Numerical analysis of abrupt heterojunction bipolar transistors

international journal of numerical modelling: electronic networks, devices and fieldsInt. J. Numer. Model.11, 221–229 (1998)

NUMERICAL ANALYSIS OF ABRUPT HETEROJUNCTIONBIPOLAR TRANSISTORS

antonio j. garcia-loureiro,1 juan m. lopez-gonzalez,2 tomas f. pena1,* and lluis prat2

1Departamento de Electrońica y Computacioń, Univ. Santiago de Compostela, Campus, Sur, 15706 Santiago deCompostela, Spain

2Departament d’Enginyeria Electrońica, Univ. Politecnica de Catalunya, Campus Nord, c/Jordi Girona, 1–3, 08034Barcelona, Spain

SUMMARYThis paper presents a physical–mathematical model for abrupt heterojunction transistors and its solution usingnumerical methods with application to InP/InGaAs HBTs. The physical model is based on the combinationof the drift–diffusion transport model in the bulk with thermionic emission and tunnelling transmission throughthe emitter–base interface. Fermi–Dirac statistics and bandgap narrowing distribution between the valence andconduction bands are considered in the model. A compact formulation is used that makes it easy to take intoaccount other effects such as the non-parabolic nature of the bands or the presence of various subbands inthe conduction process. The simulator has been implemented for distributed memory multicomputers, makinguse of the MPI message-passing standard library. In order to accelerate the solution process of the linearsystem, iterative methods with parallel incomplete factorization-based preconditioners have been used. 1998John Wiley & Sons, Ltd.

1. INTRODUCTION

Heterojunction bipolar transistors (HBTs) are nowadays an active area of research because of theinterest in their high-speed electronic circuit applications. Special attention is paid to InP/InGaAsHBTs, which have already attainedfT and fmax frequencies above 200 GHz.1 These transistors,with an abrupt heterojunction, exhibit a discontinuity in the energy bands at the emitter–base interface.2

Development of numerical models for HBTs is essential to better understand their physicalbehaviour and for design optimization. Conventional simulators of homojunction and gradualheterojunction bipolar transistors are based on drift–diffusion carrier transport.3 This model is notvalid for abrupt HBTs since the current through the energy spike formed at the emitter–baseinterface is controlled by thermionic emission and tunnelling transmission.4,5 HBTs have a veryhighly doped base region in order to improve their high-frequency performance; it is thereforenecessary to include the effects of high doping levels in the physical models of these devices.One of these important effects is the bandgap narrowing (BGN), which produces a singularcharacteristic in abrupt HBTs compared to other devices: different distributions of the totalnarrowing will determine the amount of discontinuity assigned to the conduction band and to thevalence band, therefore affecting the currents through the device.6,2

This paper presents a numerical analysis of abrupt heterojunction bipolar transistors and itsapplication to the analysis of InP/InGaAs HBTs. Section II presents the physical–mathematicalmodel of the device, including Fermi–Dirac statistics, the variation of the effective state densitiesin the bands and a bandgap narrowing model with its distribution between conduction and valencebands. The model can be generalized to include the non-parabolic nature of the bands and theexistence of more than one band in conduction processes. Section III presents the basic character-istics of the numerical implementation of the physical model described in the previous Section.

* Correspondence to: Tomas F. Pena, Department of Electronics and Computing, University of Santiago deCompostela, 15706 Santiago de Compostela, Spain. E-mail: tomasKdec.usc.es

Contract/grant sponsor: CICYT; Contract/grant numbers: TIC96-1125-C03 and TIC96-1058Contract/grant sponsor: TRACS

CCC 0894–3370/98/040221–09$17.50 Received 20 November 1997 1998 John Wiley & Sons, Ltd. Revised 23 February 1998

222 a. j. garcia-loureiro et al.

Several numerical methods have been studied in order to solve the linear system. The best choicewas considered to be the Bi-CGSTAB method. Two preconditioners based on incomplete factoriza-tions have been employed: a global version and a new one organized in blocks which tend toobtain better performance in the parallel implementation. Section IV presents the results obtainedin the simulation of InP/InGaAs HBTs.

