VBIC, HICUM and Mextram Models

1. Theory

• Definition: VBIC, HICUM and Mextram are the advanced compact models that replaced Gummel-Poon for integrated bipolar and SiGe heterojunction bipolar transistors, adding quasi-saturation, avalanche, self-heating, substrate parasitics and better high-current and high-frequency behaviour (IC-CAP; McAndrew et al.; HICUM/L2 v2.34; Mextram 504.9).
• VBIC: The vertical bipolar inter-company model, developed by the IC and CAD industries, keeps the Gummel-Poon structure but takes the Early effect from depletion charges, adds a modified Kull-Nagel quasi-saturation model, a parasitic pnp and self-heating, and reduces to Gummel-Poon with default parameters (IC-CAP; McAndrew et al.).
• HICUM: The high-current model uses a generalized integral charge-control relation with weighted charges, a transit-time model built on a critical current for high-current effects, non-quasi-static delays, avalanche and a thermal network (HICUM/L2 v2.34).
• Mextram: The Mextram model describes the main current by integral charge control and the collector epilayer by a Kull-based model with velocity saturation and the Kirk effect; it was elected a world standard model by the Compact Model Council in 2004 (Mextram 504.9).

2. Working Principle

• Weighted charge control: In HICUM each charge enters the hole-charge denominator with its own weight, so bandgap grading in SiGe bases, which makes some charges affect the current more than others, is captured (HICUM/L2 v2.34).
• Quasi-saturation: When the internal base-collector junction becomes forward biased under the emitter, holes flood the epilayer and its resistance falls; Kull-type models express the epilayer current through injection factors at its two ends (McAndrew et al.; Mextram 504.9; background knowledge for the injection factors).
• High-current roll-off: Above the critical current the collector space charge cancels the doping and the base pushes out, so the transit time rises and $f_T$ falls (HICUM/L2 v2.34; background knowledge for the Kirk mechanism).
• Self-heating: Dissipated power raises the junction temperature through a thermal resistance and capacitance, feeding back into IS and the gain (McAndrew et al.; Mextram 504.9).

3. Applications

• SiGe BiCMOS design kits: HICUM and Mextram for millimetre-wave transceivers, radar, optical drivers and high-speed analog (HICUM/L2 v2.34; Mextram 504.9; background knowledge for the applications).
• Power and analog bipolar processes: VBIC for processes needing quasi-saturation, substrate parasitics and self-heating with Gummel-Poon-like parameters (McAndrew et al.; IC-CAP).
• Breakdown and safe operating area: Avalanche and snap-back models for output stages and drivers (Mextram 504.9; HICUM/L2 v2.34).
• Thermal design: Self-heating networks for power amplifiers and thermally coupled transistor arrays (McAndrew et al.; Mextram 504.9).

