means S ? v ,? v (?) = 4D/? 2 cs I. Fourier transformation of (2.16) gives ?v = ? cs 1 +j?? cs ? v . (2.18) 23 The PSD of velocity fluctuation is therefore S ?v,?v (?) = ? 2 cs 1 +? 2 ? 2 cs S ? v ? v = 4D 1 +? 2 ? 2 cs I, (2.19) which is consistent with the ME/Fokker-Planck result in (2.15). Since the noise power spectrum density is directly related to the di?usion coe?cient, velocity fluctuation noise is also called di?u- sion noise. The PSD of velocity fluctuation in (2.15) is true only for low electric field case. For mod- ern SiGe HBTs, the built-in field in the base due to Ge gradient is quite strong. The high field impacts on di?usion, drift, and noise are no longer negligible. For example, the driving force for velocity saturation equation is approximated to be the gradient of carrier fermi level triangleinvE f , instead of the gradient of electric potential triangleinv?. (2.15) is a very rough approximation for the PSD of ve- locity fluctuation due to non-equilibrium e?ects. A self-consistent development of S ?v,?v for high electric field can be followed from the full band Monte Carlo simulation under homogeneous bulk conditions [33]. In this work, (2.15) is used for simplicity. Now the problem is to determine how the velocity fluctuation should be described as micro- scopic noise source. There are two possible kinds of descriptions, the physical vector dipole current noise source developed by Shockley [34] and the phenomenological current density Langevin noise source [35]. These are, of course, equivalent. ? Vector dipole current description Due to velocity fluctuation ? ?? v, a carrier labeled with m traveling between t 1 and t 2 has a disturbance from its ideal position as if the carrier is displaced by ? ?? r m = integraldisplay t 2 t 1 ? ?? v m dt ? ? ?? v m (t 2 ?t 1 ). 24 as shown in Fig. 2.1. This means that the carrier with charge e is taken from its ideal position and injected into a disturbed position. The current produced by this procedure ?I m is e/(t 2 ?t 1 ). 01 ,rt combarrowextenderarrowrightnosp m r combarrowextendercombarrowextenderarrowrightnosp m I 22 ,rt combarrowextenderarrowrightnosp 12 ,rt combarrowextenderarrowrightnosp ? ?? Figure 2.1: Illustration of vector dipole current. Therefore e? ?? v m = e ? ?? r m t 2 ?t 1 = e t 2 ?t 1 ? ?? r m = ?I m ? ?? r m . (2.20) (2.20) clearly reveals that velocity fluctuation has a physical meaning of vector dipole cur- rent. Now at any given instant in a small volume satisfying the mesoscopic requirement, (??) ? =?x?y?z, consider the total vector current dipole ? ? P ? = e summationdisplay m ? ?? v m (t) = ?I ? ? ?? r ? . (2.21) 25 ? ? P ? includes all the noise generated within (??) ? . For the mesoscopic domain, fluctuations of all particles are independent and the same. The PSD of ? ? P ? is thus directly obtained as S ? ? P ? ,? ? P ? = e 2 summationdisplay m S ?v,?v = 4e 2 nD 1 +? 2 ? 2 cs (??) ? , (2.22) where n is the carrier concentration at any position within the small volume (??) ? .For convenience, define the local noise source K ? ( ?? r,?)as K ? ( ?? r,?) ? 4e 2 nD 1 +? 2 ? 2 c I ? 4e 2 nDI, (2.23) so that S ? ? P ? ,? ? P ? = K ? ( ?? r,?)(??) ? . (2.24) Such vector dipole current noise representation can be very easily used to calculate its con- tribution to the terminal macroscopic noise. For example, to examine the noise voltage at terminal N, ?v N , inject noise current ?I ? at ?? r 1 , and subtract the same amount of noise cur- rent at ?? r 2 (? ?? r ? = ?? r 2 ? ?? r 1 ). Supposing the transfer impedance Z N? ? ?v N /?I ? is known for all positions within device, this gives ?v N = [Z N? ( ?? r 1 ) ?Z N? ( ?? r 2 )]?I ? = ?Z N? ?? ?? r ? ?I ? = ?Z N? ?? ? P ? . (2.25) Consequently, the total terminal noise voltage PSD can be obtained as S v N ,v N (?) = summationdisplay ? S ?v N ,?v N (r ? ,?) = summationdisplay ? |?Z Nr | 2 S ? ? P ? ,? ? P ? = integraldisplay |?Z Nr | 2 K ? ( ?? r,?)d? (2.26) This is the so-called impedance field method for noise calculation [34]. 26 ? Current density description As the treatment of noise transport is based on the drift-di?usion (DD) model, the velocity fluctuation must be expressed in terms of current density fluctuations, denoted as ?? ? (t)ata certain spatial position at time t, which can be directly inserted into the continuity equations. Again, considering a small volume (??) ? , the current density fluctuations induced by the mth carrier velocity fluctuations is represented by ? ?? j m = e (??) ? ? ?? v m . (2.27) The charge e is distributed uniformly within the whole small volume to obtain the charge density due to the mth carrier, since ? ?? j m is the current density for any point within the small volume. Then S ?j m ,?j m (?) = e 2 (??) 2 ? S ?v m ,?v m (?). (2.28) Since the velocity fluctuations of di?erent carriers within (?) ? are uncorrelated, the total power spectrum of the current density fluctuations generated by (?) ? is S ? (?) = summationdisplay m S ?j m ,?j m (?) = n(??) ? S ?j m ,?j m (?) = ne 2 (??) ? S ?v,?v (?) = K ? ( ?? r,?) (??) ? . (2.29) Curiously, the current density fluctuation is the inverse of small volume. The problem will be clear after an examination of the averaged velocity for (??) ? ??v = 1 N ? summationdisplay m ? ?? v m . (2.30) 27 As S ??v,??v = 1 N ? summationdisplay m S ?v m ,?v m = 1 n S ?v,?v (??) ? , (2.31) the averaged velocity fluctuation becomes stronger for smaller volume or fewer carriers. This is consistent with the intuitive concepts. As ?? ? = ?en ??? v, this once again gives S ? (?) = e 2 n 2 S ??v,??v = ne 2 (??) ? S ?v,?v (?) = K ? ( ?? r,?) (??) ? . (2.32) The S ? (?) in (2.33) is the PSD of auto-correlation for the current density fluctuations j( ?? r ) at any point ?? r within the small volume, i.e. . Since the current density fluc- tuation is uniform within the small volume, the current density fluctuations at any given two points within the same small volume, j( ?? r ) and j( ?? r prime ), are correlated, and = . It can be assumed that the current density fluctuations at di?erent small volumes are independent, that is, if ?? r is inside of (??) ? while ?? r prime is outside of ?? r prime then =0. Therefore the PSD of the current correlation at any given two points ?? r and ?? r prime is S ? ( ?? r, ?? r prime ,?) = K ? ( ?? r,?) (??) ?? r U (??) ?? r ( ?? r prime ) = K ? ( ?? r,?) U (??) ?? r ( ?? r prime ) (??) ?? r , (2.33) where (??) ?? r is the small volume containing ?? r , U (??) ?? r ( ?? r prime ) is a unit step function: it is one when ?? r prime is inside of (??) ?? r otherwise zero. (2.33) is already a Langevin source. However, it is not convenient to use due to the segmentation (divide the whole device into su?cient small volumes) needed before solving Langevin equations. Further, the size of the small volumes 28 should be infinite small to obtain exact results, which is the idea of integration. Note that lim (??) ?? r ?0 U (??) ?? r ( ?? r prime ) (??) ?? r = ?( ?? r ? ?? r prime ), (2.34) the infinite small limit of (2.33) gives S ? ( ?? r, ?? r prime ,?) = K ? ( ?? r,?)?( ?? r ? ?? r prime ), (2.35) where K ? ( ?? r,?) is given in (2.23). (2.35) is the Langevin source for the di?usion noise in current density representation. Since this discussion is not confined to either electrons or holes, (2.35) is applicable to both carriers. 2.1.2 GR noise GR noise induces population fluctuations within devices. Hence, the stochastic quantity pop- ulation changing rate ?(t) is a good description for such noise. ?(t) has the physical meaning of the injected current density fluctuation at a given spatial position at time t. As with the velocity fluctuation noise, the power spectrum density of ?(t), S ? ( ?? r, ?? r prime ,?), can be derived through either the ME/Fokker-Planck method or the Langevin method. Details can be found in [21,22]. By con- sidering only the band-to-band transitions and symmetric Shockley-Read-Hall (SRH) transitions, the result is S ? ( ?? r, ?? r prime ,?) = K ? ( ?? r,?)?( ?? r ? ?? r prime ),K ? ( ?? r,?) = 2(n 0 +n) ? , (2.36) where ? is the carrier life time, n 0 is the DC equilibrium carrier concentration, n is the total carrier concentration, and K ? ( ?? r,?) is the local noise source for population fluctuation. (2.36) is again applicable to both electrons and holes. 29 2.2 Electron noise of base region without distributive e?ect This section first solves a 1-D Langevin equation to obtain the electron noise PSD for base region. The impact of finite exit velocity boundary condition at CB junction is considered. The general 3-D solution derived by van Vliet is then introduced although the finite exit velocity bound- ary condition is not considered. Finally, the van Vliet model is evaluated using the 1-D solution derived for the finite exit velocity condition. The CB SCR e?ect and base distributive e?ect are not considered in either case. 2.2.1 1-D solution Assume a uniform base built-in field E induced by either the Ge gradient or the doping gra- dient, as shown in Fig. 2.2. E is measured using parameter ? as in (1.1). The minus sign in (1.1) X45 X42 X43 X30 X42 X64 X43X42 X77X64X2B X66X69X65X6CX64X45X2D X53X43X52 X42X43 Figure 2.2: Illustration of base region with built-in electric field. indicates an acceleration field for the electron from the emitter to the collector. The following are the parameters and variables used ? A E ? cross-sectional area of the 1-D device ? d B ? neutral base width. ? ? ? related to electrical field strength. ? ? n ? electron mobility. 30 ? V T ? thermal voltage. ? D n ? electron di?usion coe?cient, which is related to mobility through the Einstein relation, i.e. D n = V T ? n . ? v exit ? electron finite exit velocity at the CB junction, which is close to the electron saturation velocity. ? ? n ? electron life time. ? L n ? electron di?usion length, L n = radicalbig D n ? n . ? n 00 ? equilibrium electron concentration at the base beginning point (x = 0). n 00 = N c N v e ?E g,si +?E g0 V T N A = n 2 i e ?E g0 V T N A , (2.37) where N A is the base doping concentration. ? n ? electron concentration. ? n 0 ? equilibrium electron concentration, given by n 0 = n 00 e ? x d B . (2.38) ? ?n ? excess electron concentration, ?n = n?n 0 . ? n 1 ? ?n at x = 0. ? tildewiden ? AC electron concentration. ? tildewiden 1 ?tildewiden at x = 0. SolvingtheDC,ACcontinuityequationsandLangevinequationgivestheDCcurrent, Y-parameters and the PSDs of the intrinsic base and collector noise currents. The electron finite exit velocity 31 boundary condition is forced at the end of the base for all three cases. The high field mobility model is used ? n = ? n0 ? radicalbigg 1 + parenleftBig ? n0 E v sat parenrightBig ? . Drift-Di?usion (DD) model is applied, i.e. J n = e?En+eD n d dx n. (2.39) DC solution V BE is applied to the EB junction. Solving the continuity equation D n ? 2 ?x 2 ?n+? n E ? ?x ?n? ?n ? n = 0, (2.40) with the boundary conditions ?n| x=0 ? n 1 = n 00 parenleftBig e V BE V T ? 1 parenrightBig , ?n| x=d B = ? J n | x=d B ev exit , gives the electron concentration and terminal current densities ?n(x) = parenleftBigg e ? 1 x d B 1 ?? ?1 e 2? + e ? 2 x d B 1 ??e ?2? parenrightBigg n 1 , J C = J n | x=d B = ?en 1 v exit ??? 2 parenleftBig ? 1 ?? 2 e ?? 2 ??e ?? 1 parenrightBig , J E = J n | x=0 = ?en 1 v exit ? parenleftbigg ? 2 +? e ?? 2 +?e ?? 1 e ?? 2 ??e ?? 1 parenrightbigg , J B = J C ?J E , (2.41) 32 where ? = ? 1 ?? ? 2 ?? ,?= v exit d B ? n V T ,? 1 = ? 2 +?, ? 2 = ? 2 ??, ? = radicalBigg parenleftBig ? 2 parenrightBig 2 + parenleftbigg d B L n parenrightbigg 2 . The base transit time can be obtained from (2.41) as ? B = Q B J C = d B (??? 2 ) v exit 1?e ?? 2 ? 2 ? 1?e ?? 1 ? 1 ? ? 1 ?? 2 , (2.42) With v exit ??and d B << L n , (2.42) is reduced to the Kramer equation ? b0 = parenleftbigg 1 ? ? 1 ?e ?? ? 2 parenrightbigg d 2 B D n . (2.43) AC solution Here a small signaltildewidev be is applied to EB junction. Solving the AC continuity equation D n ? 2 ?x 2 tildewiden+? n E ? ?x tildewiden? tildewiden ? n ?j?tildewiden = 0, (2.44) with the boundary conditions tildewiden| x=0 ?tildewiden 1 = n 00 V T e V BE V T tildewidev be , tildewiden| x=d B = ? tildewide j n | x=d B ev exit . 33 leads to the electron concentration and terminal current densities as follows tildewiden(x) = ? ? e tildewide? 1 x d B 1 ?tildewide? ?1 e 2 tildewide ? + e tildewide? 2 x d B 1 ?tildewide?e ?2 tildewide ? ? ? tildewiden 1 , tildewide j c = tildewide j n | x=d B = ?etildewiden 1 v exit ??tildewide? 2 parenleftbigg tildewide? 1 ?tildewide? 2 e ?tildewide? 2 ?tildewide?e ?tildewide? 1 parenrightbigg , tildewide j e = tildewide j n | x=0 = ?etildewiden 1 v exit ? parenleftbigg ? 2 + tildewide ? e ?tildewide? 2 +tildewide?e ?tildewide? 1 e ?tildewide? 2 ?tildewide?e ?tildewide? 1 parenrightbigg , tildewide j b = tildewide j c ? tildewide j e , (2.45) where tildewide? = tildewide? 1 ?? tildewide? 2 ?? ,?= v exit d B ? n V T , tildewide? 1 = ? 2 + tildewide ?, tildewide? 2 = ? 2 ? tildewide ?, tildewide ? = radicalBigg parenleftBig ? 2 parenrightBig 2 + parenleftbigg d B L n parenrightbigg 2 +j? d 2 B D n . (2.46) From (2.45), the common-emitter Y-parameters for base region can be derived Y B 11,CE ? A E tildewide j b tildewidev be = A E e n 00 V T e V BE V T bracketleftbigg v exit ? parenleftbigg ? 2 + tildewide ? e ?tildewide? 2 +tildewide?e ?tildewide? 1 e ?tildewide? 2 ?tildewide?e ?tildewide? 1 parenrightbigg ? v exit ??tildewide? 2 parenleftbigg tildewide? 1 ?tildewide? 2 e ?tildewide? 2 ?tildewide?e ?tildewide? 1 parenrightbiggbracketrightbigg , Y B 21,CE ? ?A E tildewide j c tildewidev be = A E e n 00 V T e V BE V T v sat ??tildewide? 2 parenleftbigg tildewide? 1 ?tildewide? 2 e ?tildewide? 2 ?tildewide?e ?tildewide? 1 parenrightbigg . (2.47) An equivalent circuit is needed to model these Y-parameters. The commonly used one is shown in Fig. 2.3 (b), where the Rfractur(Y 11 ) is frequency independent. The input network of Fig. 2.3 (b) is quasi-static (QS). Of particular interest for modeling the RF noise in the base current is the frequency dependence of the real part of the input admittance (Rfractur(Y 11 )) due to the base electron transport, as shown below. The frequency dependence of Rfractur(Y 11 ) for the base was first examined by Winkel [36] using (2.47). The results show that the base minority carrier charge responds to the base emitter voltage with an input delay time ? b in , after which the collector current at the end of base region responds to the stored base minority carrier charge with another delay time ? b out . ? b in represents the input non-quasistatic (NQS) e?ect and ? b out represents the output excess phase 34 delay. In an equivalent circuit, ? b in can be modeled by an input delay resistance r b d in series with EB junction di?usion capacitanceC b bed as illustrated by Fig. 2.3 (a) [37,38]. ? b in = r b d C b bed . Here the EB depletion capacitance C bej is also included. As depletion capacitance is charged through majority carrier movement, C bej does not experience an NQS delay, and should therefore be separated from C b bed . Note that C bej = 0 and C bed = g m ? b tr for (2.47). ? b tr is the base transit time. ? b out can be included as a delay term in the transconductance. Fig. 2.4 shows the result of (2.47) together with X62X65 X67 X62X65X6A X43 X62 X62X65X64 X43 X62 X64 X72 X62 X6FX75X74 X6A X6D X76X67 X65 X77X74X2DX76 X62X65 X67 X62X65X6A X62X65X64 X43X43X2B X64 X6A X6D X65X76X67 X77X74X2D X76 X4EX51X53X29X28X61 X29X28X62 X42 X42X43 X43 X45 X45 X77X69X74X68 X69X6EX70X75X74 X65X66X66X65X63X74 X4EX51X53X77X69X74X68X6FX75X74 X69X6EX70X75X74 X65X66X66X65X63X74 X27X42 Figure 2.3: Equivalent circuit for intrinsic base of bipolar transistor without r bi . (a) With NQS input. (b) With QS input. the modeling result of Fig. 2.3 (a). Clearly, Fig. 2.3 (a) is accurate for frequencies up to f T . The input NQS e?ect becomes more important at a given frequency for a larger Ge gradient device, as shown by Rfractur(Y 11 ) in Fig. 2.4. However, the modeling error due to using the QS equivalent circuit for a real device with a base resistance becomes smaller at high current levels for larger Ge gradient devices (see Section 2.4.2). Fig. 2.5 shows the extracted delay times for the base region. The solid line is the result of finite v exit . The dashed line is the result of infinite v exit . The finite v exit will increase ? b tr by d B /v exit , as shown in Fig. 2.5 (a). Fig. 2.5 (b) shows the ? b tr normalized ? b in and ? b out . Finite v exit has a subtle e?ect on the normalized value, and the finite v exit does not change the importance of input and output NQS delay times. Note that the normalized NQS delay times increase versus ?, with the result that ? b in and ? b out are weakly dependent on the Ge gradient. For 35 0 100 200 0 1 2 x 10 4 f (GHz) Re(Y11e) 0 100 200 0 2 4 x 10 4 f (GHz) Im(Y11e) 0 100 200 0 2 4 6 x 10 4 f (GHz) Re(Y21e) 0 100 200 ?3 ?2 ?1 0 x 10 4 f (GHz) Im(Y21e) Analytical ?=5 modeling ?=5 Analytical ?=0 modeling ?=0 ?=5, f T =184 GHz ?=0, f T =83 GHz V BE =0.8 V A E =1 cm 2 ?=5 ?=0 Figure 2.4: Y-parameter modeling result using equivalent circuit with NQS input. ? 0 =270 cm/vs 2 . ? n = 1.54 ? 10 ? 7s. n 00 = 50/cm 3 . T = 300K. d B =45 nm. V BE =0.8 V. v exit = 1 ? 10 7 cm/s. f T = g m /C b bed =184 GHz for ? = 5. f T = g m /C b bed =83 GHz for ? = 0. A E =1cm 2 . input NQS delay resistance, r b d = ? b in C b bed = 1 g m ? b in ? b tr ? 1 W E L E J c /V T ? b in ? b tr ? 1 W E L E n 1 /(d B V T ) ? b in ? b tr ? 1 W E L E n 00 e V BE /V T /(d B V T ) ? b in ? b tr = d B V T W E L E n 00 e V BE /V T ? b in ? b tr = parenleftbigg d B W E L E parenrightbigg parenleftBigg n 2 i N A parenrightBiggparenleftBigg ? b in ? b tr parenrightBigg e ?E g0 /V T V T e ?V BE /V T . (2.48) The impacts of base geometry, base doping and bias on r b d are clear from (2.48). However the ? and finite v exit impacts are not clear due to the approximations made. Fig. 2.6 shows the input NQS delay resistance r b d versus ? at V BE =0.8 V. Both the finite and infinite v exit are shown. By increasing the Ge gradient, r b d is reduced. A finite CB exit velocity increases r b d . This is discussed further in Section 2.4.2. 36 0 10 20 0 0.5 1 1.5 2 2.5 3 x 10 ?12 ? ? trb (s) 0 10 20 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ? normalized NQS times v exit =1.0? 10 7 cm/s Infinite v exit ? out b /? tr b ? in b /? tr b (a) (b) Figure 2.5: Delay times of base region. ? 0 =270 cm/vs 2 . ? n = 1.54 ? 10 ? 7s. n 00 = 50/cm 3 . T = 300K. d B =45 nm. A E =1cm 2 . Noise solution The 1-D Langevin equation for the base electron noise is ?D n ? 2 ?x 2 tildewiden?? n E ? ?x tildewiden+ tildewiden ? n +j?tildewiden = ?(x,?), (2.49) where ?(x,?) = ?(x,?) + 1 e ? ?x ?(x,?). (2.50) The Langevin noise sources ?(?) is the sum of the GR noise ?(?) and the gradient of di?usion noise ?(?), which have been described in Section 2.1. The boundary condition for (2.49) is tildewiden| x=0 = 0,tildewiden| x=d B = ? tildewide j n ev exit . (2.51) 37 0 5 10 15 20 0 50 100 150 200 250 300 ? r db ( ? ) V exit =1?10 7 cm/s V exit =infinite V BE =0.8 V W E =0.24 ?m L E =20 ?m d B =45 ?m N A =2? 10 18 /cm 3 Figure 2.6: Input NQS delay resistance r b d versus ? at V BE =0.8 V. ? 0 =270 cm/vs 2 . ? n = 1.54 ? 10 ? 7s. n 00 = 50 /cm 3 (N A = 2?10 18 /cm 3 ). T = 300K. d B =45 nm. W E =0.24 ?m. L E =20 ?m. A E =1cm 2 . To solve (2.49)-(2.50), the Green function method [22] is used. First, we define a carrier density Green function G n (x,x prime ), which satisfies ?D n ? 2 ?x 2 G n (x,x prime ) ?? n E ? ?x G n (x,x prime ) + G n (x,x prime ) ? n +j?G n (x,x prime ) = ?(x?x prime ), (2.52) G n (x,x prime ) = 0| x=0 ,G n (x,x prime ) = ? tildewide j n (x) ev exit | x=d B ,x? [0,d B ],x prime ? (0,d B ). Clearly G n (x,x prime ) is the electron density change at position x responding to the unity point carrier flux density injection at position x prime , The total carrier density fluctuationtildewiden for (2.49) can then be obtained as tildewiden(x) = integraldisplay d B 0 G n (x,x prime )?(x prime ,?)dx prime . (2.53) 38 However, the PSD for terminal current instead of carrier density is needed. To provide this, operate e?E +eD n d dx on both sides of (2.53) tildewide j n (x) = integraldisplay d B 0 parenleftbigg e?E +eD d dx parenrightbigg G n (x,x prime )?(x prime ,?)dx prime . (2.54) For convenience, we define terminal carrier flux density ( current density divided by -e) Green functions (scalar) G e (x prime ) = parenleftbigg ?E +D n d dx parenrightbigg G n (x,x prime )| x=0 , (2.55) G c (x prime ) = ? parenleftbigg ?E +D n d dx parenrightbigg G n (x,x prime )| x=d B . (2.56) G e (x prime ) and G e (x prime ) are thus the emitter and collector outflow carrier flux densities responding to the unity point carrier flux density injection at position x prime , respectively. The base terminal outflow carrier flux density responding to the unity point carrier flux density injection at positionx prime , denoted as G b (x prime ) can be obtained directly from the quasi-neutral condition, that is G b (x prime ) = 1 ?G c (x prime ) ?G e (x prime ). (2.57) The terminal inflow noise current density fluctuations are tildewide j e = e integraldisplay d B 0 G e (x prime )?(x prime ,?)dx prime , tildewide j c = e integraldisplay d B 0 G c (x prime )?(x prime ,?)dx prime , tildewide j b = e integraldisplay d B 0 G b (x prime )?(x prime ,?)dx prime , (2.58) 39 The PSD for the correlation between the noise current densities of terminal ? and ? (?,? = e,c,b) can be obtained via the following integration S j ? ,j ? ? (?) ? < tildewide j ? tildewide j ? ? >= e 2 integraldisplay d B 0 integraldisplay d B 0 G ? (x) G ? ? (x prime )dxdx prime =e 2 integraldisplay d B 0 integraldisplay d B 0 G ? (x) G ? ? (x prime )dxdx prime + integraldisplay d B 0 integraldisplay d B 0 G ? (x) ? ?x ? ?x prime G ? ? (x prime )dxdx prime ?e 2 integraldisplay d B 0 integraldisplay d B 0 G ? (x) G ? ? (x prime )dxdx prime + integraldisplay d B 0 integraldisplay d B 0 ?G ? (x) ?x ?G ? ? (x prime ) ?x prime dxdx prime =e 2 integraldisplay d B 0 integraldisplay d B 0 G ? (x)K ? (x,?)?(x?x prime )G ? ? (x prime )dxdx prime + integraldisplay d B 0 integraldisplay d B 0 ?G ? (x) ?x K ? (x,?)?(x?x prime ) ?G ? ? (x prime ) ?x prime dxdx prime =e 2 integraldisplay d B 0 G ? (x prime )K ? (x prime ,?)G ? (x prime ) ? dx prime + integraldisplay d B 0 ?? G ? (x prime )K ? (x prime ,?) ?? G ? (x prime ) ? dx prime , (2.59) where ?? G ?,? (x prime ) are vector Green functions: ?? G e (x prime ) ? ? ?x prime G e (x prime ), ?? G c (x prime ) ? ? ?x prime G c (x prime ), ?? G b (x prime ) ? ? ?x prime G b (x prime ) = ? ?? G e (x prime ) ? ?? G c (x prime ). (2.60) The approximation made for the third step in (2.59) is the neglecting of two complex surface integrations when using Gauss theorem when the finite exit velocity boundary condition is not considered. (2.59) shows that the scalar Green functions should be used for GR noise and the vector Green functions should be used for di?usion noise. Now to solve G n (x,x prime ), a unity current pulse is inserted at position x prime as shown in Fig. 2.7. Note ?i=1. The boundary condition in (2.51) changes to 40 X42 X78X92 X01X69X01X69 X65 X01X69 X63 X01X69 X62 X64 X64 X64X64 X30X64 Figure 2.7: Setup for solving Langevin equation. G n (x,x prime )| x=0 = 0,G n (x,x prime )| x=x prime ? = G n (x,x prime )| x=x prime + , ?G n (x,x prime ) ?x | x=x prime + ? ?G n (x,x prime ) ?x | x=x prime ? = ?i D n ,G n (x,x prime )| x=d B = ? tildewide j n ev exit . (2.61) The third condition in (2.61) can be obtained by integrating (2.49) over the area around x prime . The solutions for (2.49) and (2.61) are G n (x,x prime ) = ? ? ? ? ? ? ? ? ? ? ? ? ? d B 2D n parenleftBigg e ? tildewide? 1 x prime d B ?tildewide?e ?2 tildewide ? e ? tildewide? 2 x prime d B parenrightBiggparenleftBigg ?e tildewide? 1 x d B +e tildewide? 2 x d B parenrightBigg tildewide ?(1?tildewide?e ?2 tildewide ? ) , if x ? [0 x prime ]; d B 2D n parenleftBigg e tildewide? 2 x d B ?tildewide?e ?2 tildewide ? e tildewide? 1 x d B parenrightBiggparenleftBigg e ? tildewide? 1 x prime d B ?e ? tildewide? 2 x prime d B parenrightBigg tildewide ?(1?tildewide?e ?2 tildewide ? ) , if x ? (x prime d B ], (2.62) wheretildewide? 1 ,tildewide? 2 ,tildewide? and tildewide ? are given in 2.46. 41 With G n (x,x prime ), the scalar and vector Green functions can be calculated from (2.55), (2.56), (2.57) and (2.60). The results are G e (x prime ) = e ? tildewide? 1 x prime d B ?tildewide?e ?2 tildewide ? e ? tildewide? 2 x prime d B 1 ?tildewide?e ?2 tildewide ? , G c (x prime ) = e ? tildewide? 1 x prime d B ?e ? tildewide? 2 x prime d B 1 ?tildewide?e ?2 tildewide ? (tildewide? 1 ?tildewide?tildewide? 2 )e tildewide? 2 tildewide? 2 ?tildewide? 1 , G b (x prime ) = 1 ?G e (x prime ) ?G c (x prime ), ?? G e (x prime ) = ? tildewide? 1 e ? tildewide? 1 x prime d B ?tildewide? 2 tildewide?e ?2 tildewide ? e ? tildewide? 2 x prime d B d B (1?tildewide?e ?2 tildewide ? ) hatwidex, ?? G c (x prime ) = ? tildewide? 1 e ? tildewide? 1 x prime d B ?tildewide? 2 e ? tildewide? 2 x prime d B d B (1?tildewide?e ?2 tildewide ? ) (tildewide? 1 ?tildewide?tildewide? 2 )e tildewide? 2 tildewide? 2 ?tildewide? 1 hatwidex, ?? G b (x prime ) = ? ?? G e (x prime ) ? ?? G c (x prime ). (2.63) Figs. 2.8 and 2.9 show the scalar Green functions and vector Green functions, respectively, plotted versus base position. For the solid lines, v exit = 1?10 7 cm/s, and for the dashed lines, v exit ??. 