VBIC Fundamentals. Colin McAndrew. Motorola, Inc East Elliot Rd. MD EL-701 Tempe, AZ USA 1 VBIC

Transcription

1 VBIC Fundamentals Colin McAndrew Motorola, Inc East Elliot Rd. MD EL-701 Tempe, AZ USA 1 VBIC

2 Outline History Review of VBIC model improvements over SGP model formulation observations and comments Parameter extraction relationship to SGP details of specific steps Practical issues geometry modeling statistical modeling Code mechanism for definition and generation 2 VBIC

3 Genesis A Direct Descendent of BCTM John Shier and Jerry Seitchik were the prime motivators Derek Bowers Mark Dunn Ian Getreu Marc McSwain Kevin Negus David Roulston Shaun Simpkins Larry Wagner Didier Celi Mark Foisy Terry Magee Shahriar Moinian James Parker Michael Schroter Paul van Wijnen many others have provided feedback and suggestions 3 VBIC

4 Junction Isolated Diffused NPN base p emitter n + collector p iso n + sinker n epi n + buried layer p substrate 4 VBIC

5 Trench Isolated Double Poly NPN base-emitter dielectric overlap capacitance n well polysilicon emitter polysilicon base base-collector dielectric overlap capacitance collector n + sinker n + buried layer p substrate trench isolation 5 VBIC

6 Doping Profile 20 emitter log10(n) (cm 3 ) x eb x bc x bl base deep collector epi x 6 VBIC

7 VBIC and SGP improved Early effect modeling physical separation of I c and I b improved HBT modeling capability improved depletion, diffusion capacitances parasitic PNP modified Kull quasi-saturation modeling constant overlap capacitances weak avalanche model base-emitter breakdown improved temperature modeling built-in potential does not become negative! self-heating incompatibilities between VBIC and SGP Early effect modeling I RB emitter crowding model 7 VBIC

8 Plot and Fit what is Important for Design Good I c Fit is Not Sufficient, I c r o is Important V o V i I c I S exp(v be /V tv ) v i v o gπ v i g m v i g o v o v i = g m g o I c V tv g o 8 VBIC

9 Early Voltage Modeling V A =I c /g o (V) data VBIC SGP V ce (V) 9 VBIC

10 Depletion Capacitance Modeling VBIC 1.2 has reach-through model 2.5 VBIC 2 SGP 1.5 C V (V) 10 VBIC

11 Problems with Kull negative output conductance at high V be is caused by velocity saturation model fixed in VBIC I c (ma) increasing V be V ce (V) 11 VBIC

12 Velocity Saturation Modeling Effect is visible for 50µm long devices! convenient to model as mobility reduction µ red ( E) = µ o µ ( E) much more informative than velocity-field plot is 1 for low field, and increase with field E common models are linear and square-root µ = red 1 + µ o V L v sat µ red = 1 + E c = v sat µ o µ o V L v sat linear model is often used for analytic simplicity (Kull), square-root model is smooth and continuous, and preferred physically 12 VBIC

13 Velocity Saturation? What does self-heating do to a resistor? resistance change is R = R 0 ( 1 + TC 1 T) temperature rise is T = R TH IV R TH V 2 R 0 R TH varies nearly as inverse area R TH = R THA ( LW) resistance varies as R 0 = R S L W putting this together gives an effective mobility reduction R R 0 = µ red 1 + R THA TC 1 V L 2 R S the parabolic variation of µ red with field is exactly what is seen in the low field data accurate velocity saturation modeling must be done with a self-consistent selfheating model 13 VBIC

14 Velocity Saturation common linear and square root models are not accurate 2.4 data linear model 2.2 square root model, varying E c asymptotic hard saturation 2 improved model 1.8 u red E (V/um) 14 VBIC

15 Electrothermal Model 10 VBIC Electrothermal HBT Modeling 9 8 VBIC data 7 6 Ic (ma) Vce (V) 15 VBIC

16 Need for Electrothermal Modeling output resistance degradation R o (Ohm) isothermal electrothermal I c (A) 16 VBIC

17 VBIC Equivalent Network s C BCO c si R S R CX I bcp Q bcp I tfp -I trp bp R BIP /q bp cx I bep Q bep ci R CI Q bcx Q bc I bc -I gc b R BX bx Qbex R BI /q b I bex bi Q be I be I txf -I tzr ei C BEO R E dt xf1 F lxf xf2 e I th R TH C TH tl thermal network Q I cxf tzf 1Ω excess phase network 17 VBIC