VBIC

Name	Formula	Related Formulae	Variables/Constants
VBIC transfer current and depletion-charge Early effect	$I_{tzf}=\frac{IS\left(e^{V_{bei}/(NF{V}_T)}-1\right)}{q_b},q_b=\frac{q_1}{2}\left(1+\sqrt{1+4q_2}\right),q_1=1+\frac{q_{je}}{VER}+\frac{q_{jc}}{VEF}$	$VEF\left(T\right)=VEF\left(T_{NOM}\right)\left[1+TCVEF\left(T-T_{NOM}\right)\right]$	$I_{tzf}:forwardtransfercurrent$ $IS:saturationcurrent$ $V_{bei}:internalbase-emittervoltage$ $NF:emissioncoefficient$ $q_b:normalizedbasecharge$ $q_1:Earlyterm$ $q_2:high-injectionterm$ $\begin{array}{r}{q}_{je},q_{jc}:normalizedbase-emitterand\\ base-collectordepletioncharges,VEF\end{array}$ $VER:forwardandreverseEarlyvoltages$ $\begin{array}{r}TCVEF:lineartemperaturecoefficient(q_1\\ form,backgroundknowledgeafterVBIC);\\ McAndrewetal.:VBICbasedonSGPwith\\ theEarlyeffectfromdepletioncharge,\\ lineartemperaturecoefficientsforVEF\\ andVERaddedinv1.3(equation1,as\\ read),quasi-neutralbaserecombination\\ modulatedthrough{q}_{1}byQNIBEIR,andan\\ excess-phasenetwork{I}_{tzf}-d(T_D{V}_{xf1})/dt-V_{xf2}\\ =0;IC-CAP:Earlyeffectfromjunction\\ depletioncharges,high-ordercontinuity,\\ andreductiontoGummel-Poonwithdefault\\ parameters;Gummel-Poonformsonthe\\ Ebers-MollandGummel-PoonModelspage\end{array}$ $(nocomputedvalue)$
Kull-Nagel quasi-saturation (epilayer) current	$I_{epi}=\frac{V_{rci}+V_T\left[K_{bci}-K_{bcx}-\ln \left(\frac{1+K_{bci}}{1+K_{bcx}}\right)\right]}{RCI},K_{bc}=\sqrt{1+GAMM{e}^{V_{bc}/V_T}}$	$I_{C,epi}=\frac{E_c+V_{C2C1}}{R_{Cv}}(\text{Mextram})$	$\begin{array}{r}{I}_{epi}:currentthroughthelightlydoped\\ collectorepilayer\end{array}$ $V_{rci}:voltageacrossit$ $\begin{array}{r}{K}_{bci},K_{bcx}:carrier-injectionfactorsatthe\\ internalandexternalendsoftheepilayer\end{array}$ $RCI:epilayerresistance$ $GAMM:epidopingparameter$ $\begin{array}{r}{V}_{bc}:localbase-collectorvoltage(form,\\ backgroundknowledge);McAndrewetal.:\\ VBICusesamodifiedKull-Nagel\\ quasi-saturationmodeland,fromv1.3,an\\ alternativedepletion-modulated{R}_{CI}for\\ verticalpnpdevices;IC-CAP:themodified\\ KullmodelvalidintotheKirkregime,\\ withquasi-saturationchargeinthe\\ diffusioncapacitance;Mextram:an\\ epilayercurrent(E_c+V_{C2C1})/R_{Cv},where{E}_c\\ accountsforconductivitymodulation,with\\ SCRCv,Ihcandaxifor\\ space-charge-limitedflow,hotcarriers\\ andtheonsetofquasi-saturation\end{array}$ $(nocomputedvalue)$
VBIC thermal and parasitic elements	$T_j=T_{amb}+\Delta T,\Delta T+R_{TH}{C}_{TH}\frac{d\Delta T}{dt}=R_{TH}{P}_{diss}$	$I_{tzf,pnp}\text{(parasitic substrate pnp, transport form)}$	$T_j:junctiontemperature$ $T_{amb}:ambienttemperature$ $\Delta T:self-heatingrise$ $R_{TH},C_{TH}:thermalresistanceandcapacitance$ $\begin{array}{r}{P}_{diss}:dissipatedpower(single-poleform,\\ backgroundknowledge);McAndrewetal.:a\\ singletime-constantself-heatingmodel,\\ withexternalthermalnetworksattachable\\ tothethermalnode,alinear{R}_{TH}(T)in\\ v1.3,fullparasiticpnpmodelling,anda\\ weakbase-collectoravalanchemodelwith\\ anaddedelementbetweennodesbxandcx;\\ IC-CAP:parasiticsubstratetransistor,\\ single-piececapacitancesforallthree\\ junctions,overlapcapacitances,baseand\\ emittercrowdingtofirstorder,weak\\ avalanche,andself-heatingdefinedonly\\ ascodehooks(seetheQ/A)\end{array}$ $(nocomputedvalue)$