0 1 2 3 4 x 10 ?6 0 0.2 0.4 0.6 0.8 1 position (cm) G e (x) G c (x) G b (x) ? with v sat effect ? ? without v sat effect Figure 2.8: Scalar Green functions. Given the Green functions in (2.63) and DC electron concentration in (2.41), all the noise PSDs can be obtained by integrating (2.59). Since the di?usion noise dominates for the base 42 0 1 2 3 4 x 10 ?6 0 1 2 3 4 5 6 x 10 5 position (cm) cm ?1 DG e (x) DG c (x) DG b (x) ? with v sat effect ? ? without v sat effect Figure 2.9: Vector Green functions. region of SiGe HBT, only the analytical results for di?usion noise are given here: S B ib ? S B ibib ? = A E bracketleftbigg |A+C| 2 ? K d B ?tildewide? 1 ?tildewide? ? 1 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 1 +? 1 ? 1 parenrightBig +|B +D| 2 ? K d B ?tildewide? 2 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 2 ?tildewide? ? 2 +? 1 ? 1 parenrightBig + (A ? +C ? )(B +D) ? K d B ?tildewide? ? 1 ?tildewide? 2 +? 1 parenleftBig e ?tildewide? ? 1 ?tildewide? 2 +? 1 ? 1 parenrightBig + (A+C)(B ? +D ? ) ? K d B ?tildewide? 1 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 2 +? 1 ? 1 parenrightBig +first4 lines,substitute? 1 ? ? 2 and ? K ? ? L +first4 lines,substitute? 1 ? ? and ? K ? KL bracketrightbigg , (2.64) 43 S B icib ? = A E bracketleftbigg ?C(A ? +C ? ) ? K d B ?tildewide? 1 ?tildewide? ? 1 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 1 +? 1 ? 1 parenrightBig ?D(B ? +D ? ) ? K d B ?tildewide? 2 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 2 ?tildewide? ? 2 +? 1 ? 1 parenrightBig ?D(A ? +C ? ) ? K d B ?tildewide? ? 1 ?tildewide? 2 +? 1 parenleftBig e ?tildewide? ? 1 ?tildewide? 2 +? 1 ? 1 parenrightBig ?C(B ? +D ? ) ? K d B ?tildewide? 1 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 2 +? 1 ? 1 parenrightBig +first4 lines,substitute? 1 ? ? 2 and ? K ? ? L +first4 lines,substitute? 1 ? ? and ? K ? KL bracketrightbigg , (2.65) S B ic ? S B icic ? = A E bracketleftbigg |C| 2 ? K d B ?tildewide? 1 ?tildewide? ? 1 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 1 +? 1 ? 1 parenrightBig +|D| 2 ? K d B ?tildewide? 2 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 2 ?tildewide? ? 2 +? 1 ? 1 parenrightBig +C ? D ? K d B ?tildewide? ? 1 ?tildewide? 2 +? 1 parenleftBig e ?tildewide? ? 1 ?tildewide? 2 +? 1 ? 1 parenrightBig +CD ? ? K d B ?tildewide? 1 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 2 +? 1 ? 1 parenrightBig +first4 lines,substitute? 1 ? ? 2 and ? K ? ? L +first4 lines,substitute? 1 ? ? and ? K ? KL bracketrightbigg , (2.66) where A = ? tildewide? 1 d B parenleftBig 1 ?tildewide?e ?2 tildewide ? parenrightBig,B= tildewide?e ?2 tildewide ? tildewide? 2 d B parenleftBig 1 ?tildewide?e ?2 tildewide ? parenrightBig,C= A (tildewide? 1 ?tildewide?tildewide? 2 )e tildewide? 2 tildewide? 2 ?tildewide? 1 ,D= ?C tildewide? 2 tildewide? 1 , ? K = 4q 2 D n n 00 e V BE V T 1 ?? ?1 e 2? , ? L = 4q 2 D n n 00 e V BE V T 1 ??e ?2? , KL= 4q 2 D n n 00 . ? 1 =tildewide? 1 | ?=0 ,? 2 =tildewide? 2 | ?=0 ,?=tildewide?| ?=0 . 44 The last two lines in each PSD represent eight terms that are obtained by a parameter substitution procedure performed on the first four lines. The velocity saturation boundary condition is involved through parameter ?. ? = 1 if such a condition is neglected. 2.2.2 General 3-D solution by van Vliet The 3-D Langevin equation for base electron noise is solved by van Vliet using Green?s func- tion method in [39]. The adiabatic (homogeneous) boundary condition, i.e. tildewiden=0 or zero electron density fluctuation, is used for both ends of the base region. The built-in field can be position dependent. It is important to note that the base distributive e?ect is not considered in van Vliet?s derivation in spite of the 3-D analysis because the AC bias voltage for the whole EB junction is assumed to be uniform. In the derivation, the Y-parameters are expressed by Green?s functions in linear fashion using the extended Green theorem. The power spectrum densities of noise are initially quadratic in Green?s functions. In order to make the connection between noise PSD and the Y-parameter, it is convenient to transform the noise PSD into a result whose main part is linear in the Green?s functions. This is accomplished using the ? theorem in [40]. Finally the base elec- tron noises are related to the Y-parameters of the base region. The detailed derivation is given in Appendix H. The original results are in common-base configuration but can be readily transformed into common-emitter configuration. The results are S B ib = 4kTRfractur(Y B 11 ) ? 2qI B b , S B ic = 4kTRfractur(Y B 22 ) + 2qI c , S B icib ? = 2kT(Y B 21 +Y B 12 ? ?g m ), (2.67) where I B b is the DC base recombination current. Here, the frequency dependence of S B ib and S B ic , as well as their correlation S B icib ? , are taken into account through the frequency dependence of the Y-parameters of baseY B . For SiGe HBTs, or modern transistors,I B b is negligible. Instead, the base 45 current due to hole injection into the emitter, I E b , dominates I b . Also van Vliet?s derivation failed to consider electron transport in the CB SCR, which is noticeable for aggressively scaled HBTs. This is because the van Vliet model was derived for early transistors in which the base current is dominated by base recombination. The 3-D Langevin equation was solved only for base minority carriers, which in the case of SiGe HBTs is the electrons. Therefore, the base DC recombination current and Y-parameters of base region should be used in (2.67). In practice, however, the total base current I b (I B b +I E b ) and the Y-parameters of whole transistor are brutally used in (2.67), and the results are recognized as the noise of whole transistor, which is not justified. These issues will be discussed in detail in Chapter 5. 2.2.3 Evaluation of van Vliet solution for finite exit velocity boundary condition Using the 1-D Langevin equation solution derived above,for cases where the finite exit veloc- ity is not included in the boundary conditions, the van Vliet model gives the base electron noise exactly, as expected. When such boundary condition is applied, however, the van Vliet model deviates from the analytical results when the base width becomes narrow. Fig. 2.10 shows the wide base case (d B =100 nm), where the van Vliet model is consistent with the 1-D solution. Fig. 2.11 shows the narrow base case (d B =20 nm). The van Vliet model clearly overestimates S ib and |Rfractur(S icib ?)| while underestimating |Ifractur(S icib ?)|. However, with a strong base built-in field, typically the case in graded SiGe HBTs, the devi- ation is significantly reduced, as shown in Fig. 2.12, where ? = 5.4. A careful inspection of the solution process does not yield an intuitive explanation for this observation, but calculations show that this is generally true for all practical values of built-in field found in modern SiGe HBTs with graded bases. Therefore, it is reasonable to continue to use the van Vliet model to describe the relationship between noise and Y-parameters of the intrinsic base for graded SiGe HBTs. This is the starting point for the analysis of the CB SCR e?ect given in Chapter 5. 46 0 0.5 1 0 1 2 x 10 ?17 f / f T S ib Analytical result van Vliet model 0 0.5 1 0 5 x 10 ?17 f / f T S ic 0 0.5 1 ?4 ?2 0 x 10 ?18 f / f T Re(S icib * ) 0 0.5 1 ?2 ?1 0 x 10 ?17 f / f T Im(S icib * ) V BE =0.8 V d B =100nm ?=0 Figure 2.10: Evaluation of van Vliet model for base region noise under d B = 100nm, ? = 5.4 (|E|=70.2 kV/cm), V BE =0.8V, where ?f T ?? g m /C be =698 GHz. A E =1cm 2 . 2.3 Extension to including emitter hole noise For modern transistors, the base current due to hole injection into the emitter, I E b , dominates the base current. At low frequency, the PSD of i E b should be 2qI E b to first order, which is much larger than the PSD of base recombination current 2qI B b . Therefore it is important to include base hole noise. 2.3.1 3-D van Vliet model To obtain the emitter hole noise induced base noise current i E b , and denote its PSD as S E ib ,itis necessary to solve a 3-D Langevin equation for emitter minority carriers (holes here as NPN HBTs are of interest) with a boundary condition ?p = 0 at both the emitter contact and the neutral to depletion boundary of the EB junction. ?p is the hole density fluctuation. ?p = 0 at the emitter contact as infinite surface recombination velocity is assumed. ?p = 0 at the neutral to depletion boundary of the EB junction, as the same adiabatic boundary condition used in [39] is assumed. 47 0 0.5 1 0 5 x 10 ?16 f / f T ,m S ib 0 0.5 1 0 1 2 x 10 ?15 f / f T ,m S ic 0 0.5 1 ?1 ?0.5 0 x 10 ?16 f / f T ,m Re(S icib * ) 0 0.5 1 ?5 0 x 10 ?16 f / f T ,m Im(S icib * ) Analytical result van Vliet model V BE =0.8 V d B =20 nm E=0 kV/cm Figure 2.11: Evaluation of van Vliet model for base region noise underd B = 20nm,? = 5.4(|E|=0 kV/cm), V BE =0.8V, where ?f T ?? g m /C be =698 GHz. A E =1cm 2 . The solution can be obtained following van Vliet?s derivation for base minority carrier noise [39]. The Langevin equation for minority holes in the emitter of NPN HBT solved here is exactly the same Langevin equation for minority holes in the base of PNP transistor solved in [39] with the same adiabatic condition?p = 0 is used in [39]. The emitter minority carrier induced noise current at the emitter-side neutral to depletion boundary of the EB junction, i E b , is analogous to the base minority carrier induced noise current at the base-side neutral to depletion boundary of the EB junction, i B e (= i B b + i B c ). Therefore, the PSD of i E b takes the functional form of the PSD of i B e in [39], that is, S E ib = 4kTRfractur(Y E 11 ) ? 2qI E b , (2.68) where Y E 11 is the input admittance seen by the base terminal due to emitter hole injection. At low frequency limit, Y E 11 ? qI E b /kT, hence S E ib ? 2qI E b . 48 0 50 100 0 0.5 1 x 10 ?16 f (GHz) S ib 0 50 100 0 x 10 ?15 f (GHz) S ic 0 50 100 ?4 ?2 0 x 10 ?17 f (GHz) Re(S icib * ) 0 50 100 ?5 0 x 10 ?16 f (GHz) Im(S icib * ) d B =20nm |E|=70.2kV/cm V BE =0.80V Analytical result [?n| x=dB =?J/qv sat ] van Vliet model [Eq (1)] Figure 2.12: Evaluation of van Vliet model for base region noise under d B = 20nm, ? = 5.4 (|E|=70.2 kV/cm), V BE =0.8V. A E =1cm 2 . The emitter hole density fluctuations induce emitter electron density fluctuations to maintain quasi-neutrality due to dielectric relaxation. The electron density fluctuations, however, induce electron current fluctuation only at the emitter contact but not at the depletion to neutral boundary, because electrons are majority carriers in the emitter. Therefore the emitter hole noise only con- tributes to the base current noise i b , but not the collector current noise i c . The PSDs of the total i b and i c can then be obtained as S EB ib = 4kTRfractur(Y EB 11 ) ? 2qI b ,S EB ic = 4kTRfractur(Y EB 22 ) + 2qI c ? 2qI c , S EB icib ? = 2kT(Y EB 21 +Y EB? 12 ?g m ) ? 2kT(Y EB 21 ?g m ), (2.69) where Y EB 11 = Y E 11 +Y B 11 ,I b = I E b +I B b , Y EB 21 = Y B 21 ,Y EB 22 = Y B 22 ,Y EB 12 = Y B 12 . 49 Interestingly, (2.69) has the same functional form as (2.67), meaning that the van Vliet model in (2.67) can be directly applied to include emitter hole noise by simply replacing Y B with Y EB , the Y-parameters of base and emitter regions. So far, the e?ect of electron transport in the CB SCR on Y-parameters and noise has not been taken into account. 2.3.2 1-D solution Similarly, the PSD for emitter hole noise S E ib of the 1-D Langevin equation solution for the emitter can be obtained from the S B ie of the 1-D Langevin equation solution for the base. That is S E ib = A E bracketleftbigg |E| 2 ? K d E ?tildewide? 1 ?tildewide? ? 1 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 1 +? 1 ? 1 parenrightBig +|F| 2 ? K d E ?tildewide? 2 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 2 ?tildewide? ? 2 +? 1 ? 1 parenrightBig +E ? F ? K d E ?tildewide? ? 1 ?tildewide? 2 +? 1 parenleftBig e ?tildewide? ? 1 ?tildewide? 2 +? 1 ? 1 parenrightBig +EF ? ? K d E ?tildewide? 1 ?tildewide? ? 2 +? 1 parenleftBig e ?tildewide? 1 ?tildewide? ? 2 +? 1 ? 1 parenrightBig +first4 lines,substitute? 1 ? ? 2 and ? K ? ? L +first4 lines,substitute? 1 ? ? and ? K ? KL bracketrightbigg , (2.70) where E = ? tildewide? 1 d E parenleftBig 1 ?tildewide?e ?2 tildewide ? parenrightBig,F= tildewide?e ?2 tildewide ? tildewide? 2 d E parenleftBig 1 ?tildewide?e ?2 tildewide ? parenrightBig, ? K = 4q 2 D p p 00 e V BE V T 1 ?? ?1 e 2? , ? L = 4q 2 D p p 00 e V BE V T 1 ??e ?2? , KL= 4q 2 D p p 00 . ? 1 =tildewide? 1 | ?=0 ,? 2 =tildewide? 2 | ?=0 ,?=tildewide?| ?=0 . tildewide? = D p tildewide? 1 ?v sr d E D p tildewide? 2 ?v sr d E , tildewide? 1 = ? E 2 + tildewide ?, tildewide? 2 = ? E 2 ? tildewide ?, tildewide ? = radicalBigg parenleftBig ? E 2 parenrightBig 2 + d 2 E D p ? p + j?d 2 E D p . (2.71) 50 Here v sr is the emitter surface recombination velocity, d E is emitter thickness, ? E represents the emitter built-in field due to non-uniform doping, D p is the emitter hole di?usion coe?cient, and ? p is the emitter hole life time. Consequently, the total base and collector current noise PSDs can be obtained by adding S E ib to S ib only. In order to evaluate (2.68), Y E 11 and I E b are derived as Y E 11 = e p 00 V T e V BE V T v sr ? parenleftbigg ? 2 + tildewide ? e ?tildewide? 2 +tildewide?e ?tildewide? 1 e ?tildewide? 2 ?tildewide?e ?tildewide? 1 parenrightbigg , (2.72) I E b = ep 00 parenleftBig e V BE V T ? 1 parenrightBig v sr ? parenleftbigg ? 2 +? e ?? 2 +?e ?? 1 e ?? 2 ??e ?? 1 parenrightbigg , (2.73) where all parameters are given in (2.71). 2.3.3 Evaluation of finite surface recombination velocity e?ect Fig. 2.13 compares two S E ib values versus frequency, which are the analytical results of (2.70) and the modeling result using (2.70), (2.72) and (2.73). Three values for the surface recombination velocity are used. Clearly the finite surface recombination velocity has only a negligible e?ect on the accuracy of (2.68). Furthermore, the frequency dependence of S E ib is weak. Since only 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 3.5 x 10 ?17 f (GHz) S ibE Analytical result modeling [4kTRe(Y 11 E )?2qI b E ] V BE =0.8 V V sr = infinite V sr = 1? 10 7 cm/s V sr = 1? 10 5 cm/s Figure 2.13: Evaluation of emitter hole noise model in (2.68). ? p = 1.54 ? 10 ? 7s. ? E =0. p 00 = 6.66/cm 3 . T = 300K. d E =120 nm. V BE =0.8 V. ? p =220 cm/cs 2 . A E =1cm 2 . 51 the di?usion noise is taken into account here, the result above is not exact for the emitter where GR noise is non-negligible. However, when the GR current is significant, the e?ect of surface recombination velocity is reduced. 2.3.4 Comparison of base electron and emitter hole contributions to S ib This comparison is based on an HBT constructed for this study with?=235 andf T =200 GHz. A CB SCR delay is included as detailed in Chapter 5. Fig. 2.14 shows a plot of S E ib and S B ib versus frequency at V BE =0.8 V. The graph shows that only for f<15 GHz is S B ib negligible. For f>30 GHz, S B ib dominates S ib . S ib has a strong frequency dependence due to base electron noise. 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 ?15 f (GHz) S ib contributions S ib B S ib E V BE =0.8 V f T =200 GHz ?=235 Figure 2.14: Comparison of base electron and emitter hole contributions to S ib . ? p = 1.54?10 ? 7s. ? E =0. p 00 = 6.66/cm 3 . T = 300K. d E =120 nm. V BE =0.8 V. ? p =220 cm/cs 2 . ? n = 1.54?10 ? 7s. ? B =5.4. n 00 = 333/cm 3 . d E =20 nm. ? n =450 cm/cs 2 . V exit = 1 ? 10 7 cm/s. ? c = 0.57ps. A E =1 cm 2 . The units of y-axis is A 2 /Hz. A E =1cm 2 . 52 2.4 Compact noise model including distributive e?ect 2.4.1 Compact noise model assuming uniform f T across EB junction The technique commonly used to deal with transistor distributive e?ects is the segmenta- tion method, where the transistor is divided into many narrow 1-D sub-transistors and these sub- transistors are then connected by divided base resistances. Fig. 2.15 shows the small signal equiv- alent circuit for this method. Y 11 and Y 21 of each sub-transistor naturally include non-quasistatic (NQS) e?ect. Y 11 includes the input NQS e?ect, while Y 21 includes the output NQS e?ect. The resistances have 4kTR thermal noise. The base and collector current noises of each segment can be described by the van Vliet model if the CB SCR e?ect is not important. 1 11 Y 1 21 Y 1 b i 1 c i 2 11 Y 2 21 Y 2 b i 2 c i 11 n Y 21 n Y n b i n c i C E B B 1 R 2 R n R 1R v 2R v 3R v Figure 2.15: Small signal equivalent circuit for a transistor divided into 1-D sub-transistors. The objective is to develop a compact noise model that includes as few noise sources as pos- sible, with their analytical expressions. Although a general analytical solution is hard to achieve, a compact noise model with only three noise sources can be derived by assuming the same f T for all of the 1-D sub-transistors according to the crowding theory in [41]. Fig. 2.16 shows the lumped small signal equivalent circuit, together with the lumped noise sources. 53 11 Y 21 Y b i c i C E B bi Z rbi i 'B b I c I Figure 2.16: Compact noise model assuming uniform f T for whole EB junction [41]. Since the output conductance g o is negligible for SiGe HBTs, the lumped Y 11 and Y 21 can be chosen to be the sum of Y 11 and Y 21 for all the sub-transistors, explicitly, Y 11 = Y 1 11 +Y 2 11 +...+Y n 11 , Y 21 = Y 1 21 +Y 2 21 +...+Y n 21 . The lumped base impedance Z bi has a complex expression, which relates to the distributive base resistances and EB capacitances. Denote the low frequency limit of the real part of Z bi as r bi , the lumped AC small signal value of the intrinsic base resistance. For transistors with double base contacts and at low current levels, r bi = R BV ? 1 12 R a50 W E L E , (2.74) where L E and W E are the base length and base width, respectively, and R a50 is the base sheet resistance. 54 The lumped base and collector current noises i b and i c are the sum of the noise currents of the 1-D sub-transistors with no crowding e?ect i b = i 1 b +i 2 b +...+i n b , i c = i 1 c +i 2 c +...+i n c . i b and i c are correlated. The noise current assigned to Z bi , i rbi , lumps together all the distributive noise e?ects, and therefore contains not only the distributive base resistance thermal noise, but also the distributive intrinsic base current noise. It is an important result of [41] that i rbi is not correlated with either i b and i c once a uniform f T across EB junction is assumed. Thus, the PSD of i rbi , S i rbi , is generally frequency dependent through the frequency dependence of the intrinsic base current noise and has a complex expression [41]. Denote the DC voltage drop across Z bi as V B x B i . For a circular emitter BJT, the low frequency limit of S i rbi is given as S irbi,cir = 4kT R BV ?f 5e V BxB i /V T + 1 6 . (2.75) For a rectangular emitter BJT, the low frequency limit of S i rbi is given as S irbi,rec = 4kT R BV ?f 5e V BxB i /V T + 4 9 . (2.76) These expressions are hard to use for noise modeling based on small signal equivalent circuits, be- cause the parameters used are not available in a small signal equivalent circuit. At low frequencies, S i rbi can be related to r bi by S i rbi = 4kT/r bi ? 2qI B /3. (2.77) 55 The derivation is given in Appendix B based on [41]. (2.77) is exact for circular emitter BJTs, and has less than a 3% error for rectangular BJTs. At low current levels or under weak crowding strength, (2.77) simplifies to become 4kT/r bi , the traditional thermal noise model for r bi . Note that (2.77) is accurate only when the input NQS e?ect is included in the intrinsic transistor model, as assumed by [41]. However, in current CAD tools, QS equivalent circuits are used, which cause some problems forr bi noise modeling. Even when the NQS equivalent circuit is used, ther bi needed for R n fitting is not always equal to r bi . These issues are discussed in more detail in Chapter 5. 2.4.2 NQS and QS base resistance Based on the crowding theory [41], Fig. 2.17 (a) with NQS input is the correct equivalent circuit for intrinsic transistor where r bi is the true lumped intrinsic base resistance. r bi is generally dependent onI b . If the carrier density modulation in base is not considered, according to Appendix B r bi ? R BV 1 +I b R BV /V T . (2.78) (2.78) implies that r bi can be modeled by R BV paralleled with g be . Clearly the I b dependence is more severe at low temperatures. An exact consideration of carrier density modulation is di?cult, but if an averaged V BE is used to measure the level of carrier density modulation, the r bi can be derived as 1 r bi ? 1 R BV ? ? ? ? ? 3 + radicalbigg 1 +I b 4n 2 i N 2 A I bs 4 ? ? ? ? ? + I b V T = 1 R BV ? ? ? 3 + radicalBig 1 + I b I bk 4 ? ? ? + I b V T , (2.79) 56 where N A is the base doping concentration, I bs is the base saturation current, and I bk ? 4n 2 i N 2 A I bs ? A E 4 ? eD n N A d B . (2.80) A E is the emitter area. Two parameters, R BV and I bk , are needed to model the bias dependence of r bi . For high speed SiGe HBTs at room temperature, R BV is quite small, and the carrier density modulation is negligible, leading to the weak I b dependence of r bi as shown in the experimental extraction in Chapter 3 and in Fig. 2.18 below by analytical calculation. be g bej C b bed C b d r b out j m vg e ??v be g bej bed CC d j m evg ?? v NQS)(a )(b B CC E E with input effect NQSwithout input effect 'B bi r B ,bi QS r Figure 2.17: Equivalent circuit for intrinsic base of bipolar transistor with r bi : (a) With NQS input; (b) With QS input. If the QS and NQS equivalent circuits in Fig. 2.17 are used to model the same Y-parameters, r bi,QS becomes a lumped resistance related to the true intrinsic resistance r bi and the NQS delay resistance r b d (r b d = ? b in /C b bed ). Applying the Taylor expansion method described in Chapter 3 yields r bi,QS ? r bi +r b d parenleftBigg C b bed C b bed +C bej parenrightBigg 2 . (2.81) At low biases r b d C b bed = ? b in is a constant. Since C b bed increases versus bias and C bej is nearly constant, the ratio C b bed /(C b bed + C bej ) 2 has a maximum value at C b bed = C bej . This means that r bi,QS should increase at low biases and fall at high biases. r bi,QS is clearly larger than r bi . Fig. 2.18 compares r bi , r bi,QS and r b d extracted from the 1-D Y-parameters in (2.47). f T is shown for reference. r bi is calculated using (2.79). Peak f T =186 GHz. Note f T roll o? is not included. 57 ?=235. ? c = 0.57ps. C bej =38 fF. W E =0.12 ?m. L E =18 ?m. These parameters are consistent with those for experimental 200 GHz SiGe HBTs. V BE =1.02 V when ?n = N A for base injection. For 200 GHz SiGe HBTs, f T rolls o? around this V BE . So the V BE <1.02 V range is concerned for AC performance, where r bi is closely bias independent. r bi,QS indeed shows a bell shape and deviates from r bi by 43% at the bias when C b bed = C bej . 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0 5 10 15 20 V be (V) Resistances ( ? ) r bi with charge modulaton r bi without charge modulaton r bi,QS r d b V BE when ?n=N A A E =0.12?18 ?m 2 Peak f T =196 GHz ?=235 f T /10 GHz Figure 2.18: Comparison between r bi , r bi,QS and r b d . ? p = 1.54 ?10 ? 7s. ? E =0. p 00 = 0.466 /cm 3 . T = 300K. d E =120 nm. ? p0 =225 cm/cs 2 . ? n = 1.54 ? 10 ? 7s. ? B =5.4. n 00 = 23.3 /cm 3 (N A = 4.3 ? 10 1 8 /cm 3 ). d E =20 nm. ? n0 =450 cm/cs 2 . V exit = 1 ? 10 7 cm/s. ? c = 0.57ps. C bej =38 fF. W E =0.12 ?m. L E =18 ?m. A E =1cm 2 . At high current levels, r bi,QS ? r bi +r b d . Thus, it is meaningful and interesting to compare r bi with r b d under di?erent parameter changes. ? W E , d B (2.48) shows that r b d ? d B /W E . On the contrary, r bi ? W E /d B . During device scaling, both W E and d B are scaled. Therefore, for di?erent generation devices, it is di?cult to compare the relative importance of r b d . For the same generation devices, however, the smaller the emitter width, the more important r b d . ? L E , N A (2.48) shows that r b d ? 1/L E N A , the same as r bi . 58 ? Ge gradient (?) As shown in Fig. 2.6, ? reduces r b d . However ? has little e?ect on r bi . Therefore, for two devices that only di?er in terms of Ge gradient, the one with the larger Ge gradient will have less modeling error using QS equivalent circuit at high current levels. Overall, unlike f T , one cannot compare the relative importance of r b d for di?erent generations of devices without knowing the design details of each generation. 2.5 Present noise models and implementation problems 2.5.1 SPICE model The default i b and i c noise models in current CAD tools are the same as those used in SPICE [42]. i b and i c are assumed to be shot like and uncorrelated. This is denoted as SPICE model and the PSDs are given by S SPICE ib = 2qI B ,S SPICE ic = 2qI C ,S SPICE icib ? = 0. (2.82) Both the theoretical analysis above and experimental data have shown that this highly simplified model is not su?cient for high frequency applications, particularly at the higher biasing currents required to achieve high speed operation [43?48]. In particular, S ib has been shown to increase with frequency, and the correlation S icib ? is significant and cannot be neglected [43,45?47,49]. 2.5.2 Transport noise model The transport noise model [43] [44] has recently been shown to work better than the SPICE model [43,45] by taking into account the correlation. The essence of this model is that the collector current noise is transported from the electron current shot noise in the emitter-base junction, with 59 a noise transit time ? n [43]: S Tran ib = 2qI B + 4qI C [1 ?Rfractur(e j?? n )], S Tran ic = 2qI C , S Tran icib ? = 2qI C (e ?j?? n ? 1). (2.83) However, in practical devices, with only a single parameter ? n , simultaneous fitting of measured S ib and S icib ? can become di?cult. Simultaneous fitting of NF min , R n , G opt and B opt is challenging in some cases. 