18 Transport Model I tf I tr I cc = q b exp N F V tv V bei I tf = I S exp N R V tv V bci I tr = I S q 2 q b = q q b q 1 1 = q je V ER q jc V EF q 2 = I tf I KF I tr I KR 18 VBIC

19 Components of Base Current Based on Recombination/Generation I b = I bi + I bn ideal and non-deal components V b I bi = I BI N I is of order 1 exp N I V tv V b I bn = I BN N N is of order 2 exp N N V tv 19 VBIC

20 Basic Modeling Equations continuity equation q( n t) J e = qg ( e R e ) carrier densities drift-diffusion relations J e = qµ e n ψ + qd e n = qµ e n ϕ e J h = qµ h p ψ qd h p = qµ h p ϕ h surface recombination J h = qs h ( p p 0 ) Shockley-Read-Hall process 2 R srh = ( np n ie ) ( τ h ( n+ n ie ) + τ e ( p+ n ie )) Auger process V tv = kt q n = n ie exp( ( ψ ϕ e ) V tv ) p = n ie exp( ( ϕ h ψ) V tv ) R aug = ( np n ie )( c e n+ c h p) 2 20 VBIC

21 1-Dimensional Base Analysis steady-state, ignore recombination/gen. J ex = const for electron quasi-fermi potential exp( ϕ e V tv ) x exp( ϕ e V tv ) = ϕe V tv x from the drift-diffusion relation exp( ϕ e V tv ) J e = qµ e n ie V tv exp( ψ V tv ) x rearranging and integrating across base ψ( x) w exp V tv J e d µ e ( x)n ie ( x) x = 0 qv tv ϕ ew exp V tv ϕ e0 exp V tv 21 VBIC

22 Gummel ICCR now ϕ h is constant across the base, so multiplying by exp( ϕ h V tv ) and noting that junction biases are ϕ h ϕ e I cc = V bei V bci I S exp exp V tv V tv q b saturation current is I S = qa e V tv G b0 normalized based charge is ( G b0 is G b at zero applied bias) q b = G b G b0 scaled base charge (Gummel number) is w 2 G b = p ( µ e n ie ) dx 0 physical basis of BJT behavior is apparent 22 VBIC

23 Quasi-Neutral Emitter Base Current in the emitter ϕ e 0 and n N de, so p«n 2 and np» n ie recombination rates are 2 p n ie exp( ϕ h V tv ) R qn, srh = ---- = τ h τ h N de R qn, aug c e n 2 2 = p = c e N de n exp( ϕ ie h V tv ) in the emitter ϕ h V bei, so integration over the emitter gives I be, qn exp( V bei V tv ) this is an ideal component of base current 23 VBIC

24 Emitter Contact Base Current recombination at the emitter contact J hec, = qs h ( p ec p ec0 ) = qs h p ec hole density at emitter side of base-emitter junction is 2 p eb, = n ie exp( V bei V tv ) N de for a shallow emitter J h = qd ( h p eb, p ) w ec e hole density at the emitter 2 p ec = n ie exp( V bei V tv ) ( N de ( 1 + S h w e D h )) surface recombination is an ideal component of base current I be, ec exp( V bei V tv ) 24 VBIC

25 Space-Charge Region Base Current in base-emitter space-charge region 2 np» n ie but both n and p are relatively small: Auger process is negligible R sc srh n ie exp( V bei V tv ), = ψ τ h exp V bei ψ + τ e exp V tv this rate is maximized when ψ = 0.5( V bei V tv log( τ h τ e )) V tv for approximately equal lifetimes R sc, srh = ( n ie ( τ h + τ e )) 0.5V bei V tv exp( ) space-charge recombination contributes to non-ideal component of base current I be, sc exp( V bei ( 2V tv )) 25 VBIC

26 Intrinsic Collector Model integrate across the epi region J ex = ϕ ew qµ e ndϕ w e ϕ e0 in the collector n = p+ N, ϕ h is constant np p( p + N) n 2 ϕ ϕ h e = = ie exp V tv differentiating w.r.t. position x gives ( 2p+ N) p 2 n ie ϕ h ϕ e ϕ e = exp x V tv V tv x ( 2p+ N)dp = np ( dϕ e ) V tv n dϕ e = V tv 2 N dp p 26 VBIC