2. PHYSICAL MODEL

The object of the physical–mathematical model is to relate electrical characteristics of the devicewith its physical and geometrical parameters. The basic equations to be solved are Poisson’sequation and electron and hole continuity, in a stationary state:

div(e=c) 5 q(p 2 n 1 N1D 2 N2

A) (1)

div(Jn) 5 qR (2)

div(Jp) 5 2qR (3)

wherec is the electrostatic potential,q is the electronic charge,e is the dielectric constant of thematerial, n and p are the electron and hole densities,N1

D and N1A are the doping effective

concentration andJn and Jp are the electron and hole current densities, respectively. The termRrepresents the volume recombination term, taking into account Schockley–Read–Hall, Auger andband-to-band recombination mechanisms.7 Any specific charge and recombination in the interfacecan be included.

The concentrations and currents of electrons and holes depend on the structure of the energybands of the HBT. Figure 1 shows the vacuum levelEo, the minimum level of the conductionbandEc and the maximum level of the valence bandEv of an emitter–base abrupt heterojunction.In this figure DEc and DEv are the energy discontinuities of theEc and Ev levels at the interface.

The energy band diagram of the device in equilibrium or in polarization can be obtained bytaking as the origin of energies the vacuum level of a reference material in thermal equilibrium,and bearing in mind the fact that the Fermi level in equilibrium is constant. In this paper thevacuum level of an intrinsic semiconductor has been taken as a reference level. The energy levelsof this reference material can be calculated as follows:

Ec,ref 5 2qxref, Ev,ref 5 2qxref 2 Eg,ref (4)

The Fermi level in equilibrium in the device is the Fermi level of the reference material; therefore

Ef,eq 5 Ei,ref 5 2qxref 2 kT ln S Nc,ref

ni,refD (5)

The quasi-Fermi potentialsfn and fp are defined as:

Figure 1. Energy band diagram in equilibrium

1998 John Wiley & Sons, Ltd. Int. J. Numer. Model.11, 221–229 (1998)

223numerical analysis of abrupt hbts

qfn 5 2 (Efn 2 Ef,eq) 5 Ei,ref 2 Efn (6)

qfp 5 2 (Efp 2 Ef,eq) 5 Ei,ref 2 Efp (7)

The energy bands at any point of the decice can be obtained from the actual electrostatic potential,c, the affinity x and the gap,Eg, of the material by means of the following equations:

Ec 5 2qc 2 qx 5 2 qc 2 qxo 2 DEbgnc (8)

Ev 5 2 qc 2 qx 2 Eg 5 2 qc 2 qxo 2 Ego 1 DEbgnv (9)

where on the right-hand side of the equations,qxo(x) is the affinity of the material for low dopinglevels andEgo is the bandgap, also for low doping levels. Equations (8) and (9) take into accountthe variations due to changes in the molar fraction in the case of compound semiconductors. Theterms DEbgn

c and DEbgnv represent the variation in theEc and Ev levels due to BGN. The total

BGN, DEbgng , is the sum ofDEbgn

c and DEbgnv .

The distribution of bandgap narrowing between the conduction and valence bands is calculatedaccording to:

DEbgnc 5 C1 S N

1018 D1/a

1 C2 S N1018 D1/2

(10)

DEbgnv 5 C3 S N

1018D1/b

1 C4 S N1018 D1/2

(11)

These expressions were obtained from the bandgap narrowing model proposed by Jain andRoulston,6 where N is the effective doping level anda, b, C1, C2, C3 and C4 are constants thatdepend on the material and type of doping.2

In all the regions of the device, except at the emitter–base interface, the carrier currents arecontrolled by drift–diffusion mechanisms, and may be expressed by:

Jn 5 2 qmnn=(fn), Jp 5 2 qmpp=(fp) (12)

where mn and mp are the mobilities of electrons and holes.The charge transport through an abrupt discontinuity is based on Grinberg’s model,5 which

includes tunnelling transmission through the spike and thermionic emission over it. The electroncurrent density through the spike in the conduction band (Figure 1) is expressed as

Jn 5 2qvneff S NcB

NcE

n(x1j ) 2 n(x2

j )e2

DEc

kT D (13)

where DEc is the discontinuity in levelEc in the heterojunction interface,NcB and NcE are theeffective density of states in the conduction band in both regions,x2

j and x1j are the two sides

of the emitter–base interface andvneff is the effective electron velocity through the interface, whichis expressed as

vneff 5 !S kT2pm*B

D (1 1 pt) (14)

where pt is the tunnelling factor through the spike andm*B is the effective electron mass in theregion of the base. Expressions similar to (13) and (14) apply for the holes current in the valenceband through the interface, although in this casept is zero since there is no spike in that band(Figure 1).

