Wideband analytical extraction technique of π-equivalent circuit model for Si/SiGe heterojunction bipolar transistor in BICMOS process

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Published in IET Microwaves, Antennas & PropagationReceived on 19th April 2013Accepted on 5th August 2013doi: 10.1049/iet-map.2013.0321

T Microw. Antennas Propag., 2014, Vol. 8, Iss. 1, pp. 57–63oi: 10.1049/iet-map.2013.0321

ISSN 1751-8725

Wideband analytical extraction technique ofπ-equivalent circuit model for Si/SiGe heterojunctionbipolar transistor in BICMOS processHany Taher1,2

1Electrical Engineering Department, Faculty of Engineering and Islamic Architecture, Umm Alqura University, 5555,

Makkah, Kingdom of Saudi Arabia2Electronic Research Institute, 12622 -Giza, Egypt

E-mail: [email protected]

Abstract: A developed analytical extraction technique of small-signal π-topology equivalent circuit model for Si/SiGeheterojunction bipolar transistor is presented. The intrinsic model parameters, including base resistance (Rb) which aredifficult to extract in the previous works, are analytically extracted by utilising a novel set of exact equations that do not needany numerical fitting, special polarisation of the device or any kind of post processing. Moreover, the substrate effect and thedistributed base-collector junction are accurately modelled and extracted. To the authors knowledge, the extracted valuesusing the presented methodology exhibit flattest and widest frequency independent behaviour among all those extracted withearlier published analytical techniques, especially for base resistance. Excellent agreement is noted between the S-parametersmeasurements and their simulated counterpart using the extracted model in the frequency range from 40 MHz–40 GHz atdifferent bias conditions.

1 Introduction

Owing to its high-speed performance, the Si/SiGeheterojunction bipolar transistor (HBT) is the preferredcandidate for integrated communications systems applications.An accurate small-signal equivalent circuit model of thedevice is very essential for reliable circuit design. Twodifferent topologies are adopted as an equivalent circuit forHBT, namely, T-topology and π–topology. The T- model isdirectly related to the physics of the device. However, theπ-model is considered as the linear form of various compactmodels such as spice gummel poon (SGB) and verticalbipolar inter company (VBIC).On the other side, the III–V HBT devices have negligible

substrate effect on their performance compared with the Si/SiGe HBT devices. Most of the previously published papersconcentrated on extracting model parameters of III–V HBTdevices, either in T-topology [1–9] or in π-topology [10–12].There are two main techniques to extract small-signal model

parameters, direct extraction [1–6, 10–12] and optimisation-based extraction [7–9]. In the former technique, the extractedvalues are obtained from the S-parameters measurements usinganalytical expressions for the model parameters. Concerningthe latter technique, it utilises numerical techniques to find thebest parameter values that generate simulation results which fitwell with the measurements data. Unfortunately, it suffersfrom local minima problems and dependency on the initialvalues of the parameters. Consequently, non-physical andnon-unique values could be extracted.

The available previous few works that tackle extraction ofπ-topology model of the Si/SiGe HBT devices problemanalytically are introduced in [13–17]. Special polarisationfor the device, frequency approximations and least squarefit are needed to extract the model parameters [13, 16]. Thetechniques presented in [16, 17] model the base-collectorjunction as a single capacitance component rather thansplitting it into internal and external ones.The single capacitance model leads to a tangible mismatch

between the measured and simulated S-parameters. Onanother note, it is important to incorporate the substrateeffect into the model to match S22 parameter measurements,especially at high frequency. However, it is not consideredin the extraction techniques that are published in [14, 15,17]. Bias independent emitter resistance is not included inthe model [14, 15] and consequently the model behaviour isnot accurate beyond 20 GHz.Concerning [14], complete equivalent circuit of the

whole device including the radio frequency (RF) probe padsis modelled and extracted. Consequently, the substrateeffect is embedded and not modelled. In a different context,Rb is very difficult to extract, and is extracted under highfrequency condition [13, 14]. In the other works [16, 17], itsuffers from noticed frequency dependent behaviour. Allthe previously introduced techniques rely on deembeddingthe effect of Rb to obtain the required relations to extractthe values of the rest of the model parameters. Therefore itis considered as an additional deembedding step to thestandard parasitic deembedding process. Sensitivity of the