2.5.3 Brutal use of van Vliet model The van Vliet model serves as the basis of several other models, e.g. [47,50?53]. Since the van Vliet derivation does not consider electron transport in the collector-base space charge region (CB SCR), characterized by transit time ? c , (2.67) cannot be automatically extended to include the CB SCR e?ect simply by replacing Y B with Y-parameters of whole intrinsic transistor [5]. In the literature [50?53], the van Vliet model is often used unphysically: ? Y B should explicitly include the input NQS e?ect, that is, Rfractur(Y B 11 ) should be frequency de- pendent, so that the frequency dependence of S B bi can be modeled. However, all the imple- mentations reported use QS equivalent circuits whose Rfractur(Y B 11 ) is frequency independent. ? The Y-parameters of the whole intrinsic transistor, including the CB SCR, are used in (2.67), the results are recognized as the noise of whole transistor without justification. For scaled bipolar transistors, CB SCR electron transport becomes more significant than base electron transport. The van Vliet model must be improved to including this e?ect. 60 2.5.4 4kTr bi for r bi noise With the uniform f T assumption, the noise PSD of i rbi can be approximated with 4kT/r bi for SiGe HBTs whose crowding e?ect is negligible. This is the base resistance noise model commonly used in CAD tools and in the literature [42, 50?53]. However, problems were encounted when using 4kTr bi for noise modeling based on either QS or NQS lumped equivalent circuits: First, the noise resistance R n cannot be well modeled, as it is sensitive to base hole noise. One has to use an empirical S ic based on noise extraction, which is unphysically larger than 2qI c for 50 GHz SiGe HBTs ( see Chapter 4). Another problem is that the absolute value of the imaginary part of the noise parameter Y opt , i.e. B opt , is overestimated by the van Vliet model based on NQS equivalent circuit. The deviation cannot be eliminated by choosing an appropriate r bi . The uniform f T assumption is not justified for high speed SiGe HBTs with narrow emitter width (? 0.24?m). The f T within the fringe transistor is smaller than that in the main transistor due to larger base width. Such non-uniformf T does not have a significant e?ect oni b andi c noises. However, the base hole noise now has to be modeled by a noise source at the input together with a correlated noise current source at the output [6]. 2.6 Methodologies to improve noise modeling Two methods are proposed to improve noise modeling in this work: ? The first method is to implement a semi-empirical model for the intrinsic transistor noise based on noise extraction for the QS equivalent circuit, as detailed in Chapter 4. Eq. (2.77) is used for S i rbi and then the intrinsic noise S ib , S icib ? and S ic are extracted from device noise parameters using standard noise de-embedding methods [54]. Equations can then be developed to model these noise sources. The deviation caused by the use of a QS input equivalent circuit, and hence the lumping of input NQS resistance into r bi , as well as the use of (2.77), is all included in the intrinsic noise. The extracted S ib and S ic are thus not 61 precisely the physical intrinsic transistor noises, which can only be obtained through higher order modeling that includes input NQS and noise crowding e?ects. The noise sources and their correlation, are first modeled as functions of frequency (?). The coe?cients are then extracted and modeled as a function of biasing current through g m . As the QS equivalent circuit is used, existing parameter extraction methods can be applied, and the proposed model can be readily implemented in current compact models. This method was verified in VBIC model using Verilog-A by Advance Design System (ADS) circuit simulator. ? The second method improves compact RF noise modeling for SiGe HBTs based on NQS equivalent circuit using new electron and hole noise models as detailed in Chapter 5. The impact of CB SCR on electron RF noise is examined to be important for scaled SiGe HBTs. The van Vliet model is then improved to account for the CB SCR e?ect. The impact of the fringe BE junction on base hole noise is further investigated. Due to the fringe e?ect, the base hole noise should be modeled with correlated noise voltage source and noise current source in hybrid representation. The base noise resistance is found to be di?erent from r bi , and cannot be explained by fringe e?ect alone. An extra parameter R bn is included for base noise resistance. With a total of four bias-independent model parameters, the combination of electron and hole noise model provides excellent noise parameter fittings for frequencies up to 26 GHz and all biases before f T roll o? for three generations of SiGe HBTs. 2.7 Summary Di?usion noise is the major noise source in SiGe HBTs. The van Vliet model is still applicable for typical SiGe HBTs with a base built-in field, and can be directly extended to include emitter hole noise. The CB SCR e?ect is important for aggressively scaled devices, and should be included in noise modeling. The fringe BE junction e?ect impacts base hole noise, and should be included for noise modeling. Present noise models are not su?ciently accurate for RF noise modeling at 62 high frequencies. Both semi-empirical and physical methods are used in this work to improve RF noise modeling. 63 CHAPTER 3 SMALL SIGNAL PARAMETER EXTRACTION Small-signal equivalent circuit accurately modeling both AC and noise characteristics of SiGe HBTs is very useful for RF circuit design as well as understanding of device physics. The topology of equivalent circuit determines the physics e?ects that can be accounted for, accuracy of final AC and noise characteristics, and a?ects circuit parameter extraction procedure as well as the physical soundness of extracted equivalent circuit parameters. Microscopic noise physics based noise models of bipolar transistors described in Chapter 2 require modeling of input NQS e?ect. This chapter examines small signal equivalent circuit modeling of input NQS e?ect including CB CSR delay, and its parameter extraction. Accurate parameter extraction is challenging in practice due to the large number of parameters involved, in spite of the various methods proposed, including both direct or analytical methods and numerical optimization based methods. Including the input NQS e?ect makes the circuit topology even more complex. Numerical methods often lead to physically meaningless values, as reviewed in [55]. The full analytical expressions of Y/Z-parameters are too complex to be directly used for extraction. This chapter presents a new direct extraction method based on Taylor series expansion analysis of Y/Z-parameter expressions [8]. The real part of Y/Z-parameters is approximated up to second order of frequency. The imaginary part is approximated up to first order. The expansion coe?cients are obtained as simple functions of equivalent circuit parameters, allowing straightfor- ward parameter extraction. The extracted parameters, such as intrinsic base resistance and excess phase delay time, show more physical bias dependences compared to conventional extraction. The 64 utility of this method is demonstrated using SiGe HBTs of di?erent sizes from di?erent technol- ogy generations, a SiGe HBT with 180 GHz peak f T is used below for illustration of extraction procedure. 3.1 Necessity of including input NQS e?ect in equivalent circuit Of particular interest to modeling of the RF noise in the base current is the frequency de- pendence of the real part of the input admittance, Rfractur(Y 11 ), due to base electron transport, which is responsible for the frequency dependence of base current noise as well as the correlation be- tween base and collector current RF noises as discussed in Chap 2. We have also show the NQS equivalent circuit for base region in Fig. 2.3. The base minority carrier charge responds to base emitter voltage by the input NQS delay time ? b in , then the collector current at the end of base region responds to the stored base minority carrier charge by the output NQS delay time ? b out . For a real device, particularly modern SiGe HBTs, the output collector current is further delayed, compared to the current at the end of the base, by the CB SCR transit time ? c . Although the circuit topology of Fig. 2.3 (a) was derived for the base region, the same circuit topology is capable of including ? b in , ? b out and ? c delays, as shown below, with proper modifications to values of its elements. However, in present BJT models e.g. SPICE Gummel-Poon and VBIC, and all the recent direct parameter extraction methods [55?61], a circuit topology of Fig. 2.3 (b) is used for the in- trinsic transistor. Although? b out has been included in Fig. 2.3 (b), the input NQS e?ect is neglected. We found problems in using such circuit topology for both AC and noise modeling of high peak f T SiGe HBTs. First of all, with Fig. 2.3 (b), the real part of Y 11 of intrinsic device, Rfractur(Y 11 ), is frequency independent [3,4]. Correct modeling of Rfractur(Y 11 ) is crucial for a physically meaningful implementation of microscopic noise physics based base and collector current RF noise models [2], 65 such as the van Vliet model, see (2.67). The increase of base current noise with frequency is di- rectly proportional to the Rfractur(Y 11 ) of the base and this part of the base current noise is correlated with the collector current noise. Another major problem with using the circuit topology with QS input is extraction of intrinsic base resistance, which we denote as r bi,QS . r bi,QS is often extracted using impedance semicircle fitting method [62], which determines base resistance from the x-axis intercept of a semicircle fitted to (Rfractur[H 11 ],Ifractur[H 11 ]) points of di?erent frequencies on a complex impedance plane for an equivalent circuit excluding the extrinsic base resistance and CB capacitance. We find that the r bi,QS extracted increases unphysically at low base currents because r b d is lumped into r bi [4]. Such unphysical result inr bi extraction was also observed by others, e.g. in [63], and is typical of existing r bi or r b extraction. The use of r bi,QS also leads to an overestimation of minimum noise figure [4]. The extracted excess phase delay time of the intrinsic device shows a strong bias-dependence even at low current levels, a clearly unphysical result. The inaccurate excess phase delay time directly a?ects the correlation between base current noise and collector current noise throughIfractur(Y 21 )invan Vliet model, see (2.67). Using an equivalent circuit based on Fig. 2.3 (a) which explicitly includes the input NQS e?ect, the abnormal bias dependence of r bi,QS can be explained and avoided. A more physical value of r bi is obtained, which also helps improving noise modeling. The extracted excess phase delay time shows a more physical bias-dependence [8]. 3.2 NQS Equivalent circuit The input NQS equivalent circuit of Fig. 2.3 (a) proposed by Winkel was based on frequency domain solution in the base only. For modern SiGe HBTs, it is necessary to include the impact of CB SCR. 66 3.2.1 CB SCR e?ect The Y-parameters of base, Y bs , can be obtained from Fig. 2.3 (a) as Y bs 11 = g be +j?C bej + j?C b bed 1 +j?? b in ,Y bs 21 = g m e ?j?? b out 1 +j?? b in . (3.1) Y bs 12 and Y bs 22 are equal to zero. With CB SCR, the Y-parameters of whole transistor, Y al , can be calculated from Y bs and ? c as [5] Y al 11 = Y bs 11 + (1??)Y bs 21 ,Y al 21 = ?Y bs 21 , (3.2) where ? = (1 ? e ?2j?? c )/(2j?? c ). (More details in (5.1)). A close inspection shows that we can still use the circuit topology of Fig. 2.3 (a) or Fig. 3.1 (a) to describe Y al with an accuracy up to the second order in frequency. The intrinsic NQS equivalent circuit including CB SCR is shown in Fig. 3.1 (b). The equivalent circuit parameters of whole transistor (C bed , r d or ? in , and ? out ) are related to those of base region (C b bed , r b d or ? b in , and ? b out ) and ? c . Denoting ? tr ? C bed /g m and ? b tr ? C b bed /g m ,wefind ? tr = ? b tr +? c , ? in = ? b in +? c ? b out + 2? c /3 ? b tr +? c , ? out = ? b out +? c ? b tr ?? b out +? c /3 ? b tr +? c . (3.3) Detailed derivation is given in Appendix C. (3.3) reveals that? in ? 2/3? tr and? out ? 1/3? tr when ? c >> ? b tr ,? b in ,? b out , the case of SiGe HBTs with ultra narrow base. To further investigate the ? c e?ect, we use analytical Y-parameter expressions of ideal 1-D base region derived from frequency domain solution of the drift-di?usion equations in (2.47). First, we extract ? b tr , ? b in and ? b out from the analytical Y-parameters. Then ? tr , ? in and ? out are evaluated 67 be g bej C b bed C b d r b out j m vg e ??v )(a B C E c without ? 'B be g bej C bed C d r out j m vg e ??v ()b B C E 'B c with ? Figure 3.1: Intrinsic NQS small signal equivalent circuit of SiGe HBTs: (a) without ? c ; (b) with ? c . using (3.3). Fig. 3.2 (a) shows ? in /? tr and ? out /? tr versus base width for di?erent ? c at ? = 6, the typical value for SiGe HBTs. ? c increases the input NQS e?ect and decreases excess phase delay time for narrow base transistor. Fig. 3.2 (b) shows ? in /? tr and ? out /? tr versus base width for di?erent ? at ? c =0.6 ps. For SiGe HBTs with higher Ge grading, i.e. larger ?, the normalized input NQS e?ect becomes larger. 0 20 40 60 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Base Width (nm) ? in / ? tr and ? out / ? tr 0 20 40 60 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Base Width (nm) ? in / ? tr and ? out / ? tr ? in /? tr ? out /? tr ? = 3, 6, 9 (b) ? c = 0.6 ps (a) ? = 6 ? c = 0.3, 0.6, 0.9 ps Figure 3.2: CB SCR e?ect on ? in and ? out . For the 1-D base region, V bs sat = 1?10 7 cm/s, ? n0 = 270 cm/Vs 2 , ? n =0.154 ?s, T=300 K. The importance of input NQS can be measured by comparing the frequency dependent part of Rfractur(Y 11 ) with g be (? I b /V T ). The frequency dependent part of Rfractur(Y 11 ) can be calculated from (3.2) 68 as ? 2 C bed ? in . For fixed frequency, bias, and emitter design, the importance of input NQS can thus be measured by C bed ? in . Fig. 3.3 shows C bed ? in versus ? for di?erent ? c . The larger ? c or the larger base Ge gradient, the more important the input NQS e?ect. 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 ?20 ? C bed ? in (F sec) ? c = 0, 0.3, 0.6 ps V BE =0.8 V Figure 3.3: CB SCR and Ge gradient impacts on the importance of input NQS e?ect. For the 1-D base region,d B =20 nm,V bs sat = 1?10 7 cm/s,? n0 = 270 cm/Vs 2 ,? n =0.154?s,T=300 K.V BE =0.8 V. 3.2.2 NQS equivalent circuit Fig. 3.4 shows the small signal equivalent circuit used. The substrate is tied to emitter to facilitate two-port RF measurements using two GSG probes. Block B X is the equivalent circuit excluding r e and substrate network from the full circuit. B M is the block obtained by excluding r c from B X . Block B IR is the intrinsic device with r bi . Block B I is the intrinsic device without r bi . Note that the control voltage for the transconductance term is the total intrinsic BE voltage drop across r d and C bed , instead of the voltage across C bed as done in Fig. 2.3 (a). This makes the total excess phase delay time ? in + ? out , designated as ? d below. Lumping ? in and ? out into ? d does not lose modeling accuracy and is advantageous for extraction, as the input NQS and output excess phase delays are now separated. C bex is the extrinsic BE capacitance, for example, the capacitance between base and emitter through spacer, and is non-negligible for small devices of 69 high f T SiGe HBTs. The output conductance r o is neglected due to the large Early voltage in SiGe HBTs. Parameter r u relating to the V CB modulation of neutral base recombination current has no significant e?ect on either Y-parameters or RF noise at frequencies above 1 GHz, and is neglected. All other extrinsic parameters have their conventional meanings. For convenience, we define total BE capacitance, total BC capacitance and its partition factor ? as C bet ? C bed +C bej ,C bct ? C bcx +C bci ,?? C bci /C bct . be g bej C bed C d r d j m evg ?? v bcx C bci C bx r bi r c r e r cs r cs C B E C X B I B bex C M B IR B Figure 3.4: Small signal equivalent circuit of SiGe HBTs with substrate tied to emitter. Now we discuss a few assumptions we will make on the bias dependence of small signal pa- rameters. Extrinsic elementr bx ,r c ,C bex are considered as bias-independent. ? is also considered as bias-independent. Strictly speaking, BE depletion capacitance C bej is a function of V BE . However, for the RF bias range across which f T is high, the variation of V BE (0.80?0.92 V) is small, and C bej can be considered bias-independent. The base charge modulation and DC crowding e?ect are 70 negligible for SiGe HBTs because of the high base doping [1]. Consequently r bi is weakly bias de- pendent before high injection base push out occurs. In our extraction method, we first consider r bi as bias-independent during extrinsic parameter extraction. Bias-independent r bi is extracted. Such assumption is justified for high f T SiGe HBTs due to the high base doping as discussed in Chapter 2. It is important to note that the bias dependent r bi,QS extracted using equivalent circuit without including input NQS e?ect is a lumped parameter involving r d and r bi [4]. The bias dependence of r bi,QS can be well reproduced by our NQS circuit with bias-independent r bi as discussed in Section IV. To make our method more general, we also give a method that can extract bias-dependent r bi , where the delay time ratios ? in /? tr and ? d /? tr are considered as constant for all biases. 3.3 Parameter extraction S-parametersaremeasuredforSiGeHBTsofdi?erentprocessgenerationswithdi?erentemit- ter geometries. Only a 180 peak f T device is used below for illustration of parameter extraction. The HBT has an emitter area A E = 0.12 ? 6 ? 1 ?m 2 . S-parameters are measured on-chip using a 8510C Vector Network Analyzer (VNA) from 1-48 GHz. The S-parameters are de-embedded using standard OPEN/SHORT structures. ?Active? measurement is made by sweeping V BE (0.80?0.98 V) with V CE =1.5 V. The f T rolls o? at V BE =0.921 V. ?Cold? measurement (V BE =0V,V CE =1.5 V) is also made to extract the substrate network and the C bc partition factor ?. The analytical Y/Z-parameter expressions for each block in Fig. 3.4 can be derived in a way very similar to the derivation in [3]. Note that C bej is not explicitly split from C bed in [3]. The results are shown in Appendix D. We also use the symbolic analysis of MATLAB to obtain these results. The source code is given in Appendix D. Some of the Y-parameters and the inverse of Z-parameters, or their linear combinations, can be used for parameter extraction. We denote them as T. Due to the complexity of equivalent circuit, the exact T expressions, however, are di?cult to use for direct parameter extraction. We notice that all of the T expressions are functions of 71 frequency or ?, and have no singularities at ?=0. Hence we can make Taylor expansions for them at ?=0. The real part of T is expanded up to the second order of ? and the imaginary part up to the first order of ?, Rfractur(T) =[Rfractur(T) 0 ] + [Rfractur(T) 2 ]? 2 +o(? 2 ), Ifractur(T) =[Ifractur(T) 1 ]?+o(?). (3.4) We emphasize that the real part expansion consists of only even orders of?terms and the imaginary part consists of only odd orders of ? terms. Any admittance, impedance or transconductance element containing ? in Fig. 3.4 contains j?. Since the real part of T only contains even order terms of j, the real part must only contain even order terms of ?. Similarly, the imaginary part of T only contains odd order terms of j, hence the imaginary part must only contain odd order terms of ?. The coe?cients directly relate to small signal equivalent circuit parameters and can then be used for parameter extraction, as detailed below. It is critical to accurately extract these coe?cients for certain Y/Z-parameter. To extract Rfractur(T) 0 and Rfractur(T) 2 , we plot Rfractur(T) versus ? 2 . A linear relation should be observed at low frequencies. With a linear fitting, the y-axis intercept gives Rfractur(T) 0 , and the slope gives Rfractur(T) 2 . Fig. 3.5 (a) illustrates the extraction of Rfractur(Y BM 11 +Y BM 12 ) 2 at low, medium and high biases. To extract Ifractur(T) 1 , we plot Ifractur(T)/? versus ? 2 . A linear relation is observed at low frequencies. With a linear fitting, the y-axis intercept gives Ifractur(T) 1 . Fig. 3.5 (b) illustrates the extraction of Ifractur(Y BM 22 ) 1 at low, medium and high biases. One could also extract Ifractur(T) 1 from the slope of Ifractur(T) versus ? at lower frequencies where the third order term is weak. This, however, is not necessary when plotting Ifractur(T)/? versus ? 2 . We now detail the extraction procedure parameter by parameter. 72 0 2 4 x 10 22 0.5 1 1.5 x 10 ?14 ? 2 (Hz 2 ) Im(Y 22BM )/ ? I C increasing 0 2 4 x 10 22 0 1 2 3 4 5 6 x 10 ?3 ? 2 (Hz 2 ) Re(Y 11BM +Y 12BM ) 0.12?6?1 ?m 2 V CE =1.5 V I C increasing Linear fitting Experiment ( a ) ( b ) Figure 3.5: Illustration of Taylor expansions coe?cient extraction for (a) Rfractur(Y BM 11 + Y BM 12 ) 2 , and (b) Ifractur(Y BM 22 ) 1 . 3.3.1 C cs , r cs , ? and r e extraction Cold measurement data is used to extract substrate network (r cs ,C cs ) using the method in [64], that is C cs = Ifractur(Y Cold 22 +Y Cold 12 ) ? ,r cs = Rfractur(Y Cold 22 +Y Cold 12 ) bracketleftbig Ifractur(Y Cold 22 +Y Cold 12 ) bracketrightbig 2 . (3.5) The CB capacitance partition factor ? can also be extracted from cold measurement data using the method in [65], explicitly ? ? Ifractur(Y Cold 11 +Y Cold 12 ) Ifractur(Y Cold 12 ) Rfractur(Y Cold 12 ) Rfractur(Y Cold 11 +Y Cold 12 ) . (3.6) 73 ? usually is small for SiGe HBTs. For example, ?=0.21 for the device used for illustration. r e is determined by the y-axis intercept of Rfractur(Z 12 ) versus 1/I c .After de-embedding the substrate network and r e , the Z-parameters of block B X are known, explicitly, Z BX 11 = Y 22 ? j?C cs 1+j?C cs r cs Y 11 Y 22 ?Y 12 Y 21 ?Y 11 j?C cs 1+j?C cs r cs ?r e ,Z BX 12 = ? Y 12 Y 11 Y 22 ?Y 12 Y 21 ?Y 11 j?C cs 1+j?C cs r cs ?r e , Z BX 21 = ? Y 21 Y 11 Y 22 ?Y 12 Y 21 ?Y 11 j?C cs 1+j?C cs r cs ?r e ,Z BX 22 = Y 11 Y 11 Y 22 ?Y 12 Y 21 ?Y 11 j?C cs 1+j?C cs r cs ?r e . (3.7) 3.3.2 C bct , C bcx , C bci , r c and g m extraction For block B X , we obtain the following Taylor expansion coe?cients using symbolic analysis in MATLAB (code is given in Appendix D) Ifractur bracketleftbig 1/(Z BX 22 ?Z BX 21 ) bracketrightbig 1 = C bct , (3.8) Rfractur bracketleftbig 1/(Z BX 22 ?Z BX 21 ) bracketrightbig 2 = C 2 bct parenleftbigg r c ??r bi C bcx +C bex C bct parenrightbigg , (3.9) Rfractur bracketleftbig 1/Z BX 12 bracketrightbig 0 = g m . (3.10) The basic idea is that r bx does not impact the above Z-parameters. Therefore, the coe?cients are not a?ected by r bx . C bct can be directly obtained from (3.8), which essentially is the method reported in [66]. With C bct and ? known, C bcx and C bci are obtained. g m can be directly obtained from (3.10). r c can be extracted from (3.9) by neglecting the term related to small ? as r c ? Rfractur bracketleftbig 1/(Z BX 22 ?Z BX 21 ) bracketrightbig 2 C 2 bct . (3.11) 74 Note that (3.11) is sensitive to substrate network de-embedding. Inaccurate C cs and r cs will result in unphysical bias dependence of r c . Now the Z-parameters of block B M can be obtained Z BM 11 = Z BX 11 ,Z BM 12 = Z BX 12 ,Z BM 21 = Z BX 21 ,Z BM 22 = Z BX 22 ?r c . (3.12) Consequently, the Y-parameters of block B M are known. 3.3.3 r bx and r bi extraction Fig. 3.6 shows the block B M of the small signal equivalent circuit. We have the following be g bej C bed C d r d j m evg ?? v bcx C bci C bx r bi r B E C bex C M B Figure 3.6: Small signal equivalent circuit of SiGe HBTs for block B M . Taylor expansion coe?cients for block B M using symbolic analysis Ifractur(Y BM 11 +Y BM 12 ) 1 ?C bet +C bex , (3.13) Ifractur(Y BM 21 ?Y BM 12 ) 1 ?g m {(r bx +r bi )(C bet +C bex ) + [? d + (r bx +?r bi )C bct ?r bi C bex ]}, (3.14) Ifractur(Y BM 22 ) 1 ?C bct +g m C bct (r bx +?r bi ). (3.15) 75 Approximation (r bx +r bi )g be << 1 is used, meaning that these expressions are less accurate at high biases. For SiGe HBTs, due to high base doping, this approximation is valid for low and medium biases, where we extract r bx . According to (3.13) and (3.14), we have Ifractur(Y BM 21 ?Y BM 12 ) 1 g m ? (r bx +r bi )Ifractur(Y BM 11 +Y BM 12 ) 1 +U, (3.16) where U ? ? d +(r bx +?r bi )C bct ?r bi C bex is bias independent at low biases since all the parameters involved are bias independent at low biases. Fig. 3.7 plots Ifractur(Y BM 21 ?Y BM 12 ) 1 /g m versus Ifractur(Y BM 11 + Y BM 12 ) 1 . A linear relation is observed. r bx +r bi is determined by the slope of fitting line according to (3.16). According to (3.15), if we plot Ifractur(Y BM 22 ) 1 versus g m C bct , a linear relation can be obtained 0 0.2 0.4 0.6 0.8 1 1.2 x 10 ?13 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 ?12 Im(Y 11 BM +Y 12 BM ) 1 Im(Y 21BM ?Y 12BM ) 1 /g m 0.12?6?1 ?m 2 V CE =1.5 V Linear fitting Experiment Figure 3.7: Ifractur(Y BM 21 ?Y BM 12 ) 1 /g m versus Ifractur(Y BM 11 +Y BM 12 ) 1 . The slope of fitting line gives r bx +r bi . as shown in Fig. 3.8. The slope gives r bx +?r bi . With ? known, r bx and r bi can then be calculated from r bx +?r bi and r bx +r bi . 76 0 0.5 1 1.5 x 10 ?15 1 1.5 2 2.5 x 10 ?14 g m C bct Im(Y 22BM ) 1 0.12?6?1 ?m 2 V CE =1.5 V Linear fitting Experiment Figure 3.8: Ifractur(Y BM 22 ) 1 versus g m C bct . The slope of fitting line gives r bx +?r bi . 