HICUM

Name	Formula	Related Formulae	Variables/Constants
HICUM generalized integral charge-control relation	$i_T=\frac{c_{10}}{Q_{p,T}}\left[\exp \left(\frac{v_{B^{\prime }{E}^{\prime }}}{m_{Cf}{V}_T}\right)-\exp \left(\frac{v_{B^{\prime }{C}^{\prime }}}{V_T}\right)\right],c_{10}=IS\cdot Q_{p0}$	$Q_{p,T}=Q_{p0}+h_{jEi}{Q}_{jEi}+h_{jCi}{Q}_{jCi}+Q_{fT}+Q_{rT},Q_{fT}=h_{f0}{Q}_{f0}+h_{fE}\Delta Q_{Ef}+\Delta Q_{Bf}+h_{fC}\Delta Q_{Cf}$	$i_T:transfercurrent$ $c_{10}:constant$ $IS:saturationcurrent$ $Q_{p,T}:weightedholecharge$ $Q_{p0}:zero-biasholecharge$ $v_{B^{\prime }{E}^{\prime }},v_{B^{\prime }{C}^{\prime }}:internaljunctionvoltages$ $m_{Cf}:forwardnon-idealitycoefficient$ $\begin{array}{r}{h}_{jEi},h_{jCi}:weightsofthebase-emitterand\\ base-collectordepletioncharges{Q}_{jEi}\end{array}$ $\begin{array}{r}{Q}_{jCi},Q_{fT},Q_{rT}:weightedforwardandreverse\\ minoritycharges\end{array}$ $\begin{array}{r}{h}_{f0},h_{fE},h_{fC}:weightsofthelow-current,\\ emitterandcollectorminoritycharges;\\ HICUM/L2v2.34:thisGICCR,thestable\\ form{i}_{Tf1}=i_{Tf}(1+i_{Tf1}/I_{Ch})replacing{c}_1=\\ c_{10}(1+i_T/I_{Ch}),andweightsthataccount\\ forbandgapgradinginSiGeHBTs\end{array}$ $(nocomputedvalue)$
HICUM transit time and critical (Kirk) current	${\tau }_f={\tau }_{f0}+\Delta {\tau }_{Ef}+\Delta {\tau }_{fh},\Delta {\tau }_{Ef}={\tau }_{Ef0}{\left(\frac{i_{Tf}}{I_{CK}}\right)}^{g_{\tau E}}$	$r_{Ci0}=\frac{w_C}{q{\mu }_{nC0}{N}_{Ci}{A}_E},V_{lim}=\frac{v_{sn}{w}_C}{{\mu }_{nC0}},V_{PT}=\frac{q{N}_{Ci}{w}_C^2}{2ϵ},I_{CK}\approx \frac{v_{ceff}/r_{Ci0}}{\sqrt{1+{\left(v_{ceff}/V_{lim}\right)}^2}}$	${\tau }_f:forwardtransittime$ ${\tau }_{f0}:low-currentpart(dependingon{v}_{B^{\prime }{C}^{\prime }})$ $\Delta {\tau }_{Ef}:emitterpartatmediumcurrent$ ${\tau }_{Ef0},g_{\tau E}:itsparameters$ $\begin{array}{r}\Delta {\tau }_{fh}:baseandcollectorpartsathigh\\ current\end{array}$ $i_{Tf}:forwardtransfercurrent$ $\begin{array}{r}{I}_{CK}:criticalcurrentfortheonsetof\\ high-currenteffects\end{array}$ $\begin{array}{r}{r}_{Ci0}:low-fieldinternalcollector\\ resistance\end{array}$ $w_C:collectorwidth$ ${\mu }_{nC0}:low-fieldelectronmobility$ $N_{Ci}:collectordoping$ $A_E:emitterarea$ $V_{lim}:voltageforvelocitysaturation$ $v_{sn}:saturationvelocity$ $V_{PT}:punch-throughvoltage$ $ϵ:permittivity$ $\begin{array}{r}{v}_{ceff}:effectiveinternalcollector-emitter\\ voltage;HICUM/L2v2.34:thesedefinitions\\ withasmoothed{I}_{CK}using{\delta }_{ck}(default2.0)\\ and{a}_{ick}(default1\times 10^{-3}),andtheminority\\ charges{Q}_{f0}={\tau }_{f0}{i}_{Tf}and\Delta Q_{Ef}=\Delta {\tau }_{Ef}{i}_{Tf}/(1+g_{\tau E});\\ simplifiedlow-voltage{I}_{CK}form,background\\ knowledge\end{array}$ $\begin{array}{r}(computed:anassumed{w}_C=0.2\mu \mathrm{m},N_{Ci}=5\times 10^{16}\\ \mathrm{c}{\mathrm{m}}^{-3},A_E=0.2by10\mu \mathrm{m},\mu =800\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})\\ and{v}_{sn}=1\times 10^{7}cm/sgive{r}_{Ci0}=156\Omega ,V_{lim}=\\ 0.25\mathrm{V},V_{PT}=1.55\mathrm{V}and{I}_{CK}of1.43,1.55\\ and1.59\mathrm{mA}at{v}_{ceff}=0.5,1and2\mathrm{V},\\ approachingq{N}_{Ci}{v}_{sn}{A}_E=1.60\mathrm{mA})\end{array}$
HICUM non-quasi-static delays and avalanche	${\tau }_2={\alpha }_{Qf}{\tau }_f,{\tau }_m={\alpha }_{iT}{\tau }_f$	$i_{AVL}\approx i_{Tf}{f}_{AVL}\left(V_{DCi}-v_{B^{\prime }{C}^{\prime }}\right)\exp \left[-\frac{q_{AVL}}{C_{jCi}\left(V_{DCi}-v_{B^{\prime }{C}^{\prime }}\right)}\right],f_{AVL}=\frac{2a_n}{b_n},q_{AVL}=\frac{b_nϵA_E}{2}$	${\tau }_2:delayoftheminoritycharge$ ${\tau }_m:delayofthetransfercurrent$ ${\alpha }_{Qf},{\alpha }_{iT}:delayfactors$ $i_{AVL}:avalanchecurrent$ $f_{AVL},q_{AVL}:avalancheparameters$ $V_{DCi}:base-collectorbuilt-involtage$ $\begin{array}{r}{C}_{jCi}:internalbase-collectordepletion\\ capacitance\end{array}$ $\begin{array}{r}{a}_n,b_n:Chynowethionizationcoefficients;\\ HICUM/L2v2.34:delaysthrougha\\ second-orderBesselnetwork,afirst-order\\ RCadjunctnetworkfortheinputNQS\\ effectsincev2.24,alateralNQS\\ capacitance{C}_{rBi}=f_{CrBi}{C}_{i}withtheorygiving\\ f_{CrBi}=0.2foralongstripe(notadvised\\ fortransientanalysis),and{i}_{AVL}setto\\ zerofor{v}_{B^{\prime }{C}^{\prime }}atorabove0withasmoothed\\ voltageterm\end{array}$ $\begin{array}{r}(computed:{\tau }_f=1\mathrm{ps}withanassumed{\alpha }_{iT}=\\ 0.2gives{\tau }_m=0.2\mathrm{ps},atransconductance\\ phaselagof7.2degreesat100\mathrm{GHz})\end{array}$

