3870 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 11, NOVEMBER 2013

Lumped Models for Assessment and Optimizationof Bipolar Device RF Noise Performance

Francesco Vitale and Ramses van der Toorn, Member, IEEE

Abstract— We present a method for simulation of RF noisecharacteristics of both intrinsic and complete Si(Ge) heterojunc-tion bipolar transistors (HBT’s), aiming at support for devicedesign and optimization. RF noise at the intrinsic device level isaddressed through an equivalent circuit based on a discretizationof partial differential equations describing the transport ofminority carriers in quasi-neutral regions. Effects of nonuniformimpurity/bandgap distribution and finite velocity recombinationat the polysilicon emitter contact are accounted for. Accuracyis verified against analytical results at intrinsic device level.Assessment of noise characteristics of a complete industrial SiGeHBT demonstrates the practical relevance of nonquasi-staticeffects on noise characteristics. Exploration of the impact ofintrinsic base doping profiles on noise performance demonstratesthe potential for device optimization.

Index Terms— Bipolar transistors, circuit simulation, noise,semiconductor device modeling, semiconductor device noise.

I. INTRODUCTION

H IGH-FREQUENCY noise performance is a key in thedesign of Si/SiGe devices for microwave applications.

Noise performance optimization relies on predictive noise sim-ulation techniques, so as to minimize the dependency on costlynoise measurements. Aiming at support for such device designand optimization, this paper presents and explores a method topredictively simulate RF noise characteristics of both intrinsicand complete Si(Ge) heterojunction bipolar transistors (HBTs).The method is founded on a mathematical mapping of thedrift-diffusion equations onto a lumped element circuit. It canefficiently compute stationary and small amplitude numericalsolutions of the time-dependent drift diffusion equation, takingtechnological parameters such as doping and bandgap profilesand mobility models as input, while needing no more than anelectrical circuit simulator. From a theoretical point of view, aswe shall discuss, this already implicitly solves the problem ofRF noise simulation, including the so-called nonquasi-static(NQS) effects. Nevertheless, we shall demonstrate how theapproach can be extended to explicitly simulate NQS RF noisecharacteristics.

Noise characterization forms the subject of an extensivebranch of the bipolar transistor literature. Published methodsinclude derivation of the noise parameters from simple small-signal equivalent circuits [1]–[4], as well as extraction of

Manuscript received July 5, 2013; revised August 22, 2013; acceptedSeptember 5, 2013. Date of publication September 20, 2013; date of currentversion October 18, 2013. The review of this paper was arranged by EditorDr. J. D. Cressler.

The authors are with the Department of Electrical Engineering, Mathematicsand Computer Science, Delft University of Technology, Delft 2628 CT,The Netherlands (e-mail: f.vitale@tudelft.nl; r.vanderToorn@tudelft.nl).

Digital Object Identifier 10.1109/TED.2013.2281237

noise parameters directly from the measured [5] or simulated[6] admittance parameters of the whole transistor. As dis-cussed in [7], however, in practice, the use of over-simplifiednoise models for the intrinsic transistor can lead to self-consistency problems and to inaccurate or unphysical results.The noise model proposed in [7] aimed at more accuracy,at the price of introducing a dedicated noise parameter, thenoise delay time, the physical meaning of which is a subjectof critical discussion [8]. As the noise model in [7] is notrooted in the y-parameters of the device, this approach lacksguaranteed consistency between the noise and the small-signal models, which, in turn, still can result in inaccuracy inpredicted transistor noise performance. The device optimiza-tion oriented work [9] deployed a detailed equivalent circuitfor the extrinsic transistor, while the intrinsic transistor wasmodeled by the conventional quasi-static (QS) small-signalcircuit; two simple uncorrelated shot noise current sourceswere used to describe the noise of the intrinsic transistor.Even though this is consistent with the small-signal modelof the intrinsic transistor, such model ceases to be valid athigher frequencies [10], [11]. An improved version of thismethod [12] replaced the intrinsic small-signal circuit by a1-D Technology Computer Aided Design device simulation.Nevertheless, still a simplified uncorrelated noise model forthe intrinsic transistor was used, inconsistent with the small-signal model of the intrinsic transistor.

