Nonlinear distortion in mm-wave SiGe HBTs: modeling and measurements

Transcription

1 Nonlinear distortion in mm-wave SiGe HBTs: modeling and measurements P. Sakalas $,#, A. Pawlak $, M. Schroter $ $ CEDIC, Technische Universität Dresden, Mommsenstrasse 13, Germany # FRLab. Semiconductor Physics Inst., Center of Physical Sciences and Technology, Lithuania paulius.sakalas@tu-dresden.de

2 Outline Outline Motivation Compact modeling approaches, requirements for mm-wave frequencies Standard compact models for SiGe HBTs, location of nonlinear effects Investigation method of distortion effects in SiGe HBTs Harmonic distortion: core model verification vs. intrinsic (1D) and internal (2D) transistors, analysis of distortion HICUM L2 verification on DC and RF characteristics of measured advanced SiGe HBTs HICUM L2 verification on harmonic distortion (FoM) on two tone load pull of SiGe HBTs Analysis of external parasitic compact model elements on harmonic distortion. Summary

3 Motivation Motivation Benefits: SiGe based HBTs have proven to be among the fastest transistors on the market today. Recent SiGe HBT technology offers f T /f max = 3/5 GHz and enables mm wave circuits for various applications: optical communications with 1 Gbit/s 77 to 15 GHz collision radar applications LNA at 26 GHz for receivers, Heterodyne receivers for operation at THz frequencies cryogenic applications Issues: =>All those applications require physics based accurate models capable to capture nonlinearities at high frequency.

4 Motivation Fundamental compact modeling approaches Goal: cover large variety of applications & circuit optimization (bias, T, f) Criteria Behavioral, X-par. Physics-based accuracy high (within narrow ranges) moderate to high over wide range numerical stability compromised outside fitting ranges fabricated devices need every possible layout used in circuits on test chip high (for standard models) only few devices (6 HF trs, 6 test structures) measurement effort moderate to high moderate to low par. extraction effort moderate (single device), very high for library, imposs. at high f moderate (single device) very low for library geometry scaling inconsistent (typically) consistent predictive capability none moderate to high statistical modeling none (very high effort) good to excellent Physics-based model (cuts design cycle & supports process dev.)

5 Motivation Compact HBT modeling at (sub-)mm-wave frequencies Circuit design related requirements: design at the physical device and process technology limit => meeting specs is very challenging large variety of applications (implemented in BiCMOS SoCs) => need careful individual and sophisticated optimization (by e.g. device sizing) accurate models up to very high frequencies incl. subharmonics (i.e. distortion) consequences for compact modeling (e.g. for foundries) physics-based => enables: geometry scaling (i.e. contact configuration & emitter size) extension into non-measurable operation (bias, frequency, temperature) ranges with reasonable accuracy predictive and statistical design geometry scalable models => enables circuit optimization large-signal capability => enables accurate small-signal and distortion modeling good modeling pays for itself if it saves just one design iteration!!

6 Existing standard large-signal BJT and HBT models Existing standard large-signal BJT and HBT models SPICE Gummel-Poon model addresses effects present in "7ies" BJT technologies, no HBT effects. HICUM/L2, MEXTRAM, VBIC include some or all SiGe HBT related effects. HICUM includes mm-wave technology related effects in SiGeC HBTs (s. DOTFIVE). S I_Rsu S R su S Q js i jsc S I_SCi i Ts C su internal transistor R Cx S I_Rcx C Q BCx Q BCx i jbcx Q ds S I_BCx C ds S I_rBi S I_BCi i jbci Q jci Q r i AVL S I_AVL B R Bx S I_RBx S I_BEp C jep i BEtp R Bi B * B Q jep S I_BEi i jbei Q jei Q f i BEti i T i jbep C BEpar1 C BEpar2 R E S I_RE noise correlation I nc =sqrt(2qi T ) I nb =sqrt(2qi jbei ) V nc 1S. V nc V nb 1S. V nb E thermal network P ΔT j T R th C th vertical NQS effects Q f,qs/τ Bf Q f,nqs α Qf R=τ Bf T b1 V nb T b2 V nc noiseless internal transistor g nc2 V nc T b1 =1+jωτ Bf sqrt(b f (2α Qf - α it ) 2 ) T b2 = jωτ Bf α it I T,qs/τ Bf V C1 V C2/τ Bf α IT I T,nqs=V C2 V C1 /τ Bf R= τ Bf α IT/3