57& The Institution of Engineering and Technology 2013

Fig. 2 Small-signal equivalent circuit model of intrinsic part ofSi/SiGe HBT

Fig. 1 Small-signal equivalent circuit model of Si/SiGe HBT

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parameters values to the measurement errors is directlyproportional to the number of deembedding steps [18].To alleviate all these deficiencies, new closed form

expressions of the π-model parameters are derived from thedeembedded small signal parameters, Y parameters, of theintrinsic part of the device. All the bias dependent modelparameters values are extracted at the bias point of interest.Moreover, no frequency approximation or numerical fittingare needed in the extraction process.One external capacitor and other internal one are incorporated

into the model to take the distributed base collector junctionphenomena into consideration. No RF probe pads model isused and their influence is deembedded from themeasurements that are used in the model extraction process.Single capacitance is effectively used to model the

substrate effect. There is no need to use more complicatednetworks such as series R-C [13] or parallel R-C with seriescapacitance [16]. As a result of taking into account the mostimportant physical phenomena, frequency independentmodel parameters values are obtained over wideband up to40 GHz. The highest addressed value in the previouspublished techniques is 30 GHz [16]. Excellent matchingbetween deembedded measured and model-calculatedS-parameters is observed at different bias points.This paper is organised as follows: the developed technique

is described in Section 2. The frequency behaviour of theextracted model parameters and validation of the extractedmodel with deembedded S-parameters measurements arepresented in Section 3. In Section 4, the conclusion is drawn.

2 Proposed extraction technique

2.1 Deembedding the extrinsic part

The small-signal model of the Si/SiGe HBT as seen from theprobe tips of the measurement system is shown in Fig. 1.

Fig. 3 Dummy structures

a Padb Short circuit andc Open circuit


It consists of two main parts, extrinsic and intrinsic. Theformer is used to model the bias independent RF probe padand transmission lines whereas the latter is used to modelthe bias-dependent core of the device.The intrinsic parameters, as depicted in Fig. 2, are Rb,

base-emitter junction capacitance (Cπ), dynamic emitterresistance (Rπ), internal base-collector junction capacitance(Cu), external base-collector junction capacitance (Cf), DCtransconductance (gmo), transient time phase delay (t),collector substrate capacitance (Cs) and base resistance (Rb).Emitter resistance (Re) is incorporated into the intrinsic partdespite it being a bias-independent parameter. It isimpossible to deembed its value from device measurementusing conventional Y− Z parameters matrix manipulationsowing to the presence of grounded intrinsic parameter, Cs.The effect of the extrinsic part is removed from the

S-parameters measurements using the pad, short and opendummy structures shown in Fig. 3 [19].

2.2 Extraction of the intrinsic parameters

The deembedded admittance parameters (Yi) can be expressedin terms of the intrinsic parameters including Re (1)–(4).These expressions are derived for the first time and havenot been mentioned before in any literature. The complete

IET Microw. Antennas Propag., 2014, Vol. 8, Iss. 1, pp. 57–63doi: 10.1049/iet-map.2013.0321

Table 1 Extracted small-signal model parameters values atdifferent bias points

Parameters Bias

No. one No. two No. three

gmo, mS 10.20 42.07 108.01Rb, Ω 36.11 41.04 40.74Rπ, kΩ 24.80 5.75 2.08Cμ, fF 6.11 6.30 6.53Cf , fF 7.94 8.19 8.49Cπ, pF 0.12 0.22 0.50Cs, fF 16.90 15.90 15.10t, pSec 2.84 2.47 2.20

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derivation is mentioned in the Appendix
Yi11 =Yp + xYm

x+ Rb Yp + xYm

( )+ Yf (1)

Yi12 =−xYm

x+ Rb Yp + xYm

( )− Yf (2)

Yi21 =gm − xYm

x+ Rb Yp + xYm

( )− Yf (3)

Yi22 =xYm + RbYm gm + Yp

( )x+ Rb Yp + xYm

( ) + Yf + Ys (4)

where

Yu,f ,s = jv Cu,f ,s Yp = 1

Rp

+ jv Cp and

x = 1+ Yp + gm( )

