lEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38. NO. I I . NOVEMBER 1991 2427 A Three-Dimensional Semiconductor Device Simulator for GaAs/AlGaAs Heterojunction Bipolar Transistor Analysis Hiang-Cheong Chan, Member, IEEE, and Tsay-Jiu Shieh, Member, IEEE A b s t r a c t - A three-dimensional semiconductor device simulator was developed to study the steady-state characteristics of heterostructure compound semiconductor devices. The semiconductor partial differential equations-Poisson’s equation and the two carrier transport equations-are solved using finite-difference discretization. The Gummel iteration method and indirect sparse matrix solution techniques are utilized for minimizing computation time and memory requirements. This simulator was applied to the analysis of heterojunction bipolar transistors. The effect of emitter grading on the current-voltage characteristic is demonstrated. A comparison between twoand three-dimensional simulations is also presented. The results show that three-dimensional analysis is indispensable, in particular for devices of small geometry. I. INTRODUCTION ETEROSTRUCTURE compound semiconductor devices such as heterojunction bipolar transistors (HBT’s) have shown a great potential for microwave and millimeter-wave circuit applications [ 11, . Unlike conventional silicon bipolar transistors, a wide energy bandgap AlGaAs emitter is used in HBT’s to minimize hole injection from the base and maintain a good emitter injection efficiency. The doping concentrations in the base and emitter can thus be optimized for low base resistance and capacitance without the constraints associated with conventional silicon bipolar transistors. The base can also be made thinner to reduce transit time and improve highfrequency performance. Numerical analysis techniques were used early in a onedimensional silicon bipolar transistor study by Gummel  in 1964. This approach was applied to heterostructure solar-cell device analysis by Sutherland 141 in 1979. Lundstrom , in 1983, presented a unified one-dimensional numerical model for both heavily doped silicon and heterojunction devices. In 1984, Kurata  extended the H Manuscript received June 26. 1990; revised May 13. 1991. This work was funded by Cray Rescarch Inc., Mendota Heights. MN 55120 The review of this paper was arranged by Associate Editor M. Shur. H. C . Chan was with the NSF Center for Advanced Electron Debiccs and Systems, Department o f Electrical Engincerihg. The Unlversity of Texas at Arlington. Arlington. TX 76019. He is now with Micron Technology, Inc.. Boise, ID 83706. T . J . Shieh is with the NSF Center for Advanced Electron Device5 and Systems. Department o f Electrical Engineering, The University of Texas at Arlington. Arlington, TX 76019. IEEE Log Number 9 10275 I , one-dimensional numerical model to include transient analysis and Yokoyama  reported on a two-dimensional numerical model for the study of AiGaAs/GaAs HBT’s. Yoshida , in 1985, investigated the aluminum composition grading effect in the emitter using a onedimensional numerical model. Recently, the effect of aluminum composition grading was again studied by Meyyappan  using a two-dimensional numerical model. These numerical analysis techniques have made a great contribution to the development of HBT device technology in the past. Now with the advances in high-performance computing facilities, it is appropriate to develop a full three-dimensional numerical analysis model for a better understanding of complete device physics. One example of the importance of using a three-dimensional analysis is the base current-crowding effect in HBT’s which do not have a self-aligned emitter. A two-dimensional simulator would not allow for an accounting of this effect. In this paper, a three-dimensional compound semiconductor device simulator which can be generalized to simulate homogeneous junction devices and other heterojunction devices is presented. In