3.3.4 C bex and C bej extraction As C bej is the low current limit of C bet , the low current limit of Ifractur(Y BM 11 + Y BM 12 ) 1 gives C bex +C bej according to (3.13). For block B M ,wehave Rfractur(Y BM 11 +Y BM 12 ) 2 ?r bx (C bej +C bex )(C bej +C bex +C bct ) +r bi C 2 bej + (?r bi C bct )C bej +g m W, (3.17) where W is a complex function of circuit parameters. Again (r bx +r bi )g be << 1 is used. Consider now the low current limit of (3.17), {r bi }C 2 bej +{(?r bi C bct )}C bej +{r bx (C bej +C bex )(C bej +C bex +C bct ) ?Rfractur(Y BM 11 +Y BM 12 ) 2 | g m ?0 } = 0. (3.18) 77 This is a quadratic equation for C bej with all the coe?cients in the curly brackets known. C bej corresponds to the positive root. C bex is then obtained. To extract the low current limits for Taylor expansion coe?cients, Ifractur(Y BM 11 + Y BM 12 ) 1 and Rfractur(Y BM 11 +Y BM 12 ) 2 , we make a linear fitting versus g m within low current domain for these coe?- cients. The y-axis intercepts give the corresponding low current limits. Fig. 3.9 illustrates the low current limit extraction of Rfractur(Y BM 11 +Y BM 12 ) 2 and Ifractur(Y BM 11 +Y BM 12 ) 1 . 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 x 10 ?13 gm (s) Im(Y 11BM +Y 12BM ) 1 low current limit ( b ) 0 0.05 0.1 0 0.5 1 1.5 2 x 10 ?25 gm (s) Re(Y 11BM +Y 12BM ) 2 low current limit ( a ) Linear fitting Experiment Figure 3.9: Illustration of low current limit extraction for (a) Rfractur(Y BM 11 +Y BM 12 ) 2 , and (b) Ifractur(Y BM 11 + Y BM 12 ) 1 . 78 So far we have extracted all the extrinsic parameters and intrinsic parameter r bi . The Y- parameters of block B IR can be obtained as Y BIR 11 = Y BM 11 1 ?r bx Y BM 11 ?j?(C bex +C bcx ),Y BIR 12 = Y BM 12 1 ?r bx Y BM 11 +j?C bcx , Y BIR 21 = Y BM 21 1 ?r bx Y BM 11 +j?C bcx ,Y BIR 22 = Y BM 12 ?r bx (Y BM 11 Y BM 22 ?Y BM 12 Y BM 21 ) 1 ?r bx Y BM 11 ?j?C bcx , (3.19) which can be transformed into Z-parameters. The Z-parameters of the intrinsic transistor, block B I , are Z BI 11 = Z BIR 11 ?r bi ,Z BI 12 = Z BIR 12 ,Z BI 21 = Z BIR 21 ,Z BI 22 = Z BIR 22 . (3.20) Consequently, the Y-parameters of block B I are now known. 3.3.5 C bed , r d , g be and ? d extraction The Y-parameter Taylor expansion coe?cients of block B I are obtained as Ifractur(Y BI 11 +Y BI 12 ) 1 = C bet , Ifractur(Y BI 21 ?Y BI 12 ) 1 = g m ? d , (3.21) Rfractur(Y BI 11 ) 0 = g be , Rfractur(Y BI 11 ) 2 = C bed ? in . (3.22) These expressions are accurate for all biases because the factor (r bx + r bi )g be does not exist any more. C bet is given by Ifractur(Y BI 11 + Y BI 12 ) 1 . Strictly speaking, C bed can be directly calculated from C bet ? C bej as C bej is known. This is, however, not accurate for low biases where C bed 1 indicates a problem with either noise measurement or equivalent circuit parameter extraction. Fig. 4.11 (a) shows the real and imaginary part of c versus frequency at I C =17.9 mA. Fig. 4.11 (b) shows the magnitude of c versus frequency at I C =17.9 mA. Fig. 4.11 (c) shows the real and imaginary part of c versus I C at f=25 GHz. Fig. 4.11 (d) shows the magnitude of c versus I C at f=25 GHz. In most cases, |c|?1 is satisfied as shown in Fig. 4.11 (b) and (d). Observe that the magnitude of c is close to unity in many cases, therefore the correlation in SiGe HBTs is important and cannot be neglected. Another related observation is that Rfractur(c) is nearly one order of magnitude smaller than Ifractur(c) in practice for frequencies less than half of f T . This is also true for noise simulation results. As a result, Rfractur(S icib ?) is much less important than Ifractur(S icib ?), which is further supported by the sensitivity analysis given below. 0 10 20 ?1 ?0.5 0 I C (mA) ? (c) and ? (c) 0 10 20 0.6 0.8 1 I C (mA) Mag(c) 0 10 20 30 ?1 ?0.5 0 f (GHz) ? (c) and ? (c) 0 10 20 30 0 0.5 1 f (GHz) Mag(c) Real Imag Real Imag Symbol marks: Experiment Solid line: New model 0.24?20?2?m 2 V CE =1.5 V f=25 GHz f=25 GHz I C =17.9 mA I C =17.9 mA (a) (b) (c) (d) Figure 4.11: Normalized correlation c of the extracted intrinsic noise for 0.24?20?2?m 2 device: (a)Rfractur(c) andIfractur(c) versus frequency atI c =17.9 mA; (b) Magnitude ofcversus frequency atI c =17.9 mA; (c) Rfractur(c) and Ifractur(c) versus I c at f=25 GHz; (d) Magnitude of c versus I c at f=25 GHz. 110 Table 4.2: Parameter values of the simplified noise model for Experiment 0.24 ? 20 ? 2 ?m 2 50 GHz SiGe HBT . Parameter Value Parameter Value ? bb ? 2 K bb ? 1.3934 ? 10 ?43 ? i cb ? 1.8 K i cb ? 6.2936 ? 10 ?32 ? cc ? 1 K cc ? 2.5782 ? 10 ?20 B bb ? 1 ? 10 ?60 B i cb ? 0 K br cb ? 3.0348 ? 10 ?22 K kr cb ? 1.0809 ? 10 ?43 B cc ? 7.8210 ? 10 ?21 4.2.6 Noise Parameter Modeling Results Using the methods described above, the 11 bias independent noise model parameters are ex- tracted. Table 4.2 lists the parameters values (in MKS units) for the measured 0.24 ? 20 ? 2 ?m 2 SiGe HBT. Fig. 4.12 shows the modeled and measured noise parameters versus frequency at I C =17.9 mA.Rfractur(S icib ?) is much smaller thanIfractur(S icib ?), making it possible to neglectRfractur(S icib ?). We thus also calculate the noise parameters with Rfractur(S icib ?)=0. The data marked with circle is the measurement. The dash line represents the SPICE model, the solid line represents the new model with Rfractur(S icib ?) and the dash dot line represents the new model with Rfractur(S icib ?)=0. Fig. 4.13 shows the noise parameters of the same device as functions of collector current at f=25 GHz. It is inconsistent to implement the van Vliet model using a QS equivalent circuit, thus noise parameters are not shown for the van Vliet model. Similarly, the transport noise model result is not shown either due to its limitations in noise source modeling. Usingtheproposednewmodel, excellentfittingisobtainedforallofthefournoiseparameters, at all frequencies and across all biasing currents. Even with Rfractur(S icib ?)=0, only Y opt is slightly a?ected at frequencies above 20 GHz. This is beneficial as we can save two noise model parameters related to Rfractur(S icib ?) by setting Rfractur(S icib ?)=0. To quantify errors from using current CAD tools, we 111 also show results obtained using the SPICE model. At low current level, the SPICE model works well, as was shown in [69]. However, at high current level, NF min , G opt and B opt are overestimated. 0 10 20 30 1 2 3 4 5 f (GHz) NF min (dB) 0 10 20 30 0.1 0.15 0.2 f (GHz) Rn (/Z0) 0 10 20 30 0 0.02 0.04 0.06 0.08 f (GHz) Gopt (S) 0 10 20 30 ?0.01 0 0.01 0.02 0.03 f (GHz) Bopt (S) Experiment SPICE This work with ?(S icib *) This work with ?(S icib *)=0 0.24?20?2 ?m 2 V CB =1.5 V I C =17.9 mA Figure 4.12: Noise parameters versus frequency for the measured noise data. I c =17.9 mA. A E =0.24 ? 20 ? 2?m 2 . So far we have used the A E =0.24 ? 20 ? 2?m 2 device as an example. A natural question is how the noise sources scale with geometry. For ideal scaling, S ib , S ic and S icib ? should all scale with the emitter area A E for the same biasing current density. This is indeed the case according to the extracted data from various geometries as shown below. Nonideal noise parameter scaling with geometry is mostly from nonideal scaling of resistances, such as r bx and r bi . 4.2.7 Sensitivity Analysis For understanding of model to data correlation and model parameter extraction, it is useful to calculate the sensitivity of noise parameters (NF min , R n and Y opt ) to the intrinsic noise model parameters. Table 4.3 gives the percentage change of noise parameters responding to 5% change 112 0 10 20 0 2 4 6 I C (mA) NF min (dB) 0 10 20 0 0.2 0.4 I C (mA) Rn (/Z0) 0 10 20 0 0.02 0.04 0.06 0.08 I C (mA) Gopt (S) 0 10 20 ?0.05 0 0.05 I C (mA) Bopt (S) Experiment SPICE This Work 0.24? 20? 2 ? m 2 V CE =1.5V f=25GHz Figure 4.13: Noise parameters versus collector current for the measured noise data. f=25 GHz. A E =0.24 ? 20 ? 2?m 2 . of the intrinsic noise model parameters. I C =17.9 mA, f=25 GHz. A E =0.24?20?2?m 2 . We list only the noise model parameters that have a large impact on the noise parameters at higher biasing currents and higher frequencies. B bb ?, B i cb ? and B cc ? mainly a?ect low bias noise parameters and thus are not listed. The sensitivity analysis shows that: ? The noise parameters are sensitive to model parameters for Ifractur(S icib ?), including K i cb ? and ? i cb ? . ? S ic (through K cc ?) is as important as S ib (through K bb ?) at high frequencies. ? The noise parameters are not sensitive to Rfractur(S icib ?) (through K br cb ? and K kr cb ? ). This explains why Rfractur(S icib ?) can be set to zero and produce good noise parameter fitting. 113 Table 4.3: Parameter sensitivity at I C =17.9mA, f=25GHz. A E =0.24 ? 20 ? 2?m 2 . Percentage variance of noise parameters responding to 5% variance of the noise model parameters. NF min R n G opt B opt K i cb ? 29.10% 10.17% 22.5% 233.3% ? i cb 22.58% 8.6% 21.25% 175.0% K bb ? 7.94% 4.94% 4.14% 78.82% K cc ? 7.94% 3.48% 5.79% 50% ? bb 7.62% 4.94% 4.72% 78.57% ? cc 3.29% 0.3% 2.92% 23.5% K kr cb ? 0.14% 0.33% 0.61% 5.25% K br cb ? 0.035% 0.12% 2.63% 1.3% 4.3 Emitter geometry scaling Optimal transistor sizing and biasing are important for high performance RF low-noise am- plifier design using SiGe HBTs. This calls for accurate understanding and modeling of the emitter geometry scaling behavior of RF noise sources, including the correlated intrinsic base and collec- tor current noises, their correlation, the thermal-like noise of intrinsic base resistance and the well know 4kTR thermal noises due to extrinsic terminal resistances. In SiGe HBTs, the crowding e?ect on noise voltage of rbi is negligible because of the high base doping, therefore the r bi noise can be approximated with 4kTr bi . The study of scaling issue is based on experimental data of SiGe HBTs with di?erent emitter geometries, indicates emitter length (LE), emitter width (WE) and emitter finger number (NE) scaling respectively. We first extract the small signal equivalent circuit parameters from measured s-parameters, and then extract the intrinsic base and collector current noises using standard noise de-embedding method [2] from measured noise parameters. With the extraction results, di?erent scaling e?ects on intrinsic noise and resistance noise are discussed. The geometry scalability of our semi-empirical model is examined. 114 4.3.1 Intrinsic noise scaling Ideally for transistors of the same vertical profile, if they are biased to have the same collector current density J C , their dc currents, ac currents and Y-parameters are proportional to emitter area A E (= W E ? L E ? N E ), where W E , L E and N E are the emitter width, length and number of fingers respectively. Using a given emitter area A E0 as a reference, I B , I C , g m and Rfractur(Y 21 ) for other emitter geometries can be calculated using corresponding emitter area scaling factor M(= A E /A E0 ). Similarly, the PSDs of the intrinsic base and collector current noise and their correlations scale linearly with A E in this ideal case. Here the 0.24 ? 20 ? 2?m 2 device is used as a reference. Our noise extraction results for di?erent emitter geometries indeed show that the intrinsic noise PSDs obey such ideal scaling rule. In Fig. 4.14, the extracted intrinsic noises for each device are divided by its M factor and plotted versus I C /M at f=15 GHz. For S ib , S ic and Ifractur(S icib ?), the normalized data of the four devices overlap well. The trend is not obvious for Rfractur(S icib ?), primarily due to extraction di?culties. Rfractur(S icib ?) is one order of magnitude lower than Ifractur(S icib ?), and has much weaker e?ect on noise parameters. Therefore Rfractur(S icib ?) is easily a?ected by measurement noise. Furthermore we found that all ? terms in the model are approximately emitter geometry independent. This leads to the following scaling rule for the K and B terms in the proposed noise model as K bb ? = K bb ? 0 M 1?? bb ? ,B bb ? = B bb ? 0 M, (4.30) K cc ? = K cc ? 0 M ?? cc ? ,B cc ? = B cc ? 0 , (4.31) K br cb ? = K br cb ? 0 ,K kr cb ? = K kr cb ? 0 /M, (4.32) K i cb ? = K i cb ? 0 M 1?? i cb ? ,B i cb ? = B i cb ? 0 M. (4.33) 115 where the subscript 0 denotes the reference transistor. Since all of the noise current PSDs scale linearly with the emitter area, the normalized correlationc does not change,|c|?1 is kept satisfied. 0 5 10 15 20 0 5 10 x 10 ?22 I C /M (mA) S ib /M (A 2 /Hz) 0 5 10 15 20 0 0.5 1 1.5 2 2.5 x 10 ?20 I C /M (mA) S ic /M (A 2 /Hz) 0 5 10 15 20 ?2 ?1 0 1 2 x 10 ?22 I C /M (mA) Re[S icib * ]/M (A 2 /Hz) 0 5 10 15 20 ?5 ?4 ?3 ?2 ?1 0 x 10 ?21 I C /M (mA) Im[S icib * ]/M (A 2 /Hz) 0.24?20?2?m 2 0.24?10?2?m 2 0.24?20?1?m 2 0.48?10?1?m 2 Frequency=15 GHz Figure 4.14: Extracted intrinsic noise divided by M vs I c /M at f=15 GHz, where M is the emitter geometry scaling factor. We now verify the geometry scaling ability of the new model with measured noise data of di?erent emitter geometries. The 0.24 ? 20 ? 2?m 2 device is used as a reference. We use 0.24 ? 10 ? 2?m 2 for emitter length scaling, 0.24 ? 20 ? 1?m 2 for number of emitter finger scaling and 0.48 ? 10 ? 1?m 2 for emitter width scaling. The model parameters for all devices satisfy the scaling rule given by (4.30)?(4.33). Note that the emitter area scaling factor M is 2 for all three scaled devices. The noise figures are shown in Figs. 5.18, 4.16, 4.17 for A E = 0.24 ? 10 ? 2?m 2 , A E = 0.24?20?1?m 2 and A E = 0.48?10?1?m 2 respectively. For each emitter geometry, a low bias and a high bias point are shown. Excellent agreement between modeling and measurement has been achieved for all of the four noise parameters and for all of the emitter geometries. Note that the collector voltage V CE of di?erent size devices are di?erent. The V CE e?ect has been taken into account by the small signal parameters. 116 0 10 20 30 0 2 4 f (GHz) NF min (dB) 0 10 20 30 0 0.2 0.4 0.6 f (GHz) Rn (/Z0) 0 10 20 30 0 0.02 0.04 f (GHz) Gopt (S) 0 10 20 30 ?0.02 ?0.01 0 0.01 f (GHz) Bopt (S) Experiment I C =1.6mA New model I C =1.6mA Experiment I C =8.0mA New model I C =8.0mA 0.24?10?2?m 2 V CE =2.0V Figure 4.15: Noise parameters versus frequency for 0.24 ? 10 ? 2?m 2 SiGe HBT at I c =1.6 mA and 8.0 mA. V CE =2.0 V. Table 4.4: Extracted r bx , r bi for 50 GHz SiGe HBTs with di?erent emitter geometries Emitter geometry M r bx r bi at peak f T r bx ?LE ?NE r bi ?LE ?NE (?m 2 )-(?)(?)(??m)(??m) 0.24 ? 20 ? 2 Ref. 2.70 2.68 108 107.2 0.24 ? 10 ? 2 1/2 3.70 5.75 74 115 0.24 ? 20 ? 1 1/2 11.5 10.5 230 210 0.48 ? 10 ? 1 1/2 13.0 21.1 - - 4.3.2 Extrinsic noise scaling Now we consider the geometry scaling of noises due to parasitic resistances. For SiGe HBT noise, r cx is less important and re is relatively small, hence only r bx and r bi are considered here. Since the intrinsic noise has shown to scale with AE ideally, if both r bx and r bi inversely scale with AE, the four normalized noise parameters, i.e., NF min , R n ?M, G opt /M and B opt /M, will nearly be geometry independent [1]. This will make optimal transistor sizing easier in LNA design. 117 0 10 20 30 0 2 4 f (GHz) NF min (dB) 0 10 20 30 0.4 0.6 0.8 f (GHz) Rn (/Z0) 0 10 20 30 0 0.01 0.02 f (GHz) Gopt (S) 0 10 20 30 ?0.01 ?0.005 0 f (GHz) Bopt (S) Experiment I C =1.5mA New model I C =1.5mA Experiment I C =8.5mA New model I C =8.5mA 0.24?20?1?m 2 V CE =3.0V Figure 4.16: Noise parameters versus frequency for 0.24 ? 20 ? 1?m 2 SiGe HBT at I c =1.6 mA and 8.0 mA. V CE =3.0 V. Table 4.4 shows the extracted r bx , r bi for SiGe HBTs with di?erent geometries. For base resistance, as WE scaling is quite di?erent with LE and NE scaling, we do not evaluate the nor- malized resistance values for WE case. For LE scaling, we compare the 0.24 ? 10 ? 2?m 2 HBT with the 0.24 ? 20 ? 2?m 2 reference device. Their normalized base resistances are close to each other as shown in Table 4.4 suggesting a near ideal LE scaling. Consequently, the four normalized noise parameters overlap with each other as shown in Fig. 4.18. Next we consider NE scaling by comparing the 0.24 ? 20 ? 1?m 2 HBT with the 0.24 ? 20 ? 2?m 2 reference. Their normalized r b values have a large di?erence due to the path resistances connected to base, leading to the discrep- ancy of normalized noise parameters. NF min is reduced using multiple emitter fingers as shown in Fig. 4.18. For the three scaling strategies, only emitter length scaling is near ideal and should be primarily considered during noise matching. 118 0 10 20 30 0 2 4 6 f (GHz) NF min (dB) 0 10 20 30 0.4 0.6 0.8 1 1.2 f (GHz) Rn (/Z0) 0 10 20 30 0 0.005 0.01 0.015 f (GHz) Gopt (S) 0 10 20 30 ?6 ?4 ?2 0 x 10 ?3 f (GHz) Bopt (S) Experiment I C =1.6mA New model I C =1.6mA Experiment I C =7.8mA New model I C =7.8mA 0.48?10?1?m 2 V CE =2.0V Figure 4.17: Noise parameters versus frequency for 0.48 ? 10 ? 1?m 2 SiGe HBT at I c =1.6 mA and 7.8 mA. V CE =2.0 V. 4.3.3 Comparison of intrinsic noise with resistance noise To compare the relative importance of noise sources in SiGe HBTs, we calculate three types ofNF min versusI C as shown in Fig. 4.19. The solid line is calculated including both intrinsic noise and resistance noise, the dash line is calculated including only intrinsic noise and the dash dot line is calculated including only resistance noise. Note these NF min do not have simple relation. For all the devices examined, the intrinsic noise contributes more noise. The bias dependence of NF min mainly comes from the bias dependence of intrinsic noise. The resistance noise adds about 1dB to NF min for all the four devices and is important. Once their values are well modeled, we can model the noise parameters accurately. 4.4 Implementation in CAD tools The semi-empirical model can be easily applied in present CAD tools. Here we demonstrate its implementation in VBIC model using Analog-A language for Advanced Design System (ADS), Agilent Technologies. 119 0 5 10 15 20 1 1.5 2 2.5 3 3.5 I C /M (mA) NF min (dB) 0 5 10 15 20 0 0.2 0.4 I C /M (mA) Rn ? M (/Z0) 0 5 10 15 20 0 0.01 0.02 0.03 0.04 0.05 I C /M (mA) Gopt/M (S) 0 5 10 15 20 ?0.025 ?0.02 ?0.015 ?0.01 ?0.005 0 I C /M (mA) Bopt/M (S) 0.24?20?2 New model 0.24?20?2 Experiment 0.24?10?2 New model 0.24?10?2 Experiment 0.24?20?1 New model 0.24?20?1 Experiment 0.48?10?1 New model 0.48?10?1 Experiment Frequency=15 GHz Figure 4.18: Normalized noise parameters versus I c /M at f=15 GHz. Fig. 4.20 illustrates the technique to introduce correlated intrinsic transistor noise sources that give PSDs in (4.26)?(4.29) for VBIC model. We add two isolated nodes v a and v b , each of them is connected to ground through a 1 Ohm noiseless conductance. Unity white noise currents i a and i b are injected into node v a and v b respectively, producing noise voltage v a and v b .Wehave S v a = S i a = 1,S v b = S i a = 1. (4.34) We add 2qI b shot noise current and g 1 ddt(v a ) noise current between base node bi and emitter node ei. Note that the time derivative operator ddt in Analog-A generates j? factor in frequency domain, leading to the frequency dependence of noise source. We then add two noise currentsg 2 v a 120 0 10 20 0 1 2 3 I C (mA) NF min (dB) 0 5 10 0 1 2 3 I C (mA) NF min (dB) 0 5 10 0 1 2 3 I C (mA) NF min (dB) 0 5 10 0 1 2 3 I C (mA) NF min (dB) Full Noise Intrinsic noise only Resistance noise only f=15 GHz (a) 0.24?20?2?m 2 (b) 0.24?10?2?m 2 (c) 0.24?20?1?m 2 (d) 0.48?10?1?m 2 Figure 4.19: NF min versus I c , determined by intrinsic noise only, resistance noise only and both of intrinsic and resistance noise for di?erent geometry SiGe HBTs: (a) 0.24 ? 20 ? 2?m 2 ; (b)0.24 ? 10 ? 2?m 2 ; (c) 0.24 ? 20 ? 1?m 2 ; (d) 0.48 ? 10 ? 1?m 2 . and g 3 v a between collector node ci and emitter node ei.Wehave S ib = 2qI b +S v a g 2 1 ? 2 = 2qI b +? 2 g 2 1 , S ic = S v a g 2 2 +S v b g 2 3 = g 2 2 +g 2 3 , S icib ? = S v a (?j?g 2 g 1 ) = ?j?g 2 g 1 . (4.35) The correlation between base and collector current noises thus is obtained by the controlled noise currents g 1 ddt(v a ) and g 2 v a . Now we need to find the expressions of g 1 , g 2 and g 3 . For simplicity, Rfractur(S icib? ) is set to zero. The frequency dependence ofS ic are neglected. Rfractur(Y 21 ) andg m are replaced with qI C /kT since g m is not referable in Analog-A of present versions. Comparing (4.35) with 121 ?1 ?1 a i b i a v b v B qI2 1 () a g ddt v a vg 2 b vg 3 ci bi ei Intrinsic transistor of VBIC Model Figure 4.20: Technique of insertion of correlated noise sources into the intrinsic transistor of VBIC model. (4.26)?(4.29), we have g 1 = radicalBig K bb ?g ? bb ? m +B bb ?, (4.36) g 2 = K i cb ? g ? i cb ? m radicalbig K bb ?g ? bb ? m +B bb ? , (4.37) g 3 = radicaltp radicalvertex radicalvertex radicalbt (K cc ?g ? cc ? m +B cc ?)Rfractur(Y 21 ) ? (K i cb ? g ? i cb ? m ) 2 K bb ?g ? bb ? m +B bb ? . (4.38) A similar method of introducing correlated i b and i c noise is given in [70], where noise cor- relation i b i ? c is introduced while i b and i c are still 2qI b and 2qI c white shot noise. This clearly is non-physical. As discussed in Chapter 2, 2qI b is mainly contributed by emitter hole noise for modern transistors. It is the base electron noise that produces significant correlation betweeni b and i c . The electron noise will inevitably produce frequency dependent excess noise current to i b .Itis this excess noise current of i b correlated with i c . In our method, the electron noise is described by g 1 , g 2 and g 3 . 122 The Analog-A code for implementation of (4.36)?(4.38) in VBIC model is given in Appendix G. Figs. 4.21, 4.22 show the noise parameters versus frequency simulated by ADS using semi- empirical noise model and SPICE noise model respectively atI C =15.1 mA. Clearly the new model improves noise modeling and gives the same results calculated by MATLAB in the previous sec- tion. Figure 4.21: Noise parameters versus frequency simulated by ADS using semi-empirical noise model at I C =15.1 mA. 4.5 Summary We have presented the noise de-embedding method for SiGe HBTs using a QS input equiva- lent circuit. The intrinsic transistor noises are then extracted through noise de-embedding method, and modeled as functions of bias and frequency based on inspection of extraction results. The modeling methodology is demonstrated using noise parameters measured from 2 to 25 GHz on SiGe HBTs featuring a 50 GHz peak f T . The imaginary part of the correlation Ifractur(S icib ?) is found 123 Figure 4.22: Noise parameters versus frequency simulated by ADS using SPICE noise model at I C =15.1 mA. to be proportional to ?. S ib and and the real part of the correlation Rfractur(S icib ?) are found to be pro- portional to ? 2 . S ic is found to be proportional to Rfractur(Y 21 ). Ifractur(S icib ?) is found to be much greater than Rfractur(S icib ?), and has a much larger impact on noise parameters. The bias dependence of all of the noise terms can all be modeled using g m . Excellent fitting of both Y-parameters and noise parameters has been achieved. The new semi-empirical model is capable of geometry scaling and can be implemented in present CAD tools. 124 CHAPTER 5 IMPROVED PHYSICAL NOISE MODEL With the technology advances, transistors are scaled, and have narrower base width and emit- ter width. Some e?ects related to BC and BE junctions are non-negligible any longer. This chapter improves compact RF noise modeling for SiGe HBTs based on NQS equivalent circuit by taking in account some of these e?ects. The impact of CB SCR on electron RF noise is examined to be important for scaled SiGe HBTs. The van Vliet model is then improved to account for the CB SCR e?ect. The impact of fringe BE junction on base hole noise is further investigated. Due to fringe e?ect, the base hole noise should be modeled with correlated noise voltage source and noise current source in hybrid representation. The base noise resistance is found to be di?erent from AC intrinsic base resistance, and thus is modeled by an extra parameter. With four bias-independent model parameters in total, the combination of electron and hole noise model provides excellent noise parameter fittings for frequencies up to 26 GHz and all biases before f T roll o? for three generations of SiGe HBTs. The new model is also capable of emitter geometry scaling. 5.1 CB SCR e?ect on electron noise As reviewed in Chapter 2, the van Vliet model solves the microscopic noise transport equa- tion for base minority carrier (electrons for NPN considered here). Van Vliet?s derivation of base and collector current noise PSDs assumed adiabatic boundary condition i.e. tildewiden=0 or zero electron density fluctuation at both ends of the base, and did not consider electron transport in the CB SCR. For scaled bipolar transistors, e.g. SiGe HBTs of 200 GHz peak f T , CB SCR electron transport 125 becomes more significant than base electron transport, calling for an investigation of its impact on transistor noise. The extremely useful result of van Vliet?s derivation is that the base and collector current noise and their correlation can be related to the Y-parameters due to intrinsic base electron transport, Y B . As shown in (2.69), the van Vliet model can be extended to include emitter hole noise by replacing Y B with the Y-parameters of base and emitter region Y EB . In the literature, the van Vliet model is often applied using Y-parameters of the whole intrinsic transistor, e.g. in [2] and [47], as opposed to Y EB , for which the model was derived. Physically speaking, both the Y-parameters and the noise parameters are modified by electron transport through the CB SCR, it is not clear at all what the relation between Y-parameters and transistor noise should be when the CB SCR is accounted for. E B C B Y EB Y CB E b i b i e i c i B c i b i Y B b i E Y SCR B e i Figure 5.1: Illustration of AC or noise current flows in ideal 1-D intrinsic SiGe HBT. Here we investigates the impact of CB SCR on transistor noise and derives an improved noise model including such impact. The CB SCR a?ects electron transport (and hence noise transport) in two ways. First, a velocity saturation boundary condition should be applied at the end of neutral base. Its e?ect on DC currents and base transit time, and noise has been investigated in Chapter 126 2. With a strong base built-in field, typically the case of graded SiGe HBTs, the van Vliet model can still correctly describe base electron noise. Therefore, we can continue to use van Vliet model for the relationship between noise and Y-parameters of the intrinsic base for graded SiGe HBTs. Secondly, electron transport through the CB SCR modifies both the Y-parameters and the noise parameters. The noise generated within CB SCR is neglected. The main CB SCR e?ect accounted is ? c delay, which was briefly discussed in [5]. We derived a new set of relationship between noise currents and Y-parameters in presence of CB SCR delay based on van Vliet model. Here we note that ? c e?ect was also included in [71]. However, the base region noise in [71] is derived from 1-D transmissionlineanalogywithoutincludingbasebuilt-infield, andneedsextraparameters (electron di?usion coe?cient D n , life time ? n and base width X B ). Therefore it is much less general than our van Vliet model based result, which is based on Y-parameters that can be measured. 