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The effect of optical injection on the gain and high frequency performance of AlGaAs/GaAs heterojunction… Lee, Chia-Nan 1995

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THE EFFECT OF OPTICAL INJECTION ON THE GAIN AND HIGH FREQUENCY PERFORMANCE OF AlGaAs/GaAs HETEROJUNCTION BIPOLAR TRANSISTORS by Chia-Nan Lee B.A.Sc., The University of British Columbia, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA JULY 1995 © Chia-Nan Lee, 1995 In presenting this thesis in partial Mfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Electrical Engineering The University of British Columbia 2356 Main Mall Vancouver, Canada ABSTRACT A comprehensive, one-dimensional, analytical model of the graded-base AlGaAs/GaAs heterojunction bipolar transistor is presented, and used to examine the influence of optical injection on the DC and high-frequency performance of a device with a conventional pyramidal structure. Absorption is limited to the base and collector regions because of the window effect of the wider bandgap emitter. Grading is considered by varying the A l mole fraction x linearly across the base to a value of zero at the base-collector boundary. Recombination in the space-charge and neutral regions of the device is modeled by considering Shockley-Read-Hall, Auger and radiative processes. Experimental devices were obtained from the Communications Research Centre, Ottawa and packaged by the author at the Alberta Microelectronics Centre and the Telecommunications Research Laboratory, Edmonton. The packaged devices were tested to evaluate the DC and high-frequency performances. Comparisons between predictions of the model and experimental data from a packaged device are presented. Optical injection is observed to improve the DC and high-frequency characteristics of the device. ii TABLE OF CONTENTS ABSTRACT i i LIST OF TABLE vi LIST OF FIGURES vii ACKNOWLEDGEMENT x 1 INTRODUCTION 1 1.1 Advantages of Using OCHBTs as Photodetectors 3 1.2 Main Feature of the Model 4 1.3 Experimental Procedure 6 1.4 Organization of the Thesis 7 2 MODEL DEVELOPMENT 8 2.1 Heterojunction Basics 8 2.2 HBT Under Illumination 16 2.2.1 Collector Hole Current 16 2.2.2 Electron Diffusion Current in the Base 20 2.2.3 Optical Generation in the Depletion Regions 27 2.2.4 Charge Flows in an HBT 28 2.3 High-Frequency Modeling 33 2.3.1 Cut-off Frequency 34 2.3.2 Maximum Frequency of Oscillation 40 2.4 Material Parameters 42 2.5 Incorporation of New Equations into DAPHNE 48 hi 3 S I M U L A T I O N R E S U L T S 49 3.1 D C Characteristics 50 3.1.1 Emitter and Collector Current Density Components 50 3.1.2 Collector and Base Current Densities 56 3.1.3 Optical Gain 61 3.2 High-Frequency Characteristics 62 3.2.1 Cut-off Frequency 64 3.2.2 Maximum Frequency of Oscillation 67 3.2.3 S2l Gain Response VO 4 E X P E R I M E N T A L R E S U L T S 76 4.1 Device Preparation 76 4.2 Experimental Setup 83 4.3 Experimental Measurements 85 4.3.1 D C Characteristics 86 4.3.2 High-Frequency Characteristics • 89 5 C O M P A R I S O N O F M O D E L A N D M E A S U R E M E N T S 95 5.1 D C Analysis 95 5.2 A C Analysis 99 6 S U M M A R Y 103 6.1 Conclusions 103 6.2 Considerations for Future Work 104 R E F E R E N C E S 106 iv A P P E N D I X A : S O U R C E C O D E F O R D A P H N E v LIST OF TABLES Parameters for the pyramidal heterojunction bipolar transistor. vi LIST OF FIGURES 1.1 The profile of the A l mole fraction (i.e., x in ALGa^As) 5 2.1 Energy-band diagram and particle current components in an HBT in the dark 12 2.2 Schematic of charge flows in an HBT in the dark 15 2.3 Energy-band diagram of the p-n base-collector junction under reverse-bias 18 2.4 Energy-band diagram and particle current components in an HBT under illumination 28 2.5 Schematic of charge flows in an HBT under illumination 29 2.6 The OCHBT structure 35 2.7 Equivalent circuit resistances for the emitter layers and emitter-base junction 37 2.8 Equivalent circuit resistances for the intrinsic and buffer regions of the collector... 