comparison to a twodimensional analysis, three-dimensional simulation requires greater computation time and memory. Therefore, this simulator was developed for use on a Cray supercomputer. The nonlinear semiconductor differential equations are discretized and linearized in three dimensions using the finite difference method. The resulting matrix from the system of linear equations may have a rank of ten thousand or higher. An efficient vectorized sparse matrix solver is very important to the project. In this work, the Nonsymmetric Preconditioned Conjugate Gradient (NSPCG) [lo] package is used. The resulting solution data are downloaded to a front-end computer where graphic presentation of the data is generated. Two examples of the application of this simulator in the analysis of HBT’s will be demonstrated. 11. PHYSICAL MODELEQUATIONS The governing heterojunction compound semiconductor device model equations used in this current version include Poisson’s equation and the two current continuity 0018-9383/91/1100-2427$01.00 0 1991 IEEE I IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38, NO. I I , NOVEMBER 1991 2428 equations eV2$ + VE V $ = -q(N,f - NA +p - n) (1) + J, V = qU,, (2) -t V J , = -qU,, (3) where E is the permittivity, $ is the electrostatic potential, q is the electronic charge, n and p are the electron and hole concentrations, NA and N , f are the ionized acceptor and donor concentrations, U, and U,, are the electron and hole recombination rates, and J , and J,, are the electron and hole current densities, respectively. Initially, the drift and diffusion model is used for the calculation of J , and J, taking into account the energy band variation of the compound semiconductors * + - Jn = -9CLnnV$, + 4DnVn TABLE I A L U M I N UMOLE M FRACTION DEPENDENT PHYSICAL PARAMETERS FOR AI,Ga, - , A s , x < 0.4 Formula Parameter Band gap (eV) Electron affinity (eV) Relative dielectric constant Effective density of energy state. conduction band Effective density of energy state, valance band + 1.25~ 1.422 4.07 - 1 . 0 6 ~ 12.9 - 2 . 9 ~ + 0.083~) 1019(0.48 + 0 . 3 1 ~ ) 2 . 5 x IO”(0.067 2.5 x where N,,,,, is the total local doping concentration, and the constants a , b, c , and d are 1500, 5700, 50, and 330 cm2/V s, respectively, for GaAs, and 500, 2000, 50, and 330 cm2/V s, respectively, for Alo,3Gao,,As.The (4) field-dependent mobility models for electrons and holes ( 5 ) are - -9CLpPV$2 - SDpVP where D , and D,, are the electron and hole diffusivity, p,, and y are the electron and hole mobility, and $, and $2 are the modified electrostatic potentials for electrons and holes, respectively. They are related to $ by J,, = x kT $l=$+-+-lnNc 4 4 where N,. and N,. are the density of the energy states in the conduction band and valence band, respectively, x is the electron affinity, Eh’ is the energy bandgap, k is Boltzmann’s constant, and Tis the absolute temperature. The energy-band discontinuity across the AlGaAs and GaAs junction is taken to be AE,. = 0.85 AE8 (8) AE!, = 0.15 AE, (9) which is known as the Dingle rule [ 111 and is somewhat controversial. However, (8) and (9) can easily be modified in the code to account for future better estimates of the energy-band discontinuity. The other material parameters which are dependent on the aluminum mole fraction x in Al,Gal - ,As can be found in ,  and are summarized in Table I. The carrier mobility is assumed to be dependent on both doping concentration and field strength . The concentration-dependent mobility model used in this work for electrons and holes is a where Eo = 4 x lo3 V/cm, Vs,, = 7.7 X lo6 cm/s, p: and U,: are the low-field mobility values obtained from (10) and ( 1 1 ) . The Einstein relationship between the mobility and diffusion coefficient is assumed. The ionized donor and acceptor concentrations in Poisson’s equation are calculated from where NO and NA are the n-type and p-type doping concentrations, & and