Re

Equations (1)–(4) can be utilised to extract the intrinsicparameters values provided that Rπ, gmo, Re and k values aredetermined beforehand. k is geometrical information and isexpressed in terms of base area (AB) and emitter area (AE)as well as the ratio between Cf and Cu as given by thefollowing equation

k = AB

AE− 1 = Cf

Cm

(5)

DC measurements, base (IB) and collector (IC) currents, areused to calculate Rπ and gmo as given by the followingequation

Rp = h vTIB

(6)

gm0 =ICh vT

(7)

where vT is the thermal voltage, and η is the ideality factor ofthe base junction.The value of the ordinate at the origin that results from

linear extrapolation of the Re(Z12) plot, of the saturateddevice, against I−1

B is the extracted value of Re [8].The rest of the parameters are ready to be extracted as

follows, combining (1)–(3) yields

Yi11 =YpX

− Yi12 (8)

Yi21 =gmX

+ Yi12 (9)

where X = x + Rb(Yπ + xYμ).Rearranging the terms of (8) and (9), dividing the two

equations and taking the magnitude of the both sides yields

Yi11 + Yi12Yi21 − Yi12

∣∣∣∣∣∣∣∣ = Yp

gm

∣∣∣∣∣∣∣∣ (10)


Using

v2C2p = Yp

∣∣ ∣∣2− 1

Rp

( )2

and gm∣∣ ∣∣ = gmo

an expression of Cπ is obtained as

Cp =gmo Yi11 + Yi12

( )/ Yi21 − Yi12( )( )∣∣ ∣∣( )2− 1/Rp

( )( )2√v

(11)

Utilising (8), an equation of X is written as

X = YpYi11 + Yi12

(12)

Using (9), an expression for t is obtained as

t = −phase X Yi21 − Yi12( )( )v

(13)

Now, the value of xis calculated from its definition, where, allits independent parameters are extracted. Furthermore, it isused with (2) and (5) to derive an expression of Cμ as follows

Cm = Im −XYi12( )

/(x+ Xk)( )

v(14)

From (5), one can obtain an expression of Cf as

Cf = kCm (15)

Using (9) and the definition of X, an expression for Rb isobtained as

Rb = Regm/Yi21 − Yi12( )− x

Yp + xYm

( )(16)

Finally, Cs is extracted using (4) as

Cs =Im Yi22 − xYm + RbYm gm + Yp

( )/(X )

( )− Yf

( )v

(17)

All the model parameters are now extracted and validation ofthe extracted values is demonstrated in the next section.


Fig. 4 Base current dependence of Re(Z12) for the saturated device

Fig. 5 Frequency behaviour of the extracted model parameters

Bias no. one (open dimond), bias no. two (open square) and bias no. three(open triangle)

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3 Validation of the extractedsmall-signal model

This study is applied to a 0.80 μm× 9.60 μm Si/SiGeHBT common emitter from AMISemiconductors. DCmeasurements of the device under test (DUT) are performed.The bias conditions are 0.80–0.90 V for base voltage and1.00–2.00 V for collector voltage. Concerning small signalmeasurements, S-parameters of the DUT and the dummystructures are measured over the frequency range 0.040–40GHz. The presented technique is validated at three differentbias points, (VCE = 1.5 V, VBE = 0.80 V and IC = 0.20 mA),(VCE = 1.5 V, VBE = 0.85 V and IC = 1.42 mA) and (VCE =1.5 V, VBE = 0.90 V and IC = 3.00 mA). The three bias pointsare given the following names, bias no. one, bias no. two andbias no. three, respectively.Utilising the DC measurements, gmo and Rπ are calculated

and listed in Table 1. Regarding k of the DUT, it has a valueof 1.3 as extracted from its layout. Re value is 15 Ω asextracted by extrapolating the Re(Z12) plot, as shown inFig. 4, towards infinite IB value.Fig. 5 depicts the extracted values of the rest of the model

parameters at the intended bias points. As shown in Fig. 5a,the frequency performance of the extracted values of Rb

exhibits independent behaviour at all bias points. It is worthnoting that, the observed flat performance over this wide