39 2.9 Equivalent circuit resistances and capacitances for the base and base-collector junction 42 3.1 Plot of emitter current density and its various components with respect to VBE 51 3.2 Plot of emitter current density and its various components with respect to VBE (zoomed in) 52 3.3 Plot of collector current density and its various components with respect to VBE.... 53 3.4 Plot of J0PTi and J0PTS with respect to VBE 54 3.5 Dependence of collector current density on VBE and optical power 56 3.6 Dependence of collector current density on VBE (linear scale) 57 3.7 Dependence of collector current density on VBE for different amounts of base grading with optical power of lmW 58 vii 3.8 Dependence of the magnitude of the base current density on VBE for varying amounts of optical power 59 3.9 Dependence of the magnitude of the base current density on VBE for varying amounts of x, 61 be 3.10 Hybrid-7t equivalent circuit for a transistor. 63 3.11 Dependence of cut-off frequency on Jc 65 3.12 Dependence of cut-off frequency on VBE 66 3.13 Dependence of cut-off frequency on Jc with varying xbe 67 3.14 Dependence of maximum frequency of oscillation on Jc 68 3.15 Dependence of maximum frequency of oscillation on VBE 69 3.16 Dependence of maximum frequency of oscillation on Jc with varying xbe 70 3.17 Plot of S21 in the dark 72 3.18 Plot of S21 under illumination 73 3.19 Equivalent circuit for calculating S2l in the dark 74 3.20 Equivalent circuit for calculating S2l under illumination 75 4.1 Layout of circuitry containing H B T devices from C R C 77 4.2 Plan view of a device block 78 4.3 Schematic of the H B T : a) Plan view, b) Cross sectional view 79 4.4 Design of the alumina substrate 80 4.5 Plan view of the aluminum casing 82 4.6 Experimental setup for D C measurements 84 4.7 Experimental setup for high-frequency measurements 85 vii i 4.8 Measured dependence of Ic on VBE (log scale) 86 4.9 Measured dependence of Ic on VBE (linear scale) 87 4.10 Measured dependence of IB on VBE 88 4.11 Measured common-emitter characteristics of the packaged device 89 4.12 Measured gain response of the packaged H B T in the dark 90 4.13 Measured gain response of the packaged H B T under illumination 91 4.14 Optical setup used in Saskatoon 93 4.15 Re-measured gain response of the packaged H B T under illumination 94 5.1 Plot of collector current vs base-emitter voltage (dark case) 96 5.2 Plot of collector current vs base-emitter voltage (illuminated case). 97 5.3 Plot of base current vs base-emitter voltage 98 5.4 Plot of S21 in the dark 99 5.5 Plot of CRC-measured S21 on the second batch of devices 100 5.6 Plot of S21 with optical injection 102 5.7 Plot of normalized S2l with optical injection 102 ix ACKNOWLEDGEMENT I would like to sincerely thank my supervisor Dr. David L . Pulfrey for his generous support and guidance during this project. As well, I would like to thank Dr. Qing Zhong L i u for his support and guidance during my work term at the Telecommunications Research Laboratory, Edmonton. Thanks to Dr. Michael K . Jackson of U B C for his suggestion on how to wire bond the device. Special thanks to A . St. Denis of U B C for supplying the H B T devices and specifying the parameters and structure of the H B T device studied in this thesis. Special thanks as well to Rob James of the Communications Research Centre, Ottawa for supplying an extra batch of H B T devices. Finally, I would like to thank my parents for their patience and encouragement. x CHAPTER 1: INTRODUCTION A Heterojunction Bipolar Transistor (HBT) is a bipolar transistor in which the emitter and the base are composed of semiconductors having different bandgap energies. The advantage of having a wide-bandgap emitter is that a potential barrier is created which suppresses the reverse injection of charges from the base to the emitter, thereby allowing near unity injection efficiencies and thus very high current gains [1,2,3]. This high injection efficiency, being independent of the base and emitter dopings, allows the base to be heavily doped and the emitter to be lightly doped to decrease the base spreading resistance and the emitter-base capacitance. As a result, the high-frequency properties of the transistor are improved. A