g D are the acceptor and donor degeneracy factors, EA and ED are the acceptor and donor energy levels, and Ef, and Ef,, are the quasi-Fermi energy levels for electrons and holes, respectively. The default values for gA and gD are 2.0. The values for EA and ED are 0.005 eV from the valance band and conduction band, respectively. For the carrier recombination rates in (2) and (3), the Shockley-Reed-Hall recombination mechanism is considered U= p n - n’ 4f( n + n;) + Gf * (p + n;) (16) where 7grf and 7$ are the effective hole and electron lifetimes, respectively, and IZ, is the intrinsic carrier concen- 2429 CHAN AND SHIEH: THREE-DIMENSIONAL SEMICONDUCTOR DEVICE SIMULATOR tration, which is calculated from N,, N,, and energy bandgap ER. If not specified, a carrier lifetime of IO-' s was used for both .Zff and Cff. 111. GENERAL ASPECTSOF THE COMPUTER CODE The computer code was written in Fortran. It consists of three major subprograms which perform input file interpretation, grid structure and initial condition setup, and general numerical solution. Most of the keyword and parameter designations used in the preparation of the input files are similar to those used in the two-dimensional homogeneous semiconductor device simulator, PISCES [ 141. The material model parameters for semiconductors such as GaAs and AlGaAs (of various aluminium mole fractions), and dielectric material such as silicon dioxide are included. These model parameters can be easily changed or modified for the analysis of devices based on other material systems such as Si/GeSi. A typical three-dimensional steady-state analysis of an HBT, shown in Fig. 1, consists of 5000 grid points and takes about 150 s of processing time for each bias voltage step. The resulting solution data include potential, field, current, energy band, and carrier concentration distributions. The current version of the computer code performs steady-state analysis only. The drift-diffusion model used in the current calculations can be modified to include the thermionic emission effect which is considered to be a more accurate model for heterojunctions [ 151. With the addition of a Schottky boundary, the computer code can be also used to simulate field-effect transistors such as the high electron mobility transistors. 0.5 A . Effect of Compositional Grading of A1 in the Emitter of an HBT To determine the effect of emitter grading, the device structure with an appropriate doping profile given in Fig. 1 was studied. Starting at the bottom, the device consists of a 0.2-pm nf-GaAs substrate for the collector layer, a 0.5-pm n-GaAs collector layer, a 0.1-pm p+-GaAs base layer, a graded n-AlGaAs emitter layer, and a uniformly doped n-Alo.3Gao,7Asemitter layer with a total emitter layer thickness of 0.7 pm. The base contacts are placed on both sides of the emitter. Four different emitter grading ranges as depicted in Fig. 2 were studied. The aluminum mole fraction changes from 0.3 to 0.0 in 1000, 500, 300, and 100 A , respectively, where it reaches the emitterbase iunction. Since the grid spacing in the base region is 0.5 UNIT : pm 3 n- 2x10%d I 0.6 pm EhmTER II 0.1 p Graded E m i ~ ~ r n -AIGaAs (x: 0.3 to 0.0) Ill 0.1 p n BASE p - GaAs 5~10'~cm.~ IV 0.5 p CXIL.LECTOR n-GaAs 5~1O'~cm.~ V 0.2 pm SUBSTRATE n - GaAs 2~1017cm-~ ~x~O~~CIII-~ Fig. 1. A three-dimensional GaAs/AIGaAs HBT device for simulation. 5 c 0.3 4 LL w $ 0.2 z 5 5 0.1 CI 0.5 IV. RESULTS ~~~htwo- and three-dimensional analyses were carried out on various device structures. The difference in device characteristics between an abrupt base-emitter junction HBT and a graded-emitter HBT was studied. The twoand three-dimensional analyses on the same device were also compared. All the devices studied are considered to be isothermal at room temperature. 