Assuming a single parabolic conduction band, the electron density can be expressed as

n 5 NcF1/2(hc) (15)

where F1/2 is the Fermi–Dirac integral of order 1/2, andhc is



hc 5Efn 2 Ec

kT(16)

From the aforementioned equations it is easy to obtain expression (17) for the electron concentrationand, using a similar procedure, equation (18) for the hole concentration:

n 5 nien expS qc 2 qfn

kT D (17)

p 5 niepexpS qfp 2 qc

kT D (18)

where nien and niep can be calculated as follows:

nien 5 ni,ref S Nc

Nc,refD expS qx 2 qxref

kT D S F1/2(hc)exp(hc)

D (19)

niep 5 ni,ref S Nv

Nv,refD expS2

(qx 2 qxref) 1 (Eg 2 Eg,ref)kT D S F1/2(hv)

exp(hv)D (20)

hc and hv are:

nc 5qc 2 qfn

kT1

qx 2 qxref

kT2 ln S Nc,ref

ni,refD (21)

hv 5qfp 2 qc

kT2

(qx 2 qxref) 1 (Eg 2 Eg,ref)kT

2 ln S Nv,ref

ni,refD (22)

note thathc and hv depend on the electrostatic potential and the quasi-Fermi potentials. Sincethese magnitudes are the unknowns in the numerical implementation of the model, an iterativesolution process is established in order to guarantee coherent results.

The formulation given by (17) and (18) for carrier concentrations is simple and compact.Parametersnien and niep may include different phenomena that affect the concentrations at highdoping levels: influence of Fermi–Dirac statistics, changes in the energy levels and variations inthe effective densities of states.

Note that the distribution of the bandgap narrowing between the conduction and valence bandsdirectly affectsnien and niep. Taking into account (8) and (9), expressions (19) and (20) become:

nien 5 F ni,ref S Nc

Nc,refD expS qxo 2 qxref

kT DG expS DEbgnc

kT D S F1/2(hc)exp(hc)

D (23)

niep 5 F ni,ref S Nv

nv,refD expS 2

(qx0 2 qxref) 1 (Eg0 2 Eg,ref)kT DG expS DEbgn

v

kT D S F1/2(hv)exp(hv)

D (24)

Notice that if the reference material were the same semiconductor at a low doping level, theterms in the first brackets of the expressions (23) and (24) would be the intrinsic concentrationof the materialnio.

This carrier concentration model can be easily extended to take new effects into account. Forexample, if a non-parabolic factorBj, for band j is considered:

gj 5Î2m3/2

dj

p2"3 (E 2 Ecj)1/2[1 1 Bj(E 2 Ecj)] (25)

where gj is the density of states function for this band, and if several subbands are included inthe conduction process, the expressions for the effective intrinsic concentrations become:



nien 5 nio Oj

Ncj

Nco

[F1/2(hcj) 132kTBjF3/2(hcj)]

exp(hcj)expS qxj 2 qxo

kT D (26)

piep 5 nio Ol

Nvl

Nvo

[F1/2(hvl) 132kTBlF3/2(hvl)]

exp(hvl)expS qxl 2 qxo 1 Egl 2 Ego

kT D (27)

where the sum is extended toj conduction subbands andl valence subbands.

3. NUMERICAL SOLUTION

The system of differential equations (1–3) presented in the previous section is discretized usingScharfertter–Gummel’s procedure,8 thereby converting it into a system of non-linear algebraicequations that is linearized by means of Newton’s method. The unknowns are the electrostaticpotential and the electron and hole quasi-Fermi potentials at each point. These potentials are usedto calculate the concentrations of carriers and the currents in a stationary state. A quasi-stationaryapproximation is used to calculate the transit times and the unity-gain frequencyfT.9

In order to reduce the time of simulation, the program was developed for distributed-memorymulticomputers, using the MIMD strategy (multiple instruction—multiple data) under the SPMDparadigm (simple program—multiple data). These machines consist of a certain number ofprocessors or nodes interconnected by a network with a certain topology. Our program wasimplemented using the MPI message-passing standard library.10 The main advantage of using thislibrary is that it is currently implemented in many computers, and this guarantees the portabilityof the code.11,12 MPI includes several communication strategies. Two of them, point-to-point andglobal communications, were used.