Heterojunction Phototransistor (HPT) is similar in structure to an H B T but is designed to utilize an optical, rather than an electrical, input signal. A 2-terminal H P T has a floating base (no base contact) while a 3-terminal H P T is exactly like an H B T . HPTs have been studied for many years because of their potential as high performance photodetectors for lightwave communications and as a possible alternative to the P I N / F E T combination in optoelectronic integrated circuits (OEIC) [4,5]. The first monolithic integrated photodetector-preamplifier implemented with an AlGaAs /GaAs H P T (3-terminal) and H B T s was reported by Wang and Ankr i [6], and had -9 a minimum detectable power of -30dBm at 140Mb/s for an error rate of 10 . Later Chandrasekhar et al. [7] reported an all-bipolar monolithic photoreceiver, using InP/InGaAs H P T (3-terminal) and H B T s , at 2Gb/s with high sensitivity. The performance 1 of the all-bipolar monolithic photoreceiver by Chandrasekhar et al. was comparable to that of the best hybrid P I N photoreceiver to date at that time. 2-terrninal HPTs are able to provide large optical gain without excess noise due to avalanching. However, being a two terminal device, the average incident light provides the quiescent bias current for the transistor. A t low optical power, the optical gain of the 2-terminal H P T is generally small. This is due to recombination at the emitter-base heterojunction. As a result, the collector current is small and this leads to a longer charging time for the junction capacitance. Consequently, the gain-bandwidth product is small. This product increases as the incident optical power is increased. However, a larger optical power causes a large penalty in sensitivity at high speeds. To increase the speed of the 2-terminal H P T , a base terminal is essential to provide the additional charging current [5]. This was first demonstrated by Fritzsche et al. [8]. Later Chandrasekhar et al. also demonstrated the enhanced performance of the InP/InGaAs H P T with a base terminal [5]. Chandrasekhar and co-workers reported an improvement of more than five times in the optical gain of a three-terminal device over that of a two-terminal device (the three-terminal device being optimally biased) over a 17dB range of input optical power. The small signal 3dB bandwidth of the three-terminal device was also enhanced 15 times over that of the two-terminal device over the same range of input optical power. In recent years, H B T s w i t h / r in excess of 160GHz have been reported in discrete devices [9]. Since H B T s have a structure similar to that of HPTs, it would be reasonable to expect that i f H B T s can operate as photodetectors, the frequency performance of the photodetector can be improved. The gain and frequency response of HPTs have already been studied in detail in the past, and several models have also been developed [10,11,12]. 2 There also exist several models for the H B T [13,14,15], but all of these only analyze the H B T s in the absence of light. So far, there is not yet a model developed that would provide a detailed analysis of the D C and high-frequency response of the H B T under iUumination. In this thesis, a comprehensive Ebers-Moll model that incorporates optical excitation, base grading, and the various generation and recombination processes wi l l be presented. Using this model, a detailed analysis of the optical characteristics of the n-p-n AlGaAs /GaAs H B T s wil l be presented. Note that from now on, to avoid confusion, a 3-terminal H P T wil l be referred to as an optically controlled H B T ( O C H B T ) and a 2-terminal H P T wi l l be called an HPT. 1.1 Advantages of Using OCHBTs as Photodetectors The H P T is very attractive for its sensitivity at high bit rates compared to other photodetectors like P I N / F E T and A P D [16]. The wide-gap emitter provides the design flexibility to tailor the emitter and base dopings to obtain high injection efficiency, high current gain, low junction capacitance, low base series resistance, and also a transparent window for incident light. This directly translates to an