Mextram

Name	Formula	Related Formulae	Variables/Constants
Mextram main current and Early effect	$I_N=\frac{I_s}{q_B}\left[\exp \left(\frac{V_{B2E1}}{V_T}\right)-\exp \left(\frac{V_{B2C2}^{\ast }}{V_T}\right)\right]$	$q_B=q_1\frac{1+\sqrt{1+4\left(\dots \right)/I_k}}{2},q_1=1+\frac{V_{tE}}{V_{er}}+\frac{V_{tC}}{V_{ef}}$	$I_N:maincollectorcurrent$ $I_s:saturationcurrent$ $q_B:normalizedbasecharge$ $V_{B2E1},V_{B2C2}^{\ast }:internaljunctionvoltages$ $I_k:high-injectionkneecurrent$ $q_1:Earlyfactor$ $\begin{array}{r}{V}_{tE},V_{tC}:depletion-chargecurvatureterms\\ ofthebase-emitterandbase-collector\\ junctions\end{array}$ $\begin{array}{r}{V}_{er},V_{ef}:reverseandforwardEarlyvoltages\\ (q_{B}shapeasread,detailselided);\\ Mextram504.9:theintegralchargecontrol\\ relationinthisform,abias-dependent\\ Earlyeffect,highinjection,a\\ graded-bandgapEarlyeffectforSiGe,\\ currentcrowdingandconductivity\\ modulationofthebaseresistance,\\ optionaldistributedhigh-frequency\\ effectsintheintrinsicbase,anda\\ referencetemperatureof25{}^{\circ }C\end{array}$ $(nocomputedvalue)$
Mextram avalanche and self-heating	$I_{avl}=I_{C1C2}{\int }_{SCR}{\alpha }_n\left(E\right)dx,{\alpha }_n\left(E\right)=a_n\exp \left(-\frac{b_n}{E}\right)$	$T=T_{amb}+DTA+\Delta T_{dyn},\Delta T_{dyn}=R_{th}{P}_{diss}(\text{static})$	$I_{avl}:avalanchecurrent$ $I_{C1C2}:epilayercurrent$ $\begin{array}{r}{\alpha }_n:electronionizationcoefficient\\ (Chynowethlaw)\end{array}$ $\begin{array}{r}E:electricfieldinthebase-collector\\ space-chargeregion(SCR)\end{array}$ $a_n,b_n:ionizationconstants$ $T:devicetemperature$ $T_{amb}:ambient$ $DTA:localtemperatureoffset$ $\Delta T_{dyn}:self-heatingrise$ $\begin{array}{r}{R}_{th},C_{th}:thermalresistanceand\\ capacitance;Mextram504.9:theavalanche\\ currentastheepilayercurrenttimesthe\\ integratedChynowethlawwithalinearized\\ field,parametersWavlandVavl,optional\\ EXAVLaboveIhcwithsnap-back,Zener\\ tunnellingintheemitter-basejunction\\ from504.8,andan{R}_{th},C_{th}networkdriven\\ bythetotaldissipatedpower;HICUMand\\ VBICalsoincludethermalnetworks(static\\ form,backgroundknowledge)\end{array}$ $\begin{array}{r}(computed:anassumed{R}_{th}=5000K/Wat3\\ \mathrm{mW}raisesthejunctionby15\mathrm{K},whichat\\ fixed{V}_{BE}multipliesISbyabout8.9;with\\ anassumed{C}_{th}=1\times 10^{-10}J/Kthethermaltime\\ constantis0.5\mu \mathrm{s})\end{array}$

Important Questions

Ques: What does VBIC add to the Gummel-Poon model?

IC-CAP lists the additions: an Early effect from junction depletion charges for better output conductance, a modified Kull model for quasi-saturation valid into the Kirk regime, a parasitic substrate transistor, single-piece capacitances for all three junctions, improved temperature scaling, first-order base and emitter crowding, quasi-saturation charge in the diffusion capacitance, overlap capacitances, weak base-collector avalanche and infinite continuity of the equations. With default parameters it reduces as closely as possible to Gummel-Poon, which was essentially unchanged for over 20 years. McAndrew et al. add the v1.3 changes: temperature coefficients for VEF and VER, quasi-neutral base recombination modulated through $q_1$, a depletion-modulated collector resistance for vertical pnp devices and an IBM-compatible avalanche element. VBIC therefore suits users who want Gummel-Poon-like parameters with the missing physics added (my reading). Reference: Agilent IC-CAP 2008 documentation, Nonlinear Device Models Volume 2, ‘VBIC Model’ (fetch-tool summary); C. C. McAndrew, T. Bettinger, L. Lemaitre and M. Tutt, ‘BJT Modeling with VBIC, Basics and V1.3 Updates’, NXP-hosted white paper MSM2003 (venue and year not stated; references to 2002) (equations partly garbled in the extract) (fetch-tool summary).