27 Intrinsic Collector Model (cont d 1) substitute in integral J ex = p w qv tv µ e w 2 p 0 N dp p I epi0 = qav tv µ e p w ( p w w p 0 ) + Nlog p 0 n = p+ N implies p w p 0 = n w n 0 so p w p 0 = n 0 V bci exp n w V bcx V tv standard quasi-neutrality gives 2 N 4n ie V bci n 0 = N 2 exp V tv and similarly for n w as a function of V bcx 27 VBIC

28 Intrinsic Collector Model (cont d 2) Final model K bci log K bcx V rci V tv K bci K bcx I epi0 = = + R CI exp( ) K bci 1 G AMM V bci V tv exp( ) K bcx = 1 + G AMM V bcx V tv I rci = I epi R CI I epi0 V O V rci ( 2V H ) O RCF empirical modification of velocity saturation model to prevent negative and include high bias effects g o 28 VBIC

29 Quasi-saturation Modeling I c R c-mod Ic Rcr Vcei Vcbi Vce (V) V cei V cbi 29 VBIC

30 Passivity in forward operation, ignoring base current, q b modulation, and parasitics, the power dissipated is I S V be exp N F V tv V bc exp N R V tv Vce this is proportional to 1 V be 1 exp V tv N R N F V ce exp N R V tv Because V be and V ce are positive, passivity requires , i.e. N N R N F N R F 30 VBIC

31 Passivity (cont d) Similar analysis for reverse operation leads to N R N F This implies N R N F The analysis is more restrictive than necessary, and HBTs most definitely have N R N F, however this shows that with certain parameter values VBIC (and SGP, and any other BJT model with separate forward and reverse ideality factors) can be non-passive! it is preferred if a model enforces physically realistic behavior regardless of parameter values N F N R VBIC has and for SGP compatibility and HBT modeling accuracy 31 VBIC

32 Electrothermal Modeling significantly increases complexity of model simpler approaches have proposed (e.g. V be and β correction) but these are not physically consistent all branch constitutive relations become functions of local temperature as well as branch voltages self-consistent with temperature model I th is computed as the sum over the nonstorage elements of the product of current and voltage for every branch cannot simply multiply terminal currents and voltages because of energy storage elements 32 VBIC

33 VBIC Parameters from SGP Core Model Similar to SGP program sgp_2_vbic available simple translations from SGP to VBIC R BX = R BM R BI = R B R BM C JC = C JC X JC C JEP = C JC ( 1 X JC ) T D = πt F P TF VBIC

34 VBIC Parameters from SGP (cont d) Early Effect Model is Different specify V be and V be for matching g o g o f I c g o r I e = = 1 V AF f f 1 V be V AR V bc V AF 1 V AR r r 1 V be V AR V bc V AF from analysis of output conductance in forward and reverse operation q bcf c bcf q f bef ( g o ) I c c ber q bcr q ber r ( g o ) I e V EF V ER = VBIC

35 What is a Parameter? specified devices model parameters measured data can t compare simulated data physical parameters can compare physical parameters 35 VBIC

36 Direct Extraction vs. Optimization direct extraction is uses simplifications of a model, it does not give the best fit of the un-simplified model to data direct extraction is based on manipulation of a model and data, the quantities fitted are NOT the most important quantities as far as circuit performance is concerned the goal of characterization is accurate simulation of important measures of performance of circuits, not of unrelated metrics of individual devices (at sometimes strange biases) models are approximations and trade-offs must be made during characterization to reflect the modeling needs of target circuits for a technology to optimize the trade-offs you need to fit multiple targets (g m /I d or I c /g o ), and these must be weighted over geometry and bias 36 VBIC

37 VBIC Parameter Extraction based on a combination of parameter initialization (extraction) and optimization both steps use a subset of data and subset of the model parameters you can NEVER directly equate a number derived from simple manipulation of measured data to a a model parameter V A = SGP V AF Vbe 1 SGP V AF SGP V AR proper equivalence involves EXACTLY simulating and processing data initial parameters are refined when the approximate model used as a basis to determine a parameter is not sufficiently accurate 37 VBIC