increase in the signal/noise ratio configuration has over the P IN/FET configuration is the better materials compatibility which would make the fabrication process easier and more cost effective. Personick [17] reported that for an H P T to be used as a detector in an optical fiber link, it has to meet the following requirements: a) high quantum efficiency, b) low input capacitance, c) large current gain even at low collector current, and d) biased operation by reducing the main noise parameter Another advantage the H P T / H B T 3 for setting the D C collector current at the optimum level, depending on the required bandwidth. As mentioned earlier, the current gain of an H P T is generally small at low input optical power. While it is not possible to bias an H P T electrically via the base current, an O C H B T does allow for an electrical bias to be applied to the base terminal to achieve a large optical gain even at a low input optical power. This ability to allow for optimum external bias through the third terminal is the main advantage that the O C H B T has over the H P T . As a result, O C H B T s have, in general, higher fT, higher fmax, and larger current gains compared to HPTs, making them better candidates than HPTs for applications as photodetectors in high-speed lightwave communication systems. 1.2 Main Feature of the Model The main feature of the model is the inclusion of the optical generation terms in the emitter-base space-charge region on the base side, the base-collector space-charge region, the quasi-neutral base region, and the quasi-neutral collector region in an Ebers-Moll representation of the H B T . The reason for placing the optical generation terms in these areas is because, for an optical source of 0.8/im wavelength, the emitter being A l ^ G a ^ A s wil l be transparent to the light, while the base and collector regions, being GaAs, wi l l absorb the incident light. The model also includes base grading and grading of the emitter conduction band spike. The base composition profiles which are considered are illustrated in Figure 1.1. For simplicity, absorption in the base is assumed to take place even when the base is graded. Since the absorption is predominantly in the base-collector space-charge region, this assumption is justified. 4 AlMole ' Fraction 0.3 0.2 0.1 0.0 EMITTER ND=5xl017cm COLLECTOR ND=3xl016cmV2xl018cm'3 Distance Figure 1.1 The profile of the A l mole fraction (i.e., x in A l G a j x As) . The four cases of base grading are referred to as xbe = 0.3, 0.2, 0.1, and 0 where xbe is the A l mole fraction at the emitter-base metallurgical boundary. The derivations of the emitter and collector currents are based on the thermionic and tunneling current representation of Grinberg et al. [13], but extended to incorporate optical excitation and base grading. As well, recombination in both the quasi-neutral base region and the emitter-base space-charge region, and generation in the base-collector space-charge region are also included in the calculation of the other currents in the device. Inclusion of these current components in the model allows a useful extension of Lundstrom's [14] Ebers-Moll formulation for H B T s to be realized. Three processes of recombination-generation in the space-charge and quasi-neutral regions of the device are considered, namely: Shockley-Read-Hall, Auger, and radiative. Many of the material parameters for A l G a l x A s are taken from the device analysis program S E D A N III [18]. As well, Fermi-Dirac statistics are used. Note that most of the equations were previously 5 formulated by Ho [19]. This thesis extends Ho's work on the effect of base grading on the gain and high-frequency performance of AlGaAs/GaAs HBTs, by including the effect of optical excitation. 