The classical studies [13], [14] offer a thorough foundationfor the theory of RF noise of intrinsic bipolar transistors.The well-known Van Vliet formula [14] establishes a linearrelationship between the power spectral densities (PSDs) ofthe noisy port currents and the admittance parameters of theintrinsic device. This shows that the noise characteristics areimplicitly established by a sufficiently detailed and accurateac small signal model of the intrinsic device. Pioneeringtheoretical work in the direction of explicit noise simulationwas published in [15], which derived an equivalent circuitof an intrinsic transistor from a discretization of the small-signal transport equations. All the physical noise sources wereaccounted for by formally assigning Nyquist thermal noise toeach resistor in the network. The presence of negative resis-tances in the network, however, then required negative spectraldensities. This limits the implementability to contexts thatsupport negative PSDs, excluding, e.g., the modern standardmodeling language Verilog-A.

Following a line of attack inspired by [15] and [16], butaiming for implementability in present day’s standard com-mercial circuit simulation contexts, in this paper, we developa practical, lumped element model for noise simulation of

0018-9383 © 2013 IEEE

VITALE AND VAN DER TOORN: LUMPED MODELS FOR ASSESSMENT AND OPTIMIZATION 3871

bipolar transistors, focusing on noise performance of RFHBTs. Our model will be systematically derived from adiscretization of the partial differential equations describingthe transport of minority carriers in quasi-neutral (QN) regions.This results in algebraic equations, which can be representedas an equivalent lumped network [16]–[19]. The effect of abuilt-in drift field and finite surface recombination velocity atthe polysilicon emitter contact is considered. The effect of theadditional delay in the base–collector (BC) space charge region[20] is outside the scope of this paper; [8] showed this effectto be negligible for SiGe HBTs, although it may be relevantfor III–V HBTs.

In Section II, our lumped network is introduced along withthe local noise sources. In Section III, analytical expressionsof the intrinsic admittance parameters for an ideal 1-D SiGeHBT are derived, which serve as a reference in Section Vfor the verification of the model at the intrinsic device level.The lumped network is then supplemented with an extrinsicnetwork to predict the noise characteristics of an industrialSiGe HBT; in this practical application, we also demonstratethe significance to industry of simulation of NQS intrinsictransistor noise. Finally, we explicitly demonstrate that ourapproach captures the impact of intrinsic base doping profileson transistor noise performance, hence that our approach cansupport device optimization.

II. MODEL DESCRIPTION

A. Equivalent Network

Consider a volume element V of an n-type semiconductorregion with inhomogeneous doping concentration in whichholes have been injected from a p-type contact. Assume thatthe injection levels are low compared with majority concen-trations and that the bulk recombination rate is proportionalto the excess minority-carrier concentration. A subdivision ofspace into small volume elements of area A and width �xthen leads to the following discretization of the differentialequations describing the transport of holes in the n-region:

i p(x) = − ADp

�x[qp′(x +�x)−qp′(x)]+ Aμp Eqp′(x)

(1)

i p(x + �x) = i p(x) − A�x∂qp′(x)

∂ t− A�x

τpqp′(x) (2)

where p′ is the excess hole concentration, i p is the holecurrent, Dp , μp , and τp , are the diffusion coefficient, themobility, and the lifetime of holes, respectively, and E isthe internal drift field determined by inhomogeneous impuritydistribution and/or by nonuniform bandgap.

With the introduction of the quantities listed in Table I, itis readily verified that (1) and (2) can be mapped into theequivalent network shown in Fig. 1(a).

Arbitrarily accurate numerical solutions of the drift-diffusion (DD) equations can be achieved by cascading anumber N of networks of the type shown in Fig. 1, alongwith the boundary conditions

qp′(0) = qp0n

[exp

(Va

nVT

)− 1

](3)

qp′(W ) = 0 (4)

TABLE I

ANALOGOUS QUANTITIES FOR THE LUMPED NETWORK AND

MINORITY-CARRIER TRANSPORT EQUATIONS [16], [21]

Fig. 1. Equivalent network of a section �x of a QN region. (a) Noiseless.(b) With noise sources.

with Va the applied voltage and n the nonideality factor.Condition (4) corresponds to a short circuit connected atthe end of the cascaded two ports. For finite recombinationvelocity at the metal or polysilicon contact (or for finite exitvelocity of minority carriers at the BC junction of bipolartransistors), the boundary condition (4) becomes

i p(W ) = q ASp p′(W ). (5)

In the equivalent network, condition (5) corresponds to aconductance (combinance) Hcs = ASp connected at the outputterminals of the cascaded two ports.