Ques: Does VBIC include a self-heating model?

The sources disagree because they describe different versions. IC-CAP’s 2008 documentation says self-heating is defined only through code hooks, with the model itself planned for a future release. McAndrew et al., describing VBIC up to v1.3, state that VBIC has a single time-constant self-heating model, that a more complex network was left out because it adds system variables, that users can attach an external thermal network to the thermal node, and that v1.3 adds a linear $R_{TH}(T)$. The VBIC releases themselves contain the thermal node and the RTH and CTH parameters (background knowledge), so self-heating is part of the model, and the IC-CAP statement most likely reflects the extraction tool’s support at the time rather than the model (my reading). For an assumed 5000 $K/W$ at 3 $\mathrm{mW}$ the rise is 15 $\mathrm{K}$ (computed), too large to ignore. Reference: Agilent IC-CAP 2008 documentation, Nonlinear Device Models Volume 2, ‘VBIC Model’ (fetch-tool summary); C. C. McAndrew, T. Bettinger, L. Lemaitre and M. Tutt, ‘BJT Modeling with VBIC, Basics and V1.3 Updates’, NXP-hosted white paper MSM2003 (venue and year not stated; references to 2002) (equations partly garbled in the extract) (fetch-tool summary).

Ques: How do Kull-type models describe quasi-saturation?

By treating the lightly doped collector epilayer as a resistor whose conductivity rises when carriers are injected into it. In the Kull-Nagel form the epilayer current is $[V_{rci}+V_T(K_{bci}-K_{bcx}-\ln ((1+K_{bci})/(1+K_{bcx})))]/RCI$, with injection factors $K=\sqrt{1+GAMM{e}^{V/V_T}}$ at the two ends (background knowledge). When the internal junction is reverse biased the factors are 1 and the epilayer is a plain resistance; when it becomes forward biased, injected holes modulate its conductivity and the current saturates more gradually. VBIC uses a modified Kull-Nagel model and, for vertical pnp devices, an alternative depletion-modulated resistance (McAndrew et al.). Mextram writes the epilayer current as $(E_c+V_{C2C1})/R_{Cv}$, adding space-charge-limited flow, hot-carrier current Ihc and the smoothing parameter axi. Reference: C. C. McAndrew, T. Bettinger, L. Lemaitre and M. Tutt, ‘BJT Modeling with VBIC, Basics and V1.3 Updates’, NXP-hosted white paper MSM2003 (venue and year not stated; references to 2002) (equations partly garbled in the extract) (fetch-tool summary); R. van der Toorn, J. C. J. Paasschens and W. J. Kloosterman, ‘The Mextram Bipolar Transistor Model, level 504.9’, NXP Semiconductors and Delft University of Technology (24 November 2010) (fetch-tool summary).

Ques: How does HICUM’s transfer current differ from the Gummel-Poon form?