38 STEPS: C be, C bc, C cs Junction Depletion Capacitance Extraction for each junction measure capacitance (forward bias is important for C be ) use optimization to fit model to data data 120 VBIC 100 C J (ff) V (V) J 38 VBIC

39 STEP: V A Coupled Early Voltage Extraction measure forward output (FO) at a moderate V be, high enough to get reasonable data but below where series resistance and high-level injection effects are dominant measure reverse Gummel (RG) data at two values of V eb (0 and 1) differentiate FO data to get g o = I c V ce compute I c g o and find its maximum value from RG data compute I el ( I eh I el ) where added subscripts mean low and high V eb select one point from the RG data in the region between where noise and high bias effects are observable (the flat region ) should be at V bc < F C P C if possible 39 VBIC

40 STEP: V A (cont d) compute junction depletion charges and capacitances at the selected forward and reverse biases form and solve the equations q bcf c bcf q f bef ( g o ) I c ( q q bcr q berl q berh )I el berh ( ) I eh I el V EF V ER = 1 1 this directly follows from the transport current formulation you can mix and match derivatives or differences for each component, picking maximum I c g o is useful for FO data as it avoids data that has a significant avalanche component 40 VBIC

41 STEP: V A (cont d) Forward Early Effect V A =I c /g o (V) data VBIC V ce (V) avalanching not included 41 VBIC

42 STEP: V A (cont d) Reverse Early Effect data VBIC V A R=I c /g o (V) V bc (V) poor data high bias effects 42 VBIC

43 STEP: I BCIP Substrate Current Analysis in Forward Gummel (FG) data at V cb = 0 the parasitic base-collector turns on at high V be because of R C de-biasing under these conditions I c R C I s = I BCIP exp N CIP V tv at this stage assume that N CIP = 1 from the ideal region of I s ( I c ) the intercept and slope give I BCIP and R C from the deviation from ideality at high the substrate resistance R S can be calculated really the sum R S + R BIP I c 43 VBIC

44 STEP: I BCIP (cont d) High Bias FG Data 10 2 data used to get initial estimates of I BCIP, R C, and R S I (A) I c I b I s V be (V) 44 VBIC

45 STEP: I BCIP (cont d) High Bias FG Data, I s (I c ) I s R S /R C I s (A) 10 6 slope R C /V tv data 10 9 ideal region model I c (ma) 45 VBIC

46 STEP: I S FG Data Analysis for I S and N F select ideal region FG I c data, calculate q 1, form q 1 I c (accounts for Early effect) extract I S and N F from slope and intercept d(log(i c ))/d(v be ) (/V) data model V (V) be 46 VBIC

47 STEP: I B From FG I b or RG I b -I s Data select ideal region data, I BI and N I follow from slope and intercept at low bias, subtract ideal component to get non-ideal component, fit this I b (A) data model V be (V) 47 VBIC

48 STEP: I SP from ideal region RG I e data extract from the slope of q 1 I e from ideal region RG I e data extract and N FP from slope and intercept N R I SP there is no Early effect in the model for the reverse transport current, so no correction for this is necessary 48 VBIC

49 STEP: I KF Same Technique is Used for I KR and I KP select FG high bias data to avoid the region where the non-ideal component of base current is significant from the ideal component of base current calculate the intrinsic base-emitter voltage V bei calculate q 1 from intrinsic biases approximation: do not know base-collector debiasing, so assume = V bci V bc from I c = I S exp( V bei ( N F V tv )) calculate q b q b calculate q 2 = q b ( q b q 1 ) then I KF = I S exp( V bei ( N F V tv )) q 2 49 VBIC

50 STEP: I KF fitted at knee B f 25 data 20 model V be (V) resistive effects not yet characterized 50 VBIC

51 STEP: R B Base and Emitter Resistance Characterization the open collector (Giacolleto) method is used to determine R E must be done at very high Ib initial estimates of R BX and R BI are made from Ning-Tang analysis R BX, R BI, I IKF, R C and N CIP are optimized to fit high bias FG data all of I c, I b and I s are fitted the residual is y ŷ y + ŷ, which is symmetric with respect to relative error, and insensitive to outliers proper residual calculation is a key to robust optimization 51 VBIC

52 Open Collector Method for R E Done at high bias physical analysis V ce = I b R E + Kln( 1 + I b I OS ) derivative is V ce 0.5K K = R I E R E b I b ( I b + I OS ) 2I b take derivative, plot versus 1 I b abscissa intercept gives R E, slope gives K optimization is used to refine parameters 52 VBIC