B. Noise Sources

Microscopic noise sources within semiconductor devices arerecognized to be diffusion and recombination-generation noisesources [22]. In the lumped model, they can be accounted forby adding a current noise source in parallel to each Hd andHc, respectively, as shown in Fig. 1(b). Their mean squarevalue in the i th cell of the cascaded network can be written as

ı2dx(i) = 2q2[ p′(xi−1) + p′(xi ) + 2 pn0 ]Hdd f (6)

ı2rx(i) = 2q2[ p′(xi ) + 2 pn0 ]Hcd f (7)

where (6) represents a numerically more accurate implemen-tation of the counterpart given in [18]. When finite velocity

3872 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 11, NOVEMBER 2013

recombination is considered, an additional current noise sourcemust be added in parallel to Hcs, which accounts for fluctu-ations of the recombination at the contact. The mean squarevalue of this noise source can be written as

ı2rs = 2q2 p′(W )Hcsd f. (8)

The local noise sources (6)–(8) are uncorrelated, hence theirimplementation in a circuit network is trivial. The transfer ofthe local noise sources is evaluated by the circuit simulatorthrough the calculation of the transfer function between thelocation of the internal noise source and the specified boundaryof the QN regions [23]. The same transfer is used by the circuitsimulator to compute the admittance parameters: hence noiseand small-signal parameters are evaluated in a self-consistentand unified manner.

III. SMALL-SIGNAL ANALYTICAL SOLUTIONS

In the limit of N → ∞ and �x → 0, the lumped networkreturns by constructing the exact solution of the DD equations.Closed-form solutions under small-signal operation can bederived if spatially constant values for τp and μp are assumed.Assuming an exponentially varying doping concentration and auniform badgap in the base region (F = const.), the common-base input and transadmittance of a bipolar transistor can becalculated following the method described in [24] yielding:

ycb11b = g0b

(η/2 + ζη coth ζη

)(9)

ycb21 = −g0beη/2ζη csch ζη (10)

where ζη = √(η/2)2 + 2 jω/ω0b, ω0b = 2Dn/W 2

B , η is thedrift-field factor, and

g0b = q ADnn0p

WB VTexp

(VBE

nbVT

).

Similar to the notation used in [25] and [26], ycb11b represents

the contribution of the base region to the total input common-base (cb) admittance, ycb

11. The common-emitter (ce) input andtransadmittance follow directly from (9) and (10): yce

11b =ycb

11b + ycb21, yce

21 = −ycb21. The emitter region is included by

adding its contribution y11e to the input admittance. Assuminga constant doping profile in the emitter region and a finiterecombination velocity at the contact (Hcs �= 0), one readilyobtains from the cascaded network

y11e = g0eζ1 + νspζ tanh ζ

νspζ + tanh ζ(11)

with νsp = Dp/Sp WE , ζ = √2 jω/ω0e and

g0e = q ADp p0n

WE VTexp

(VBE

neVT

).

Common-emitter noise PSDs [27] then follows:

Sib = 4kT Re(yce11b + y11e) − 2q IB (12)

Sic = 2q IC (13)

Sibic∗ = 2kT(y∗ce

21 − gm0)

(14)

where IB and IC are the dc BC currents, respectively, andgm0 is the low-frequency transconductance. In the aboveexpressions, y12 and y22 have been omitted as they are usuallynegligible compared with y11 and y21 and do not appreciablyaffect the transistor noise parameters.