5.1.1 Model equation derivation We denote the AC electron current injected into CB SCR as i B c and the AC collector current as i c . The electrons inside the CB SCR induce base hole accumulation at the SCR side of the base region and electron depletion at the SCR side of the collector region. The first part adds an extra base hole current ?i b , which is i B c ?i c , to original base current i B b . Note that i c and i b take positive signs when they flow into the electrodes. Physics analysis [72] shows that i c and i B c can be related by ?(?) ? i c i B c = 1 ?e ?2j?? c 2j?? c , where ? c is the collector transit time. The total AC/noise base and collector currents can be derived as i b = (i E b +i B b ) + (1??)i B c ,i c = ?i B c . (5.1) 127 With (5.1) and by neglecting Y EB 12 and Y EB 22 , we obtain the Y-parameters of the whole intrinsic transistor including CB SCR as Y 11 = Y EB 11 + (1??)Y EB 21 ,Y 21 = ?Y EB 21 . (5.2) The noise PSDs including CB SCR transport are derived from (5.1) as S ib ?= S EB ib + 2Rfractur[(1??)S EB icib ? ] +|1 ??| 2 S EB ic , S ic ?= |?| 2 S EB ic . S icib ? ?= ?S EB icib ? +?(1?? ? )S EB ic . (5.3) Here S EB is given in (2.69). Fig. 5.2 shows the e?ect of ? c on noise, ? c =0 for the dash lines and ? c =0.75? tr for the solid lines. Emitter hole noise is not included. Clearly the base current noise is significantly enlarged due to CB SCR electron transport, particularly with increasing frequency. It is a direct result of increase of AC base current caused by CB SCR e?ect. 0 50 100 0 0.5 1 x 10 ?15 f (GHz) S ib 0 50 100 0 x 10 ?15 f (GHz) S ic 0 50 100 ?5 0 x 10 ?16 f (GHz) Re(S icib * ) 0 50 100 ?2 ?1 0 x 10 ?15 f (GHz) Im(S icib * ) d B =20nm |E|=70.2kV/cm V BE =0.80V Eq. (9) with ? c =0 Eq. (9) with ? c =0.75? tr Figure 5.2: Comparison of the intrinsic noise with ? c =0 and ? c =0.75? tr .For? c =0.75? tr , f T =174 GHz. Emitter hole noise is not included. 128 For various reasons discussed above, it is highly desirable to express the noise PSDs in (5.3) in terms of the Y-parameters for the whole intrinsic transistor Y. A set of such expressions are derived below S ib =4kTRfractur(Y EB 11 ) ? 2qI b + 4kTRfractur[(1??)Y EB 21 ] ? 4kTRfractur(1??)g m +|1 ??| 2 S EB ic =4kTRfractur(Y 11 ) ? 2qI b ? 4kTRfractur(1??)g m +|1 ??| 2 2qI c ={4kTRfractur(Y 11 ) ? 2qI b }+ 2qI c |1 ??| 2 ? 4kTg m Rfractur(1??), S ic ={2qI c }|?| 2 , S icib ? =?2kTY EB 21 ??2kTg m +?(1?? ? )S EB ic ={2kTY 21 ??2kTg m }+?(1?? ? )2qI c ={2kT(Y 21 ?g m )}+ 2qI c (??|?| 2 ) + 2kTg m (1??). (5.4) We illustrate the S ib derivation as an example. The first step is obtained directly from (5.3) and (2.69). The second step is obtained using (5.2). Note that the terms enclosed by {} in (5.4) are the noise expressed by van Vliet model using the Y-parameters of whole transistor, a brutal force application of van Vliet model (using Y despite that it needs Y EB ? often used without justification). The additional terms in our new model, (5.4), represent the error introduced by using the van Vliet model with the overall transistor Y-parameters. Fig. 5.3 compares the improved model, the brutal use of van Vliet model and the exact result, that is, Langevin equation solution used with (5.3). The improved model works very well, and gives results nearly identical to the exact result. S ib and |Rfractur(S icib ?)| are overestimated by the brutal use of van Vliet model, while |Ifractur(S icib ?)| is correctly modeled for the analytical result where g m = I c /V T . For practical SiGe HBTs, g m is typically smaller than I c /V T at high current levels [1]. Consequently, the brutal use of van Vliet model cannot correctly model |Ifractur(S icib ?)| at high I c . The inconsistent modeling of S ib and |Ifractur(S icib ?)| results in an overestimation of NF min for the brutal 129 Table 5.1: Extracted delay time from DESSIS simulation data Peak f T (GHz) Device (?m 2 ) ? c (ps) ? tr (ps) ? c /? tr 65 0.5 ? 1 0.57 2.1 27% 85 0.2 ? 1 0.55 1.5 37% 183 0.12 ? 1 0.58 0.86 67% use of van Vliet model. The magnitude of derivation depends on the ratio ? c /? tr , which increases with scaling. Table 5.1 shows the extracted ? c /? tr ratio from DESSIS simulated three generations of SiGe HBTs using the method of [73]. The ratio increases with device scaling, indicating that the BC SCR has more significant impact on higher f T devices. Even though the di?erences look small on the plots shown, the resulting di?erences in noise parameters of the intrinsic transistor (NF min , R n , and Y opt ) are significant, making them important to model. For transistors in which base resistance is large, the final impact on overall transistor noise parameters is smaller, simply because of the less importance of intrinsic transistor noise. 0 50 0 5 x 10 ?16 f (GHz) S ib 0 50 0 x 10 ?15 f (GHz) S ic 0 50 ?2 ?1 0 x 10 ?16 f (GHz) Re(S icib * ) 0 50 ?1 ?0.5 0 x 10 ?15 f (GHz) Im(S icib * ) d B =20nm |E|=70.2kV/cm V BE =0.80V ? c =0.75? tr Eq. (9), exact noise Eq. (1), with Y?=Y Eq. (10), improved model Figure 5.3: Comparison between the brutal used van Vliet model and the improved model under ? c =0.75? tr . f T =174 GHz. 130 5.1.2 Verification and discussion To verify our derivations, we examine the new model using hydrodynamic DESSIS noise simulation. The device has 184 GHz peak f T with e?ective d B =20nm. At V BE = 0.79V, ? c =0.75(? b +? c ), f T =155 GHz. Fig. 5.4 compares the improved model with the extracted ? c , the brutal use of van Vliet model and the extracted intrinsic base electron noise. The new model improves S ib and Rfractur(S icib ?) modeling. The DESSIS simulated S ic < 2qI C is a direct result of hydrodynamic simulation. 0 50 0 5 x 10 ?25 f (GHz) S ib 0 50 0 1 2 x 10 ?23 f (GHz) S ic 0 50 ?4 ?2 0 x 10 ?25 f (GHz) Re(S icib* ) 0 50 ?3 ?2 ?1 0 x 10 ?24 f (GHz) Im(S icib* ) Eq. (10), improved model DESIS simulated electron noise Eq. (1), with Y?=Y V BE =0.79V f T =155 GHz Figure 5.4: Comparison between van Vliet model, new model and the extracted intrinsic noise from DESIS simulation results. ? c =0.75(? b +? c ) is used in the new model. E?ective d B =20nm, ?=5.4, |E|=70.2 kV/cm. We approximate (5.4) up to the second order of ?. g be = I b /V T are assumed. The frequency dependence of S ib caused by emitter hole noise is negligible compared to that of base electron 131 noise (see Fig. 2.14). We have S ib ?2qI b +? 2 bracketleftbigg 4kTg m parenleftbigg ? tr ? in ? 2 3 ? 2 c parenrightbigg + 2qI c ? 2 c bracketrightbigg , S ic ?2qI c ?? 2 (2/3qI c ? 2 c ), S icib ? ??j?[2kTg m (? in +? out ?? c ) + 2qI c ? c ] ?? 2 bracketleftbigg 2kTg m parenleftbigg ? 2 in +? in ? out + 1 2 ? 2 out ? 2 3 ? 2 c parenrightbigg + 2 3 qI c ? 2 c bracketrightbigg . (5.5) Clearly for general case where g m negationslash= I c /V T , (5.5) cannot be simplified using three or four lumped model parameters. Now we consider an extreme case, i.e. ? c >> ? b or ? c ? ? tr . This eventually becomes the physical scenario described by the transport noise model [44] [43]. Under such condition, ? in ? 2/3? tr and ? out ? 1/3? tr as discussed in Chapter 3. We then have S ib ?2qI b +? 2 (2qI c ? 2 c ), S ic ?2qI c ?? 2 (2/3qI c ? 2 c ), S icib ? ??j?(2qI c ? c ) ?? 2 parenleftbigg 1 9 kTg m + 2 3 qI c parenrightbigg ? 2 c . (5.6) Comparing with the Taylor expression of transport model equations in Chapter 4, we found that S ib ?S tran ib , S ic ?S tran ic + 2/3Rfractur(S tran icib ? ), S icib ? ?S tran icib ? ? 4/9Rfractur(S tran icib ? ). (5.7) This shows that under ? c >> ? b condition, the transport noise model does not well model the intrinsic noise. However, it is a good approximation as S ib and Ifractur(S icib ?) have been correctly 132 modeled. The improved model thus provides a means of ?bridging? the van Vliet model and the transport noise model. 5.2 Fringe BE junction e?ect on base hole noise Base hole noise is another major noise source for SiGe HBTs. Traditionally this noise is modeled by the thermal noise of r bx and r bi , the small signal base resistance for the extrinsic and intrinsic region respectively. r bx is the resistance of a true resistor whose noise can be well modeled with 4kTr bx . However, r bi is a lumped resistance. There are two kinds ofr bi , depending on whether QS equivalent circuit or NQS equivalent circuit is used. As discussed in Chapter 2 and Chapter 3, r bi,nqs is more physical and also smaller than r bi,qs . We found problems of using 4kTr bi for noise modeling based on either QS or NQS lumped equivalent circuit. Firstly, noise resistance R n cannot be well modeled, which is sensitive to base hole noise. One has to use an empirical S ic based on noise extraction, which is unphysically larger than 2qI c for 50 GHz SiGe HBTs [2]. Another problem is that the absolute value of the imaginary part of noise parameter Y opt , i.e. B opt , is overestimated by van Vliet model based on NQS equivalent circuit. The deviation cannot be eliminated by choosing appropriate r bi . This work aims to solve these two problems by modeling the distributive e?ect of base hole noise. The distributive e?ect is a significant feature of intrinsic base hole noise [48]. The best way to examine this e?ect is through microscopic noise simulation. There exist two kinds of distributive e?ect, the fringe e?ect associated with the edge transistor and the crowding e?ect associated with the intrinsic transistor. To account for these e?ects, we divide the BE/BC junction into four seg- ments A 1?4 , leading to five equivalent base resistances of three types as shown in Fig. 5.5. Further analysis shows that at least four segments (five resistors) are needed. Type I resistances are for the edge transistors. Type III resistances are for the main intrinsic transistor. Type II resistances are a combination of resistances from the main and edge transistors. Because of the narrow emitter width 133 X6CX65X66X74 X49 X6CX65X66X74 X49X49 X72X69X67X68X74 X49 X72X69X67X68X74 X49X49X49X49X49 X45 X42 X43 1 X41 2 X41 3 X41 4 X41 Figure 5.5: Illustration of base distribution e?ect by dividing the base resistances into five segments of three types. Double base contact is used. and high base doping, DC crowding e?ect is negligible in practice. Hence the traditional 4kTr bi description is theoretically true only for the main intrinsic transistor without the fringe region [41]. In lumped equivalent circuit based modeling, the fringe region or edge transistor is not explicitly separated from the main intrinsic region [2,51,52]. However it is unknown how the fringe e?ect a?ects base hole noise and how important the e?ect is. We will show that the base hole noise should be modeled by a noise voltage source at the input and a correlated noise current source at the output due to the fringe e?ect. The f T is no longer assumed to be uniform across the whole BE junction as opposite to [41]. The fringe transistor has lower f T because of wider base at the edge of emitter and smaller V BE . It is the correlation of the two noise sources that cause the B opt ) problem described above. The base noise resistance needed to fit noise data from both microscopic noise simulation and measurements is found to be not the same as r bi , which cannot be explained by fringe e?ect. Such observation based on simulation was also reported in [46]. We hence use an extra parameter R bn as base noise resistance to improve R n fitting. DESSIS device simulation is used as guidance, as base hole and electron noises can be separated in simulation. Experimental data are used to verify the new model. 134 5.2.1 Physical considerations The five resistance model has captured both the fringe e?ect and crowding e?ect of base hole noise in a lumped fashion. Fig. 5.6 shows the small signal equivalent circuit that corresponds to Fig. 5.5. The five resistances correspond to those five segments. The four capacitors and X45 X42 X43 2 X52 2 X52 3 X52 1 X43 1 X43 2 X43 2 X43 2 2 X6D X67X76 1 1 X6D X67X76 1 X52 1 X52 1 2 3 4 1 4 X6D X67X76 2 3 X6D X67X76 2 X52 X76 Figure 5.6: Small signal equivalent circuit of five segments model. Only the noise voltage source of left R 2 is shown. g be is neglected. Four nodes are labeled. transconductances correspond to segmentA 1?4 . Note thatg m2 >> g m1 ,C 2 >> C 1 . g be is neglected in Fig. 5.6 which is only used for base hole noise derivation. The g be in the small signal equivalent circuit of SiGe HBT is not neglected. All the small signal components are connected through four inner nodes. The resulting equivalent circuit is symmetric. Although the DC base-emitter bias is the same for A 1?4 segments, the local f T varies along the emitter junction. A 2 and A 3 have the same f T . A 1 and A 4 , however, have lower f T because of wider base of the edge transistor, meaning that g m1 /C 1 = 4kTR 2 [? 2 c /4], S ih,R2 == 4kTR 2 (g m1 +g m2 ) 2 (1?? c ?? gm ) 2 , S ihvh ? ,R2 == 4kTR 2 (g m1 +g m2 )(1?? c )(? c ?? gm ). (5.14) The right R 2 has the same noise as the left R 2 , therefore the two R 2 contribute two times of the noise shown in (5.14). Now the overall noise can be obtained by adding the contributions of two R 1 and two R 2 in (5.14) as S vh = 4kTR bn , S ih = 4kTR bn g 2 m K 1 , S ihvh ? = 4kTR bn g m K 2 , (5.15) where g m = 2(g m1 +g m2 ),R bn = [R 1 +R 2 (1?? c ) 2 ]/2, K 1 = R 2 /2 R bn (? c ?? gm ) 2 ,K 2 = R 2 /2 R bn 2(1?? c )(? c ?? gm ). (5.16) We have lumped R 1 , R 2 , ? gm and ? c into three model parameters R bn , K 1 and K 2 . The following observations are noted: 138 ? The thermal resistance R bn defined in (5.16) is actually the lumped intrinsic base resistance r bi of Fig. 5.6 . To provide this, we examine the BE input impedance of Fig. 5.6 Z BE = 1 2 bracketleftbigg R 1 + 1 +j?C 2 R 2 j?(C 1 +C 2 ) ?? 2 C 1 C 2 R 2 bracketrightbigg ? 1 2 bracketleftBigg R 1 +R 2 parenleftbigg C 2 C 1 +C 2 parenrightbigg 2 bracketrightBigg + 1 j?(2C 1 + 2C 2 ) = bracketleftbig R 1 +R 2 (1?? c ) 2 bracketrightbig /2 + 1 j?(2C 1 + 2C 2 ) = r bi + 1 j?(2C 1 + 2C 2 ) . (5.17) (5.17) means that Z BE can be modeled by a resistance in series with total BE capacitance. Such resistance essentially is r bi , and clearly equal to the R bn in (5.16). However, the R bn needed to fit experimental noise data is di?erent from the r bi extracted either based on QS or NQS equivalent circuit as detailed below. ? According to stochastic physics, the normalized correlation should not exceed unity [2], meaningK 2 2 ? K 1 . If BE fringe e?ect is not taken in account, i.e. ? c = ? gm , thenK 1 = K 2 = 0. The new model reduces to 4kTR bn ? 4kTr bi . Fig. 5.7 shows the simulated base hole noise in hybrid representation with the new model at V BE =0.90V. Note that Rfractur(S ihvh? ) > 0, which is consistent with g m1 /C 1 ? b . The initial value can be estimated with ? tr . Its final value can be determined by NF min fitting. ? R bn mainly a?ects R n for all frequencies and all biases. How R bn compares to r bi depends on technology generation as shown by Fig. 5.12 (b) and Fig. 5.15 (b). Its initial value can be estimated with r bi at low current levels, and finally determined by R n fitting. ? K 2 reduces B opt for high biases and has little e?ect on other parameters as shown by Fig. 5.12. K 2 , which involves? C , ? gm andR 2 /R bn can be estimated with simulation. Experimen- tally, a small value, e.g. 0.01, should be used as initial guess? Its final values is determined by high current level B opt fitting. 150 0 10 20 0 2 4 f (GHz) NF min (dB) 0 10 20 0.2 0.4 0.6 0.8 f (GHz) Rn (/Z0) 0 10 20 0 0.01 0.02 0.03 f (GHz) Gopt (S) 0 10 20 ?15 ?10 ?5 0 5 x 10 ?3 f (GHz) Bopt (S) Experiment I C =1.6 mA New model I C =1.6 mA Experiment I C =8.0 mA New model I C =8.0 mA 50 GHz SiGe HBT 0.24?10?2 ?m 2 V CE =2.0V Figure 5.18: Noise parameters versus frequency for scaled 50 GHz SiGe HBTs (A E = 0.24?10? 2?m 2 ). ? K 1 can increase NF min and R n as shown by Fig. 5.14 (a) and (b). However, in model im- plementation, we choose K 1 = K 2 2 to simplify parameter determination, which is generally satisfactory. Table 5.2 summarizes the model parameters for the previous results. r bi and r bx are also shown. Table 5.2: Model parameters, r bi * and r bx * for reference f T Device ? c K 1 K 2 R bn r bi * r bx * GHz ?m 2 ps - - ??? 50 0.24 ? 10 ? 2 0.8 0.002 0.045 6.2 5.4 3.75 0.24 ? 20 ? 2 4.2 2.7 2.4 90 0.12 ? 8 ? 4 0.8 0.02 0.1 3.0 3.0 1.52 0.12 ? 20 ? 4 1.2 1.2 3.67 160 0.12 ? 12 0.5 0.02 0.1 7.5 11.1 6.10 0.12 ? 18 4.0 7.0 4.77 151 0 10 20 0 1 2 f (GHz) NF min (dB) 0 10 20 0 0.1 0.2 f (GHz) Rn (/Z0) 0 10 20 0 0.02 0.04 f (GHz) Gopt (S) 0 10 20 ?0.02 ?0.01 0 f (GHz) Bopt (S) Experiment I C =3.1 mA New model I C =3.1 mA Experiment I C =16.5 mA New model I C =16.5 mA 90 GHz SiGe HBT 0.12?8?4 ?m 2 V CB =1.0 V Figure 5.19: Noise parameters versus frequency for scaled 90 GHz SiGe HBTs (A E =0.12 ? 8 ? 4?m 2 ). 5.4 Summary We have presented an improved RF noise model for SiGe HBTs using NQS equivalent circuit. The van Vliet model has been extended to include both emitter hole noise and CB SCR e?ect for modern BJTs. The CB SCR delay time decreases high frequency NF min for high biases. The base hole noise is modeled by a noise voltage source and a correlated noise current source in hybrid representation due to fringe BE junction e?ect. The correlation between two noise sources decreasesB opt . The base noise resistanceR bn is not always the same as the intrinsic base resistance r bi , which cannot be explained by fringe e?ect. Model parameter extraction guidelines are given. The utility of the model has been demonstrated using experimental data of SiGe HBTs from three generations. 152 0 10 20 0 1 2 3 f (GHz) NF min (dB) 0 10 20 0.2 0.4 0.6 0.8 f (GHz) Rn (/Z0) 0 10 20 0 0.005 0.01 0.015 f (GHz) Gopt (S) 0 10 20 ?10 ?5 0 5 x 10 ?3 f (GHz) Bopt (S) Experiment I C =0.6 mA New model I C =0.6 mA Experiment I C =11 mA New model I C =11 mA 160 GHz SiGe HBT 0.12?12?1 ?m 2 V CB =0.5V Figure 5.20: Noise parameters versus frequency for scaled 160 GHz SiGe HBTs (A E =0.12 ? 12?m 2 ). 153 BIBLIOGRAPHY [1] J. D. Cressler and G. 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Auburn University: PhD. dissertation, August 2005. 159 APPENDIX A REPRESENTATION TRANSFORMATION FOR TWO-PORT NETWORK A.1 T-matrix for noise representation transformation Table A.1: Transformation Matrices to Calculate Noise Matrices Original Representation T Y- Z- A- Y- bracketleftbigg 10 01 bracketrightbigg bracketleftbigg Y 11 Y 12 Y 21 Y 22 bracketrightbigg bracketleftbigg ?Y 11 1 ?Y 21 0 bracketrightbigg Z- bracketleftbigg Z 11 Z 12 Z 21 Z 22 bracketrightbigg bracketleftbigg 10 01 bracketrightbigg bracketleftbigg 1 ?Z 11 0 ?Z 21 bracketrightbigg Resulting A- bracketleftbigg 0 A 12 1 A 22 bracketrightbigg bracketleftbigg 1 ?A 11 0 ?A 21 bracketrightbigg bracketleftbigg 10 01 bracketrightbigg A.2 Derivation of Noise Parameters According to (1.4), we have NF = S s /N s S o /N o = N o N s G = 1 + N added N s G = 1 + N added /G N s . (A.1) This means that NF can be calculated at any point of circuit by 1 plus a ratio (Noise added by two-port network divided by noise from signal source). Fig. A.1 shows the circuit configuration used for derivation. We denote the input admittance of the two-port network asY I . Y S = G S +jB S . We choose node B for derivation. 160 Figure A.1: Noise Figure We have v added = v a Y S Y I +Y S + i a Y I +Y S ,v s = i s Y I +Y S . which means N added ?= S v vextendsingle vextendsingle vextendsingle vextendsingle Y S Y I +Y S vextendsingle vextendsingle vextendsingle vextendsingle 2 + S i |Y I +Y S | 2 + 2Rfractur(Y ? S S iv ?) |Y I +Y S | 2 , N s ?= 4kTRe(Y S ) |Y I +Y S | 2 = 4kTG S |Y I +Y S | 2 . Therefore NF = 1 + S v 4kTG S bracketleftbigg S i S v + 2Rfractur parenleftbigg Y ? S S iv ? S v parenrightbigg +|Y S | 2 bracketrightbigg (A.2) With some algebra maniplutation, (A.2) can be simplified into (1.5) with relations in (1.8). Note that generally we have two opposite values for G opt from a square root, however only the positive value is chosen because of the resistivity of signal source admittance. 161 APPENDIX B APPROXIMATION OF INTRINSIC BASE RESISTANCE NOISE CONSIDERING CURRENT CROWDING EFFECT B.1 General Principles The approximation for the intrinsic base resistance noise current PSD, S irbi , is based on J. C. J. Paasschens?s theorectical analysis of BJT noise considering both dc and ac crowding [41]. In [41], the intrinsic base resistance noise was described using V B x B i , the dc voltage drop cross the equivalent base resistance representing dc current crowding. To avoid V B x B i extraction, we relate the intrinsic base resistance noise to its small signal value r bi and dc current I B . This is helpful when modeling noise based on small signal equivalent circuit. Carefully observing the I B and S irbi expressions for both circular and rectangular emitter BJTs, (see (37), (41), (52) and (56) in[41]), wefoundthatallthesetermsareinverselyproportional to R BV and the remaining part of these expressions excluding R BV only depends on V B x B i . Here R BV is the low current limit of V B x B i /I B . Therefore we can make an approximation for S irbi using a linear combination of 4kTg bi and qI B as S irbi,appr = (? 1 4kTg bi ?? 2 qI B )?f. (B.1) where g bi = 1/r bi . Note that the error is independent of R BV . Therefore the two coe?cients are general for any crowding strength. In the following we will obtain these two coe?cients for circular and rectangular emitters respectively. 162 B.2 Circular Emitter BJT For circular emitter BJT [41], I B = V T (e V BxB i /V T ? 1) R BV . (B.2) g bi is obtained by g bi = dI B dV B x B i = e V BxB i /V T R BV . (B.3) The base resistance noise is given in (2.75) by [41] Substitute (B.2), (B.3) and (2.75) into (B.1), one has 4? 1 +? 2 = 10/3, (B.4) ? 2 = 2/3. (B.5) This gives ? 1 = 1 and ? 2 = 2/3. Therefore, we get an exact expression for S irbi as S irbi,appr = bracketleftbig 4kT/r bi ? 2qI B /3 bracketrightbig ?f. (B.6) To make (B.6) positive, we need r bi < 6(V T /I B ) = 6r ?,e . This is easily satisfied in practice. 163 B.3 Rectangular Emitter BJT For rectangular emitter BJT [41], I B = 2V T (e V BxB i /V T ? 1) +V B x B i 3R BV . (B.7) g bi is thus g bi = dI B dV B x B i = 2e V BxB i /V T + 1 3R BV . (B.8) The noise S irbi is given in (2.76) by [41] If V B x B i >> V T , S irbi will be proportional to e V BxB i /V T , the same as I B and g bi . Therefore the error using approximation (B.1) at large V B x B i will saturate to a constant. Because of the exponential term, V B x B i = 10V T is su?cient to cause such saturation of error. Therefore we only need to consider V B x B i ? [0 10V T ]. We optimize ? 1 and ? 2 to minimize the error defined by Err= |(S irbi,theory ?S irbi,appr )/S irbi,theory |. We found ? 1 = 1.0149 and ? 2 = 0.6772. The solid line in Fig. B.1 shows the error from approxi- mations. The error is smaller than 1.5% for all V B x B i . These two coe?cients are very close to the values of the circular case. The dash line in Fig. B.1 shows the error using ? 1 = 1 and ? 2 = 2/3. The error is less than 3%. Therefore we can unify these two cases using (B.6) with little loss in accuracy. Fig. B.2 shows the ratio of calculated noise to the theoretical noise using this method and traditional 4kT/r bi method. The error has been much decreased using the proposed method. For the SiGe HBTs used in this work, V B x B i /V T is less than 0.8 at peak f T , making the crowding e?ect indeed unimportant for practical purposes. This is in part by design, as the HBTs are typically designed to keep the crowding e?ect under control. 164 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 V B x B i /V T Error (%) ? 1 =1.0149, ? 2 =0.6772 ? 1 =1, ? 2 =2/3 Figure B.1: Approximation induced error versus V B x B i for rectangular emitter BJT. 0 2 4 6 8 10 0.95 1 1.05 1.1 1.15 1.2 1.25 V B x B i /V T S irbi,modeling /S irbi,theory Traditional 4KT/r bi ? 1 =1.0149, ? 2 =0.6772 ? 1 =1, ? 