53 R E Extraction Plot dv ce /di b (Ohm) data linear fit /I b (1/mA) 53 VBIC

54 Base Resistance DC Ning-Tang Analysis determine V bei from I b I bc avoid region where is significant calculate V I b at three high bias points V=I b (R BX +R BI /q b )+I e R E I b (A) V/I b =R E +R BX +R BI /q b +βr E V be (V) 54 VBIC

55 Base Resistance DC (cont d) get system of linear equations to solve V I b V I b V I b = β 1 q b, β 2 q b, β 3 q b, 3 R E + R BI R E R BX however, at high bias β β low q b, therefore the matrix is poorly conditioned need additional information AC data physical calculation for (Ning-Tang) R E R BI from open collector method 55 VBIC

56 Base Resistance AC Impedance Circle Method indeterminacy in DC data can be broken with AC data adds orthogonal information simple theory gives low frequency asymptote as R B + r π +( 1 + β ac )R E and high frequency asymptote as R B + R E because of deviations at high frequency from a circle the high frequency asymptote is extrapolated from low frequency data however, detailed simulations with a realistic small-signal model show that the radius of the low frequency data depends strongly on all components of the smallsignal model simple extraction from the impedance circle is not possible still should look at this data for extraction 56 VBIC

57 Robust, Symmetric Residuals y data y data + R r n n y data ny data n R k ( n) ( 1 + n) R k 2 R r 1 Residual y model 57 VBIC

58 STEP: R B (cont d) I c 10 4 I (A) I b 10 7 data 10 8 I s VBIC V be (V) 58 VBIC

59 STEP: A V Avalanche Parameter Extraction compute initial A VC1 and A VC2 from FO data at highest V ce optimize to fit V A = I c g o V A =I c /g o (V) data VBIC V (V) ce 59 VBIC

60 Activation Energy Characterization I S T ( ) = I S ( T nom ) T X IS E A exp k q T -- T nom T nom N F select a target current level in the ideal operating region, calculate at each T I S = q 1 I c exp( V be ( N F V tv )) q 1 account for temperature dependence slope of N F logi S X IS logt vs. 1 T is qe A k from which E A can be calculated X IS kept at default value 60 VBIC

61 V be vs. Temperature Important for Bandgap Design data VBIC 0.75 V c =1.0uA (V) T (K) 61 VBIC

62 f T (I c ) Characterization initialize T F and I TF from 1 f T vs. 1 I c optimize to fit data c be include shape parameters and Q TF 2.5 data 2 VBIC 1.5 f T (GHz) I (A) c 62 VBIC

63 Miscellaneous further optimization is used to fit VBIC to data over the complete bias range temperature coefficients are determined by extracting temperature dependent parameters at different temperatures and then extracting TC s from the parameters resistance TC s from sheet resistance test structures and measurements all standard extraction data should be without self-heating low bias or pulsed measurements over temperature thermal resistance is extracted by optimization to fit high I c V ce data the transit time model is identical to that of SGP and so existing techniques can be directly used for f T ( I c ) or s-parameter modeling 63 VBIC

64 FG Current Modeling data 10 4 VBIC 10 5 I (A) V be (V) 64 VBIC

65 FG Beta Modeling Bf 25 data 20 VBIC V be (V) 65 VBIC

66 DC Quasi-saturation initialize collector resistance parameters G AMM V O smart way to initialize and? optimize to fit saturation characteristics 6 data 5 VBIC 4 I c (ma) V (V) ce 66 VBIC

67 DC Quasi-saturation quasi-saturation region behavior cannot be modeled well with SGP 6 5 lo R C SGP 4 I c (ma) 3 hi R C SGP V (V) ce 67 VBIC

68 Geometry Modeling all capacitances and currents should be modeled using area and perimeter components (and corner components if necessary), e.g. I S = A e I SA + P e I SP unless there is a very substantial difference between a masked dimension and an effective electrical dimension, the area and perimeter can be computed as A e = W e L e, P e = 2( W e + L e ) it is not possible to distinguish between a delta between masked and effective dimensions and variations in the perimeter component forward transit time is also modeled with area and perimeter components Q diff A e I SA T + P so FA e I SP T = I FP S T F A e I SA T + P FA e I SP T FP T F = A e I SA + P e I SP 68 VBIC