IV. MODEL IMPLEMENTATION

In our exploration of the approach, we have made use ofthe following trial functions for the doping profile in the baseregion [28]:

N(x) = Np exp

[−

(x

WB

)α

ln

(Np

Nc

)](15)

N(x) = Np exp

[−

(x − x p

WB − x p

)2

ln

(Np

Nc

)](16)

where Nc represents the doping concentration in the baseregion at the collector side and α in (15) is a parameter forthe base doping profile. Once the doping profiles are defined,the minority carrier mobility can be calculated from usualempirical models (see [29] and references therein)

μp,n = μ1p,n + μ0p,n

1 + (N(x)/N0p,n

)γp,n(17)

where μ1p = 130 cm2/V, μ0p = 370 cm2/V, N0p = 8 ×1017 cm−3, γp = 1.25, μ1n = 232 cm2/V, μ0n =1180 cm2/V, N0n = 8 × 1016 cm−3, γn = 0.9. The circuitelements Hd at each section of the QN region are calculatedusing the definition listed in Table I. We neglected recombi-nation, hence Hc = 0.

Elements F depend on the mobility and the internal driftfield E , which is given in [30]1

E(x) = VT

(1

N(x)

d N (x)

dx− 1

n2i (x)

dn2i (x)

dx

). (18)

The second term in (18) is null for uniform bandgap. For agraded Ge profile throughout the base, the bandgap varieslinearly and ni (x) can be calculated according to [31]. Thefirst term in (18) is constant for exponentially varying dopingprofiles (α = 1) and null for homogenous doping concentra-tion (α → ∞). In all other cases, it can be derived with theapproximation of exponential impurity concentration betweenthe adjacent nodes such that its contribution is constant at eachsection

1

N(x)

d N(x)

dx≈ VT

�xln

N(xi )

N(xi+1)= const.

Finally, the noise sources are added to the network. Theirevaluation requires the knowledge of node voltages (i.e., ofthe excess minority carriers within the QN region). To avoid apreliminary dc simulation, the excess minority charge can becalculated from the doping profile (e.g., [28] and [30])2

n(x) = Jnn2i (x)

q N(x)

∫ WB

x

N(x)

Dn(x)n2i (x)

dx (19)

where

Jn = qn2i0 exp

(V0

nbVT

) / ∫ WB

0

N(x)n2i0

Dn(x)n2i (x)

dx (20)

is the electron current density and ni0 is the intrinsic carrierconcentration in lightly doped silicon. In Fig. 2, the analytical

1Signs must be exchanged in (18) when a drift field acting on minorityholes, instead of electrons, is considered.

2Expressions (19) and (20) need an extra term when the boundary condition(5) is considered instead of (4).

VITALE AND VAN DER TOORN: LUMPED MODELS FOR ASSESSMENT AND OPTIMIZATION 3873

Fig. 2. Three different base doping profiles (left-hand side) and the corre-sponding normalized excess minority charge distribution (right-hand side) foran applied junction voltage V0 = 0.9 V. Solid and dashed lines: analyticalcurves calculated from (19). Symbols: numerical values extracted from a dcsimulation of an equivalent network consisting of N = 100 segments.

minority carrier distribution (19) is plotted against the numer-ical values extracted from a dc simulation of the equivalentnetwork, for three different doping profiles. The demonstratedagreement justifies the use of (19) in the place of a preliminarydc simulation for the calculation of the local noise sources.

V. RESULTS AND DISCUSSION

As a first exploration of our approach, an ideal 1-Dn-p-n SiGe HBT in common-emitter configuration was ana-lyzed with the specifications listed in Table II. Heavy dopingeffects have been accounted for in the emitter region [32].For a direct comparison with analytical results (12)–(14), anexponentially varying acceptor concentration was assumed forthe base region and a homogeneous impurity concentration forthe emitter as well as position-independent device parameters,such as mobility and lifetime. Excellent agreement between thetheoretical and simulated admittance parameters (not showndue to lack of space) and noise characteristics is achieved, asshown in Fig. 3.

To explore the practical uses of the lumped network anddemonstrate the relevance of intrinsic NQS effects and impliedcorrelated noise, we have supplemented our model with anetwork representing the parasitics of a planar HBT. In Fig. 5,we confront the resulting model (Fig. 4) of a complete HBTwith noise measurements taken on an industrial 0.25-μmSiGe-BiCMOS process [33], featuring fT ,peak = 70 GHz.The bias points have been selected so as to satisfy therequirement of lowest Fmin (VBE = 0.82 V) or highest cutofffrequency (VBE = 0.9 V). The technological parameters shownin Table II have been modified to resemble those of the realdevice considered here, while the emitter area, the base (WB )and the emitter width (WE ) are kept the same. The extractionof the parasitics surrounding the intrinsic device have beenperformed based on the MEXTRAM library parameter set ofthe selected device, further optimized directly on ac measure-ments (e.g., [34]).