In the denominator. Gummel-Poon divides by a normalized base charge built from Early voltages and knee currents. HICUM’s generalized integral charge-control relation divides $c_{10}=IS\cdot Q_{p0}$ by a weighted hole charge $Q_{p,T}=Q_{p0}+h_{jEi}{Q}_{jEi}+h_{jCi}{Q}_{jCi}+Q_{fT}+Q_{rT}$, in which the depletion charges and the low-current, emitter, base and collector minority charges each carry their own weight. The weights account for bandgap grading in SiGe HBTs, where charge near the emitter affects the current differently from charge near the collector, so the same charges that set the capacitances also set the DC current. HICUM also uses a forward non-ideality coefficient $m_{Cf}$ and, in the current implementation, a numerically stable high-current factor $i_{Tf1}=i_{Tf}(1+i_{Tf1}/I_{Ch})$. Reference: M. Schroter, A. Pawlak and A. Mukherjee, ‘HICUM/L2 Model Documentation’, version 2.34, TU Dresden (March 2017) (some equations garbled in the extract) (fetch-tool summary); Ebers-Moll and Gummel-Poon Models page of this vault.

Ques: What sets the critical current for high-current effects in HICUM?

The collector design. HICUM defines a low-field collector resistance $r_{Ci0}=w_C/(q{\mu }_{nC0}{N}_{Ci}{A}_E)$, a velocity-saturation voltage $V_{lim}=v_{sn}{w}_C/{\mu }_{nC0}$ and a punch-through voltage $V_{PT}=q{N}_{Ci}{w}_C^2/(2ϵ)$, and builds a smoothed critical current $I_{CK}$ from them. For an assumed 0.2 $\mu \mathrm{m}$ collector doped to $5\times 10^{16}$ $\mathrm{c}{\mathrm{m}}^{-3}$ under a 0.2 by 10 $\mu \mathrm{m}$ emitter, with 800 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ and $1\times 10^7$ $cm/s$, $r_{Ci0}=156$ $\Omega$, $V_{lim}=0.25$ $\mathrm{V}$ and $V_{PT}=1.55$ $\mathrm{V}$ (computed). The low-voltage form $(v_{ceff}/r_{Ci0})/\sqrt{1+(v_{ceff}/V_{lim}{)}^2}$ (background knowledge) gives 1.43 to 1.59 $\mathrm{mA}$ from 0.5 to 2 $\mathrm{V}$, approaching the Kirk limit $q{N}_{Ci}{v}_{sn}{A}_E=1.60$ $\mathrm{mA}$ (computed). Above $I_{CK}$ the emitter, base and collector transit-time terms rise, and $f_T$ peaks just below it. Heavier collector doping raises $I_{CK}$ but lowers the breakdown voltage, the basic speed-breakdown trade-off of HBTs (background knowledge). Reference: M. Schroter, A. Pawlak and A. Mukherjee, ‘HICUM/L2 Model Documentation’, version 2.34, TU Dresden (March 2017) (some equations garbled in the extract) (fetch-tool summary).

Ques: How do the models describe avalanche breakdown?

All three model weak avalanche at the base-collector junction, with different detail. VBIC has a weak avalanche current and, from v1.3, an extra element between nodes bx and cx (McAndrew et al.). HICUM writes $i_{AVL}\approx i_{Tf}{f}_{AVL}(V_{DCi}-v_{B^{\prime }{C}^{\prime }})\exp [-q_{AVL}/(C_{jCi}(V_{DCi}-v_{B^{\prime }{C}^{\prime }}))]$ with $f_{AVL}=2a_n/b_n$ and $q_{AVL}=b_nϵA_E/2$, linking the field to the depletion capacitance, and sets it to zero in forward bias. Mextram multiplies the epilayer current by the integral of Chynoweth’s law $a_n\exp (-b_n/E)$ over the space-charge region with a linearized field, using Wavl and Vavl, and its optional EXAVL extension reaches currents above Ihc and reproduces snap-back. Mextram 504.8 also added Zener tunnelling in the emitter-base junction. Reference: C. C. McAndrew, T. Bettinger, L. Lemaitre and M. Tutt, ‘BJT Modeling with VBIC, Basics and V1.3 Updates’, NXP-hosted white paper MSM2003 (venue and year not stated; references to 2002) (equations partly garbled in the extract) (fetch-tool summary); M. Schroter, A. Pawlak and A. Mukherjee, ‘HICUM/L2 Model Documentation’, version 2.34, TU Dresden (March 2017) (some equations garbled in the extract) (fetch-tool summary); R. van der Toorn, J. C. J. Paasschens and W. J. Kloosterman, ‘The Mextram Bipolar Transistor Model, level 504.9’, NXP Semiconductors and Delft University of Technology (24 November 2010) (fetch-tool summary).