1.0 e,-- 0.55 0.6 0.65 63.7 0.75 0.8 DISTANCE FROM EMITTER CONTACTLumI Fig. 2 . Aluminum mole fraction as a function of distance from the emitter contact, shown for four different emitter gradings. A 100 in the simulation, the last case is considered to be numerically equivalent to an abrupt junction. The collector current as a function of the base-toemitter voltage (Gummel plot) for the four different grading cases is shown in Fig. 3. The results show that the collector cuFent is insensitive to the linear grading range above 300 A. The turn-on voltage of the abrupt-junction HBT, which has a uniform aluminum mole fraction of 0.3 in the emitter, is much higher than that of other gradedemitter HBT's. Fig. 4 shows the conduction band along the device center line close to the base-emitter junction. The spike observed on the conduction band of the abrupt-junction HBT can be considered as a large injection barrier for electrons as compared to the other graded emitter cases. It is also evident from the electron concentration profile along the device center line close to the base-emitter junction, shown in Fig. 5, that the number of electrons injected into the base region is much lower in the abrupt-junction HBT than that for other graded-emitter HBT's under the same IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38, NO. 11, NOVEMBER 1991 2430 102 I t 1.6 I 2 L? 1 . 4 > z 1.2 p 7 A - I vbe=l.4u v L =1.0v 4 2 n + 0.8 5 M ; I O.b 10-1 U @ 0.4 0 0.2 0.8 1 1.2 1.4 1.6 1.8 n 2 0.55 BASE-EMITTER VOLTAGE(Vo1 t 5 ) 0.59 0.63 0.71 0.67 0.75 DISTANCE FROM EMITTER CONTACT(um1 Fig. 3. C o k c t o r current as a function of v,><, for the four different emitter grading cases. Fig, 6, Conduction-bandenergy along the center line perpendicularto the base-emitter interface plane of the simulated abrupt-junction HBT for four different bias voltages. 5 1.46 & 1.42 9 m z 0 1.38 - 0.6 g 3 4 1.34 8 e 1.3 ' , 0.55 2 0 I ' ' " " 0.59 " " " 0.63 " " 0.67 ' " " I : 0.4 : - ' , ' 0.71 0.75 DISTANCE FROM EMITTER CONTACT(I" Fig. 4. Conduction-band energy along the center line perpendicular to the base-emitter interface plane for Vhr = 1.4 V and V,,, = 2.1 V. 0.55 0.59 0.63 0.67 0.71 0.75 DISTANCE FROM EMITTER CONTACT(w) Fig. 7. Conduction-band energy along the center line perpendicular to the base-emitter interface plane of the 300-A emitter-grading HBT for four different bias voltages. t I-IBBB A' LNI EAR GRADING 1-568 A' LINEAR Wlffi 3-366 A' LINEAR GW16 4-WI JMIM EMITTER~BASE 0.55 0.59 0.63 0.67 0.71 0.75 DISTANCE FROM EMITTER CONTACTiumI Fig. 5 . Electron concentration along the center line perpendicular to the base-emitter interface plane for V,,, = 1.4 V and V , , = 2. I V . base-emitter bias voltage of 1.4 V and collector-emitter bias voltage of 2.1 V. This observation is indeed consistent with results reported in the literature . Figs. 6 and 7 show how the conduction band changes with the base-emitter bias yoltage V,, for the abrupt-junction case and for the 300-A graded emitter case, respectively. In the abrupt-junction case, the conduction band spike incdeases in height with increasing bias voltage. In the 300-A graded-emitter case, no significant spike is observed. Instead, a dip occurs close to the junction, which becomes an electron trap as pointed out by Hayes [ 161. r -- - 108 1 , 0.55 " " " 0.59 " ' ' , 0.63 , I,' ' 0.67 . " ' " " 0.71 ' I ' 0.75 DISTANCE FROM EMITTER CONTACTipm) Fig. 8. Carrier concentration along the center line perpendicular to the base-emitter interface plane of the abrupt-junction HBT for four different bias voltages. Figs. 8 and 9 show the corresponding carrier distribution in this region. In the abrupt-junction case, the injected electton concentration in the base is relatively low. In the 300-A graded-emitter case, carrier distribution profiles are quite different. When the bias voltage is greater than 1.0 V, electrons tend to accumulate at the conductionband dip, and on average, the electron concentration is much higher than that for the abrupt-junction case. CHAN AND SHIEH: THREE-DIMENSIONAL SEMICONDUCTOR DEVICE SIMULATOR 1.55 0.59 0.63 0.67 0.71 0.75 DISTANCE FROM EMITTER CONTACT(um1 Fig. 9 . Carrier concentration along the center line perpendicular to the base-emitter interface plane of the 300-A emitter-grading HBT for four different bias voltages E 0.5 1 1.5 2 2.5 3 x (uml Fig. 1 1 . Vector plot of the current flow on the base-emitter interface plane obtained from three-dimensional simulation with V,,,, = 1.2 V . dimensional analysis would not be able to take this into account, in particular when the device dimension along the y-axis is comparable to the other device dimensions. 