The external terminal noise characteristics follow directlyfrom the two-port method described in [35]. To quantifythe impact of the internal correlation, Fig. 5 shows ourresults considering NQS and implied correlated noise effects(solid curves) together with the classical Spice model (dashedcurves, [5]), which neglects both noise correlation and thefrequency dependency of Sib. At medium bias conditions

TABLE II

DEVICE PARAMETERS OF 1-D SiGe HBT WITH EMITTER AREA

A = 0.5 μm × 20.3 μm(a)

Fig. 3. Intrinsic base and collector current PSDs and their correlationversus the normalized frequency ω(τb + τe) at VBE = 0.9 V for anideal SiGe-HBT with technological parameters listed in Table II. Symbols:simulated data. Solid lines: analytical curves calculated from (12)–(14). Here,τb = W2

B (η − 1 + e−η)/Dnη2, τe = W2E /2Dp + WE /Sp , and η = 3.

( fT fT ,peak), not shown here due to lack of space, thenoise behavior of the transistor is dominated by the timeconstants of parasitic resistances and depletion capacitances sothat the details of the noise in the intrinsic device are actuallynot manifest at terminal level and the traditional spice modelappears to suffice. Under high-speed bias conditions, however,(approaching fT ,peak) the contribution of the diffusion chargesto the total transit time is significant and intrinsic effects arestriking. Detailed modeling of the intrinsic device is clearlyneeded in this regime, as is manifest from Fig. 5.

The lumped network is a suitable tool for the optimization oftechnological parameters, like base doping profile, with respectto noise performance. A similar exercise was done in [36].However, in [36], a conventional noise analysis technique wasused, which can bring to inconsistent results, as observed laterin [7].

Here, we have considered three different base doping pro-files, shown in Fig. 2: the exponential, the Gaussian, andthe epitaxial Gaussian profile [see (15) and (16)]. To makesuch analysis meaningful, the sheet resistance must be keptconstant: this condition ensures that the base resistance and

3874 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 11, NOVEMBER 2013

Fig. 4. Intrinsic model supplemented with parasitic base and emitterresistances (RBx, RBi and RE ) and intrinsic as well as extrinsic depletioncapacitances (CBE, CBCi, CBCx, and XCBCx).

Fig. 5. Simulated (solid and dashed lines) versus measured noise FOMsas function of the frequency at fixed VCE = 2 V and VBE = 0.9 V(JC = 0.6 mA/μm2). Solid lines: simulated data from the presented equiva-lent network. Dashed lines: noise simulation from a QS small-signal circuitfor which the SPICE noise model has been applied.

the base Gummel number do not change substantially when adifferent base doping profile is considered. From the expres-sion of the depletion capacitance of a linearly graded junction[37], it is possible to demonstrate that also the depletioncapacitances do not change appreciably; hence, the sameextrinsic network can be used to analyze the impact of differentbase doping profiles on the noise parameters of the wholedevice. The result of the analysis is shown in Fig. 6, wherealso the simulation curves calibrated with measured data areshown (thick solid line). The constancy of the sheet resistancethroughout the series of experiments is mostly reflected bythe constancy of the noise resistance Rn in the low-frequencylimit. The epitaxial Gaussian profile with the peak dopingconcentration deep in the base (x p = 20 nm) shows lesscorrelation between the BC noise currents than the otherprofiles. This is due to the presence of two internal electricdrift fields in the base region: the first one (from x = 0 tox = x p) decelerates the minority carriers injected in the baseregion, whereas the second one (from x = x p to x = WB )accelerates and pushes them toward the collector. Such electricfields facilitate the decoupling of the emitter and collectorsides of the base region, attenuating the correlation effect. As a

Fig. 6. Device noise parameters versus frequency for several differentintrinsic base doping profiles at VBE = 0.9 V.

consequence, Fmin and Gopt are clearly higher when the peakof the doping concentration resides deep into the base. Forthe same reason, Bopt is lower due to a reduced correlationeffect. As a conclusion, a strong electric field accelerating theinjected carriers toward the collector side is desirable sinceit enhances the correlation, improving both the noise and thespeed performances of the transistor. This is certainly achievedwhen the peak of the base doping resides as close as possibleto the emitter edge, like for the exponential or the Gaussiandoping profile.