2 =2/3 Figure B.2: Comparison between approximation method and the traditional 4kT/r bi method for rectangular emitter BJT. 165 APPENDIX C DERIVATION OF NQS DELAY TIME WITH CB SCR We can drive (3.3) by approximating (3.2) with the functional form of (3.1) using Taylor expansion analysis method. We use Arabic numeral subscripts to indicate the order of the Taylor expansion coe?cients, as was done in Section III. The first order coe?cients for Ifractur(Y 11 ) without and with CB SCR delay are Ifractur(Y bs 11 ) 1 = C bej +C b bed , Ifractur(Y al 11 ) 1 = C bej +C b bed +g m ? c ? C bej +C bed , (C.1) respectively. (C.1) means C bed = C b bed +g m ? c . With the definitions of ? tr and ? b tr , we obtain the ? tr expression in (3.3). The second order coe?cients of Rfractur(Y 21 ) for the base and the whole intrinsic transistor are Rfractur(Y bs 21 ) 2 = g m ? b tr ? b in , Rfractur(Y al 21 ) 2 = g m ? tr bracketleftBigg ? b in +? c ? b out + 2? c /3 ? tr bracketrightBigg ? g m ? tr ? in . (C.2) (C.2) directly gives the ? in expression in (3.3). Similarly, by comparing Ifractur(Y bs 21 ) 1 and Ifractur(Y al 21 ) 1 ,we have ? in +? out = ? b in +? b out +? c . With the ? in expression already known, the ? out expression in (3.3) can then be obtained through substraction. 166 APPENDIX D ANALYTICAL Y-PARAMETERS In the following, we derive the analytical Y-parameters for di?erent blocks related to the small signal equivalent circuit in Fig. 3.4 using two methods. D.1 Manual Derivation of Analytical Y-parameters We define Y I and G M as Y I ? g be +j?C bej + j?C bed 1 +j?C bed r d ,G M ? g m e ?j?? d . (D.1) We have Y BI 11 = Y I +j?C bci ,Y BI 12 = ?j?C bci ,Y BI 21 = G M ?j?C bci ,Y BI 22 = j?C bci . (D.2) then Y BIR 11 = Y I +j?C bci 1 +Y I r bi +j?C bci r bi ,Y BIR 12 = ?j?C bci 1 +Y I r bi +j?C bci r bi , Y BIR 21 = G M ?j?C bci 1 +Y I r bi +j?C bci r bi ,Y BIR 22 = j?C bci (1+Y I r bi +G M r bi ) 1 +Y I r bi +j?C bci r bi . (D.3) We further define ? ? Y BIR 11 Y BIR 22 ?Y BIR 12 Y BIR 21 +j?C bcx (Y BIR 11 +Y BIR 22 +Y BIR 12 +Y BIR 21 ) +j?C bex Y BIR 22 ?? 2 C bex C bcx T ? 1 +Y BIR 11 r bx +Y BIR 22 r c +?r bx r c +j?(r bx C bex +r bx C bcx +r c C bcx ), (D.4) 167 we have Y BM 11 = Y BIR 11 +j?C bex +j?C bcx +?r c T ,Y BM 12 = Y BIR 12 ?j?C bcx T , Y BM 21 = Y BIR 21 ?j?C bcx T ,Y BM 22 = Y BIR 22 +j?C bcx +?r bx T . (D.5) D.2 MATLAB Code for Analytical Y-parameters Derivation For the Taylor expansion analysis, g be is neglected. % w -- omega % gbe -- EB conductance % Cd -- EB diffusion capacitance % Cj -- EB depletion capacitance % Cs -- Cbci, Ci is a reserved symbol of MATLAB % Cx -- Cbcx % Cbex -- extrinsic EB capacitance % gm -- transconductance % t -- total output delay time (tau_in+tau_out) %Rb --rbi % Rd -- delay resistance rd % Rc -- rci+rcx %Rx --rbx clear all; syms w gbe Cd Cj Cs Cx Cbex gm t Rb Rd Rc Rx real YI=j*w*Cd/(1+j*w*Rd*Cd)+j*w*Cj;%+gbe; GM=gm*exp(-j*w*t); Ybci=j*w*Cs; Ybcx=j*w*Cx; Y=[YI+Ybci -Ybci GM-Ybci Ybci]; % Y -- Y-parameters of BI block Z=inv(Y); Z(1,1)=Z(1,1)+Rb; % Z -- Z-parameters of BIR block YY=inv(Z); YY(1,1)=YY(1,1)+Ybcx+j*w*Cbex; YY(1,2)=YY(1,2)-Ybcx; YY(2,1)=YY(2,1)-Ybcx; YY(2,2)=YY(2,2)+Ybcx; simple(YY); % YY -- Y-parameters of BM block without rbx detYY=YY(1,1)*YY(2,2)-YY(1,2)*YY(2,1); T=1+Rx*YY(1,1); YYx(1,1)=YY(1,1)/T; YYx(2,2)=(YY(2,2)+Rx*detYY)/T; YYx(1,2)=YY(1,2)/T; 168 YYx(2,1)=YY(2,1)/T; % YYx -- Y-parameters of BM block with rbx detYY=YY(1,1)*YY(2,2)-YY(1,2)*YY(2,1); T=1+Rx*YY(1,1)+Rc*YY(2,2)+detYY*Rx*Rc; YYY(1,1)=(YY(1,1)+Rc*detYY)/T; YYY(2,2)=(YY(2,2)+Rx*detYY)/T; YYY(1,2)=YY(1,2)/T; YYY(2,1)=YY(2,1)/T; ZZZ=inv(YYY); % YYY/ZZZ -- Y/Z-parameters of BX block D.3 MATLAB code for Taylor expansion For (3.8)-(3.10), AA=1/(ZZZ(2,2)-ZZZ(2,1)); BB=1/ZZZ(1,2); imAA=taylor(imag(AA),w,2); simple(imAA); pretty(imAA); % imAA -- the first order coefficient of Im(AA) reAA=taylor(real(AA),w,3); simple(reAA); pretty(reAA); % reAA -- the second order coefficient of Re(AA) reBB=taylor(real(BB),w,1); simple(reBB); % reBB -- the zero order coefficient of Re(BB) For (3.13)-(3.15) and (3.17) imAA=taylor(imag(YYx(1,1)+YYx(1,2)),w,2); simple(imAA); pretty(imAA); % imAA -- the first order coefficient of Im(Y11^BM+Y12^BM) imAA=taylor(imag(YYx(2,1)-YYx(1,2)),w,2); simple(imAA); pretty(imAA); % imAA -- the first order coefficient of Im(Y21^BM-Y12^BM) imAA=taylor(imag(YYx(2,2)),w,2); simple(imAA); pretty(imAA); % imAA -- the first order coefficient of Im(Y22^BM) reAA=taylor(real(YYx(1,1)+YYx(1,2)),w,3); simple(reAA); pretty(reAA); % imAA -- the second order coefficient of Re(Y11^BM+Y12^BM) For (3.21)-(3.22), g be should be added, then the Y-parameters should be re-calculated. 169 imAA=taylor(imag(Y(1,1)+Y(1,2)),w,2); simple(imAA); pretty(imAA); % imAA -- the first order coefficient of Im(Y11^BI+Y12^BI) imAA=taylor(imag(Y(2,1)-Y(1,2)),w,2); simple(imAA); pretty(imAA); % imAA -- the first order coefficient of Im(Y21^BI-Y12^BI) reAA=taylor(real(Y(1,1)),w,1); simple(reAA); pretty(reAA); % imAA -- the zero order coefficient of Re(Y11^BI) reAA=taylor(real(Y(1,1))-gbe,w,3); simple(reAA); pretty(reAA); % imAA -- the second order coefficient of Re(Y11^BI) 170 APPENDIX E MATLAB CODE FOR SMALL SIGNAL PARAMETER EXTRACTION The following MATLAB code extracts the small signal parameters for a 50 GHz SiGe HBT with A E = 0.24 ? 20 ? 2 ?m 2 . HasCbi=0; % Cbi is the crowding cap paralleled with rbi. 0: no Cbe HasCrowdingnoise=0; % 0: Sirbi=4kT/rbi, otherwise 4kT/rbi-2qIb/3 % Data: Fixed Vce=1.5 V, Vbe=0.77-0.869 V, freq 2-26 GHz, num_bias=20; % Data: peakfT 50GHz at bias 19. Ae=0.24x20x2 um^2. Open de-embedded. load_data_26G; f=Y_cell_exp{step}(:,1); Omega=2*pi*f; %=============Deembedding Lb Lc Le ===== for step=1:1:num_bias Lb(step)=4.8e-11; Lc(step)=4.8e-11; Le(step)=1.12e-11; Ydlc=Y_cell_exp{step}(:,2:5); Zdlc=Y_to_Z(Ydlc); Zdlc(:,1)=Zdlc(:,1)-j*Omega*Lb(step)-j*Omega*Le(step); Zdlc(:,4)=Zdlc(:,4)-j*Omega*Lc(step)-j*Omega*Le(step); Zdlc(:,2)=Zdlc(:,2)-j*Omega*Le(step); Zdlc(:,3)=Zdlc(:,3)-j*Omega*Le(step); Ydlc=Z_to_Y(Zdlc); Y_cell_exp{step}(:,2:5)=Ydlc; end; %===De Ccs rcs ============= for step=1:1:num_bias Ccs(step)=1.80e-14; % obtained from cold measurement Rcs(step)=180; % obtained from cold measurement Ccs2(step)=0; Y=Y_cell_exp{step}(:,2:5); Ys=j.*Omega.*Ccs(step)./(1+j.*Omega.*Ccs(step).*Rcs(step)); Y(:,4)=Y(:,4)-Ys; Y_cell_exp{step}(:,2:5)=Y; end; %== extract Re ========== for step=1:1:num_bias Ybeta=Y_cell_exp{step}(:,2:5); Zbeta=Y_to_Z(Ybeta); z12(step)=real(Zbeta(1,2)); 171 y21(step)=real(Ybeta(1,3)); %plot(Omega,real(Ybeta(:,3))); hold on; end; % ben=17; enn=18; % xx=1./Ic_tmp;%; % k_b=polyfit(xx(ben:enn),z12(ben:enn),1); %intercept gives Re % plot(xx,re_gm,xx,xx.*k_b(1)+k_b(2),?+-?); hold on; %== De-embed Re, extract Cbct tt gm ========== for step=1:1:num_bias Re(step)=0.66; % determined from above method Y=Y_cell_exp{step}(:,2:5); Y=com_Y_re(Y,-Re(step)); Y_cell_exp{step}(:,2:5)=Y; Z=Y_to_Z(Y); H=Z(:,4)-Z(:,3); G=Z(:,1)-Z(:,2); Q=Z(:,2)-Z(:,3); k_b=linefit(Omega,imag(1./H),1,20,1,0,?+-?,?r?); Cbct(step)=k_b(1); k_b=linefit(Omega.^2,real(1./H),1,20,-1,0,?+-?,?r?); tt(step)=k_b(1)/Cbct(step)^2; k_b=linefit(Omega.^2,real(1./Z(:,2)),5,15,1,0,?+-?,?r?); gm0(step)=k_b(2); end; % smooth tt (for rcx) tt(1)=tt(2); tt(5:num_bias)=tt(5:num_bias).*0+tt(4); tt=fitcurv(tt.?,0.3)?; %===== De Rcx ======== for step=1:1:num_bias Rcx(step)=tt(step); Y=Y_cell_exp{step}(:,2:5); Y=com_Y_rc(Y,-Rcx(step)); Y_cell_exp{step}(:,2:5)=Y; end; %===taylor method for Rbx ==== for step=1:1:num_bias Y=Y_cell_exp{step}(:,2:5); %--imag y12 k_b=linefit(Omega.^2,imag(Y(:,2))./Omega,1,5,1,0,?+-?,?r?); Cbctt(step)=-k_b(2); %--imag y11 k_b=linefit(Omega.^2,imag(Y(:,1))./Omega,1,3,1,0,?+-?,?r?); Ct(step)=k_b(2); %--real Y12 k_b=linefit(Omega.^2,real(Y(:,2)),1,2,-1,0,?+-?,?r?); ReY12_2(step)=-k_b(1); %--imag Y22 k_b=linefit(Omega.^2,imag(Y(:,4))./Omega,1,4,1,0,?+-?,?r?); ImY22_1(step)=k_b(2); %--real Y21 k_b=linefit(Omega.^2,real(Y(:,3)),1,4,1,0,?+-?,?r?); gm(step)=k_b(2); %--imagY2112 172 k_b=linefit(Omega,imag(Y(:,3)-Y(:,2)),1,5,-1,0,?+-?,?r?); ImY2112(step)=-k_b(1); %--real Y11+Y12 k_b=linefit(Omega.^2,real(Y(:,1)+Y(:,2)),1,5,1,0,?+-?,?r?); ReY1112(step)=k_b(1); end; %===== Extract Rbx Cbex Cbcx Cbci ==== k_b=linefit(gm0.*Cbct,ImY22_1,1,8,1,0,?+?,?r?); Rx_rRi=k_b(1); k_b=linefit(Ct-Cbct,ImY2112./gm0,1,num_bias,1,0,?+?,?b?);Rx_Ri=k_b(1); r=0.3; % Cbci/Cbct, extracted from cold measurement Ri=(Rx_Ri-Rx_rRi)/(1-r); Rx=Rx_Ri-Ri; Cbcx=Cbct.*(1-r); Cbci=Cbct.*r; k_b=linefit(gm0,Ct-Cbct,1,8,1,0,?+?,?b?); cj=k_b(2); k_b=linefit(gm0,ReY1112,2,3,1,0,?+?,?r?); test=k_b(2); aa=Ri; bb=r.*Ri.*Cbctt; cc=-test+Rx.*cj.*(cj+Cbctt); xx=(-bb+sqrt(bb.^2-4.*aa.*cc))./(2.*aa); Cbex=(Cbct-Cbct+cj-xx(1)); Cbej=cj-Cbex; %==== De Rbx Cbcx Cbex ====== for step=1:1:num_bias Rbx(step)=Rx; Y=Y_cell_exp{step}(:,2:5); Y=com_Y_rb(Y,-Rbx(step)); Y=Y-[j*Omega*Cbcx(step) -j*Omega*Cbcx(step)... -j*Omega*Cbcx(step) j*Omega*Cbcx(step)]; rbi_QSx(step)=circle(1./Y(:,1)); Y(:,1)=Y(:,1)-j.*Omega.*Cbex(step); rbi_QS(step)=circle(1./Y(:,1)); Y_cell_exp{step}(:,2:5)=Y; end; %==== Extract and de go ===== for step=1:1:num_bias Y=Y_cell_exp{step}(:,2:5); go(step)=0;%real(Y(1,4)); %figure(100); plot(Omega,real(Y(:,4)),Omega,Omega-Omega+go(step)); Y(:,4)=Y(:,4)-go(step); Y_cell_exp{step}(:,2:5)=Y; end; %%%==== Extraction for QS rbi and intrinsic para====== %--de-embed rbi_QS, then extract gbe, gm ,Cbet and Taud for step=1:1:num_bias Y=Y_cell_exp{step}(:,2:5); Y=com_Y_rb(Y,-rbi_QS(step)); y11=Y(:,1)+Y(:,2); y21=Y(:,3)-Y(:,2); %--- Cbet_QS ben=1; enn=15; k_b=polyfit(Omega(ben:enn).^2,imag(y11(ben:enn))./Omega(ben:enn),1); %plot(Omega.^2,imag(y11)./Omega,Omega.^2,Omega.^2*k_b(1)+k_b(2)); Cbet_QS(step)=k_b(2); %--- gm_QS ben=1; enn=15; 173 k_b=polyfit(Omega(ben:enn).^2,real(y21(ben:enn)),1); %plot(Omega.^2,real(y21),Omega.^2,Omega.^2*k_b(1)+k_b(2)); hold on; gm_QS(step)=k_b(2); %--- gbe_QS ben=1; enn=5; k_b=polyfit(Omega(ben:enn).^2,real(y11(ben:enn)),1); %plot(Omega.^2,real(y11),Omega.^2,Omega.^2*k_b(1)+k_b(2)); hold on; gbe_QS(step)=k_b(2); %--- Taud_QS ben=1; enn=15; k_b=polyfit(Omega(ben:enn).^2,imag(y21(ben:enn))./Omega(ben:enn),1); %plot(Omega.^2,imag(y21)./Omega,Omega.^2,Omega.^2*k_b(1)+k_b(2)); Taud_QS_old(step)=-k_b(2)/gm_QS(step); %plot(Omega,-imag(log(y21))./Omega); hold on; k_b=slopefit(Omega(ben:enn),imag(y21(ben:enn))); Taud_QS_old(step)=-k_b/gm_QS(step); %plot(Omega,imag(y21),Omega,Omega*k_b); hold on; end; Taud_QS_old(1)=Taud_QS_old(2); Taud_QS=fitcurv(Taud_QS_old?,1)?; gbe_QS=Ib_tmp./0.026; %%%===Extraction for QS end=== %==== NQS tau extraction ====== for step=1:1:num_bias Y=Y_cell_exp{step}(:,2:5); Z=Y_to_Z(Y); H=Z(:,4)-Z(:,3); Q=Z(:,2)-Z(:,3); k_b=linefit(Omega.^2,real(H./Q),1,20,1,0,?+-?,?r?); ReHQ(step)=-k_b(1).*gm0(step); k_b=linefit(Omega,imag(H./Q),1,20,-1,0,?+-?,?r?); ImHQ(step)=k_b(1).*gm0(step); end; k_b=linefit(gm0,ImHQ,1,6,1,0,?+?,?r?); Cbei0=gm0.*k_b(1); Cbej0=ImHQ-Cbei0; fn=num_bias-7; Cbej0(fn:num_bias)=Cbej0(fn:num_bias).*0+Cbej0(fn-1); Cbei0=ImHQ-Cbej0; %plot(gm0,Cbei0,gm0,Cbej0); k_b=linefit(Cbei0(1:10),ReHQ(1:10),1,6,1,1,?ks?,?k?); Tautr0=Cbei0./gm0; Taud0=k_b(2)/Cbej0(8)/Tautr0(1).*Tautr0; Tauin0=(Taud0(1)-k_b(1))/Tautr0(1).*Tautr0; %plot(gm0,Tautr0,gm0,Tauin0,gm0,Taud0); %====== extract rbi ========== gbe=Ib_tmp./0.026; for step=1:1:num_bias Y=Y_cell_exp{step}(:,2:5); Y11=gbe(step)+j.*Omega*(Cbej0(step)+Cbci(step))... +j.*Omega.*Cbei0(step)./(1+j.*Omega.*Tauin0(step)); Ybi=1./(1./Y(:,1)-1./Y11); Rbi(step)=sum(1./real(Ybi(11:15)))/(15-11+1); 174 % subplot(1,2,1); % plot(Omega,1./real(Ybi),Omega,Omega+Rbi(step)-Omega); hold on; % subplot(1,2,2); % plot(Omega,imag(Ybi)./Omega,Omega,Omega.*0+(Cbei0(step))/5); end Rci=Rbi-Rbi; % set to be zero %=== If rbi bias dependent method, use the following block %Tau_in=Tauin0; Rd=Tau_in./Cbei; %Tau_out_all=Taud0; %Taud=Tau_out_all-Tau_in; %Cbei=Cbei0; %gm=gm0; %Cbej=Cbej0; % ====== over ==== %=== If rbi bias independent, cnt ==== %=== de Rbi Cbi, Rci ====== for step=1:1:num_bias rbi(step)=Ri; % bias independent value Y=Y_cell_exp{step}(:,2:5); if(HasCbi==1) Cbi(step)=(Cbci(step)+Cbet_QS(step))./5; Zrbi=Rbi(step)./(1+j.*Omega.*Rbi(step).*Cbi(step)); else Zrbi=Omega-Omega+Rbi(step); end; Y=com_Y_rb(Y,-Zrbi); Y=com_Y_rc(Y,-Rci(step)); Y_cell_exp_sav{step}(:,2:5)=Y; end; %=== Extracion of the intrinsic transistor % for bias independent rbi method ======= for step=1:1:num_bias ytmp=Y_cell_exp_sav{step}(:,2:5); %--- Cbci ben=1; enn=6; k_b=polyfit(Omega(ben:enn).^2,imag(ytmp(ben:enn,2))./Omega(ben:enn),1); %plot(Omega.^2,imag(ytmp(:,2))./Omega,Omega.^2,Omega.^2*k_b(1)+k_b(2)); %Cbci(step)=-k_b(2); %--- ru ben=1; enn=4; k_b=polyfit(Omega(ben:enn).^2,real(ytmp(ben:enn,2)),1); Ru(step)=abs(1./k_b(2)); Ru(step)=1.5e6; %plot(Omega.^2,real(ytmp(:,2)),Omega.^2,Omega.^2*k_b(1)+k_b(2)); %=== de Cbci (Y12)=== ytmp=ytmp-[-ytmp(:,2) ytmp(:,2) ytmp(:,2) -ytmp(:,2)]; Y_cell_exp{step}(:,2:5)=ytmp; 175 %plot(Omega,real(ytmp(:,1))); hold on; %--- gbe ben=1; enn=5; k_b=polyfit(Omega(ben:enn).^2,real(ytmp(ben:enn,1)),1); %plot(Omega.^2,real(ytmp(:,1)),Omega.^2,Omega.^2*k_b(1)+k_b(2)); gbe_ex(step)=k_b(2); %--- Cd*Cd*Rd ben=1; enn=15; k_b=polyfit(Omega(ben:enn).^2,real(ytmp(ben:enn,1)),1); %plot(Omega,real(ytmp(:,1)),Omega,Omega.^2*k_b(1)+k_b(2)); hold on; CdCdRd(step)=k_b(1); %--- Cbet ben=1; enn=6; k_b=polyfit(Omega(ben:enn).^2,imag(ytmp(ben:enn,1))./Omega(ben:enn),1); %plot(Omega.^2,imag(ytmp(:,1))./Omega,Omega.^2,Omega.^2*k_b(1)+k_b(2)); Cbet(step)=k_b(2); CdCdCdRdRd(step)=-k_b(1); %--- gm ben=1; enn=6; k_b=polyfit(Omega(ben:enn).^2,real(ytmp(ben:enn,3)),1); %plot(Omega.^2,real(ytmp(:,3)),Omega.^2,Omega.^2*k_b(1)+k_b(2)); gm_ex(step)=k_b(2); %--- Taud+Tau_in = Tau_out_all ben=10; enn=20; k_b=polyfit(Omega(ben:enn).^2,imag(ytmp(ben:enn,3))./Omega(ben:enn),1); %plot(Omega.^2,imag(ytmp(:,3))./Omega,Omega.^2,Omega.^2*k_b(1)+k_b(2)); Tau_out_all(step)=-k_b(2)/gm_ex(step); gm(step)=gm_ex(step); %gbe(step)=gbe_ex(step); gbe(step)=Ib_tmp(step)./(k*bias_cell{step}(5)/q);%gbe_ex(step); end; %=== split Cbet into Cbei and Cbej k_b=polyfit(gm(1:10),Cbet(1:10),1); %plot(gm,Cbet,gm,gm*k_b(1)+k_b(2)); for step=1:1:num_bias; Cbej(step)=k_b(2); Cbei(step)=Cbet(step)-Cbej(step); Tau_in(step)=CdCdRd(step)/Cbei(step); end; Tau_in(1)=Tau_in(3); Tau_in(2)=Tau_in(3); Rd=Tau_in./Cbei; Taud_old=Tau_out_all-Tau_in; Taud=fitcurv(Taud_old?,1)?; %plot(Vbe(1:num_bias),Tau_in,?r*?,Vbe(1:num_bias),Tau_out_all,?rs?,... % Vbe(1:num_bias),Taud_QS,?gv?,Vbe(1:num_bias),Cbei./gm); hold on; 176 APPENDIX F MATLAB CODE FOR INTRINSIC NOISE EXTRACTION The following MATLAB code is used to extract the intrinsic noise for a 50 GHz SiGe HBT with A E = 0.24 ? 20 ? 2 ?m 2 . N_exp_adm_cell_intr is the full PSD of intrinsic base and collector current noises. N_exp_adm_cell_base is the PSD of base electron noise, i.e. the 2qI b removed version of N_exp_adm_cell_intr. F.1 MATLAB code load_data_26G; num_b=1; num_e=20; Omega=Y_cell_exp{step}(:,1).*2.*pi; index=[149141924]; %Noise frequency points for step=num_b:1:num_e %de-embeded Y parameters N_Y_exp=select_row(Y_cell_exp{step},index); f_N=N_Y_exp(:,1); Omega_N=f_N*2*pi; N_Y_6=N_Y_exp(:,2:5); N_Y_5=com_Y_rc(N_Y_6,-j*Omega_N*Lc(step)-Rcx(step)); %Lc Rcx N_Y_5=com_Y_rb(N_Y_5,-j*Omega_N*Lb(step)); %Lb N_Y_5=com_Y_re(N_Y_5,-j*Omega_N*Le(step)); %Le N_Y_4=N_Y_5; N_Y_4(:,4)=N_Y_4(:,4)... -j*Omega_N.*Ccs(step)./(1+j*Omega_N*Ccs(step)*Rcs(step));%Rcs Ccs N_Y_4=N_Y_4-[j*Omega_N*Cbco(step) ... -j*Omega_N*Cbco(step) ... -j*Omega_N*Cbco(step) ... j*Omega_N*Cbco(step)]; %Cbco N_Y_3=com_Y_rb(N_Y_4,-Rbx(step)); %Rbx N_Y_3=com_Y_rc(N_Y_3,-Rci(step)); %Rci N_Y_3=com_Y_re(N_Y_3,-Re(step)); %Re N_Y_2=N_Y_3; N_Y_2(:,1)=N_Y_2(:,1)-j*Omega_N*Cbex(step)-gbex(step); %Cbex 177 N_Y_2=N_Y_2-[j*Omega_N*Cbcx(step) ... -j*Omega_N*Cbcx(step) ... -j*Omega_N*Cbcx(step) ... j*Omega_N*Cbcx(step)]; %Cbcx if(HasCbi==1) Zrbi=Rbi(step)./(1+j.*Omega_N.*Rbi(step).*Cbi(step)); else Zrbi=Rbi(step); end; N_Yintrinsic=com_Y_rb(N_Y_2,-Zrbi); %Rbi Cbi N_Ybase=N_Yintrinsic; N_Ybase(:,1)=N_Ybase(:,1)-gbe(step)-j*Omega_N*Cbej(step); N_Yintr_cell_exp{step}=[f_N N_Yintrinsic]; N_Ybase_cell_exp{step}=[f_N N_Ybase]; %Noise de-embedding T=bias_cell{step}(5); N_exp_6=FRY_to_Svi(10.^(N_cell_exp{step}(:,2)./10),... N_cell_exp{step}(:,3)*get_Z0,N_cell_exp{step}(:,4),T_noise); N_exp_5=cha_to_imp_noise(N_exp_6,Y_to_Z(N_Y_6)); N_exp_5(:,4)=N_exp_5(:,4)-4*k*T*Rcx(step); %Rcx N_exp_4=imp_to_adm_noise(N_exp_5,N_Y_5); N_exp_4(:,4)=N_exp_4(:,4)-... 4*k*T*real(j*Omega_N.*Ccs(step)./(1+j*Omega_N*Ccs(step)*Rcs(step)));%Rcs N_exp_3=adm_to_imp_noise(N_exp_4,Y_to_Z(N_Y_4)); N_exp_3(:,1)=N_exp_3(:,1)-4*k*T*(Re(step)+Rbx(step)); % Rbx re N_exp_3(:,4)=N_exp_3(:,4)-4*k*T*(Re(step)+Rci(step)); % Rci re N_exp_3(:,2)=N_exp_3(:,2)-4*k*T*(Re(step)); %re N_exp_3(:,3)=N_exp_3(:,3)-4*k*T*(Re(step)); %re N_exp_2=imp_to_adm_noise(N_exp_3,N_Y_3); N_exp_2(:,1)=N_exp_2(:,1)-2.*k.*T.*gbex(step); %gbex N_exp_1=adm_to_imp_noise(N_exp_2,Y_to_Z(N_Y_2)); if(HasCbi==1) Zrbi=Rbi(step)./(1+j.*Omega_N.*Rbi(step).*Cbi(step)); else Zrbi=Rbi(step); end; if(HasCrowdingnoise==1) noiserbiv=(4.*k.*T./Rbi(step)-2.*q.*Ib_tmp(step)./3).*Zrbi.*conj(Zrbi); else noiserbiv=4.*k.*T./Rbi(step).*Zrbi.*conj(Zrbi); end; N_exp_1(:,1)=N_exp_1(:,1)-noiserbiv; %Rbi N_exp_intrinsic=imp_to_adm_noise(N_exp_1,N_Yintrinsic); N_exp_base=N_exp_intrinsic; N_exp_base(:,1)=N_exp_base(:,1)-2*q*Ib_tmp(step); N_exp_adm_cell{step}=[f_N N_exp_intrinsic]; % for noise calculateion 178 N_exp_adm_cell_intr{step}=[f_N N_exp_intrinsic]; % intrinsic N_exp_adm_cell_base{step}=[f_N N_exp_base]; % base end; F.2 Data of S-parameters and noise The following is the S-parameters and noise data de-embedded with OPEN structure for the 50 GHz SiGe HBT with A E = 0.24 ? 20 ? 2 ?m 2 . ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 1 !Bias Values Read: !Vb:.770 V, Ib:.002 mA !Vc:1.500 V, Ic:.520 mA !Date: 27 Mar 2003 !Time: 15:52:30 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .978 -24.1 1.921 160.8 .051 74.7 .978 -7.7 3.0000 .957 -35.8 1.755 152.0 .075 67.5 .959 -10.9 4.0000 .935 -46.7 1.724 143.6 .095 60.9 .937 -13.8 5.0000 .919 -56.8 1.630 136.2 .113 54.9 .914 -16.6 6.0000 .897 -66.4 1.457 128.2 .129 49.4 .896 -18.5 7.0000 .878 -75.0 1.472 122.3 .141 44.0 .869 -21.3 8.0000 .858 -83.4 1.403 115.8 .151 39.3 .848 -23.3 9.0000 .843 -90.4 1.312 110.1 .161 35.2 .831 -25.0 10.0000 .826 -97.5 1.233 104.7 .167 31.3 .814 -26.7 11.0000 .820 -102.1 1.200 99.8 .177 28.2 .794 -28.5 12.0000 .797 -109.1 1.110 94.8 .179 24.7 .783 -29.8 13.0000 .790 -114.9 1.047 90.0 .182 21.5 .770 -31.2 14.0000 .780 -119.2 1.000 85.6 .186 18.6 .758 -32.7 15.0000 .770 -124.1 .938 81.7 .187 16.1 .750 -34.1 16.0000 .760 -127.8 .931 77.8 .190 13.6 .733 -35.7 17.0000 .751 -131.9 .876 74.2 .191 11.6 .725 -37.0 18.0000 .749 -135.7 .834 70.6 .193 9.2 .718 -38.4 19.0000 .739 -137.6 .815 67.8 .195 7.4 .706 -40.1 20.0000 .741 -141.7 .787 64.1 .195 4.8 .698 -41.3 21.0000 .737 -145.4 .750 61.0 .188 2.8 .693 -42.6 22.0000 .734 -147.3 .724 57.6 .192 .5 .686 -43.8 23.0000 .722 -150.4 .697 55.4 .189 .1 .678 -45.6 24.0000 .723 -152.8 .668 52.1 .191 -2.1 .670 -46.7 25.0000 .721 -155.3 .644 49.8 .189 -3.5 .666 -48.6 26.0000 .723 -157.5 .618 47.4 .186 -5.1 .665 -49.7 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 179 2.0000 .23 .770 8.2 .66 15.57 5.0000 .66 .801 36.8 .70 9.72 10.0000 1.25 .704 77.5 .61 6.92 15.0000 2.72 .614 106.4 .56 4.47 20.0000 3.19 .600 130.1 .39 2.86 25.0000 4.30 .625 145.5 .34 .93 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 2 !Bias Values Read: !Vb:.796 V, Ib:.008 mA !Vc:1.500 V, Ic:1.400 mA !Date: 27 Mar 2003 !Time: 15:52:32 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .949 -34.5 4.789 155.8 .050 69.6 .952 -12.2 3.0000 .912 -50.5 4.283 145.9 .070 60.5 .909 -16.8 4.0000 .880 -64.5 4.020 136.6 .086 52.9 .863 -20.7 5.0000 .859 -76.6 3.686 129.1 .099 46.3 .818 -24.0 6.0000 .832 -87.8 3.233 121.7 .109 40.5 .785 -25.7 7.0000 .814 -97.1 3.122 115.7 .116 35.6 .741 -28.6 8.0000 .796 -106.0 2.885 109.9 .121 31.0 .708 -30.3 9.0000 .782 -112.9 2.640 105.1 .126 27.7 .685 -31.6 10.0000 .766 -119.7 2.433 100.5 .128 24.4 .664 -32.8 11.0000 .764 -123.4 2.310 96.3 .135 22.1 .637 -34.3 12.0000 .748 -130.2 2.111 92.5 .134 19.3 .626 -35.1 13.0000 .743 -135.2 1.968 88.8 .134 16.9 .612 -36.1 14.0000 .738 -138.9 1.853 85.5 .136 14.6 .600 -37.1 15.0000 .731 -143.3 1.723 82.3 .136 12.9 .592 -38.0 16.0000 .726 -146.3 1.676 79.0 .138 10.9 .571 -39.3 17.0000 .720 -150.0 1.566 76.2 .138 9.3 .565 -40.3 18.0000 .720 -152.8 1.483 73.6 .138 7.6 .559 -41.5 19.0000 .711 -153.9 1.426 71.4 .140 6.7 .547 -42.8 20.0000 .714 -158.0 1.367 68.2 .138 4.5 .539 -43.8 21.0000 .710 -161.5 1.294 65.6 .132 3.9 .537 -44.6 22.0000 .714 -162.5 1.242 63.2 .136 1.7 .529 -45.7 23.0000 .701 -165.1 1.192 61.3 .135 1.6 .522 -47.2 24.0000 .704 -167.2 1.132 58.9 .134 .2 .516 -48.1 25.0000 .701 -168.9 1.092 56.5 .133 .1 .514 -49.8 26.0000 .705 -171.3 1.036 54.8 .131 -1.7 .514 -51.1 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 -0.00 .905 7.2 .37 14.10 5.0000 .65 .608 33.7 .39 13.81 10.0000 1.02 .545 78.9 .32 10.06 15.0000 2.14 .476 109.7 .30 7.54 20.0000 2.56 .466 136.9 .23 5.98 25.0000 3.54 .509 151.5 .22 4.13 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 3 !Bias Values Read: !Vb:.811 V, Ib:.014 mA !Vc:1.500 V, Ic:2.560 mA !Date: 27 Mar 2003 !Time: 15:52:35 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang 180 ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .914 -45.0 7.683 150.5 .048 64.7 .917 -16.9 3.0000 .869 -64.5 6.708 139.5 .064 54.2 .849 -22.6 4.0000 .836 -80.2 6.041 129.7 .076 46.3 .781 -26.9 5.0000 .815 -93.5 5.388 122.1 .085 39.7 .721 -30.1 6.0000 .791 -104.9 4.658 115.2 .091 34.4 .680 -31.4 7.0000 .776 -114.1 4.355 109.5 .096 30.0 .629 -34.0 8.0000 .763 -122.5 3.952 104.4 .098 26.2 .594 -35.2 9.0000 .754 -128.7 3.579 100.1 .101 23.4 .571 -36.0 10.0000 .743 -134.9 3.267 96.1 .101 20.9 .550 -36.8 11.0000 .742 -138.0 3.060 92.3 .107 19.1 .522 -37.9 12.0000 .732 -144.0 2.787 89.2 .105 17.1 .513 -38.4 13.0000 .731 -148.4 2.584 86.0 .105 15.3 .501 -38.9 14.0000 .728 -151.5 2.420 83.2 .106 13.4 .489 -39.7 15.0000 .723 -155.3 2.250 80.5 .105 12.3 .483 -40.4 16.0000 .716 -157.9 2.165 77.6 .106 10.8 .463 -41.6 17.0000 .714 -160.8 2.021 75.3 .106 10.1 .458 -42.2 18.0000 .714 -163.2 1.911 73.0 .106 8.9 .452 -43.3 19.0000 .709 -164.3 1.829 71.1 .108 8.1 .441 -44.3 20.0000 .712 -167.5 1.748 68.3 .106 6.8 .432 -45.2 21.0000 .713 -170.3 1.652 66.3 .101 7.2 .433 -45.8 22.0000 .710 -171.4 1.581 64.0 .104 5.2 .426 -46.7 23.0000 .701 -173.1 1.509 62.4 .104 5.3 .419 -48.2 24.0000 .702 -175.0 1.440 60.3 .103 4.5 .417 -49.1 25.0000 .708 -177.1 1.385 58.6 .103 5.6 .413 -50.5 26.0000 .709 -178.4 1.319 56.8 .100 3.5 .414 -51.8 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .21 .893 5.9 .29 15.54 5.0000 .65 .533 32.1 .30 15.35 10.0000 .97 .460 80.5 .25 11.51 15.0000 1.92 .407 114.1 .23 9.02 20.0000 2.33 .397 143.3 .19 7.42 25.0000 3.19 .448 157.7 .20 5.66 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 4 !Bias Values Read: !Vb:.819 V, Ib:.022 mA !Vc:1.500 V, Ic:3.360 mA !Date: 27 Mar 2003 !Time: 15:52:37 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .891 -52.0 9.538 147.1 .046 61.4 .889 -20.0 3.0000 .844 -73.3 8.175 135.5 .060 50.7 .806 -26.2 4.0000 .814 -89.7 7.186 125.6 .070 42.5 .726 -30.4 5.0000 .793 -103.2 6.308 118.1 .077 36.2 .661 -33.5 6.0000 .773 -114.4 5.420 111.7 .081 31.4 .618 -34.4 7.0000 .761 -123.0 4.980 106.2 .084 27.4 .566 -36.7 8.0000 .753 -131.0 4.484 101.4 .086 24.2 .532 -37.6 9.0000 .744 -136.7 4.042 97.5 .088 21.9 .510 -38.1 10.0000 .737 -142.3 3.673 93.8 .089 19.7 .490 -38.7 11.0000 .736 -145.3 3.418 90.3 .093 18.1 .464 -39.6 12.0000 .729 -150.6 3.115 87.5 .091 16.5 .456 -39.8 13.0000 .729 -154.6 2.884 84.6 .091 15.4 .444 -40.3 14.0000 .725 -157.3 2.695 81.9 .092 13.8 .434 -40.9 15.0000 .722 -160.7 2.505 79.5 .091 13.1 .428 -41.5 16.0000 .719 -163.1 2.397 76.8 .092 11.9 .409 -42.5 17.0000 .715 -165.7 2.240 74.7 .091 11.4 .405 -43.0 18.0000 .717 -168.0 2.115 72.7 .091 10.1 .401 -44.2 181 19.0000 .706 -169.0 2.017 70.8 .093 9.7 .390 -45.1 20.0000 .714 -171.9 1.933 68.2 .092 8.5 .383 -45.8 21.0000 .711 -174.6 1.828 66.3 .087 9.4 .383 -46.4 22.0000 .712 -175.3 1.746 64.2 .090 7.9 .377 -47.2 23.0000 .699 -177.3 1.662 62.7 .090 9.0 .372 -48.6 24.0000 .714 -178.9 1.590 60.6 .089 7.5 .367 -49.4 25.0000 .715 179.6 1.531 58.8 .089 8.4 .366 -50.8 26.0000 .708 178.4 1.453 57.3 .087 7.3 .364 -51.9 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .25 .908 8.2 .26 15.64 5.0000 .71 .498 32.6 .27 16.11 10.0000 1.02 .419 81.8 .22 12.15 15.0000 1.85 .366 116.6 .21 9.66 20.0000 2.26 .384 147.3 .16 8.09 25.0000 3.05 .432 160.7 .18 6.35 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 5 !Bias Values Read: !Vb:.829 V, Ib:.028 mA !Vc:1.500 V, Ic:4.860 mA !Date: 27 Mar 2003 !Time: 15:52:39 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .855 -63.1 12.354 141.9 .043 56.4 .840 -24.8 3.0000 .810 -86.6 10.258 129.7 .054 45.3 .734 -31.3 4.0000 .787 -103.3 8.721 119.9 .061 37.6 .644 -35.3 5.0000 .771 -116.4 7.502 112.8 .066 31.9 .576 -37.8 6.0000 .758 -126.7 6.392 107.0 .068 28.0 .533 -38.2 7.0000 .750 -134.5 5.758 102.0 .070 24.9 .484 -40.0 8.0000 .745 -141.5 5.137 97.8 .071 22.3 .453 -40.4 9.0000 .740 -146.6 4.608 94.2 .072 20.6 .432 -40.5 10.0000 .735 -151.4 4.172 90.9 .073 19.1 .415 -40.8 11.0000 .734 -154.1 3.856 87.8 .076 18.1 .391 -41.5 12.0000 .730 -158.6 3.516 85.3 .074 16.4 .384 -41.5 13.0000 .730 -161.8 3.250 82.7 .074 16.1 .375 -41.8 14.0000 .728 -164.2 3.031 80.3 .075 14.8 .365 -42.3 15.0000 .727 -167.2 2.817 78.1 .074 14.6 .359 -42.6 16.0000 .724 -169.4 2.680 75.7 .075 14.1 .343 -43.6 17.0000 .726 -171.5 2.508 73.8 .075 13.6 .340 -44.0 18.0000 .723 -173.6 2.361 71.9 .075 13.4 .336 -45.2 19.0000 .718 -174.6 2.257 70.2 .075 13.4 .328 -45.9 20.0000 .719 -176.9 2.150 67.9 .075 12.3 .320 -46.5 21.0000 .723 -179.1 2.038 66.0 .072 14.2 .321 -47.0 22.0000 .717 -179.8 1.944 64.1 .074 11.8 .316 -47.6 23.0000 .714 178.5 1.859 62.9 .074 13.1 .310 -49.3 24.0000 .722 177.0 1.772 60.9 .075 13.1 .305 -50.2 25.0000 .716 175.7 1.700 59.4 .073 13.1 .304 -51.1 26.0000 .719 174.6 1.621 57.7 .075 13.6 .304 -52.8 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .11 .941 11.0 .25 14.47 182 5.0000 .78 .450 38.8 .22 