7 Existing standard large-signal BJT and HBT models Where are the nonlinear effects in HBTs? I C ~ exp(v BE /V T ); Q mobile ~ I C γ (γ 1); I aval ~ I C exp(v CB /V B ); Q depl ~ (1-V j /V D ) z... schematic cross-section typical profile 25 Doping ( ) x (μm) Mol fraction ( ) (%) 1D (or intrinsic) transistor (here: f T,peak = 25 GHz) internal transistor (includes internal base resistance) device nonlinearity is mostly defined by intrinsic transistor

8 Existing standard large-signal BJT and HBT models Goals of this presentation consider intrinsic model based on 1D device simulation (as reference) start with single tone signal at fundamental frequencies of 1Hz, (1, 5, 2) GHz => verify compact model distortion behavior up to subharmonics of several peak f T (i.e. far above measurable frequencies!!) extension to internal transistor (with 2D device simulation as reference) => impact of nonlinear internal base resistance extension to complete (3D) transistor structure and experimental characteristics => impact of external regions.

9 Investigation method of distortion effects in SiGe HBTs Investigation method of distortion effects in SiGe HBTs Start with 1D transistor => use 1D device simulation as reference define suitable DC bias point => simple amplifier circuit (R G = 5 Ω, R L = 97 Ω) 25 J C V B E /V.9.88 R G C B V B L B T 1D V C L C CC i RL 5 v in R L v out V (V) CE T 1D : use either 1D numerical transistor or compact model (HICUM) => same numerical "environment" for compact model and reference run mixed-mode device simulator with single-tone input signal

10 Investigation method of distortion effects in SiGe HBTs Model evaluation and verification procedure... time domain transient simulation is employed input voltage amplitude from small-signal (1 mv) to large-signal is varied (2 mv) => input power P in = [-4, ] dbm (= [.1μW, 1mW]) fundamental frequencies f of input signal are selected: - f = 1 Hz (no capacitive effects, guaranteed quasi-static (qs) operation) - f = 1 GHz (some capacitive effects, qs operation) - f = 5 GHz (5 th harmonic close to f T,peak, capac. effects, some non-qs operation) - f = 2 GHz (close to f T,peak, 5th harmonic = 1THz, significant non-qs effects) to ensure stead-state at least 1 periods were simulated for each simulation: time domain waveform of last periods for node voltages, branch currents and charges were recorded Fourier transformation of waveforms of desired variables was applied => frequency components (=> harmonic distortion) of variables and FoMs single-tone and resulting harmonic distortion (HD) was considered => reveals fundamental model limitations evolution of FoMs {P out (P in ), P 1dB (J C ), OIP 3 (J C )} with f is analyzed

11 Investigation method of distortion effects in SiGe HBTs f = 1 Hz 1D: Harmonic distortion: P out vs. P in V BE =.88V, V CE 2= 1V P out D sim. f1 1D sim. f2 1D sim. f3 1D sim. f4 1D sim. f5 HICUM/L2 (tr) P avs P out D sim. f1 1D sim. f2 1D sim. f3 1D sim. f4 1D sim. f5 HICUM/L2 (tr) f = 1 GHz P avs 2 2 f = 2 GHz P out D sim. f1 1D sim. f2 1D sim. f3 1D sim. f4 1D sim. f5 HICUM/L2 (tr) f = 5 GHz P avs P out D sim. f1 1D sim. f2 1D sim. f3 1D sim. f4 1D sim. f5 HICUM/L2 (tr) P avs very good agreement up to f = f T,peak /5

12 Investigation method of distortion effects in SiGe HBTs J C D: Harmonic distortion: bias point shift vs. P in 1D simulation HICUM/L2 (HB) HICUM/L2 (tr) 1D simulation HICUM/L2 (HB) HICUM/L2 (tr) P 1dB P avs V BE =.88V, V CE = 1V f = 1 Hz f = 5 GHz J C P avs D simulation HICUM/L2 (HB) HICUM/L2 (tr) 1D simulation HICUM/L2 (HB) HICUM/L2 (tr) P 1dB f = 1 GHz f = 2 GHz J C 2 15 P 1dB J C 2 15 P 1dB P avs P avs excellent agreement (also for harmonics) even beyond P 1dB