VI. CONCLUSION

A method for dedicated high-frequency noise simulationsof bipolar devices was presented. The approach is based on alumped network, which returns by constructing an arbitrarilyaccurate solution of the Langevin drift-diffusion equations inthe QN regions of bipolar devices. This enables assessmentof noise characteristics and admittance parameters in a unifiedand intrinsically consistent manner. Although in [38] and [39],it was shown that higher order models, like hydrodynamic,yields somewhat more reliable results under quasi-ballistictransport conditions, the DD approach can be still consideredvalid (and simpler) for a large variety of cases, includingsubmicrometer devices [40].

The lumped network has been verified against analyticalresults at the intrinsic device level, in special cases in whichanalytical results can be achieved. Furthermore, we havesupplemented the model with an RC network representingthe parasitics of an industrial SiGe HBT. Simulated datawere compared with the measured ones at two different biaspoints. At medium bias conditions, intrinsic NQS effects,as correlation, are mostly screened by the extrinsic networkand, e.g., the traditional Spice noise model appears to suffice.At higher biases, the contribution of the diffusion charges tothe dynamic performances of the whole device are significantand intrinsic effects are manifest.

The proposed lumped model facilitates the analysis of thenoise performances of a bipolar device as function of its

VITALE AND VAN DER TOORN: LUMPED MODELS FOR ASSESSMENT AND OPTIMIZATION 3875

technological parameters (e.g., doping profiles), making thisapproach suitable for device analysis and optimization. Toillustrate these capabilities, we have analyzed the influenceof three different base doping profiles on the high-frequencynoise parameters. We have observed that to achieve low-noise performances, the peak of the base doping concentrationshould reside as close as possible to the base–emitter junction.This is achieved at best with an exponential doping profilecharacterized by a strong internal drift field accelerating theminority carriers toward the collector contact.

ACKNOWLEDGMENT

The authors would like to thank NXP semiconductors fornoise data and in-kind and financial support.

REFERENCES

[1] G. Hanson and A. Van der Ziel, “Shot noise in transistors,” Proc. IRE,vol. 45, no. 11, pp. 957–963, Nov. 1957.

[2] E. Nielsen, “Behavior of noise figure in junction transistors,” Proc. IRE,vol. 45, no. 7, pp. 957–963, Jul. 1957.

[3] R. Hawkins, “Limitations of nielson’s and related noise equationsapplied to microwave bipolar transistors, and a new expression for thefrequency and current dependent noise figure,” Solid-State Electron.,vol. 20, no. 3, pp. 191–196, 1977.

[4] L. Escotte, J.-P. Roux, R. Plana, J. Graffeuil, and A. Gruhle, “Noisemodeling of microwave heterojunction bipolar transistors,” IEEE Trans.Electron Devices, vol. 42, no. 5, pp. 883–889, May 1995.

[5] S. Voinigescu, M. Maliepaard, J. Showell, G. Babcock, D. Marchesan,M. Schroter, P. Schvan, and L. Harame, “A scalable high-frequency noisemodel for bipolar transistors with application to optimal transistor sizingfor low-noise amplifier design,” IEEE J. Solid-State Circuits, vol. 32,no. 9, pp. 1430–1438, Sep. 1997.

[6] F. Herzel and B. Heinemann, “High-frequency noise analysis of Si/SiGeheterojunction bipolar transistors,” Int. J. Electron., vol. 81, no. 1,pp. 37–48, 1996.

[7] G. Niu, J. Cressler, S. Zhang, W. Ansley, C. Webster, and D. Harame,“A unified approach to RF and microwave noise parameter modelingin bipolar transistors,” IEEE Trans. Electron Devices, vol. 48, no. 11,pp. 2568–2574, Nov. 2001.