17.31 10.0000 1.07 .354 85.9 .20 13.04 15.0000 1.80 .319 122.4 .20 10.45 20.0000 2.18 .352 154.5 .16 8.83 25.0000 2.97 .414 164.8 .17 7.04 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 6 !Bias Values Read: !Vb:.831 V, Ib:.036 mA !Vc:1.500 V, Ic:5.060 mA !Date: 27 Mar 2003 !Time: 15:52:42 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .844 -66.3 13.111 140.5 .042 55.1 .825 -26.1 3.0000 .802 -90.2 10.793 128.1 .052 44.0 .714 -32.7 4.0000 .781 -106.8 9.103 118.5 .059 36.4 .622 -36.5 5.0000 .767 -119.7 7.789 111.5 .063 31.1 .554 -38.9 6.0000 .755 -129.8 6.624 105.8 .065 27.1 .511 -39.2 7.0000 .748 -137.3 5.938 100.9 .067 24.2 .463 -40.7 8.0000 .744 -144.1 5.288 96.8 .067 21.9 .433 -41.1 9.0000 .739 -148.9 4.739 93.4 .069 20.3 .413 -41.1 10.0000 .735 -153.6 4.289 90.2 .069 19.2 .397 -41.2 11.0000 .735 -156.3 3.958 87.1 .072 18.1 .374 -41.9 12.0000 .729 -160.5 3.606 84.7 .070 17.1 .367 -41.9 13.0000 .732 -163.6 3.337 82.2 .070 16.4 .358 -42.1 14.0000 .728 -165.9 3.108 79.9 .071 15.6 .349 -42.6 15.0000 .730 -168.7 2.891 77.7 .070 15.1 .343 -43.0 16.0000 .725 -170.7 2.745 75.4 .071 15.1 .328 -43.8 17.0000 .725 -173.0 2.564 73.5 .071 14.5 .325 -44.3 18.0000 .725 -174.8 2.421 71.7 .071 14.1 .322 -45.4 19.0000 .722 -176.0 2.311 70.0 .072 13.9 .313 -46.1 20.0000 .722 -178.1 2.198 67.8 .072 13.8 .305 -46.9 21.0000 .723 180.0 2.087 66.0 .068 14.4 .307 -47.4 22.0000 .718 178.8 1.991 64.1 .071 13.7 .302 -47.9 23.0000 .711 177.6 1.899 62.9 .070 15.1 .296 -49.3 24.0000 .720 176.1 1.809 61.0 .071 14.3 .292 -50.0 25.0000 .721 175.0 1.744 59.4 .071 16.2 .291 -51.5 26.0000 .719 173.8 1.662 57.9 .071 15.7 .287 -52.6 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .27 .845 15.6 .24 19.41 5.0000 .89 .386 29.1 .24 17.42 10.0000 1.08 .338 87.0 .19 13.24 15.0000 1.79 .312 123.9 .19 10.63 20.0000 2.17 .347 156.5 .16 9.00 25.0000 2.95 .417 165.2 .16 7.26 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 7 !Bias Values Read: !Vb:.836 V, Ib:.042 mA !Vc:1.500 V, Ic:6.180 mA !Date: 27 Mar 2003 !Time: 15:52:44 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang 183 ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .825 -72.9 14.618 137.6 .040 52.4 .793 -28.8 3.0000 .787 -97.4 11.808 125.1 .049 41.4 .673 -35.3 4.0000 .771 -113.7 9.801 115.7 .054 34.1 .579 -38.8 5.0000 .760 -126.0 8.305 109.0 .057 29.3 .512 -40.8 6.0000 .751 -135.5 7.044 103.7 .059 26.0 .470 -40.8 7.0000 .745 -142.5 6.261 99.0 .060 23.4 .425 -42.0 8.0000 .743 -148.7 5.559 95.2 .060 21.5 .397 -42.2 9.0000 .740 -153.2 4.976 92.0 .062 20.4 .379 -42.0 10.0000 .735 -157.5 4.496 88.9 .062 19.3 .363 -42.1 11.0000 .737 -160.1 4.137 86.0 .064 18.6 .342 -42.6 12.0000 .734 -163.9 3.775 83.8 .063 18.1 .337 -42.5 13.0000 .735 -166.6 3.486 81.4 .063 17.5 .328 -42.7 14.0000 .731 -168.8 3.247 79.1 .063 16.8 .320 -43.1 15.0000 .731 -171.3 3.023 77.1 .063 17.1 .314 -43.5 16.0000 .729 -173.3 2.862 74.8 .064 16.6 .300 -44.2 17.0000 .728 -175.3 2.677 73.1 .064 16.3 .296 -44.5 18.0000 .729 -177.2 2.523 71.3 .064 16.6 .293 -45.8 19.0000 .725 -178.2 2.408 69.6 .064 16.7 .286 -46.3 20.0000 .725 179.7 2.292 67.6 .065 16.4 .278 -47.2 21.0000 .733 177.3 2.174 65.7 .062 17.7 .279 -47.6 22.0000 .726 177.1 2.069 63.9 .064 16.6 .275 -47.9 23.0000 .719 175.7 1.980 62.6 .063 17.7 .271 -49.6 24.0000 .723 174.1 1.889 61.0 .065 17.6 .266 -50.3 25.0000 .724 172.9 1.803 59.5 .064 19.2 .266 -51.7 26.0000 .723 172.1 1.728 58.0 .065 18.4 .262 -53.0 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .50 .649 17.4 .23 23.42 5.0000 .84 .390 42.0 .21 18.13 10.0000 1.16 .319 89.9 .18 13.59 15.0000 1.80 .299 127.7 .18 10.98 20.0000 2.15 .337 161.1 .16 9.33 25.0000 3.03 .406 169.3 .16 7.52 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 8 !Bias Values Read: !Vb:.841 V, Ib:.050 mA !Vc:1.500 V, Ic:7.380 mA !Date: 27 Mar 2003 !Time: 15:52:47 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .805 -80.0 16.115 134.5 .038 49.8 .757 -31.5 3.0000 .775 -104.6 12.772 122.0 .045 39.0 .630 -37.8 4.0000 .763 -120.4 10.449 113.0 .049 32.3 .536 -40.8 5.0000 .755 -132.1 8.774 106.7 .052 27.8 .471 -42.5 6.0000 .748 -140.9 7.425 101.6 .053 25.0 .431 -42.3 7.0000 .745 -147.4 6.552 97.3 .054 23.0 .390 -43.1 8.0000 .743 -153.1 5.802 93.7 .054 21.6 .363 -43.1 9.0000 .740 -157.1 5.185 90.6 .055 20.4 .347 -42.8 10.0000 .738 -161.0 4.680 87.7 .056 19.9 .332 -42.7 11.0000 .739 -163.5 4.293 85.0 .057 19.2 .313 -43.1 12.0000 .735 -166.9 3.917 82.8 .057 19.0 .308 -43.0 13.0000 .739 -169.5 3.620 80.6 .057 18.7 .299 -43.1 14.0000 .735 -171.5 3.368 78.5 .057 18.3 .292 -43.3 15.0000 .735 -173.9 3.137 76.5 .057 18.7 .287 -43.7 16.0000 .732 -175.8 2.963 74.4 .058 18.4 .273 -44.6 17.0000 .730 -177.7 2.773 72.6 .058 19.4 .271 -44.6 18.0000 .733 -179.1 2.615 70.9 .058 19.3 .269 -45.8 184 19.0000 .729 179.8 2.493 69.4 .059 19.2 .261 -46.8 20.0000 .730 177.9 2.367 67.3 .059 19.4 .253 -47.4 21.0000 .732 175.6 2.252 65.6 .057 20.3 .255 -48.1 22.0000 .728 175.4 2.145 63.7 .059 19.7 .252 -48.2 23.0000 .725 173.5 2.046 62.6 .058 22.2 .247 -49.2 24.0000 .725 172.7 1.950 60.7 .060 22.2 .243 -50.7 25.0000 .729 171.6 1.877 59.7 .060 23.4 .242 -52.0 26.0000 .728 170.7 1.796 58.2 .059 24.0 .240 -53.7 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .66 .529 19.5 .23 25.02 5.0000 1.00 .298 31.2 .23 18.48 10.0000 1.23 .294 94.4 .18 13.97 15.0000 1.85 .296 132.2 .17 11.31 20.0000 2.21 .331 165.0 .16 9.59 25.0000 3.01 .406 171.9 .16 7.88 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 9 !Bias Values Read: !Vb:.844 V, Ib:.054 mA !Vc:1.500 V, Ic:8.100 mA !Date: 27 Mar 2003 !Time: 15:52:49 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .795 -83.7 16.853 133.0 .036 48.4 .739 -32.9 3.0000 .769 -108.3 13.226 120.5 .043 37.8 .608 -39.1 4.0000 .759 -123.7 10.747 111.7 .047 31.2 .515 -41.8 5.0000 .753 -135.0 8.983 105.5 .049 27.4 .451 -43.3 6.0000 .747 -143.5 7.597 100.6 .050 24.7 .412 -42.9 7.0000 .744 -149.8 6.679 96.4 .051 22.6 .372 -43.6 8.0000 .743 -155.1 5.908 92.9 .051 21.6 .348 -43.5 9.0000 .742 -159.1 5.277 89.9 .052 20.8 .331 -43.0 10.0000 .738 -162.8 4.766 87.1 .053 20.1 .318 -42.9 11.0000 .741 -165.1 4.362 84.4 .054 19.9 .299 -43.2 12.0000 .736 -168.5 3.984 82.3 .054 19.8 .294 -43.1 13.0000 .740 -170.9 3.678 80.1 .054 19.9 .285 -43.2 14.0000 .738 -172.9 3.422 78.0 .055 19.4 .279 -43.5 15.0000 .738 -175.2 3.185 76.1 .054 19.8 .274 -43.7 16.0000 .734 -177.0 3.007 74.0 .055 19.4 .262 -44.7 17.0000 .733 -178.8 2.815 72.3 .055 20.0 .259 -44.9 18.0000 .734 179.8 2.658 70.6 .055 20.6 .256 -46.1 19.0000 .730 178.6 2.530 69.1 .056 21.2 .250 -47.0 20.0000 .731 177.0 2.405 67.0 .056 21.0 .244 -47.6 21.0000 .734 174.9 2.288 65.5 .054 23.4 .245 -48.0 22.0000 .733 174.5 2.173 63.8 .055 21.2 .241 -48.6 23.0000 .725 172.2 2.078 62.3 .056 22.3 .235 -49.8 24.0000 .726 172.0 1.985 60.7 .057 23.6 .233 -50.9 25.0000 .737 170.6 1.908 59.0 .057 24.8 .229 -52.5 26.0000 .727 169.9 1.827 58.1 .058 25.0 .227 -53.0 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .72 .486 21.1 .22 25.56 185 5.0000 1.02 .277 32.6 .22 18.73 10.0000 1.26 .280 97.3 .17 14.17 15.0000 1.88 .291 135.6 .17 11.48 20.0000 2.23 .330 166.5 .16 9.71 25.0000 2.98 .414 174.1 .15 8.10 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 10 !Bias Values Read: !Vb:.849 V, Ib:.066 mA !Vc:1.500 V, Ic:9.680 mA !Date: 27 Mar 2003 !Time: 15:52:52 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .778 -91.2 18.241 129.9 .034 45.6 .699 -35.5 3.0000 .759 -115.4 14.048 117.6 .040 35.6 .566 -41.3 4.0000 .754 -130.0 11.269 109.2 .042 29.6 .474 -43.5 5.0000 .750 -140.5 9.351 103.4 .044 26.3 .414 -44.5 6.0000 .747 -148.3 7.898 98.8 .045 24.2 .377 -43.9 7.0000 .745 -154.0 6.901 94.8 .046 22.8 .341 -44.2 8.0000 .745 -158.9 6.093 91.5 .046 21.8 .319 -44.0 9.0000 .744 -162.4 5.438 88.7 .047 21.5 .304 -43.4 10.0000 .741 -165.9 4.906 86.0 .047 21.4 .291 -43.2 11.0000 .744 -168.1 4.480 83.5 .049 20.9 .274 -43.4 12.0000 .740 -171.0 4.099 81.5 .048 21.2 .270 -43.3 13.0000 .742 -173.2 3.783 79.5 .048 21.8 .263 -43.4 14.0000 .740 -175.1 3.513 77.5 .049 21.7 .256 -43.5 15.0000 .741 -177.1 3.272 75.6 .049 22.2 .252 -43.9 16.0000 .740 -179.0 3.081 73.6 .050 22.6 .240 -44.7 17.0000 .738 179.5 2.890 72.0 .050 23.2 .238 -44.9 18.0000 .738 178.1 2.726 70.4 .050 23.6 .236 -46.3 19.0000 .735 176.8 2.594 68.8 .051 24.2 .228 -46.8 20.0000 .736 175.4 2.461 66.9 .052 23.9 .223 -47.6 21.0000 .729 173.6 2.332 65.3 .050 24.9 .226 -48.1 22.0000 .733 173.0 2.239 63.5 .051 24.4 .221 -48.1 23.0000 .730 171.3 2.136 62.4 .052 27.5 .216 -49.5 24.0000 .735 170.5 2.029 60.9 .054 27.2 .211 -50.5 25.0000 .731 169.5 1.962 59.1 .056 30.3 .213 -52.9 26.0000 .736 168.5 1.866 58.0 .055 29.5 .209 -53.3 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .84 .411 25.4 .22 26.47 5.0000 1.01 .259 43.3 .21 19.24 10.0000 1.33 .258 104.1 .17 14.52 15.0000 1.94 .287 140.3 .17 11.75 20.0000 2.29 .335 170.4 .16 9.95 25.0000 3.06 .401 176.4 .16 8.25 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 11 !Bias Values Read: !Vb:.851 V, Ib:.074 mA !Vc:1.500 V, Ic:10.620 mA !Date: 27 Mar 2003 !Time: 15:52:54 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang 186 ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .770 -95.1 18.885 128.3 .033 44.3 .679 -36.8 3.0000 .755 -118.9 14.414 116.2 .038 34.7 .545 -42.3 4.0000 .752 -133.0 11.499 108.0 .040 29.0 .455 -44.2 5.0000 .751 -143.2 9.509 102.3 .042 25.9 .396 -45.0 6.0000 .748 -150.5 8.027 97.9 .043 24.1 .361 -44.3 7.0000 .746 -155.9 6.993 94.0 .043 23.0 .327 -44.5 8.0000 .746 -160.7 6.170 90.9 .043 22.4 .306 -44.1 9.0000 .746 -164.1 5.501 88.1 .044 21.9 .292 -43.4 10.0000 .743 -167.4 4.966 85.5 .045 22.1 .279 -43.1 11.0000 .745 -169.4 4.529 83.0 .046 21.9 .263 -43.4 12.0000 .743 -172.4 4.146 81.1 .046 22.0 .259 -43.4 13.0000 .746 -174.3 3.822 79.0 .046 22.2 .252 -43.1 14.0000 .742 -176.4 3.554 77.0 .047 22.5 .246 -43.7 15.0000 .743 -178.2 3.314 75.2 .047 23.6 .241 -43.8 16.0000 .741 -180.0 3.111 73.3 .047 23.7 .231 -44.8 17.0000 .740 178.3 2.917 71.6 .048 24.5 .228 -44.9 18.0000 .742 177.1 2.753 70.1 .049 25.0 .227 -46.2 19.0000 .740 175.7 2.619 68.4 .050 26.1 .220 -47.2 20.0000 .739 174.7 2.481 66.7 .049 25.4 .216 -47.7 21.0000 .736 172.5 2.362 65.0 .050 28.0 .216 -48.2 22.0000 .739 172.3 2.251 63.4 .050 27.8 .212 -48.0 23.0000 .730 170.3 2.148 62.2 .050 28.1 .207 -49.8 24.0000 .736 169.8 2.047 60.6 .051 29.0 .203 -50.3 25.0000 .731 168.5 1.969 59.4 .052 31.0 .202 -51.9 26.0000 .741 167.6 1.893 57.8 .052 30.5 .199 -53.2 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .90 .376 27.4 .21 26.83 5.0000 1.05 .233 46.2 .20 19.49 10.0000 1.36 .246 107.9 .17 14.68 15.0000 1.96 .287 143.4 .16 11.90 20.0000 2.34 .341 172.2 .15 10.07 25.0000 3.14 .402 178.4 .17 8.28 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 12 !Bias Values Read: !Vb:.854 V, Ib:.082 mA !Vc:1.500 V, Ic:11.580 mA !Date: 27 Mar 2003 !Time: 15:52:57 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .763 -98.7 19.462 126.8 .032 42.9 .659 -38.0 3.0000 .752 -122.1 14.733 114.9 .036 33.5 .524 -43.2 4.0000 .751 -135.8 11.686 107.0 .038 28.6 .437 -44.8 5.0000 .750 -145.4 9.640 101.5 .040 25.7 .381 -45.4 6.0000 .748 -152.5 8.132 97.2 .041 24.1 .346 -44.5 7.0000 .747 -157.7 7.066 93.4 .041 23.2 .314 -44.6 8.0000 .747 -162.1 6.229 90.3 .041 22.7 .294 -44.1 9.0000 .747 -165.4 5.555 87.7 .042 22.5 .280 -43.4 10.0000 .744 -168.6 5.011 85.1 .043 22.7 .269 -43.1 11.0000 .747 -170.6 4.565 82.6 .044 22.8 .254 -43.3 12.0000 .745 -173.3 4.180 80.8 .044 22.9 .250 -43.2 13.0000 .748 -175.3 3.857 78.7 .044 23.3 .242 -43.3 14.0000 .744 -177.1 3.584 76.8 .045 23.9 .237 -43.6 15.0000 .745 -178.9 3.343 75.1 .045 24.7 .232 -43.7 16.0000 .743 179.3 3.135 73.1 .045 25.1 .222 -44.8 17.0000 .742 177.7 2.939 71.5 .046 26.3 .220 -44.8 18.0000 .745 176.5 2.774 70.0 .047 25.9 .218 -46.1 187 19.0000 .739 175.2 2.634 68.6 .047 28.1 .213 -46.7 20.0000 .740 174.0 2.498 66.7 .048 27.7 .207 -47.6 21.0000 .742 172.1 2.382 65.2 .048 30.4 .209 -47.9 22.0000 .737 171.6 2.273 63.3 .048 29.4 .204 -47.8 23.0000 .734 170.2 2.175 62.2 .048 31.9 .201 -49.4 24.0000 .734 169.4 2.072 60.7 .050 31.9 .196 -50.7 25.0000 .729 167.7 1.971 59.6 .052 31.8 .195 -51.8 26.0000 .739 167.4 1.906 57.7 .051 33.8 .192 -53.9 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 .96 .349 29.4 .21 27.08 5.0000 1.08 .218 51.2 .20 19.72 10.0000 1.41 .242 111.5 .17 14.81 15.0000 1.99 .290 146.1 .16 12.03 20.0000 2.39 .343 173.9 .16 10.14 25.0000 3.26 .398 -179.6 .18 8.28 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 13 !Bias Values Read: !Vb:.856 V, Ib:.084 mA !Vc:1.500 V, Ic:12.700 mA !Date: 27 Mar 2003 !Time: 15:52:59 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .756 -102.5 19.995 125.3 .030 41.7 .639 -39.1 3.0000 .750 -125.4 15.013 113.6 .034 32.7 .505 -44.0 4.0000 .750 -138.5 11.847 105.9 .036 28.0 .419 -45.3 5.0000 .751 -147.8 9.747 100.5 .038 25.2 .365 -45.6 6.0000 .749 -154.6 8.218 96.4 .039 24.2 .332 -44.7 7.0000 .748 -159.6 7.126 92.7 .039 23.1 .301 -44.5 8.0000 .748 -163.8 6.278 89.7 .039 23.1 .283 -44.1 9.0000 .748 -166.8 5.595 87.1 .040 23.1 .269 -43.3 10.0000 .745 -169.9 5.046 84.6 .041 23.3 .259 -42.9 11.0000 .748 -171.9 4.592 82.2 .042 23.7 .245 -43.1 12.0000 .746 -174.6 4.207 80.3 .042 24.1 .240 -43.0 13.0000 .750 -176.3 3.880 78.3 .042 25.2 .233 -43.1 14.0000 .745 -178.1 3.609 76.5 .043 25.8 .229 -43.3 15.0000 .748 -179.9 3.363 74.7 .043 25.6 .225 -43.6 16.0000 .746 178.5 3.151 72.8 .043 26.7 .215 -44.6 17.0000 .743 176.4 2.957 71.0 .044 26.9 .211 -44.5 18.0000 .747 175.8 2.791 69.7 .045 28.2 .211 -45.9 19.0000 .740 174.2 2.641 68.2 .045 29.4 .205 -46.9 20.0000 .743 173.5 2.512 66.3 .046 28.9 .199 -47.4 21.0000 .741 171.2 2.394 64.8 .045 30.6 .202 -47.8 22.0000 .741 171.0 2.284 63.1 .047 30.7 .198 -47.5 23.0000 .741 168.8 2.184 61.7 .048 33.5 .194 -49.6 24.0000 .741 168.6 2.081 60.2 .049 31.3 .185 -50.0 25.0000 .740 167.2 1.991 59.1 .050 34.6 .188 -52.4 26.0000 .736 166.5 1.914 58.0 .050 34.4 .184 -54.0 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.02 .317 32.0 .21 27.35 188 5.0000 1.13 .197 55.6 .20 19.91 10.0000 1.46 .233 115.9 .17 14.93 15.0000 2.04 .284 148.9 .17 12.10 20.0000 2.43 .351 175.2 .16 10.23 25.0000 3.23 .415 -177.8 .17 8.51 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 14 !Bias Values Read: !Vb:.859 V, Ib:.096 mA !Vc:1.500 V, Ic:13.920 mA !Date: 27 Mar 2003 !Time: 15:53:01 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .750 -106.4 20.474 123.8 .029 40.6 .618 -40.2 3.0000 .748 -128.6 15.251 112.3 .033 32.0 .485 -44.7 4.0000 .749 -141.2 11.977 104.8 .035 27.5 .402 -45.6 5.0000 .751 -150.0 9.831 99.6 .036 25.3 .350 -45.8 6.0000 .750 -156.5 8.287 95.6 .036 24.2 .319 -44.7 7.0000 .750 -161.2 7.167 92.1 .037 23.7 .290 -44.4 8.0000 .750 -165.1 6.312 89.1 .037 23.8 .272 -43.7 9.0000 .750 -168.2 5.623 86.6 .038 23.8 .260 -43.0 10.0000 .747 -171.1 5.073 84.1 .039 24.4 .250 -42.7 11.0000 .750 -173.0 4.607 81.8 .040 24.8 .236 -42.8 12.0000 .748 -175.6 4.225 80.0 .040 25.6 .232 -42.7 13.0000 .751 -177.2 3.899 78.0 .040 25.8 .225 -42.8 14.0000 .748 -179.1 3.621 76.1 .041 26.5 .222 -43.0 15.0000 .751 179.4 3.379 74.4 .041 27.5 .216 -43.4 16.0000 .748 177.6 3.163 72.5 .042 28.0 .209 -44.3 17.0000 .745 175.8 2.968 70.9 .042 28.9 .205 -44.0 18.0000 .748 175.0 2.803 69.4 .043 29.4 .204 -45.8 19.0000 .745 173.6 2.664 68.0 .044 30.9 .199 -47.0 20.0000 .745 172.8 2.522 66.2 .045 31.2 .193 -47.0 21.0000 .741 170.5 2.397 64.6 .044 33.0 .195 -47.8 22.0000 .743 170.5 2.295 62.9 .045 32.5 .192 -47.2 23.0000 .743 168.6 2.186 61.7 .046 33.6 .187 -49.6 24.0000 .743 168.1 2.091 60.2 .047 33.9 .182 -50.1 25.0000 .746 167.0 1.999 58.5 .048 35.2 .183 -52.2 26.0000 .735 166.0 1.926 57.8 .049 38.1 .175 -52.2 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.06 .284 34.7 .21 27.59 5.0000 1.17 .181 61.2 .19 20.08 10.0000 1.51 .228 120.9 .16 15.06 15.0000 2.08 .290 151.9 .16 12.22 20.0000 2.47 .354 176.1 .16 10.28 25.0000 3.26 .422 -177.3 .17 8.60 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 15 !Bias Values Read: !Vb:.859 V, Ib:.096 mA !Vc:1.500 V, Ic:13.840 mA !Date: 27 Mar 2003 !Time: 15:53:04 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang 189 ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .750 -106.4 20.471 123.8 .029 40.7 .618 -40.2 3.0000 .748 -128.6 15.248 112.3 .033 31.8 .485 -44.7 4.0000 .749 -141.2 11.976 104.8 .035 27.5 .402 -45.6 5.0000 .750 -150.0 9.830 99.6 .036 25.4 .351 -45.7 6.0000 .750 -156.5 8.284 95.6 .036 24.0 .319 -44.7 7.0000 .750 -161.2 7.168 92.0 .037 23.5 .290 -44.4 8.0000 .750 -165.2 6.312 89.1 .038 23.4 .272 -43.7 9.0000 .750 -168.2 5.622 86.6 .038 23.8 .260 -43.1 10.0000 .746 -171.1 5.069 84.1 .039 24.2 .250 -42.7 11.0000 .751 -173.0 4.608 81.8 .040 24.8 .236 -42.8 12.0000 .748 -175.5 4.226 80.0 .040 25.3 .232 -42.7 13.0000 .752 -177.3 3.898 78.0 .040 26.3 .225 -42.9 14.0000 .749 -179.1 3.622 76.2 .041 26.8 .222 -43.1 15.0000 .751 179.3 3.380 74.4 .042 27.5 .216 -43.3 16.0000 .748 177.6 3.163 72.6 .042 28.4 .208 -44.1 17.0000 .744 175.9 2.964 70.9 .041 29.1 .205 -44.0 18.0000 .748 175.1 2.804 69.5 .043 29.9 .203 -45.9 19.0000 .745 173.8 2.659 68.0 .043 31.7 .199 -46.3 20.0000 .745 172.8 2.519 66.2 .045 30.7 .194 -47.2 21.0000 .747 170.7 2.408 64.6 .044 32.9 .195 -47.8 22.0000 .743 170.8 2.291 63.0 .045 31.7 .192 -47.0 23.0000 .738 168.3 2.183 61.6 .046 34.4 .187 -48.9 24.0000 .741 168.0 2.082 60.3 .046 35.0 .180 -50.0 25.0000 .737 167.3 2.009 59.3 .048 36.5 .182 -51.7 26.0000 .739 166.6 1.919 57.9 .048 37.3 .179 -54.1 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.05 .282 34.8 .21 27.60 5.0000 1.17 .181 60.9 .19 20.08 10.0000 1.51 .228 120.2 .16 15.04 15.0000 2.07 .288 151.8 .16 12.21 20.0000 2.49 .356 175.9 .15 10.28 25.0000 3.25 .414 -178.0 .17 8.55 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 16 !Bias Values Read: !Vb:.861 V, Ib:.106 mA !Vc:1.500 V, Ic:15.120 mA !Date: 27 Mar 2003 !Time: 15:53:06 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .746 -110.0 20.851 122.4 .028 39.5 .599 -41.1 3.0000 .747 -131.6 15.427 111.1 .031 31.2 .467 -45.2 4.0000 .749 -143.7 12.063 103.9 .033 27.2 .386 -45.9 5.0000 .751 -152.1 9.879 98.8 .034 25.0 .337 -45.7 6.0000 .751 -158.2 8.328 94.9 .035 24.4 .307 -44.5 7.0000 .751 -162.7 7.187 91.5 .035 24.1 .280 -44.0 8.0000 .752 -166.4 6.326 88.6 .035 24.1 .264 -43.5 9.0000 .752 -169.3 5.634 86.1 .036 24.5 .252 -42.6 10.0000 .750 -172.1 5.083 83.7 .037 25.2 .242 -42.2 11.0000 .753 -174.0 4.614 81.4 .038 25.9 .228 -42.4 12.0000 .748 -176.4 4.232 79.7 .038 26.4 .226 -42.4 13.0000 .753 -178.1 3.904 77.8 .039 27.3 .219 -42.4 14.0000 .751 -179.7 3.629 75.9 .039 27.9 .216 -42.6 15.0000 .752 178.7 3.384 74.2 .039 29.2 .211 -42.9 16.0000 .749 177.1 3.164 72.3 .040 29.7 .204 -43.9 17.0000 .747 175.4 2.974 70.7 .041 31.1 .200 -44.0 18.0000 .751 174.6 2.808 69.3 .041 31.5 .200 -45.3 190 19.0000 .748 173.3 2.661 67.9 .043 32.3 .195 -46.6 20.0000 .746 172.2 2.521 66.0 .044 32.2 .189 -47.0 21.0000 .748 169.8 2.405 64.4 .043 34.2 .190 -47.8 22.0000 .744 170.1 2.288 62.9 .045 35.0 .186 -47.5 23.0000 .740 168.3 2.196 61.6 .045 36.4 .181 -48.7 24.0000 .746 167.6 2.088 60.1 .047 36.3 .178 -49.9 25.0000 .748 167.1 2.001 58.7 .047 37.1 .177 -51.8 26.0000 .739 165.7 1.924 57.5 .048 37.6 .173 -52.5 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.12 .255 37.1 .21 27.74 5.0000 1.22 .166 67.0 .19 20.22 10.0000 1.55 .228 125.7 .16 15.17 15.0000 2.12 .293 154.4 .16 12.28 20.0000 2.54 .360 177.4 .16 10.32 25.0000 3.31 .431 -177.2 .17 8.65 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 17 !Bias Values Read: !Vb:.864 V, Ib:.114 mA !Vc:1.500 V, Ic:16.560 mA !Date: 27 Mar 2003 !Time: 15:53:09 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .741 -113.7 21.133 120.9 .027 38.4 .578 -41.9 3.0000 .745 -134.6 15.540 109.9 .030 30.4 .448 -45.6 4.0000 .750 -146.1 12.096 102.9 .031 27.0 .371 -45.7 5.0000 .752 -154.0 9.893 98.1 .032 25.4 .325 -45.4 6.0000 .752 -159.8 8.338 94.3 .033 24.8 .296 -44.2 7.0000 .752 -164.1 7.180 91.0 .033 24.9 .271 -43.5 8.0000 .754 -167.7 6.317 88.2 .034 24.9 .255 -42.9 9.0000 .753 -170.4 5.628 85.7 .035 25.8 .244 -42.0 10.0000 .752 -173.1 5.071 83.4 .035 26.1 .235 -41.6 11.0000 .754 -174.8 4.599 81.2 .036 26.9 .222 -41.6 12.0000 .752 -177.0 4.225 79.4 .036 27.5 .219 -41.6 13.0000 .755 -178.8 3.896 77.5 .037 28.9 .214 -41.8 14.0000 .752 179.5 3.621 75.7 .037 29.1 .210 -42.0 15.0000 .753 178.1 3.378 74.1 .038 30.8 .206 -42.3 16.0000 .752 176.4 3.152 72.2 .039 31.1 .197 -43.1 17.0000 .748 174.9 2.968 70.7 .039 32.6 .195 -43.1 18.0000 .752 173.9 2.804 69.2 .040 33.2 .194 -44.9 19.0000 .749 172.9 2.656 67.8 .041 33.4 .189 -45.6 20.0000 .748 171.8 2.513 66.0 .042 34.3 .184 -46.0 21.0000 .748 170.3 2.398 64.5 .042 36.9 .185 -47.1 22.0000 .748 169.7 2.286 62.9 .042 36.7 .184 -47.1 23.0000 .749 167.7 2.186 61.7 .044 37.6 .177 -48.5 24.0000 .745 167.5 2.087 59.9 .046 38.6 .174 -49.2 25.0000 .749 166.3 1.987 58.8 .047 40.4 .176 -50.7 26.0000 .746 165.8 1.925 58.0 .048 41.3 .168 -52.9 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.20 .228 39.9 .21 27.84 191 5.0000 1.27 .154 74.1 .19 20.33 10.0000 1.61 .231 130.0 .16 15.22 15.0000 2.18 .299 157.1 .16 12.33 20.0000 2.62 .365 178.5 .16 10.32 25.0000 3.42 .436 -174.8 .17 8.65 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 18 !Bias Values Read: !Vb:.866 V, Ib:.124 mA !Vc:1.500 V, Ic:17.960 mA !Date: 27 Mar 2003 !Time: 15:53:11 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .738 -117.3 21.263 119.5 .026 37.3 .558 -42.5 3.0000 .745 -137.3 15.569 108.9 .029 30.0 .432 -45.7 4.0000 .750 -148.2 12.064 102.1 .030 26.6 .358 -45.4 5.0000 .753 -155.8 9.863 97.4 .031 25.1 .313 -44.8 6.0000 .754 -161.3 8.315 93.8 .031 25.3 .287 -43.5 7.0000 .753 -165.3 7.142 90.5 .032 25.4 .263 -42.7 8.0000 .757 -168.8 6.280 87.8 .032 25.7 .249 -42.0 9.0000 .755 -171.4 5.594 85.4 .033 26.0 .239 -41.1 10.0000 .754 -173.8 5.042 83.2 .034 27.5 .230 -40.8 11.0000 .755 -175.6 4.565 80.9 .035 28.1 .219 -40.8 12.0000 .753 -177.5 4.197 79.3 .035 29.4 .215 -40.8 13.0000 .755 -179.3 3.877 77.5 .035 30.2 .211 -40.7 14.0000 .754 179.0 3.598 75.7 .036 31.0 .206 -41.0 15.0000 .754 177.5 3.357 74.0 .037 32.2 .202 -41.2 16.0000 .752 176.2 3.125 72.3 .037 32.5 .194 -42.4 17.0000 .750 175.1 2.950 70.9 .039 34.7 .194 -42.9 18.0000 .750 173.5 2.784 69.4 .040 34.7 .191 -43.8 19.0000 .749 173.3 2.649 68.0 .040 36.5 .187 -44.4 20.0000 .750 171.5 2.487 66.0 .041 35.8 .182 -45.5 21.0000 .750 171.0 2.390 64.7 .040 39.5 .181 -45.5 22.0000 .751 169.4 2.265 63.0 .042 37.9 .179 -46.4 23.0000 .750 169.0 2.179 62.3 .043 40.9 .176 -47.4 24.0000 .750 167.3 2.066 60.5 .045 40.5 .171 -48.9 25.0000 .747 167.2 2.009 59.2 .047 41.7 .173 -51.3 26.0000 .747 166.1 1.901 57.8 .047 41.6 .168 -52.0 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.28 .203 43.4 .21 27.88 5.0000 1.32 .146 81.9 .18 20.40 10.0000 1.66 .236 134.0 .16 15.26 15.0000 2.24 .302 159.5 .16 12.33 20.0000 2.72 .373 179.2 .16 10.28 25.0000 3.37 .432 -176.7 .18 8.69 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 19 !Bias Values Read: !Vb:.866 V, Ib:.124 mA !Vc:1.500 V, Ic:17.880 mA !Date: 27 Mar 2003 !Time: 15:53:13 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang 192 ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .738 -117.3 21.255 119.5 .026 37.4 .558 -42.5 3.0000 .745 -137.2 15.570 108.9 .029 30.0 .432 -45.7 4.0000 .751 -148.1 12.059 102.2 .030 26.7 .358 -45.4 5.0000 .753 -155.6 9.866 97.5 .031 25.4 .314 -44.8 6.0000 .754 -161.1 8.312 94.0 .031 25.1 .287 -43.4 7.0000 .753 -165.2 7.144 90.7 .032 25.3 .264 -42.5 8.0000 .757 -168.5 6.277 88.0 .032 25.9 .248 -41.8 9.0000 .754 -171.3 5.597 85.6 .033 26.4 .238 -41.0 10.0000 .756 -173.4 5.039 83.5 .034 27.4 .229 -40.7 11.0000 .755 -175.4 4.569 81.2 .035 28.5 .219 -40.6 12.0000 .753 -177.0 4.199 79.7 .035 30.0 .216 -40.5 13.0000 .753 -179.2 3.879 77.8 .036 30.3 .212 -40.6 14.0000 .755 179.5 3.596 76.0 .036 31.9 .206 -40.8 15.0000 .754 177.6 3.359 74.4 .037 32.1 .202 -41.1 16.0000 .751 176.5 3.125 72.5 .037 33.1 .194 -42.0 17.0000 .752 175.6 2.956 71.3 .038 35.5 .195 -42.1 18.0000 .749 173.8 2.788 69.7 .039 35.5 .191 -43.0 19.0000 .744 174.1 2.651 68.8 .040 36.7 .188 -43.8 20.0000 .750 171.8 2.490 66.3 .040 36.7 .180 -45.2 21.0000 .756 171.6 2.383 65.5 .041 39.8 .180 -44.9 22.0000 .755 169.9 2.260 63.4 .041 38.5 .175 -47.5 23.0000 .747 170.7 2.182 62.8 .043 42.3 .177 -47.1 24.0000 .756 167.8 2.075 60.9 .043 41.5 .175 -48.9 25.0000 .742 167.6 1.992 60.1 .046 43.8 .175 -49.5 26.0000 .755 166.4 1.918 58.4 .046 42.7 .174 -51.8 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.29 .204 43.2 .21 27.88 5.0000 1.32 .146 80.8 .19 20.39 10.0000 1.67 .246 133.3 .16 15.28 15.0000 2.24 .303 159.1 .16 12.33 20.0000 2.70 .373 178.3 .16 10.28 25.0000 3.46 .428 -177.3 .18 8.56 ! ! S-Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! !Bias# 20 !Bias Values Read: !Vb:.869 V, Ib:.136 mA !Vc:1.500 V, Ic:19.440 mA !Date: 27 Mar 2003 !Time: 15:53:16 !Deembedding: ON ! ! Freq S11 Mag S11 Ang S21 Mag S21 Ang S12 Mag S12 Ang S22 Mag S22 Ang ! (GHz) (Deg) (Deg) (Deg) (Deg) !________ ________ ________ ________ ________ ________ ________ ________ ________ ! 2.0000 .735 -121.1 21.178 118.0 .025 36.3 .537 -42.9 3.0000 .745 -140.0 15.478 107.8 .027 29.2 .415 -45.5 4.0000 .752 -150.3 11.930 101.3 .028 26.8 .344 -44.7 5.0000 .753 -157.4 9.761 96.9 .029 25.7 .304 -43.9 6.0000 .755 -162.5 8.232 93.4 .030 25.9 .279 -42.4 7.0000 .754 -166.3 7.052 90.2 .030 26.0 .257 -41.4 8.0000 .758 -169.6 6.194 87.7 .031 26.7 .243 -40.7 9.0000 .756 -172.2 5.523 85.3 .032 27.4 .234 -39.8 10.0000 .759 -174.2 4.972 83.2 .032 28.9 .225 -39.6 11.0000 .755 -176.2 4.497 81.0 .033 29.7 .216 -39.4 12.0000 .755 -177.5 4.140 79.5 .033 31.0 .212 -39.2 13.0000 .755 -179.8 3.831 77.7 .034 32.1 .209 -39.6 14.0000 .757 179.0 3.550 75.9 .035 32.5 .204 -39.7 15.0000 .753 177.3 3.319 74.3 .036 34.0 .201 -40.3 16.0000 .753 176.4 3.071 72.6 .036 35.4 .192 -40.9 17.0000 .755 175.8 2.914 71.4 .037 36.2 .193 -41.4 18.0000 .749 173.4 2.749 69.7 .038 37.5 .189 -41.5 193 19.0000 .747 174.6 2.625 69.1 .038 38.5 .187 -42.4 20.0000 .752 171.4 2.446 66.3 .040 38.4 .179 -44.0 21.0000 .767 172.5 2.357 65.7 .040 40.3 .178 -44.6 22.0000 .756 169.8 2.223 63.5 .040 40.5 .173 -46.8 23.0000 .748 171.6 2.152 63.2 .043 45.0 .176 -47.0 24.0000 .762 167.6 2.037 60.9 .042 43.2 .174 -47.4 25.0000 .742 167.8 1.972 60.5 .045 45.3 .177 -46.9 26.0000 .757 166.3 1.879 58.9 .045 45.6 .173 -51.6 ! ! Noise Parameters vs Frequency vs Bias !M NOISE: M:PS;A:16;C:16;DC:1;H:0;P:1;DOT:DUT_NF_M; ! ! Freq F(min) Gamma Opt Gamma Opt Normalized Associated ! (GHz) Fitted Fitted Fitted Rn Z0=50 Gain (dB) ! (dB) Mag Angle Fitted Fitted !