Fig. 6 Comparison between the measured (open circle) andsimulated (thick line) S-parameters [40 MHz – 40 GHz] at

a Bias no. oneb Bias no. twoc Bias no. three


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frequency band has not been obtained using all the previouslypublished methodologies. The same excellent behaviour isnoted for all the other extracted parameters over all the biaspoints. The extracted parameters values are listed in Table 1.As a result of the robustness of the presented technique,
excellent matching between the measured and simulatedS-parameters is obtained, as shown in Figs. 6a–c. Theerrors between the measurements and the simulation resultsare quantified using the following equation

e %[ ] = 100× 1

4N

∑2i,j=1

∑Nk=1

Smij fk( )− Ssij fk

( )Smij fk

( )∣∣∣∣∣

∣∣∣∣∣2

(18)

where N is the number of frequency points, Smij fk( )

andSsij fk( )

are the measured and simulated S-parameters atfrequency fk, respectively. The calculated residual error isabout 0.5% at the three bias points.

4 Conclusion

In this paper, a new set of equations is developed toanalytically extract π-topology small-signal modelparameters of the Si/SiGe HBT. The presented techniqueuses the S-parameters measurements of the DUT anddeembedding dummy, pad-short-open, structures. All thebias dependent model parameters are extracted on theintended bias condition. The extracted values utilising thistechnique are reliable, physical and frequency independentover wide frequency band. Excellent agreement is obtainedbetween the S-parameters measurements and the simulatedresults of the extracted model over different bias points.Therefore the proposed method can be considered as aperfect candidate for standard extraction technique of HBTdevices in BICMOS process.

Fig. 7 Terminal vlotages and currents definitions used to drive Y-parameters expressions of small-signal equivalent circuit model

5 References

1 Spiegel, S.J., Ritter, D., Ham, R.A., Feygenson, A., Smith, P.R.:‘Extraction of the InP/GaInAs heterojunction bipolar transistorsmall-signal equivalent circuit’, IEEE Trans. Electron. Devices, 1995,42, (6), pp. 1059–1064

2 Schaper, U., Holzapfl, B.: ‘Analytical parameter extraction of the HBTequivalent circuit with T-like topology from measured S-parameters’,IEEE Trans. Microw. Theory Tech., 1995, 43, (3), pp. 493–498

3 Rudolph, M., Doerner, R., Heymann, P.: ‘Direct extraction of HBTequivalent-circuit elements’, IEEE Trans. Microw. Theory Tech.,1999, 47, (1), pp. 82–84

4 Tasker, P.J., Fernindez-Barciela, M.: ‘HBT small signal T and π: modelextraction using a simple, robust and fully analytical procedure’. Proc.IEEE MTT-S Int. Microwave Symp. on Dig., June 2002, pp. 2129–2123

5 Sotoodeh, M., Sozzi, L., Vinay, A., et al.: ‘Stepping toward standardmethods of small-signal parameter extraction for HBT’s’, IEEE Trans.Electron Devices, 2000, 47, (6), pp. 1139–1151

6 Chen, H.-Y., Chen, K.-M., Huang, G.W., Chang, C.-Y.: ‘A novelapproach for parameter determination of HBT small-signal equivalentcircuit’, IEICE Trans. Electron., 2005, E88-C, (6), pp. 1133–1142

7 Samelis, A., Pavlidis, D.: ‘DC to high-frequency HBT-model parameterevaluation using impedance block conditioned optimization’, IEEETrans. Microw. Theory Tech., 1997, 45, (6), pp. 886–897

8 Gobert, Y., Tasker, P.J., Bachem, K.H.: ‘A physical, yet simple,small-signal equivalent circuit for the heterojunction bipolartransistor’, IEEE Trans. Microw. Theory Tech., 1997, 45, (1),pp. 149–135

9 Ooi, B.L., Zhou, T.S., Kooi, P.S.: ‘AlGaAs/GaAs HBT modelestimation through the generalized pencil-of-function method’, IEEETrans. Microw. Theory Tech., 2001, 49, (7), pp. 1289–1294

10 Tseng, H.C., Chou, J.H.: ‘A pure analytic method for direct extraction ofcollector-up HBTs small-signal parameters’, IEEE Trans. ElectronDevices, 2004, 51, (12), pp. 1972–1977