[8] J. Herricht, P. Sakalas, M. Ramonas, M. Schroter, C. Jungemann, andA. Mukherjee, “Systematic compact modeling of correlated noise inbipolar transistors,” IEEE Trans. Microw. Theory Tech., vol. 60, no. 11,pp. 3403–3412, Nov. 2012.

[9] T. Vanhoucke, J. J. T. M. Donkers, G. A. M. Hurkx, P. H. C. Magnee,R. van Dalen, J. H. Egbergs, and D. B. M. Klaassen, “Layout and spaceroptimization for high-frequency low-noise performance in HBT’s,” inProc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Oct. 2010,pp. 53–56.

[10] J. Paasschens, R. Havens, and L. Tiemeijer, “Modeling the correlationin high-frequency noise of (hetero-junction) bipolar transistors usingcharge-partitioning,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol.Meeting, Oct. 2003, pp. 221–224.

[11] P. Sakalas, J. Herricht, M. Ramonas, and M. Schroter, “Noise model-ing of advanced technology high speed SiGe HBTs,” in Proc. IEEEBipolar/BiCMOS Circuits Technol. Meeting, Oct. 2010, pp. 169–172.

[12] T. Vanhoucke, D. Klaassen, H. Mertens, J. Donkers, G. Hurkx, H. Huiz-ing, P. Magnee, E. Hijzen, R. van Dalen, E. Gridelet, and J. Slotboom,“Virtual technology for RF process and device development,” in Proc.IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Oct. 2011, pp. 37–44.

[13] A. Van der Ziel, “Theory of shot noise in junction diodes and junctiontransistors,” Proc. IRE, vol. 43, no. 11, pp. 1639–1646, Nov. 1955.

[14] K. Van Vliet, “General transport theory of noise in PN junction-likedevices–I. Three-dimensional green’s function formulation,” Solid-StateElectron., vol. 15, no. 10, pp. 1033–1053, 1972.

[15] D. Polder and A. Baelde, “Theory of noise of transistor-like devices,”Solid-State Electron., vol. 6, pp. 103–110, Mar. 1963.

[16] J. Linvill, “Lumped models of transistors and diodes,” Proc. IRE, vol. 46,pp. 1141–1152, Jun. 1958.

[17] G. Liebmann, “Solution of partial differential equations with a resistancenetwork analogue,” Brit. J. Appl. Phys., vol. 1, pp. 92–103, Apr. 1950.

[18] R. Beatie, “A lumped model analysis of noise in semiconductor devices,”IRE Trans. Electron Devices, vol. 6, pp. 133–140, Apr. 1959.

[19] G. Baldwin and B. Wooley, “Computer-aided analysis of solid-stateanalog delay lines using conventional circuit analysis programs,” IEEEJ. Solid-State Circuits, vol. 9, no. 1, pp. 20–26, Feb. 1974.

[20] K. Xia, G. Niu, and Z. Xu, “A new approach to implementing high-frequency correlated noise for bipolar transistors compact modeling,”IEEE Trans. Electron Devices, vol. 59, no. 2, pp. 302–308, Feb. 2012.

[21] D. Peterson, “Noise performance of transistors,” IRE Trans. ElectronDevices, vol. 9, pp. 296–303, May 1962.

[22] K. Van Vliet, “Noise sources in transport equations associated withambipolar diffusion and Shockley-read recombination,” Solid-State Elec-tron., vol. 13, no. 5, pp. 649–657, 1970.

[23] R. Rohrer, L. Nagel, R. Meyer, and L. Weber, “Computationally efficientelectronic-circuit noise calculations,” IEEE J. Solid-State Circuits, vol. 6,no. 4, pp. 201–213, Apr. 1971.

[24] J. T. Winkel, “Transmission-line analogue of a drift transistor,” Phil.Res. Rep., vol. 14, pp. 52–64, Jan. 1959.

[25] J. Hamel, “An accurate charge-control approach for modeling excessphase shift in the base region of bipolar transistors,” IEEE Trans.Electron Devices, vol. 43, no. 7, pp. 1092–1098, Jul. 1996.