69 Geometry Modeling (cont d) similarly from the SGP model the area and perimeter components of collector current are V be A e I SA exp V tv V bc V AFA V be P e I SP exp V tv V bc V AFP want I c V be = I exp S V tv V bc V AF equating these gives V AF = A e I SA V AFA P e I SP V AFP A e I SA + P e I SP 69 VBIC

70 Always Verify Geometry Models Expect the Unexpected! non-ideal component of base current does not always scale with perimeter have seen collector resistance increase as emitter length increases I BEN observed expected L e 70 VBIC

71 Resistance Geometry Modeling This is the most difficult think in terms of conductance, not resistance R BI = k n ρ sbe W e L e ( ) number of 1st order physical base contacts theory simulation reality n=1 1/3? n=2 1/12 1/12? n=4 1/32 1/28.6? for small emitters the n=4 model is reasonable for highly doped base rings as emitter length increases the effective number of base contacts changes use 1 ( 1 k 1 + ( 1 k 4 1 k 1 )( W e L e )) rather than k 1 ( k 1 k 4 )( W e L e ) for monotonic decrease in R BI with L e 71 VBIC

72 R BI Variation with Geometry R BI increases as emitter length increases! /12-(1/12-1/28.6)W/L 140 R /(12+( )W/L) W/L increasing length 72 VBIC

73 Base Structure Effective Base Structure Varies with Geometry effectively 4 base contacts effectively single base contact 73 VBIC

74 Conductance Increases with Width Expect a 1/L Dependence for R BX R BX = R 0 + R L L???? δi δg δl G BX = 1 R BX = G 0 + G L L asymptotically correct as L gets large 74 VBIC

75 Statistical Modeling must be based on process and geometry dependent models process dependence defined by Ida and Davis, BCTM89 most efficient and accurate way is backward propagation of variance (BPV) provides Monte Carlo (distributional) and case models for inter-die variation defined in BCTM97 and SISPAD98 exactly the same modeling basis and BPV characterization methodology can be applied to intra-die variation (mismatch), see Drennan BCTM98 75 VBIC

76 VBIC Code Complete VBIC Code is Available the only rational way to define a compact model is in terms of a high-level symbolic description not through 40 SPICE subroutines historically there have been several proprietary languages and model interfaces MAST for Saber (Analogy) ADMIT for Advice (Lucent) ASTAP/ASX (IBM) TekSpice (Tektronix/Maxim) MDS (Hewlett-Packard) probably others VBIC had its own symbolic definition VHDL-A looked promising, but appears to have been overtaken by Verilog-A in the industry 76 VBIC

77 VBIC Code (cont d) VBIC is Defined in Verilog-A well, really in a subset of Verilog-A conditional definition of 8 separate versions, 3- and 4-terminal, iso-thermal and electro-thermal, and constant- and excess-phase automated tools generate implementable code FORTRAN and C probably Perl looking at XSPICE DC and AC solvers are provided transient and other types of analyses depend on simulator algorithms can use Verilog-A description this can be very slow institute/iafgp/neu/vbic/ index.html 77 VBIC

78 VBIC s11 Modeling 0.8 Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC 0.6 Re(S11) F (GHz) Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC Im(S11) F (GHz) 78 VBIC

79 VBIC s12 Modeling 0.2 Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC 0.15 Re(S12) F (GHz) Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC 0.12 Im(S12) F (GHz) 79 VBIC

80 VBIC s21 Modeling 0-2 Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC -4 Re(S21) F (GHz) 8 7 Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC 6 Im(S21) F (GHz) 80 VBIC

81 VBIC s22 Modeling Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC Re(S22) F (GHz) Vbe=0.766 data Vbe=0.766 VBIC Vbe=0.927 data Vbe=0.927 VBIC Vbe=1.082 data Vbe=1.082 VBIC Im(S22) F (GHz) 81 VBIC

82 VBIC s-parameter Modeling s-parameter fits have been done to SGP and VBIC to the same data in the same optimization tool comparison of RMS errors over bias and frequency s-parameter SGP RMS % error VBIC RMS % error Re(s 11 ) Im(s 11 ) Re(s 12 ) Im(s 12 ) Re(s 21 ) Im(s 21 ) Re(s 22 ) Im(s 22 ) VBIC