________ __________ __________ __________ __________ __________ ! 2.0000 1.41 .176 46.2 .21 27.80 5.0000 1.38 .141 89.5 .18 20.40 10.0000 1.73 .254 137.6 .16 15.28 15.0000 2.30 .308 160.7 .17 12.27 20.0000 2.82 .383 179.4 .16 10.19 25.0000 3.51 .433 -177.4 .18 8.51 194 APPENDIX G VERILOG-A CODE OF VBIC MODEL FOR SEMI-EMPIRICAL NOISE MODEL IMPLEMENTATION Only the noise block is given, which is relavant to the semi-empirical noise model. Branch n_ia and n_ib consist of 1 Ohm resistance respectively. Branch b_bei is the intrinsic BE diode. Branch b_cei is the intrinsic CB current flow path. ... // begin noise block n_gm = abs(Itzf)/(nf_t*Vtv); // added by kejun n_cSib = n_Kbb*pow(n_gm,n_abb)+n_Bbb+1e-60; // added by kejun n_cSic = (n_Kcc*pow(n_gm,n_acc)+n_Bcc)*n_gm; // added by kejun n_cSicib = n_Kcb*pow(n_gm,n_acb)+n_Bcb; // added by kejun n_cVib = n_cSic-n_cSicib*n_cSicib/n_cSib; // added by kejun n_cVib = n_cVib>0.0 ? sqrt(n_cVib) : 0.0; // added by kejun I(n_ia) <+ white_noise(1); // added by kejun I(n_ib) <+ white_noise(1); // added by kejun I(n_ia) <+ V(n_ia); // added by kejun I(n_ib) <+ V(n_ib); // added by kejun I(b_bei) <+ ddt(V(n_ia))*sqrt(n_cSib); // added by kejun I(b_cei) <+ V(n_ia)*n_cSicib/sqrt(n_cSib); // changed by kejun I(b_cei) <+ V(n_ib)*n_cVib; // changed by kejun I(b_bei) <+ white_noise(2*?QQ*abs(Ibe)) +flicker_noise(kfn*pow(abs(Ibe),afn),bfn); I(b_bex) <+ white_noise(2*?QQ*abs(Ibex)) +flicker_noise(kfn*pow(abs(Ibex),afn),bfn); I(b_bep) <+ white_noise(2*?QQ*abs(Ibep)) +flicker_noise(kfn*pow(abs(Ibep),afn),bfn); I(b_rcx) <+ white_noise(4*?KB*Tdev*Gcx); I(b_rci) <+ white_noise(4*?KB*Tdev*((abs(Irci) +1.0e-10*Gci)/(abs(Vrci)+1.0e-10))); I(b_rbx) <+ white_noise(4*?KB*Tdev*Gbx); I(b_rbi) <+ white_noise(4*?KB*Tdev*qb*Gbi); I(b_re) <+ white_noise(4*?KB*Tdev*Ge); I(b_rbp) <+ white_noise(4*?KB*Tdev*qbp*Gbp); I(b_cep) <+ white_noise(2*?QQ*abs(Iccp)); I(b_rs) <+ white_noise(4*?KB*Tdev*Gs); // end noise block ... 195 APPENDIX H DERIVATION OF LOW INJECTION VAN VLIET MODEL IN ADMITTANCE REPRESENTATION H.1 Fundamentals H.1.1 Operator We define inner product ? integraldisplay fg ? dv. For operator L, its adjoint operator, tildewide L, is defined as ? = contintegraldisplay C[f,g] ?d?. (H.1) That is, there is only a surface integration for the di?erence of inner products. The surface integral is along the inner surface. Note that tildewide L = (L T ) ? , where L T is the transpose of L, superscript * denotes conjugate. If L = tildewide L, L is called self-adjoint operator. If the surface integration of a self-adjoint operator vanishes, L is a Hermitian operator. For carrier transport in semiconductor, carrier continuity equations like (2.40), (2.44) and (2.49) should be satisfied. The carrier changing rate operators for electron and hole in frequency domain are L n = s+ 1/? n ?triangleinv?? ?? E ?Dtriangleinv 2 , (H.2) L p = s+ 1/? p +triangleinv?? ?? E ?Dtriangleinv 2 , (H.3) 196 wheres = j?. The di?usion coe?cientD is assumed to be position independent, while the electric field is not subjected to such constraint. Their adjoint operators are tildewide L n = ?s+ 1/? n +? ?? E ?triangleinv?Dtriangleinv 2 , (H.4) tildewide L p = ?s+ 1/? p ?? ?? E ?triangleinv?Dtriangleinv 2 . (H.5) The adjoint operators are simple because the electrical field is in front of triangleinv. Further, in accord with [39], we write ? the spatial parts of these operators, i.e. L =?+s and tildewide L = tildewide ?+s ? .For example, the ? of hole carrier is ? p = 1/? p +triangleinv?? ?? E ?Dtriangleinv 2 , (H.6) tildewide ? p = 1/? p ?? ?? E ?triangleinv?Dtriangleinv 2 . (H.7) H.1.2 Green?s theorem for L p The Green?s theorem for L p is that for any two functions ?(r) and ?(r), (H.1) is satisfied, explicitly integraldisplay ? ? L p ?dv? integraldisplay ? tildewide L ? p ? ? dv = contintegraldisplay [?? ? (Dtriangleinv 0 ?? ?? E 0 )? +?Dtriangleinv 0 ? ? ] ?d?. (H.8) The surface integral is along the inner surface. H.1.3 Dirac delta function The bulk delta function ?(r?r prime ) is zero at any position except r prime and integraltext ?(r?r prime )dv = 1. There are many analytical?(r?r prime ) functions, for example?triangleinv 2 [1/(4?|r?r prime |)]. ?(r?r prime ) has the following 197 properties: ?(r prime )?(r ?r prime ) = ?(r)?(r ?r prime ), triangleinv?(r ?r prime ) = ?triangleinv prime ?(r ?r prime ), ?(r prime ) triangleinv?(r ?r prime ) = ?(r) triangleinv?(r ?r prime ) + [triangleinv?(r)]?(r ?r prime ), integraldisplay ?(r 0 ?r)?(r)dv = ?(r 0 )/2, (H.9) where ?(r) can be any function, and r 0 is a point on the smooth boundary surface of given integra- tion volume. H.1.4 ? theorem ? is the spatial part of operator L. The ? theorem derived by van Vliet in [40] gives the connection between the covariance ?(r,r prime ) ?< ?p(r,t)?p(r prime ,t) > and the noise source ?(r,t) with strength ?(r,r prime ) = 1 2 S bulk , that is (? r +? r prime)?(r,r prime ) = 1 2 S bulk (r,r prime ). (H.10) Note ?(r,r prime ) =?(r prime ,r) and S bulk (r,r prime ) = S bulk (r prime ,r). Certain boundary conditions can be stated which indicate that?often satisfies delta-type singularities at the surface of volumeV. The solution inside V stemming from the volume sources only will be denoted as ? prime . H.2 Problem setup for base low injection noise of PNP transistor We consider the low injection minority carrier (i.e. hole) noise for the base region of a PNP transistor as shown in Fig. H.1. S ? and S ? are the neutral base ending surface at EB and CB junc- tions. S f is the free surface of base. S c is the base contact surface. We allow position dependent 198 built-in field E(r) and life time ?(r), while a position independent di?usion coe?cient D. E(r) should not depend on the carrier density. B E C f S c S , S ?? P P N Figure H.1: Schematic geometry of a PNP transistor. The hole continuity equation with homogenous boundary condition is L p p = ?(r,s),?(r,s) = ?(r,s) +triangleinv??(r,s), (H.11) p| ?=S ? ,S ? = 0, (H.12) where L p is given in (H.3), p is the hole density fluctuation. ?(r,s) is the di?usion noise with PSD (in flux density representation, i.e. no charge units e) S ? (r,r prime ) = 4Dp s (r)?(r ?r prime )I, (H.13) according to (2.35), where p s (r) is the total DC hole density. ?(r,s) is the GR noise with PSD S ? (r,r prime ) = braceleftbigg 4p s (r) ? p ? 2Dtriangleinv 2 p s (r) + 2 triangleinv?[? ?? E(r)p s (r)] bracerightbigg ?(r ?r prime ) (H.14) 199 Note (H.14) is equivalent to (2.36) once p s (r) >> p s0 (r), a condition well satisfied when the transistor is forward biased. p s0 (r) is the hole density at zero bias. To provide this, we consider the DC continuity equation, which is ? p p s (r) = 0. Hence, S ? (r,r prime ) = bracketleftbigg 2p s (r) ? p ? 2? p p s (r) bracketrightbigg ?(r ?r prime ) = 2p s (r) ? p ?(r ?r prime ) ? 2[p s (r) +p s0 (r)] ? p ?(r ?r prime ). The reason of using (H.14) instead of (2.36) will be clear when applying ? theorem below. The total noise spectrum is S ? (r,r prime ) =S ? (r,r prime ) +triangleinv?triangleinv prime ?S ? (r,r prime ) = braceleftbigg 4p s (r) ? p ? 2Dtriangleinv 2 p s (r) + 2 triangleinv?[? ?? E(r)p s (r)] bracerightbigg ?(r ?r prime ) + 4Dtriangleinv?triangleinv prime [p s (r)?(r ?r prime )]. (H.15) Due to the assumptionp s (r) >> p s0 (r), the van Vliet model is not correct at zero bias. Clearly finite exit velocity e?ect at CB junction is not considered as of the homogenous boundary condition used. H.3 Green?s function of homogeneous boundary Define the homogeneous boundary Green?s function G s as L p G s (r,r prime ,s) = ?(r ?r prime ),G s (r 0 ,r prime ,s) = 0| r 0 ?S c ,S f ,S ? ,S ? . (H.16) If the surface recombination velocity of S c and S f are not infinite, boundary conditions in (H.16) are not correct. However, theoretical analysis shows that the noise results will not change [39]. Define the adjoint Green?s function tildewide G s as tildewide L p tildewide G s (r,r primeprime ,s) = ?(r ?r primeprime ). (H.17) 200 Here r prime and r primeprime are source positions within the volume of integration. The Green?s functions are not defined when the source positions are on the boundary yet. A ?good? boundary condition for tildewide G s should be chosen so that reciprocity G s (r,r prime ,s) = tildewide G s? (r prime ,r,s) holds. This means that contintegraltext C[G s (r 0 ,r prime ,j?), tildewide G s? (r 0 ,r primeprime ,s)]d? must vanish. To provide this, tildewide G s should satisfy the boundary condition according to (H.8) tildewide G s (r 0 ,r primeprime ,s) = 0| r 0 ?S c ,S f ,S ? ,S ? . Due to the reciprocity condition, G s (r prime ,r 0 ,s) = 0| r 0 ?S c ,S f ,S ? ,S ? , tildewide G s (r primeprime ,r 0 ,s) = 0| r 0 ?S c ,S f ,S ? ,S ? . So far the Green?s functions are fully defined. G(r,r prime ,s) has the following property: for r = r ? and r prime ? r + ? , contintegraldisplay (Dtriangleinv ? ?? ?? E ? )G s (r ? ,r + ? ,s) ?d? ? = ?1, (H.18) where r is a point on surface ?, and r prime approaches to the surface ? from the inside. To provide this, consider an infinite small volume ?V enclosed by surface ? and surface t as shown in Fig. H.2. The surface t is a auxiliary surface infinitely close to surface ?. r prime ? r + ? is contained by ?V. According to the boundary condition of G(r,r prime ,s), G(r ? ,r + ? ,s) = 0,G(r t ,r + ? ,s) = 0. (H.19) 201 and then take a volume integral of (H.16) inside ?V, integraldisplay triangleinv?(? ?? E ?Dtriangleinv)G(r,r prime ,s)dv = integraldisplay ?(r ?r prime )dv = 1. (H.20) By using the Gauss theorem for the left side of (H.20), the volume integral can be transformed into a surface integral. Noticing that the surface integral on surface t is infinite small, (H.18) is then obtained. Physically, (H.18) means that when the delta current injection position is very close to surface ?, then all the injected current will be collected by surface ?. S ? t S r ? r ? t r Figure H.2: Illustration of surface integral. H.4 Hole concentration fluctuation and its spectrum The Green?s theorem with ? = p(r,s) and ? = tildewide G s (r,r prime ,s)gives p(r prime ,s) = integraldisplay tildewide G s? (r,r prime ,s)?(r,s)dv + contintegraldisplay [ tildewide G s? (r 0 ,r prime ,s)(Dtriangleinv 0 ?? ?? E 0 )p(r 0 ,s) ?p(r 0 ,s)Dtriangleinv 0 tildewide G s? (r 0 ,r prime ,s)]?d?. (H.21) 202 Making the changes r prime ? r, r ? r primeprime and using the reciprocity, the hole concentration fluctuation can be obtained as p(r,s) = integraldisplay G s (r,r primeprime ,s)?(r primeprime ,s)dv + contintegraldisplay [G s (r,r 0 ,s)(Dtriangleinv 0 ?? ?? E 0 )p(r 0 ,s) ?p(r 0 ,s)Dtriangleinv 0 G s (r,r 0 ,s)]?d?. (H.22) (H.22) is valid for G s with any boundary condition. Especially for the homogeneous boundary condition defined for G s , (H.22) reduces to p(r,s) = integraldisplay G s (r,r primeprime ,s)?(r primeprime ,s)dv. (H.23) From (H.23), the noise spectrum of the correlation betweenp(r,s) andp(r prime ,s) can be obtained S p (r,r prime ,?) ?

= integraldisplayintegraldisplay G s (r,r 1 ,?s) G s (r prime ,r 2 ,s)dv 1 dv 2 = integraldisplayintegraldisplay G s (r,r 1 ,?s)S ? (r 1 ,r 2 )G s (r prime ,r 2 ,s)dv 1 dv 2 . (H.24) (H.24) is quadratic in Green?s function, and should be transformed to be linear in Green?s function using the ? theorem. H.4.1 van Vliet - Fasset form of noise spectrum According to (H.10), the bulk covariance ? prime (r 1 ,r 2 ) satisfies (L ? p,r 1 +L p,r 2 )? prime (r 1 ,r 2 ) = (? p,r 1 +? p,r 2 )? prime (r 1 ,r 2 ) = 1 2 S ? (r 1 ,r 2 ), (H.25) 203 so that S p (r,r prime ,?) =2 integraldisplayintegraldisplay G s (r,r 1 ,?s)G s (r prime ,r 2 ,s)[L ? p,r 1 +L p,r 2 ]? prime (r 1 ,r 2 )dv 1 dv 2 . (H.26) Changing r ? r 2 and making ? =? prime (r 1 ,r 2 ) and ? ? = G s (r,r 1 ,?s)G s (r prime ,r 2 ,s), the Green?s theorem gives integraldisplay G s (r,r 1 ,?s)G s (r prime ,r 2 ,s)L p,r 2 ? prime (r 1 ,r 2 )dv 2 = integraldisplay ? prime (r 1 ,r 2 ) tildewide L ? p,r 2 G s (r,r 1 ,?s)G s (r prime ,r 2 ,s)dv 2 + contintegraldisplay [?G s (r,r 1 ,?s)G s (r prime ,r 0 ,s)(Dtriangleinv 0 ?? ?? E 0 )? prime (r 1 ,r 0 )]?d? + contintegraldisplay [? prime (r 1 ,r 0 )Dtriangleinv 0 G s (r,r 1 ,?s)G s (r prime ,r 0 ,s)]?d?. (H.27) Similarly, Changing r ? r 1 and making ? ? =? prime (r 1 ,r 2 ) and ? = G s (r,r 1 ,?s)G s (r prime ,r 2 ,s), the conjugate Green?s theorem gives integraldisplay G s (r,r 1 ,?s)G s (r prime ,r 2 ,s)L ? p,r 1 ? prime (r 1 ,r 2 )dv 1 = integraldisplay ? prime (r 1 ,r 2 ) tildewide L p,r 1 G s (r,r 1 ,?s)G s (r prime ,r 2 ,s)dv 1 + contintegraldisplay [?G s (r,r 0 ,?s)G s (r prime ,r 2 ,s)(Dtriangleinv 0 ?? ?? E 0 )? prime (r 0 ,r 2 )]?d? + contintegraldisplay [? prime (r 0 ,r 2 )Dtriangleinv 0 G s (r,r 0 ,?s)G s (r prime ,r 2 ,s)]?d?. (H.28) 204 Now using (H.27) and (H.28), (H.26) becomes S p (r,r prime ,?) =2 integraldisplayintegraldisplay ? prime (r 1 ,r 2 )[ tildewide L p,r 1 + tildewide L ? p,r 2 ] tildewide G s (r 1 ,r,s) tildewide G s? (r 2 ,r prime ,s)dv 1 dv 2 + 2 contintegraldisplayintegraldisplay ?G s (r,r primeprime ,?s)G s (r prime ,r 0 ,s)(Dtriangleinv 0 ?? ?? E 0 )? prime (r primeprime ,r 0 )dv primeprime ?d? + 2 contintegraldisplayintegraldisplay ? prime (r primeprime ,r 0 )Dtriangleinv 0 G s (r,r primeprime ,?s)G s (r prime ,r 0 ,s)dv primeprime ?d? + 2 contintegraldisplayintegraldisplay ?G s (r,r 0 ,?s)G s (r prime ,r primeprime ,s)(Dtriangleinv 0 ?? ?? E 0 )? prime (r 0 ,r primeprime )dv primeprime ?d? + 2 contintegraldisplayintegraldisplay ? prime (r 0 ,r primeprime )Dtriangleinv 0 G s (r,r 0 ,?s)G s (r prime ,r primeprime ,s)dv primeprime ?d?. (H.29) Note that the bulk part of (H.29) can be further reduced using the definition of Green?s functions S p (r,r prime ,?)| bulk =2 integraldisplayintegraldisplay tildewide G s? (r 2 ,r prime ,s)? prime (r 1 ,r 2 )[ tildewide L p,r 1 tildewide G s (r 1 ,r,s)]dv 1 dv 2 + 2 integraldisplayintegraldisplay tildewide G s (r 1 ,r,s)? prime (r 1 ,r 2 )[ tildewide L ? p,r 2 tildewide G s? (r 2 ,r prime ,s)]dv 1 dv 2 =2 integraldisplayintegraldisplay tildewide G s? (r 2 ,r prime ,s)? prime (r 1 ,r 2 )?(r 1 ?r)dv 1 dv 2 + 2 integraldisplayintegraldisplay tildewide G s (r 1 ,r,s)? prime (r 1 ,r 2 )?(r 2 ?r prime )dv 1 dv 2 =2 integraldisplay [? prime (r,r primeprime )G s (r prime ,r primeprime ,s) +? prime (r primeprime ,r prime )G s (r,r primeprime ,?s)dv primeprime , (H.30) so that S p (r,r prime ,?) =2 integraldisplay [? prime (r,r primeprime )G s (r prime ,r primeprime ,s) +? prime (r primeprime ,r prime )G s (r,r primeprime ,?s)dv primeprime + 2 contintegraldisplayintegraldisplay ?G s (r,r primeprime ,?s)G s (r prime ,r 0 ,s)(Dtriangleinv 0 ?? ?? E 0 )? prime (r primeprime ,r 0 )dv primeprime ?d? + 2 contintegraldisplayintegraldisplay ? prime (r primeprime ,r 0 )Dtriangleinv 0 G s (r,r primeprime ,?s)G s (r prime ,r 0 ,s)dv primeprime ?d? + 2 contintegraldisplayintegraldisplay ?G s (r,r 0 ,?s)G s (r prime ,r primeprime ,s)(Dtriangleinv 0 ?? ?? E 0 )? prime (r 0 ,r primeprime )dv primeprime ?d? + 2 contintegraldisplayintegraldisplay ? prime (r 0 ,r primeprime )Dtriangleinv 0 G s (r,r 0 ,?s)G s (r prime ,r primeprime ,s)dv primeprime ?d?. (H.31) 205 This is the van-Fasset form whose main part is linear in Green?s functions. (H.31) is valid for G s with any boundary condition for the hole density fluctuation caused by bulk noise sources. Particularly, for the homogeneous boundary condition of G s , G s (r,r 0 ,s) = 0. All surface integrals in (H.31) vanish, therefore S p (r,r prime ,?) =2 integraldisplay [? prime (r,r primeprime )G s (r prime ,r primeprime ,s) +? prime (r primeprime ,r prime )G s (r,r primeprime ,?s)dv primeprime . (H.32) H.4.2 Solution for ? theorem at low injection For low injection, the ? theorem with the source (H.15) and ? (H.7) is bracketleftbigg 2 ? p +triangleinv?? ?? E(r) ?Dtriangleinv 2 +triangleinv prime ?? ?? E(r prime ) ?Dtriangleinv prime2 bracketrightbigg ? prime (r,r prime ) = braceleftbigg 2p s (r) ? p ?Dtriangleinv 2 p s (r) +triangleinv?[? ?? E(r)p s (r)] bracerightbigg ?(r ?r prime ) + 2Dtriangleinv?triangleinv prime [p s (r)?(r ?r prime )]. (H.33) Using the properties of delta function in (H.9), one finds that (H.33) admits the solution ? prime (r,r prime ) = p s (r)?(r ?r prime ). (H.34) The derivation manifests the value of using ?p s (r) = 0 in the GR noise source as discussed in Section H.2. With (H.34) and (H.32), the spectrum of hole fluctuation for homogeneous boundary condition is S p (r,r prime ,?) = 2p s (r)G s (r prime ,r,s) + 2p s (r prime )G s (r,r prime ,?s). (H.35) 206 H.5 Terminal noise current spectrum The operator to transform carrier density into current density is (e? ?? E ?eDtriangleinv) for hole. The total hole current density at given point r is j(r,t) = e(? ?? E ?Dtriangleinv)p(r,t) ?e?(r,t) = j o (r,t) ?e?(r,t), (H.36) where j o (r,t) is the response fluctuation current density j o (r,t) = e(? ?? E ?Dtriangleinv)p(r,t). (H.37) Therefore, the spectrum of is S j (r,r prime ,?) = S j o(r,r prime ,?) ?S j o? ,e? (r,r prime ,?) ?S e? ? ,j o(r,r prime ,?) +e 2 S ? (r,r prime ,?). (H.38) The current spectrum due to ? is S j ? (?) = e 2 contintegraldisplaycontintegraldisplay S ? (r ? ,r + ? ,?) ?d? ? ?d? + ? = e 2 contintegraldisplaycontintegraldisplay 4Dp s (r ? )?(r ? ?r + ? )d? ? ?d? + ? = 0. (H.39) In the following, the rest three components are calculated. 207 H.5.1 Spectrum due to j o The correlation spectrum of j o , ,is S j o(r,r prime ,?) =2e 2 (? ?? E ?Dtriangleinv)(? ?? E prime ?Dtriangleinv prime )[p s (r)G s (r prime ,r,s) +p s (r prime )G s (r,r prime ,?s)] =2e 2 p s (r prime )(Dtriangleinv?? ?? E)Dtriangleinv prime G s (r,r prime ,?s) + 2e 2 braceleftBig (Dtriangleinv prime ?? ?? E prime )p s (r prime ) bracerightBig (Dtriangleinv?? ?? E)G s (r,r prime ,?s) + 2e 2 p s (r)(Dtriangleinv prime ?? ?? E prime )DtriangleinvG s (r prime ,r,s) + 2e 2 braceleftBig (Dtriangleinv?? ?? E)p s (r) bracerightBig (Dtriangleinv prime ?? ?? E prime )G s (r prime ,r,s) (H.40) The reason of separation will be clear when it is connected to Y-parameters below. Now the correlation PSD of two di?erent terminal ? and terminal ? (? negationslash= ?) is, S i o? ? i o ? (?) = contintegraldisplaycontintegraldisplay S j o(r,r prime ,?) ?d? ? ?d? ? vextendsingle vextendsingle vextendsingle vextendsingle r=r ? ,r prime =r ? = contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv ? G s (r ? ,r ? ,?s) ?d? ? ?d? ? + contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv ? G s (r ? ,r ? ,s) ?d? ? ?d? ? . (H.41) For the auto-correlation PSD of terminal ?, r = r ? and r prime = r ? cannot be set simultaneously, since the derivative of G(r ? ,r ? ,s) cannot be defined. The trick is to set r = r ? and let r prime ? r + ? (ortoset r prime = r ? and let r ? r + ? ). Here the superscript + indicates that the surface position is approached from the inside (consistent with the Green?s theorem). Note that e(Dtriangleinv ? ?? ?? E ? )p s (r ? ) = ?J DC ? (r ? ), contintegraldisplay J DC ? (r ? ) ?d? ? = I ? 208 Then the auto-correlation PSD of terminal ? is S i o? ? i o ? (?) = contintegraldisplaycontintegraldisplay S j o(r,r prime ,?) ?d? ? ?d? ? | r=r ? ,r prime ?r + ? = contintegraldisplaycontintegraldisplay 2e 2 p s (r + ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv + ? G s (r ? ,r + ? ,?s) ?d? ? ?d? + ? + contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv + ? ?? ?? E + ? )Dtriangleinv ? G s (r + ? ,r ? ,s) ?d? ? ?d? + ? ? contintegraldisplay 2eJ DC ? (r ? ) contintegraldisplay (Dtriangleinv ? ?? ?? E ? )G s (r ? ,r + ? ,?s) ?d? ? ?d? + ? = contintegraldisplaycontintegraldisplay 2e 2 p s (r + ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv + ? G s (r ? ,r + ? ,?s) ?d? ? ?d? + ? + contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv + ? ?? ?? E + ? )Dtriangleinv ? G s (r + ? ,r ? ,s) ?d? ? ?d? + ? + 2eI ? . (H.42) H.5.2 Spectrum due to correlation of j o and ? Similar to the derivation ofS p (r,r prime ,?), using (H.23) and (2.35),S p ? ,? (r,r prime ,?) can be obtained as S p ? ,? (r,r prime ,?) = 4D integraldisplay G s (r,r primeprime ,?s) triangleinv primeprime [p s (r prime )?(r primeprime ?r prime )] = ?4Dp s (r prime ) triangleinv prime G s (r,r prime ,?s), (H.43) since the surface integral vanishes. Further, S j o? ,? (r ? ,r ? ,?) = 4eDp s (r ? )(Dtriangleinv ? ?? ?? E ? ) triangleinv ? G s (r ? ,r ? ,?s), (H.44) and finally S i o? ? ,e integraltext ? ? ?d? ? (?) = e 2 contintegraldisplaycontintegraldisplay 4Dp s (r ? )(? ?? E ? ?Dtriangleinv ? ) triangleinv ? G s (r ? ,r ? ,?s) ?d? ? ?d? ? . (H.45) 209 Similarly, S e integraltext ? ? ? ?d? ? ,i o ? (?) = e 2 contintegraldisplaycontintegraldisplay 4Dp s (r ? )(? ?? E ? ?Dtriangleinv ? ) triangleinv ? G s (r ? ,r ? ,s) ?d? ? ?d? ? . (H.46) H.5.3 Terminal total noise current density spectrum The correlation PSD of two di?erent terminal ? and terminal ? (? negationslash= ?) is, S i ? ? ,i ? (?) =S i o? ? i o ? (?) ?S i o? ? ,e integraltext ? ? ?d? ? (?) ?S e integraltext ? ? ? ?d? ? ,i o ? (?) =? contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv ? G s (r ? ,r ? ,?s) ?d? ? ?d? ? ? contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv ? G s (r ? ,r ? ,s) ?d? ? ?d? ? . (H.47) The auto-correlation PSD of terminal ? is, S i ? ? ,i ? (?) =S i o? ? i o ? (?) ?S i o? ? ,e integraltext ? ? ?d? ? (?)| r ? =r + ? ?S e integraltext ? ? ? ?d? ? ,i o ? (?)| r ? =r + ? =? contintegraldisplaycontintegraldisplay 2e 2 p s (r + ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv + ? G s (r ? ,r + ? ,?s) ?d? ? ?d? + ? ? contintegraldisplaycontintegraldisplay 2e 2 p s (r ? )(Dtriangleinv + ? ?? ?? E + ? )Dtriangleinv ? G s (r + ? ,r ? ,s) ?d? ? ?d? + ? + 2eI ? . (H.48) H.6 Y-parameters in homogeneous Green?s function We supply small signal v ? at surface ? and measure the small signal current that flows into surface ?, then Y ?? ? i ? /v ? . Denote the small signal hole carrier density p(r). We need to solve p(r) for equation L p p(r) = 0,p(r ? ) = 0,p(r ? ) = ep s (r ? )/kTv ? . 210 To do this, we insert ? = p(r), ? = tildewide G s (r,r prime ,s) into (H.8), with boundary condition of both p(r) and Green?s function, we at once obtain (r prime is substituted with r) p(r) = ? contintegraldisplay p(r ? )Dtriangleinv ? G s (r,r ? ,s) ?d? ? . (H.49) With the current operator (e? ?? E ?eDtriangleinv) for hole carrier, we obtain the Y-parameter as Y ?? = ? e 2 kT contintegraldisplaycontintegraldisplay p s (r ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv ? G s (r,r ? ,s) ?d? ? ?d? ? . (H.50) Let ? ? ? + for (H.50), Y ?? can be obtained as Y ?? =? e 2 kT contintegraldisplaycontintegraldisplay p s (r + ? )(Dtriangleinv ? ?? ?? E ? )Dtriangleinv + ? G s (r,r + ? ,s) ?d? ? ?d? + ? =? e 2 kT contintegraldisplaycontintegraldisplay p s (r ? )(Dtriangleinv + ? ?? ?? E + ? )Dtriangleinv ? G s (r,r ? ,s) ?d? ? ?d? + ? . (H.51) The second step of (H.51) follows from symmetry. H.7 Relation between Y-parameter and noise spectrum Comparing the Y-parameters in (H.51) and (eq:Yab) with the noise spectrum in (H.48) and (H.47), their relation can be summarized as Si ? ? ,i ? (?) = 2kT(Y ?? +Y ? ?? ) +? ?? 2eI ? , (H.52) where ? ?? is the Kronecker delta. (H.52) is the van Vliet model in common-base configuration. Note the terminal DC current I ? takes the positive sign when it flows outward from the device and the e should be ?e for NPN transistor. 211 H.7.1 Common-base noise for BJTs The PSD of i CB e and i CB c noise currents for BJTs in common-base configuration as shown in Fig. H.3 (a) can be obtained from (H.52) directly S CB ie = 4kTRfractur(Y CB 11 ) ? 2qI E , S CB ic,ie ? = 2kT(Y CB 21 +Y ?CB 12 ), S CB ic = 4kTRfractur(Y CB 22 ) + 2qI C . (H.53) E C B Noiseless BJT CB e i CB c i 1 V 2 V 1 I 2 I E CB Noiseless BJT CE b i CE c i () a common base () b common emitter Figure H.3: Admittance representation for BJT noise: (a) Common-base; (b) Common-emitter. H.7.2 Common-emitter noise for BJTs Comparing Fig. H.3 (a) with Fig. H.3 (a), we have i CE b = ?i CB e ? i CB c and i CE c = i CB c . Therefore the PSD of i CE b and i CE c is S CE ib = S CB ie +S CB ic + 2Rfractur[S CB ic,ie ? ] = 4kTRfractur(Y CB 11 +Y CB 21 +Y CB 12 +Y CB 22 ) ? 2qI B , S CE ic,ib ? = ?S CB ic ?S CB ic,ie ? = ?2kT(Y CB 21 +Y CB 22 +Y ?CB 12 +Y ?CB 22 ) ? 2qI C , S CE ic = S CB ic = 4kTRfractur(Y CB 22 ) + 2qI C . (H.54) 212 Now derive the Y-parameter relations between the common-base and common-emitter configura- tions. V 1 , V 2 , I 1 and I 2 defined in Fig. H.3 (a) satisfy ? ? ? I 1 I 2 ? ? ? = ? ? ? Y CB 11 Y CB 12 Y CB 21 Y CB 22 ? ? ? ? ? ? V 1 V 2 ? ? ? . (H.55) V 1 , V 2 , I 1 and I 2 also satisfy ? ? ? ?I 1 ?I 2 I 2 ? ? ? = ? ? ? Y CE 11 Y CE 12 Y CE 21 Y CE 22 ? ? ? ? ? ? ?V 1 V 2 ?V 1 ? ? ? . (H.56) According to (H.55) and (H.56), ? ? ? Y CE 11 Y CE 12 Y CE 21 Y CE 22 ? ? ? = ? ? ? Y CB 11 +Y CB 12 +Y CB 21 +Y CB 22 ?Y CB 12 ?Y CB 22 ?Y CB 21 ?Y CB 22 Y CB 22 ? ? ? . (H.57) Finally, with (H.57) and (H.54), the van Vliet model in common-emitter configuration can be obtained S CE ib = 4kTRfractur(Y CE 11 ) ? 2qI B , S CE ic,ib ? = 2kT(Y CE 21 +Y ?CE 12 ) ? 2qI C , S CE ic = 4kTRfractur(Y CE 22 ) + 2qI C . (H.58) 213