11 Bousnina, S., Mandeville, P., Kouki, A.B., Surridge, R., Ghannouchi,F.M.: ‘Direct parameter-extraction method for HBT small-signalmodel’, IEEE Trans. Microw. Theory Tech., 2002, 50, (2), pp. 529–536

12 Degachi, L., Ghannouchi, F.M.: ‘An augmented small-signal HBTmodel with its analytical based parameter extraction technique’, IEEETrans. Electron Devices, 2008, 55, (4), pp. 968–972

13 Lee, K., Choi, K., Kook, S.-H., Cho, D.-H., Park, K.-W., Kim, B.:‘Direct parameter extraction of SiGe HBTs for the VBIC bipolarcompact model’, IEEE Electron Device Lett., 2005, 52, (3), pp. 375–384

14 Taher, H., Schreurs, D., Nauwelaers, B.: ‘Extraction of small-signalequivalent circuit model parameters for Si/SiGe HBT usingS-parameters measurements and one geometrical information’,Int. J. Electron. Commun. (AEU), 2006, 60, (8), pp. 567–572

15 Taher, H.: ‘Direct extraction technique of π-topology small-signalequivalent circuit model for Si/SiGe HBT’, Microw. Opt. Technol.Lett., 2012, 54, (3), pp. 584–589

16 Yang, T.-R., Tsai, J.M.-L., Ho, C.-L., Hu, R.: ‘SiGe HBT’s small-signalPi modeling’, IEEE Trans. Microw. Theory Tech., 2007, 55, (7),pp. 1417–1424

17 Olvera-Cervantes, J.-L., Cressler, J.D., Medina-Monroy, J.-L.,Thrivikraman, T., Banerjee, B., Laskar, J.: ‘A new analytical methodfor robust extraction of the small-signal equivalent circuit for SiGeHBTs operating at cryogenic temperatures’, IEEE Trans. Microw.Theory Tech., 2008, 56, (3), pp. 1417–1424

18 Masood, S.M., Johansen, T.K., Vidkj, J., Krozer, V.: ‘Uncertaintyestimation in SiGe HBT small-signal modeling’. Proc. GAAS Conf.,2005, pp. 393–396

19 Tiemeijer, L.F., Havens, R.J.: ‘A calibrated lumped-elementde-embedding technique for on-wafer RF characterization ofhigh-quality inductors and high-speed transistors’, IEEE Trans.Electron Devices, 2003, 50, (3), pp. 822–829

6 Appendix

See Fig. 7.

6.1 Y11 derivation

By definition, Y11 = (i1/v1)|v2=0. Therefore by shorting thecollector to the ground, the expression of Y11 can beobtained as follows

i1 = i1i + i1f

where i1f is the current passing through Cf

v1 = i1iRb + v/p (19)

v/p = vp + vpYp + gmvp( )

Re = vp 1+ Yp + gm( )

Re

( )(20)


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where Yp = (1/Rp)+ jvCp and v/p is the potential of thenode between Rb and Cu
vp = ipZp (21)

where iπ is the current passing through Zπ and equals

ip = i1i − v/pYm (22)

where Yμ = jωCμ

Substituting from (22) into (21) yields vp = i1iZp−v/pYuZp, substituting with the expression of v/p from (20),the result is

vp = i1iZp − vp 1+ Yp + gm( )

Re

( )YuZp (23)

vp = i1iZp

1+ 1+ Yp + gm( )

Re

( )YuZp

(24)

Substituting with this expression into (20) to obtain anexpression of v/p as follows

v/p = i1iZp 1+ Yp + gm

( )Re

( )1+ 1+ Yp + gm

( )Re

( )YuZp

(25)

From (25) into (19)

v1= i1i RB+Zp 1+ Yp+gm

( )Re

( )1+ 1+ Yp+gm

( )Re

( )YuZp

( )

i1i=v11

RB+ 1+ Yp+gm( )

Re

( )/ Yp+ 1+ Yp+gm

( )Re

( )Yu

( )( )(26)

On the other hand

i1f = viYf (27)

Finally, adding (26) and (27) results in the complete form ofY11 as followsLet x = 1(Yπ + gm)Re, abstracted form for Y11 is obtained

Y11 =Yp + xYu

x+ Rb Yp + xYu( )+ Yf (29)