[26] N. Rinaldi, H. de Graaf, and J. Tauritz, “Analysis and modeling of small-signal bipolar transistor operation at arbitrary injection levels,” IEEETrans. Electron Devices, vol. 45, no. 8, pp. 1817–1825, Aug. 1998.

[27] A. Van der Ziel and G. Bosman, “Accurate expression for the noisetemperature of commone emitter microwave transitors,” IEEE Trans.Electron Devices, vol. 31, no. 9, pp. 1280–1283, Sep. 1984.

[28] K. Suzuki, “Optimum base doping profile for minimum base transittime,” IEEE Trans. Electron Devices, vol. 38, no. 9, pp. 2128–2133,Sep. 1991.

[29] P. Ashburn, SiGe Heterojunction Bipolar Transistors. New York, NY,USA: Wiley, 2003.

[30] R. van Overstraeten, H. DeMan, and R. Mertens, “Transport equationsin heavy doped silicon,” IEEE Trans. Electron Devices, vol. 20, no. 3,pp. 290–298, Mar. 1973.

[31] H. Kroemer, “Two integral relations pertaining to the electron transportthrough a bipolar transistor with a nonuniform energy gap in the baseregion,” Solid-State Electron., vol. 28, no. 11, pp. 1101–1103, Feb. 1985.

[32] J. Slotboom and H. de Graaf, “Measurements of bandgap narrow-ing in Si bipolar transistors,” Solid-State Electron., vol. 19, no. 10,pp. 857–862, Oct. 1976.

[33] P. Deixler, R. Colclaser, D. Bower, N. Bell, W. de Boer, D. Szmyd,S. Bardys, W. Wilbanks, P. Barre, M. V. Houdt, J. Paasschens,H. Veenstra, E. V .D. Heijden, J. Donkers, and J. Slotboom, “QUBiC4G:An fT / fmax = 70/100 GHz 0.25 μm low power SiGe-BiCMOSproduction technology with high quality passives for 12.5 Gb/s opticalnetworking and emerging wireless applications up to 20 GHz,” inProc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Feb. 2002,pp. 201–204.

[34] F. Vitale, R. Pijper, and R. van der Toorn, “Base resistance distribution inbipolar transistors: Relevance to compact noise modeling and extractionfrom admittance parameters.” in Proc. IEEE Bipolar/BiCMOS CircuitsTechnol. Meeting, Sep. 2010, pp. 161–164.

[35] H. Hillbrand, “An efficient method for computer aided noise analysisof linear amplifier networks,” IEEE Trans. Circuits Syst., vol. 23, no. 4,pp. 235–238, Apr. 1976.

[36] G. Niu, S. Zhang, J. Cressler, A. Joseph, J. Fairbanks, L. Larson,C. Webster, W. Ansley, and D. Harame, “Noise modeling and SiGe pofiledesign tradeoffs for RF applications,” IEEE Trans. Electron Devices,vol. 47, no. 11, pp. 2037–2044, Nov. 2000.

[37] S. Sze, Physics of Semiconductor Devices. New York, NY, USA: Wiley,1969.

[38] C. Jungemann, B. Neinüs, and B. Meinerzhagen, “Hierarchical 2-D DDand HD noise simulations of Si and SiGe devices - Part I: Theory,”IEEE Trans. Electron Devices, vol. 49, no. 7, pp. 1250–1257, Jul. 2002.

[39] C. Jungemann, B. Neinüs, and B. Meinerzhagen, “Hierarchical 2-DDD and HD noise simulations of Si and SiGe devices - Part II:Results,” IEEE Trans. Electron Devices, vol. 49, no. 7, pp. 1258–1264,Jul. 2002.

[40] F. Bonani and G. Ghione, Noise in Semiconductor Devices,Modeling and Simulation. New York, NY, USA: Springer-Verlag,2001.

3876 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 11, NOVEMBER 2013

Francesco Vitale is currently pursuing the Ph.D.degree in electronics from the Delft University ofTechnology, Delft, The Netherlands.

His current research interests include RF noisemodeling of SiGe HBTs.

Ramses van der Toorn (M’06) received the Ph.D.degree cum laude in geophysical fluid dynamicsfrom Utrecht University, Utrecht, The Netherlands,in 1997.

He is a Professor with Delft University, Delft, TheNetherlands.