105 nl 243 I 104 t 2 103 w Q 102 w t 10‘ w 26 100 [L ?? 10-1 10-2 1 1.05 1.; 1.15 1.2 1.25 1 . 3 1.35 BASE-EMITTER VOLTAGE i V o l t51 Fig. 10. Comparison of the collector and base currents per unit emitter area as a function of Vhr obtained from two- and three-dimensional simulations. B. Comparison of Two-and Three-Dimensional Analyses The three-dimensional device structure used in this analysis is the same as the one shown in Fig. 1. The grid structure for the numerical calculation consists of 13 x 16 X 30 nonuniform meshes. The two-dimensional device topology for comparison is obtained by cutting through the center of the emitter on the x-z plane. The device plane is thus normal to the y-axis. The collector and base currents per unit emitter area as a function of base-to-emitter voltage (Gummel plot) for the two cases are shown in Fig. 10. A higher collector current is predicted from the three-dimensional analysis for the same bias conditions. In this case, the fringing field and current are accurately taken into account. In the case of the twodimensional analysis, the field and current are confined to the cutting device plane. The base resistance and currentcrowding effects are thus overestimated and a lower collector current is obtained. To demonstrate the effect of the fringing field and current in the three-dimensional analysis, a vector plot of the current density on the base-emitter interface plane is shown in Fig. 11. It shows that the base current flows into the active emittedbase interface region from all directions. Neglecting the fringing current and field in a two- V . CONCLUSIONS In this paper, a three-dimensional numerical simulator for the analysis of heterojunction compound semiconductor devices is reported. It numerically solves Poisson’s equation and the two current continuity equations in a three-dimensional grid structure using the finite difference method. The alternating iteration method (Gummel method) is implemented to find a self-consistent solution. The computer code was developed for use on a Cray supercomputer. A user-friendly input processor provides easy user interface. The effects of various emitter gradings were studied and demonstrated. Simulation results have shown that the tumon voltage of the abrupt-junction HBT, which has a uniform aluminum mole fraction of 0.3 in the emitter, is higher than that for the graded-emitter HBT’s due to the conduction band spike. Comparison between two- and three-dimensional simulations is presented. The calculated collector current is higher from a three-dimensional analysis than that from a two-dimensional analysis. In the two-dimensional analysis, base resistance and emitter crowding effects are shown to be overestimated because fringing fields and currents are not taken into account. ACKNOWLEDGMENT The authors would like to acknowledge both Cray Research Inc. for continuous funding of this project and The Center for High Performance Computing of The University of Texas System for technical support. The authors would also like to thank D. Klemer and P. Fort for proofreading of this manuscript. REFERENCES [ I ] H. Kroemer, “Heterostructure bipolar transistor and integrated circuit,” Proc. IEEE, vol. 7 0 , p. 13, 1982.  