13 Investigation method of distortion effects in SiGe HBTs 1D: Harmonic distortion: P out vs. J C at P 1dB 2 f = 1 Hz variable R L 2 f = 1 GHz P out D sim. f1. 1D sim. f2. 1D sim. f3. 1D sim. f4. 1D sim. f5 HICUM/L2 (tr) J C peak f T P out D sim. f1. 1D sim. f2. 1D sim. f3. 1D sim. f4. 1D sim. f5 HICUM/L2 (tr) J C peak ft 2 f = 5 GHz 2 f = 2 GHz P out D sim. f1. 1D sim. f2. 1D sim. f3. 1D sim. f4. 1D sim. f5 HICUM/L2 (tr) J C peak f T P out D sim. f1. 1D sim. f2. 1D sim. f3. 1D sim. f4. 1D sim. f5 HICUM/L2 (tr) J C peak f T very good agreement up to f = f T,peak /5, some Δ at f f T,peak & high J C

14 Investigation method of distortion effects in SiGe HBTs OIP 3 OIP f = 1 Hz ~J C 1D: Harmonic distortion: OIP 3 vs. J C 1 1D simulation HICUM/L2 (tr) f = 5 GHz J C V CE = 1V OIP 3 variable R L OIP D simulation HICUM/L2 (tr) f = 1 GHz J C f = 2 GHz 2 1D simulation HICUM/L2 (tr) J C 2 1D simulation HICUM/L2 (tr) J C model tracks trend very well, some deviation beyond J C (f T,peak )

15 Investigation method of distortion effects in SiGe HBTs 1D: Time domain results at P 1dB : V B E =.88V J RL dbm -23 dbm f = 1 Hz 1D Simulation HICUM/L t ( s) 1-12 dbm f = 5 GHz -1 dbm J RL 5 5 f = 1 GHz 1D Simulation HICUM/L t (ps) f = 2 GHz J RL 5 5 1D Simulation 1 1 HICUM/L2 2 t (ps) 3 4 J RL 5 5 1D Simulation 1 2 HICUM/L2 4 6 t (ps) 8 1 excellent agreement up to f = f T,peak /5, some deviation at f f T,peak

16 1D: Analysis of distortion effects 1D: Analysis of distortion effects Impact of i T and g m modeling on bias point shift classical approximation i T = I S v B'E' exp m C V T => no compression due to missing impact of mobile charge I C (ma) D simulation HICUM/L2 classical ICCR P avs simplified GICCR approximation (ICCR) i T = c 1 ( v B'E' V T ) v B'C' V T Q p + Q j, T + h f τ f i T exp exp( ) Δi C (ma) D simulation HICUM/L2 simple (5th) ICCR => deviations, but correct trend P avs simple form of GICCR can already explain trends

17 1D: Analysis of distortion effects P out D: Impact of high-current effects: P out vs. P in 1D sim. on, on off, on on, off off, off I T (ΔQ f,t ), Q f (ΔQ f ) f = 1 Hz P avs 2 1 I (ΔQ ), Q (ΔQ ) T f,t f f V BE =.88V, V CE = 1V high-current effect related model parameters turned on/off separately for charge (ΔQ f ) and transfer current (ΔQ f,t ) low f: deviations from ΔQ f,t at f 2 and P 1dB high f: deviations from ΔQ f at f 3 and P 1dB I T (ΔQ f,t ), Q f (ΔQ f ) P out D sim. on, on off, on on, off off, off f = 5 GHz P avs P out D sim. on, on off, on on, off off, off f = 2 GHz P avs high-current effects should be considered

18 1D: Analysis of distortion effects J RL 1D: Impact of high-current effects: time domain at P 1dB D sim. on, on off, on on, off off, off I T (ΔQ f,t ), Q f (ΔQ f ) f = 1 Hz V BE =.88V, V CE = 1V high-current effect related model parameters turned off/on separately for charge (ΔQ f ) and transfer current (ΔQ f,t ) low f: amplitude deviations from ΔQ f,t J RL t ( s) D sim. on, on off, on on, off off, off I T (ΔQ f,t ), Q f (ΔQ f ) f = 5 GHz high f: only small deviations from ΔQ f J RL D sim. on, on off, on on, off off, off I T (ΔQ f,t ), Q f (ΔQ f ) f = 2 GHz t (ps) t (ps) high-current effect impact mainly on amplitude, less on phase