Choosing a method for solving the resulting system of linear equations is a key task, sincemost of the computing time is spent in that part. There are basically two types of methods tosolve this system: direct methods and iterative methods. The matrices corresponding to the linearsystems being solved are large, unsymmetric, highly sparse, not diagonally dominant, and, therefore,very badly conditioned, and the iterative methods appear to be ideally suited for this class ofmatrices. The termiterative methodrefers to a wide range of techniques that use successiveapproximations to obtain more accurate solutions to a linear system at each step. In the last fewyears, a set of techniques based on projection processes onto Krylov subspaces have appeared asthe most effective methods to solve large, sparse systems.13–15 Some of the best known methods,such as Conjugate Gradient, are only effective for symmetric positive definite systems. Forunsymmetric ones, the best technique depends on each particular problem. We have studied thebehaviour of some of the most important iterative methods,16 and our conclusion is that, for theproblem at hand, the best choice is the Bi-CGSTAB17 method, when it is used with anappropriate preconditioner.

In general, for iterative methods, the convergence rate depends on spectral properties of thecoefficient matrix. Hence one may attempt to transform the linear system into one that is equivalentin the sense that it has the same solution, but that it has more favourable spectral properties. Apreconditioneris a matrix that effects such a transformation. So, if we have a linear systemAx5 b, and if a matrixM approximates the coefficient matrixA in some way, the transformed system,

M21Ax 5 M−1b (28)

has the same solution as the original one, but the spectral properties of its coefficient matrix maybe better. The best choice forM depends on each particular problem, and is a trade-off betweenthe cost of constructing and applying the preconditioner, and the gain in convergence speed.

We have studied different preconditioning alternatives,16 and the conclusion is that incompletefactorization preconditioners are the most suitable for our problem. These preconditioners arebased on an approximate LU factorization of the coefficient matrixA, where certainfill elements,zero positions that would be non-zero in an exact factorization, are ignored. Such a preconditioneris then given in factored formM 5 LU with L lower andU upper triangular. In this work, we



have used a variant called ILU (fill,t). This preconditioner implies an incomplete LU factorization,in which it is possible to control the fill-in level of the matrix. This fill-in is controlled throughtwo parameters,fill , which refers to the numerical size of the fill-ins introduced, andt, whichindicates the minimum value for which fill-in is possible.

The main problem with this preconditioner is in its parallel implementation. In each iterationof the method a lower and an upper triangular system must be solved, and the dependence onthe external loop of the solution process cannot be avoided easily. In order to improve this lowlevel of parallelism, we have implemented a block-organized ILU version,18 Block ILU (fill,t,ovp),where the original matrix is subdivided into a number of overlapping blocks, nullifying someelements. So each block is assigned to a processor. The ILU of this block matrix is then computedin parallel, and the fill-in is again controlled through parametert. This setup produces a partitioningeffect such as the one represented in Figure 2, for the case of four processors. Due to thecharacteristics of this preconditioner, there is a certain loss of information in the regions that areeliminated. This implies that the number of iterations will increase as the number of blocksincreases (as a direct consequence of increasing the number of processors). This loss can becompensated to a certain extent by the information provided by the overlapping zones. The amountof overlap is controlled through a parameterovp.

In addition to the ILU (fill,t) preconditioner, other versions such as MILU (modified incompleteLU), including some of its versions,13 were tried. However, we did not obtain any effectiveimprovements.

All the simulator code, from mesh generation and refinement to the construction and solutionof the systems of equations with different preconditioners, has been parallelized, and tested on aCray T3E multicomputer using the MPI communications library. This is a very powerful andflexible parallel scalable system.11 It consists of anywhere from 16 to 2048 processors connectedby a high bandwidth bidirectional 3D torus network. Each cell includes a Dec Alpha 21164microprocessor, local memory and control logic. The memory system is logically shared andphysically distributed (NUMA—non-uniform memory access), i.e. all of the processors have theirown local memory but they can access the memory of the other processors with no need to usea message-passing protocol. Therefore, the T3E multiprocessor can be programmed followingeither a message-passing model or a shared-memory programming model.

Figure 3 plots the speed-up obtained with a 5176 mesh points in the Cray T3E. Figure 3(a)represents global ILU, while Figure 3(b) corresponds to block-ILU preconditioning. This speed-up is defined asT(1)/T(N), whereT(i) is the total execution time of the program on a multiprocessorcomposed ofi processors.

With global ILU preconditioning, a uniform and stable behaviour is observed. A minimum fill-in level of 1 was required to ensure convergence. As long as the fill-in level is increased thespeed-up is improved, and the total time is decreased, mainly because the linear system ofequations is solved in a smaller iteration count.