6.2 Y12 derivation

By definition, Y12 = (i1/v2)|v1=0. Therefore by shorting thebase to the ground, the expression of Y12 can be obtained asfollows

v2 = vu + v/p (30)

vu = iuZu (31)

i1 = v11

RB + 1+ Yp + gm( )

Re

( )/ Y((

(

Y11 =1

RB + 1+ Yp + gm( )

Re

( )/ Yp +((


where iu and vu are the current passing through andthe voltage drop across Zu, respectively. On the other hand,Zu = (1/jωCu)

i1 = i1i + i1f

iu = ip − i1i = vpYp − i1i(32)

From (32) into (31), an expression of vu is obtained as follows

vu = vpYpZu − i1iZu (33)

Substituting with the value of vu into (30)

v2 = vpYpZu − i1iZu + v/p (34)

On the other hand, and by using the definition of x

v/p = −i1iRb = vp + vp Yp + gm( )

Re = xvp (35)

vp = − i1iRb

x(36)

Substituting from (35) into (36) a new expression of vu isobtained as follows

vu = − i1iRbYpZux

− i1Zu (37)

From (35), (36) and (37) into (33), yielding to

v2 = − i1iRbYpZux

− i1iZu − i1iRb

= −i1iRbYpZu + xZu + xRb

x

( )

i1i = −v2x

RbYpZu + xZu + xRb(38)

i1f = −v2Yf (39)

Finally, adding (38) and (39) results in the complete form ofY12 as follows

Y12 =−xYu

x+ RB Yp + xYu( )− Yf (40)

6.3 Y21 derivation

By definition, Y21 = (i2/v1)|v2=0. Therefore by shorting thecollector to the ground, the expression of Y21 can be

p + 1+ Yp + gm( )

Re

( )Yu))+ Yf

)

1+ Yp + gm( )

Re

( )Yu))+ Yf

(28)


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obtained as follows
i2 = i2i + i2f

i2i = gmvp + iu (41)

iu = −v/pYu (42)

v/p = xvp (43)

From (43) into (42) with the value of v/p

iu = −xvpYu (44)

Substituting into (41), yielding to

i2i = gmvp − xvpYu = vp gm − xYu( )

(45)

v1 = i1iRb + v/p = ip − iu( )

Rb + xvp (46)

Substituting with the value of iu from (43)

v1 = vpYp + xvpYu( )

Rb + xvpv1 = vp Yp + xYu

( )Rb + x

[ ] (47)

Substituting with the value of vπ into (45) an equation of i2i asa function of v1 is obtained

i2i =v1

Yp + xYu( )

Rb + x[ ] gm − xYu

( )(48)

i2f = −v1Yf (49)


Y21 =gm − xYu

x+ RB Yp + xYu( )− Yf (50)

6.4 Y22 derivation

By definition, Y22 = (i2/v2)|v1=0. Therefore by shorting thebase to the ground, the expression of Y22 can be obtained asfollows

i2 = i2i + i2f

i2i = v2YS + gmvp + iu (51)


where YS = jωCS

iu = ip + v/pRb

(52)

Substituting with the expression of v/p and ip from (35) and(21), respectively, into (51), results in

iu = vp Yp + x

Rb

( )(53)

v2 = iuZu + v/p (54)

Substituting with the expression of v/p and iu from (35) and(52), respectively, into (53), results in

v2 = vp Yp + x

Rb

( )Zu + x

[ ]

vp = v2Yp + (x/Rb)( )

Zu + x[ ] (55)

Substituting with the expression of vπ from (54) into (52),results in

iu =v2

Yp + (x/Rb)( )

Zu + x[ ] Yp + x

Rb

( )(56)

Substituting with the expression of vπ and iu from (54) and(55), respectively, into (50), results in

i2i = v2YS +v2gmRb

YpRbZu + xZu + xRb

+ v2YpRbZu + xZu + xRb

YpRb + x( )

(57)

i2f = v2Yf (58)


Y22 =xYu + RbYu gm + Yp

( )x+ Rb Yp + Yu

( ) + YS + Yf (59)


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Wideband analytical extraction technique of π-equivalent circuit model for Si/SiGe heterojunction bipolar transistor in BICMOS process