19 1D: Analysis of distortion effects P out D: Impact of BC depletion capacitance: P out vs. P in 1D sim. HICUM/L2 Q jci, lin Q jci, 2nd Q jci, 5th f = 1 GHz V BE =.8V, V CE = 1V P out D sim. HICUM/L2 Q jci, lin Q jci, 2nd Q jci, 5th f = 5 GHz P avs P avs Q jci (V B C ) approximated by 5th-order polynomial over relevant voltage range => consider only linear or 2nd-order approximation for charge term (keep using complete analytical term in GICCR to keep bias current unaffected) linear Q jci (i.e. C jci =const): visible impact from f 3 on & already at low power level 2nd-order polynomial: visible impact from f 4 on & at low power level

20 1D: Analysis of distortion effects Impact of BC depletion capac.: time domain results at P 1dB J RL D sim. HICUM/L2 Q jci, lin Q jci, 2nd Q jci, 5th f = 1 GHz V BE =.8V, V CE = 1V J RL D sim. HICUM/L2 Q jci, lin Q jci, 2nd Q jci, 5th f = 5 GHz t (ps) t (ps) low f: mostly amplitude and some shape deviation for linear Q jci high f: - amplitude and shape deviation increase, also incorrect phase - 2nd-order approximation of Q jci is worse than first-order approximation need to have accurate Q jci (V B C ), do not need Q jci (J C )

21 1D: Analysis of distortion effects B l E i jbci i BEi 2D: Impact of nonlinear internal base resistance E C Q jci Q jei E emitter window b E /2 B B* T i Z Bi n + p Q r Q f n-sic n + buried layer I Bi /2 V BE =.88V, V CE = 1V i AVL p + i T C emitter perimeter junction internal transistor P out P out D 2D P avs D 2D f = 2 GHz f = 5 GHz P avs significant impact of R Bi on higher order distortion at high frequencies

22 Measured devices Measured devices SiGe HBTs: IHP, Frankfurt/Oder, f T = 3 GHz, f max = 5 GHz, A E = 8*.12 μm*.96 μm ST Microelectronics, f T = 32 GHz, f max = 37 GHz, A E =.12 μm*9.94 μm

23 3D: model verification: IV characteristics 3D: model verification: IV characteristics J C, J B (ma/µm 2 ) SiGe HBT A E =.12 um*9.94 um V BC = V J C J B HICUM a) V BE (V) b) J C (ma/µm 2 ) SiGe HBT, A E =.12 um*9.94 um Meas. HICUM V BE [.6-.99, 5m] V V CE (V) a) Forward Gummel plot with V BC = V. => High collector current linearity and current gain β =3. b) Output characteristic with a V BE drive. => HICUM is in a perfect agreement.

24 3D: model verification: IV characteristics 3D: model verification: f T, f max, C BC f T, f max (GHz) 4 SiGe HBT A E =.12 um*9.94 um 3 V BC = V 2 1 f T f max HICUM C BC (ff) SiGe HBT A =.12um*9.94um E A =.12um*.96um*8 E HICUM a) b) J C (ma/µm 2 ) V BC (V) => good agreement in a whole bias range. => weakly nonlinear depletion capacitance behavior. => HICUM is in good agreement.

25 5 GHz SiGe HBT, harmonic distortion: Pout (Pavs) 5 GHz SiGe HBT, harmonic distortion: P out (P avs ) Fundamental frequency of 1 GHz, five harmonics were measured. 1 GHz 1 GHz 2 GHz 3 GHz 4 GHz 5 GHz a) b) SiGe HBT A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V. The 5 th harmonic is on the limit of network analyzer frequency, 5 GHz. From the linearized capacitance simulations it was found a negligible impact of base/ collector junction capacitance C BC on nonlinearity. HICUM is in a good agreement.