For the local ILU factorizations, the speed-up is represented in Figure 3(b), taking the overlapfactor as a parameter (keeping constant the fill-in level). If the values of fill-in or overlap levelsare low, then the preconditioner does not display uniform behaviour, in the sense that there arecases for which the system has to carry out many more iterations because the preconditioner is

Figure 2. Matrix splits into blocks



Figure 3. Speed-up for ILU and Block-ILU in Cray T3E

Table I. Doping profile of InP HBT

W (mm) Neff (cm23)

Emitter (n-InP) 0·20 1018

Base (p-InGaAs) 0·05 1019

Collector (n-InGaAs) 0·25 531016

of lower quality. However, as would be expected, for medium or large values of both parametersthe behaviour is uniform. In all of the cases shown, the speed-up is slightly inferior than in thecase of global ILU, although the total time required to carry out the simulation is slightly inferior.This is due to the fact that the Block ILU(fill,t,ovp), requires fewer operations than the ILU(fill,t).

4. APPLICATION TO ABRUPT InP/InGaAs HBT

The simulator described was applied to the analysis of an abrupt InP/InGaAs HBT. The dopingprofile of the simulated InP HBT is shown in Table I. Physical parameter values used in themodelling of this HBT can be found in the literature.2

As can be seen in Figure 4(a) a spike appears in the conduction band at the emitter–baseinterface. Although the discontinuity of theEC level at the interface is constant,DEC 5 0·199 eV,the shape of the spike changes with the bias voltage. For low values ofVBE the spike is verythin and the tunnelling through it is high (Figure 4(b)). Otherwise, whenVBE takes high valuesthe spike is wider and the tunnelling factor decreases. ForVBE greater than 0·9 V the tunnellingis approximately zero. Since the collector current is determined by transport through this interface,as shown in Figure 6(a), it is necessary to calculate the energy band diagram in order to obtainthe electrical characteristics of these kind of transistors.

Figure 4. Conduction band energy in the base–emitter and tunnel factor versus base–emitter voltage



Figure 5. Electron concentration versus position

The Fermi–Dirac statistic is required to simulate these transistors because of the high dopinglevels used in their base. Figure 5 presents the electron density distribution in the base calculatedusing Fermi–Dirac statistics (curve b) and Boltzmann approximation (curve a). As can be seen,differences greater than 10 per cent are found. Base contribution to diffusion capacitance can beexpressed by

CSB 5 dQnB/dVBE (29)

where QnB is the electron charge density in the base, which depends on electron density alongthis region. Consequently, differences in this magnitude will affect the values ofCSB, and thereforethe device frequency response.

Figure 6(a) shows the collector current density versusVBE voltage. Curve b was obtained withthe model described in this work. If this model is only modified in the distribution of bandgapnarrowing (the total bandgap narrowing is equally distributed between conduction and valencebands) curve (a) is obtained. Finally, curve (c) would be obtained if the conventional drift–diffusion transport model were used instead of the interface transport model described in Section2. Note that differences between curves (b) and (c) are greater than one order of magnitude.

Unity-gain frequency versus collector current density is presented in Figure 6(b) for the samecases as in Figure 6(a). Again, the differences between the drift–diffusion transport model andthe one described in this work are very important.

5. CONCLUSIONS

In this work a numerical model for abrupt HBTs has been presented. This model integrates theconventional drift-diffusion charge transport mechanism in the bulk of the device with chargetransport by thermionic emission and tunnelling transmission through the emitter–base interface,which are characteristic of this type of abrupt HBT. The model is therefore applicable to thestudy of InP/InGaAs HBTs.

In this physical model, the importance of the distribution of the bandgap narrowing betweenthe conduction and valence bands has been highlighted, since this determines the discontinuities

Figure 6. Collector current density and unity-gain frequency



in the energy levels and, consequently, the currents in the device. We have presented a compactformulation that is able to integrate different effects such as bandgap narrowing, the non-parabolicnature of the bands and the change in the effective densities of states.

The numerical model has been implemented to run on distributed-memory multicomputers,therefore allowing a fast and efficient study of these devices. Furthermore, the parallel implemen-tation has been developed using message-passing standard libraries, making the code highlyportable.