Ques: How large is self-heating in a small SiGe HBT?

Large enough to change the bias point. All three models include a thermal network: VBIC a single time constant (McAndrew et al.), HICUM a network with $R_{th}$ and $C_{th}$ driven by the dissipated power, and Mextram a temperature equal to ambient plus DTA plus a dynamic rise from $R_{th}$ and $C_{th}$. For an assumed 5000 $K/W$ and 3 $\mathrm{mW}$ (2 $\mathrm{mA}$ at 1.5 $\mathrm{V}$) the junction rises by 15 $\mathrm{K}$, which at fixed $V_{BE}$ multiplies IS by about 8.9 (computed), so a fixed-voltage bias would run away without emitter degeneration. With an assumed $C_{th}$ of $1\times 10^{-10}$ $J/K$ the thermal time constant is 0.5 $\mu \mathrm{s}$ (computed), so the temperature follows slow signals but not RF, which makes output conductance frequency dependent. Mextram clips $R_{th}$ at zero to keep the network valid. Reference: C. C. McAndrew, T. Bettinger, L. Lemaitre and M. Tutt, ‘BJT Modeling with VBIC, Basics and V1.3 Updates’, NXP-hosted white paper MSM2003 (venue and year not stated; references to 2002) (equations partly garbled in the extract) (fetch-tool summary); M. Schroter, A. Pawlak and A. Mukherjee, ‘HICUM/L2 Model Documentation’, version 2.34, TU Dresden (March 2017) (some equations garbled in the extract) (fetch-tool summary); R. van der Toorn, J. C. J. Paasschens and W. J. Kloosterman, ‘The Mextram Bipolar Transistor Model, level 504.9’, NXP Semiconductors and Delft University of Technology (24 November 2010) (fetch-tool summary).

References

1. Agilent Technologies, “VBIC Model,” IC-CAP 2008 documentation, Nonlinear Device Models Volume 2. Available: https://people.ece.ubc.ca/robertor/Links_files/Files/ICCAP-2008-doc/icmdlhf/icmdlhf09.html
2. C. C. McAndrew, T. Bettinger, L. Lemaitre and M. Tutt, “BJT Modeling with VBIC, Basics and V1.3 Updates,” white paper MSM2003 (venue and year not stated). Available: https://www.nxp.com/docs/en/white-paper/MSM2003_BJT_COLIN_MCANDREW.pdf
3. M. Schroter, A. Pawlak and A. Mukherjee, “HICUM/L2 Model Documentation,” version 2.34, TU Dresden, Mar. 2017. Available: https://www.iee.et.tu-dresden.de/iee/eb/forsch/Hicum_PD/Hicum23/hicum_L2V2p34_manual.pdf
4. R. van der Toorn, J. C. J. Paasschens and W. J. Kloosterman, “The Mextram Bipolar Transistor Model, level 504.9,” NXP Semiconductors and Delft University of Technology, Nov. 2010. Available: https://www.eng.auburn.edu/~niuguof/mextram/_downloads/cde3edee01fa3f4fe9c3fa26d909862f/Mextramdefinition_504.9.pdf

Related pages in this vault: Ebers-Moll and Gummel-Poon Models, Bipolar Transistors (BJT and SiGe HBT), Bipolar Junction Transistor (BJT), Power Device Compact Models, Diode Compact Models, ASM-HEMT Model, Angelov and Empirical FET Models, Phototransistor (HPT and Photo-FET), Insulated Gate Bipolar Transistor (IGBT).