26 5 GHz SiGe HBT, harmonic distortion: Pout (Pavs) 5 GHz SiGe HBT, HD: I C (P avs ), load circles Fundamental frequency 1 GHz a) HICUM: solid line SiGe HBT A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V. I C (A) b) V CE (V) Pin -21dBm HICUM I C Pin -11dBm V BE =.9V Deviation from quiescent collector current with input power is due to self-biasing. At high input power non quiescent bias can may trigger impact ionization (V CE >1.6 V) and pronounced selfheating => increase of HD Avl.

27 5 GHz SiGe HBT, HD: Impact of external capacitances 5 GHz SiGe HBT, HD: Impact of external capacitances Time domain, excitation with 1 GHz and -21 dbm CW power. HICUM: solid line Excluded external parasitic capacitances a) b) SiGe HBT: A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V. Exclusion of the external parasitic base/emitter and base/collector capacitances only slightly impacts the dynamic collector current. At higher frequencies an impact may be more evident.

28 5 GHz SiGe HBT, HD: Impact of external base resistance 5 GHz SiGe HBT, HD: Impact of external base resistance A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V, excitation with 1 GHz. Rbx/2 Rbx/2 a) b) Reduced external base resistance from 11 Ω to 5.5 Ω increase f max value. Standard IV and RF characteristics are preserved. No impact on HD since external parasitic resistance is constant with bias.

29 5 GHz SiGe HBT, HD: Impact of external collector resistance 5 GHz SiGe HBT, HD: Impact of external collector resistance A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V, excitation with 1 GHz. Rcx/2 Rcx/2 a) b) Reduced external collector resistance from 1 Ω to 5 Ω. =>increase f max and f T values, improves IV and RF performance. negligible impact on HD.

30 5 GHz SiGe HBT, HD: Impact of emitter resistance 5 GHz SiGe HBT, HD: Impact of emitter resistance A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V, excitation with 1 GHz Re/2 Re/2 a) b) Reduced external emitter resistance from 2.9 to 1.45 Ω. => slight increase f T, f max. => decrease of J C at peak f T, f max. => base and collector current increase at high V BE. =>HD increase, G T increase due to lower negative feedback, ie. lower linearization by constant resistor.

31 5 GHz SiGe HBT, HD: modeling at high input power 5 GHz SiGe HBT, HD: modeling at high input power Excitation with 5 and 1 GHz Pin -11 dbm and +1 dbm CW power. 5 GHz, -11 dbm 5 GHz, -11 dbm a) 1 GHz, 1 dbm b) c) d)

32 5 GHz SiGe HBT, GT and PAE% 5 GHz SiGe HBT, G T and PAE% A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V, excitation with 1 GHz Linear gain up to -11 dbm is observed. 35% of maximum power added efficiency at 1 GHz is achieved.

33 5 GHz SiGe HBT, IIP3, OIP3 versus IC 5 GHz SiGe HBT, IIP3, OIP3 versus I C A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.9 V, excitation with 1 GHz Good agreement with HICUM of third order input IIP3 and output OIP3 interception points.

34 37 GHz SiGe HBT, HD: Pout (Pavs) 37 GHz SiGe HBT, HD: P out (P avs ) Fundamental frequency 1 GHz a) b) SiGe HBT A E =.12 μm*9.94 μm, V CE = 1 V, V BE =.9 V. HICUM is in fair agreement.

35 37 GHz fmax SiGe HBT, HD: IC,DC (Pavs), load circles 37 GHz f max SiGe HBT, HD: I C,DC (P avs ), load circles Fundamental freq. 1 GHz 4 E V BE [.6-.9, 5m] V J C (ma/µm 2 ) Meas. -11 dbm -21 dbm HICUM SiGe HBT A E =.12 μm*9.94 μm, V CE = 1 V, V BE =.9 V Collector current drops with input power with a following increase. V CE (V) Dynamic current circles at P in = -11 dbm hit the region of impact ionization BV CE = 1.6V. HICUM is in fair agreement but not for the load circles.

36 37 GHz SiGe HBT, HD: GT (Pavs), PAE% 37 GHz SiGe HBT, HD: G T (P avs ), PAE% Fundamental frequency 1 GHz a) b) SiGe HBT A E =.12 μm*9.94 μm, V CE = 1 V, V BE =.9 V. PAE reaches 4%. Good agreement with HICUM.