Global ILU has proved to be a good choice among the preconditioners that have been consideredin this work; not only for its better speed-up but also for a smaller computing time. However, itshould be noted that, for not so badly conditioned matrices or for very large problems (3Dsimulations), block preconditioning is expected to outperform global preconditioning, since theblocks method is better suited for the increase in parallelism in the new problem.

Results obtained in numerical analysis of InP/InGaAs HBT for base carrier density, collectorcurrent density and unity-gain frequency show the importance of the Fermi–Dirac statistics, thedistribution of bandgap narrowing between conduction and valence bands, and the field-emissiontransport trough spike at the emitter–base interface, in the simulation of these transistors.

ACKNOWLEDGEMENTS

The work described in this paper was supported in part by the Ministry of Education and Science(CICYT) of Spain under projects TIC96–1125–C03 and TIC96–1058, and by Training and Researchon Advanced Computing Systems (TRACS) at the Edinburgh Parallel Computing Centre (EPCC).We want to thank CIEMAT (Madrid) for providing us access to the Cray T3E multicomputer.

REFERENCES

1. S. Yamahata, K. Kurishima, H. Ito and Y. Matsuoka, ‘Over-220 Ghz-fT-and-fmax InP/InGaAs double-heterojunctionbipolar transistors with a new hexagonal-shaped emitter’, inTech. Diag. GaAs IC Symp., 1995, pp. 163–166.

2. J. M. Lopez-Gonzalez and Ll. Prat, “The importance of bangap narrowing distribution between the conduction andvalence bands in abrupts HBTs,”IEEE Trans.,Ed–44 (7), 1046–1051 (1997).

3. P. A. Mawby, Physics and Technology of Heterojunction Devices, D. V. Morgan and R. H. Williams (Eds.), PeterPeregrinus Ltd., 1991, Chap. 3.

4. K. Horio and H. Yanai, ‘Numerical modeling of heterojunctions including the heterojunction interface,’IEEE Trans.,ED–37(4), 1093–1098 (1990).

5. A. A. Grinberg, M. S. Shur, R. J. Fischer and H. Morkoc, “An investigation of the effect of graded layers andtunnelling on the performance of AlGaAs/GaAs heterojunction bipolar transistors,”IEEE Trans., ED–31(12), 1758–1765 (1984).

6. S. C. Jain and D. J. Roulston, “A simple expression for band gap narrowing (BGN) in heavily doped Si, Ge, GaAsand GexSi12x strained layers,”Solid-State Electron.,34(5), 453–465 (1991).

7. C. M. Wolfe, N. Holonyak and G. E. Stillman,Physical Properties of Semiconductors, Prentice Hall, 1989, Chap. 8.8. D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator,”IEEE Trans., ED-16,

64–77 (1969).9. H. Gummel, “On the definition of cutoff frequencyft, Proc. IEEE, 57, 2159 (1969).

10. University of Tennessee,MPI: A Message-Passing Interface Standard, June 1995.11. S. L. Scott, “Synchronization and communication in the T3E multiprocessor,”Tech. Rep., Cray Research Inc., 1996.12. D. Sitsky, “Implementation of MPI on the Fujitsu AP1000: Technical details,”Tech. Rep., Dept. of Computer Science,

Australian National University, Sept. 1994.13. R. Barrett, M. Berry,et al., Templates for the Solution of Linear Systems. Building Blocks for Iterative Methods,

SIAM, 1994.14. A. Van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric

linear systems,”SIAM J. Sci. Statist. Comput., 13, 631–644 (1992).15. Y. Saad,Iterative Methods for Sparse Linear Systems, PWS Publishing Co., 1996.16. A. J. Garcıá-Loureiro, T. Fernańdez Pena, J. M. Lo´pez-Gonza´lez and Ll. Prat, “Preconditioners and nonstationary

iterative methods for semiconductor device simulation,” inConferencia de Dispositivos Electrońicos (CDE-97),Universitat Politecnica de Catalunya, Barcelona, Feb. 1997, pp. 403–409.

17. T. F. Pena, J. D. Bruguera and E. L. Zapata, “Finite element resolution of the 3D stationary semiconductor deviceequations on multiprocessors,”J. Integr. Comput.-Aided Eng., 4(1), 66–77 (1997).

18. G. Radicati di Brozolo and Y. Robert, “Parallel conjugate gradient-like algorithms for solving sparse nonsymmetriclinear systems on a vector multiprocessor,”Parallel Comput., 11, 223–239 (1989).


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Numerical analysis of abrupt heterojunction bipolar transistors