P. M. Asbeck er a l . , “Heterojunction bipolar transistors for microwave and millimeter wave integrated circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-35, p. 1462, 1987. I 2432 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38, NO. I I . NOVEMBER 1991 131 H. K. Gummel, “A self-consistent iterative scheme for one dimensional steady state transistor calculations.“ IEEE Truns. Elec,tron Dovices. vol. ED-I I , p. 455. 1964. (41 J . E. Sutherland and J . R. Hauser. “A computer analysis of heterojunction and graded composition solar cells.” IEEE T r m s . Electron Devices, vol. ED-24, p. 363. 1977. [SI M. S . Lundstrom and R. J . Schuelke, “Numerical analysis of heterostructure semiconductor devices.” IEEE Trans. Electron Devices, vol. ED-30. p. 1151, 1983. 161 M. Kurata and J . Yoshida. “Modeling and characterization for high speed GaAs-AIGaAs n-p-n HBT,” IEEE Trans. Electron Devices, vol. ED-31, p. 467, 1984.  K. Yokoyama et a l . . “Accurate modelling of AIGaAs/GaAs heterostructure bipolar transistor by two-dimensional computer simulation,” IEEE Trans. Electron Devices. vol. ED-31, p. 1222. 1984. 181 J . Yoshida et a l . , “Emitter base bandgap grading effects on GaAs/AIGaAs heterojunction bipolar transistor characteristics.” IEEE Trans. Electron Deijices, vol. ED-32, p. 1714. 1985. 191 M. Meyyappan et a l . . “Numerical study of an AIGaAs/GaAs heterostructure bipolar transistor: emitter design and compositional grading,” Solid-State Electron.. vol. 31, p. 161 I , 1988. [ I O ] T. Oppe et a l . , NSPCG User’s Guide, Version I . Austin. TX: Center for Numerical Analysis. Univ. of Texas at Austin, 1988. [ I I ] R. Dingle et a l . . “Quantum state of confined carriers in very thin AIGaAs-GaAs-AIGaAs heterostructure.” Phy. Rev. Lett.. vol. 33, p. 827, 1984. 1121 H. C. Casey and M . B. Panish. Heterostructure Laser. New York: Academic Press. 1978. 1131 S . Adachi. “GaAs. AIAs. and AI,Ga, _,As: Material parameters for use in research and device applications,” J . Appl. Phys.. vol. 5 8 , no. 5 . p. RI. Aug. 1985. 1141 M. R. Pinto et a l . . PISCES User’s Guide. Stanford. CA: Stanford Univ.. 1986. 1151 K. Horio and H . Yanai. “Numerical modeling of heterostructure including the thermionic emission mechanism at the heterojunction interface,” IEEE Trans. Electron Devices, vol. 37. p. 1093. Apr. 1990. 1161 J . R. Hayes et a l . , “Optimum emitter grading for heterojunction bipolar transistors,” Appl. Phys. Lett.. vol. 43, p. 949, Nov. 1983. Hiang Cheong (Charlie) Chan (S’87-M’90) received the B.S. (highest honors), M.S.. and Ph.D. degrees in electrical engineering from The University of Texas at Arlington in 1984. 1986, and 1990. respectively. From June 1986 to May 1990, he was a Graduate Research Assistant with the NSF/Center for Advanced Electron Devices and Systems, The University of Texas at Arlington. where he was involved in the development of a three-dimensional heterostructure compound semiconductor device simulator. In May 1990. he joined Micron’Technology, Inc. in Boise, ID, working on advanced CMOS DRAM technology development. His interests include research and development of ultra large scale integrated circuit technology and semiconductor memories. Dr. Chan is a member of Tau Beta Pi and Eta Kappa Nu. Tsay-Jiu Shieh (S’SI-M’83-S’83-M’85) received the B.S. degree in physics from National Taiwan University in 1976 and the M.S. and Ph.D. degrees in electrical engineering from the University of Cincinnati, Cincinnati, OH, in 1983 and 1985, respectively. He joined The University of Texas at Arlington in 1985 as an Assistant Professor of Electrical Engineering, teaching courses in VLSI design and technology. He was one of the founding members of the NSF Industry/University Cooperative Research Center for Advanced Electron Devices and Systems at The University of Texas at Arlington. He is presently an Associate Professor conducting research in MBE growth. rapid thermal processing of compound semiconductor ohmic contacts. semiconductor device modeling, and vacuum microelectronics technology development.