37 37 GHz SiGe HBT, HD: dynamic base and collector current and voltage waves and IIP3 and 37 GHz SiGe HBT, HD: dynamic base and collector current and voltage waves and IIP3 and OIP3 versus bias. SiGe HBT A E =.12 μm*9.94 μm, V CE = 1 V, V BE =.9 V. a) b) c) a) base dynamic current and voltage at 5 GHz and P in = 5 dbm. b) collector dynamic current and voltage at 5 GHz and P in = 5 dbm. IIP3 and OIP3 at 1 GHz.

38 SiGe HBT, measured load pull at 45 GHz SiGe HBT, measured load pull at 45 GHz a) V in =.88V, V out =1V Γ in =.15+j.88, P in = 15dBm HICUM P out in dbm b) P in = 15dBm, V out =1V meas. sim. HICUM V BE in V P out in dbm c) V in =.88V, V out =1V meas. sim. HICUM P in in dbm SiGe HBT A E = 8*.12 μm*.96 μm, V CE = 1 V, V BE =.88 V. a) Simulated load isopower lines fairly matches the measured data at 45 GHz. Measurements were performed with Maury impedance automated tuner (MT984A) system ATS. b) The power of the sum of harmonics versus V BE. HICUM is solid line. c) Output power versus input power.

39 Summary HBT modeling approaches Summary physics-based vs. behavioral v: clear advantages for physics-based models => exclusively used in SiGe HBT design kits Detailed comparison of core compact model formulation with 1D device simulation typical distortion characteristics over wide range of input power, bias, and frequency => - very good agreement up to at least P 1dB, J C (f T, peak ), and f = f T,peak /4 - still reasonable agreement at f =.8 f T, peak (with f 5 = 1THz!!) Analysis of distortion in advanced SiGe HBTs impact of high-current effects, BC depletion capacitance, internal base resistance => identification of the contribution of transfer current and charge-storage nonlinearity Detailed comparison of compact model with measurements (3D) Perfect agreement of DC, RF characteristics of high speed SiGe HBTs (f T / f max = 32/37 GHz), SiGe HBTs (f T /f max = 3 /5 GHz).

40 Summary Nonlinear characteristics were measured on wafer with a 5 GHz PNA -X. Load pull single and two tone measurements were carried out with automated impedance tuner system from Maury. HICUM model agrees fairly well with measured nonlinear data up to measure 5 GHz. External parasitic bias independent capacitances for SiGe HBTs have negligible influence on the harmonics. Acknowledgment European Union's Seventh Programme for research, technological development and demonstration under grant agreement No European CATRENE Office (project RF2THz) Keysight Technologies (PNA-X loan) and constant support. Steffen Lehmann is acknowledged for the load pull simulation.

41 Summary References [1] MTT IMS 214, WMF workshop, "Physiscs-based Nonlinear Compact Transistor modeling for mm-wave Applications", Drs. P. Sakalas, M. Schroter. [2] S. Narayanan, Transistor distortion analysis using Volterra Series representation, BSTJ, pp , [3] S. Chisholm and L. Nagel, Efficient computer simulation of distortion in electronic circuits, IEEE Trans. Circ. Theory, pp , Nov [4] D. Weiner and J. Spina, Sinusoidal analysis and modeling of weakly nonlinear circuits, Van Nostrand, NY, 198. [5] R. Meyer, Intermodulation in high-frequency bipolar transistor integrated-circuit mixers, IEEE. J. Sol.-St. Circ., Vol. 21, pp , [6] S. Maas et al., Intermodulation distortion in heterojunction bipolar transistors, IEEE Trans. MTT, Vol. 4, pp ,1992. [7] M. Schroter, et al., Compact modeling of high-frequency distortion in Si integrated bipolar transistors, IEEE Trans. on Electron Dev., Vol. 47, pp , 2. [8] M. Vaidyanathan et al., "High-frequency distortion in bipolar transistors", IEEE Trans. MTT, Vol. 51, pp , 23. [9] P. Sakalas, et al., Modeling of SiGe power HBT intermodulation distortion using HI- CUM, Proc. ESSDERC 23, Lisboa, pp , 23.