1.3 Experimental Procedure Experimental HBTs were obtained from the Communications Research Centre (CRC), Ottawa for the experiment. These devices had been previously DC- and RF-characterized at CRC using on-wafer probing techniques. In order to package the devices for testing, the wafer was diced to obtain the individual devices. Microwave design was performed at the Alberta Microelectronics Centre (AMC). The design was then laid out on alumina substrates onto which the devices were attached and wire-bonded. Once the packaging was completed, Gummel plots and common-emitter characteristics were measured using a parameter analyzer, first in the dark and then under illumination to observe the difference. The optical source used was a 780nm laser which delivers lmW of optical power. Next, the devices were RF-characterized by biasing them at a suitable point and modulating the laser driver with a network analyzer. Results were obtained up to a frequency of 2GHz. Finally, the devices were taken to the University of Saskatchewan in Saskatoon where the optical source was modulated by an external optical modulator (EOM) up to a frequency of 10GHz. 6 1.4 Organization of the Thesis In Chapter 1, a brief introduction to the development of HPTs and O C H B T s over the past few years was presented. The advantages of an H P T / H B T configuration as a photoreceiver over a P I N / F E T configuration, and the advantages of an O C H B T over an H P T were discussed. The main features of the extended model were also summarized. As well, the experimental procedure was described. In Chapter 2, the model is presented in detail. In Chapter 3, the J -V characteristics, the optical gain, and the high-frequency figures-of-merit computed from the model are presented. In Chapter 4, the experimental setup, and the measurement results such as I -V characteristics and 3dB bandwidths are presented. In Chapter 5, the experimental data is compared with the theoretical values calculated from the model. Finally, conclusions and recommendations are presented in Chapter 6. 7 CHAPTER 2: MODEL DEVELOPMENT 2.1 Heterojunction Basics The model is based on the 'Theirniomc-Field-Diffusion" model of Grinberg et al. [13] but extended to include non-iiifinite surface recombination velocities at the contacts, base grading, and a more accurate formulation of the space-charge region recombination-generation currents. The effects due to series resistance, high-level injection, and hot-electrons are neglected. The basic Ebers-Moll current-voltage relationship for the H B T wi l l be presented. For simplicity, the dielectric constants of the emitter and the base are assumed to be constant in the derivation of the depletion-layer width and the capacitance. This is clearly not the case in a graded-base or a graded-junction O C H B T . However, since the base is usually heavily doped, the base side depletion-layer width wil l be very small. As a result, the change of the dielectric constant at the base side due to base grading is negligible. In fact, as the A l mole fraction varies from 0.1 to 0.3, the dielectric constant changes by less than 10% [19]. The thermionic-field-eniission model is used in the treatment of the carrier transport across the abrupt emitter-base junction. Hole carrier transport across the emitter and collector, and electron carrier transport across the base are governed by the simple diffusion process and the proper boundary conditions. Following the work of Ho [19], and with reference to Figure 2.1, the various current components present in an H B T in the dark are as follows: 8 Jp (xE), dark = [fBE//w - 1 ] cosh (2.1) / p ( x c ) , d a r k = JpC cosh \L?cj 1VBI /kT (2.2) / „ (0 ) ,da rk = - 1 - — 2 ? - ( kT _i •-nj (2.3) J„(W),dark = 2te™yn 1 + nso e / k T - 1 -2te 2.sW HBW\ e / k T -1 (2.4) where sinhl y /LpE L p E S p E / r , + taJW*/T ' /  UpE V / pE J LpESpE/ + t a n h V /  LPEJ RE =1 + -WTPEPB -exp cosh and 9 «„ ={ri-fY'w -{h-fY -AE"/T z„ = qvTnEy„e /kT bn=(r1-f)er^w-{r2-fyw rx = s +1 r2 =s-t s /2 «Jf2L2„B+4 2LnB qAE f kTXB Wc = Wcc - xc Here low-level injection has been assumed. AE is the hole potential energy barrier at the emitter-base junction; u is the average x-directed hole thermal velocity in the emitter. yn which is never less than 1, is the tunneling factor [13]; AEn is the electron potential energy barrier at the emitter-base junction; x>TnE is the average x-directed electron thermal velocity pE in the emitter; LpE = ^DpE^PE ^ m e n ° l e minority carrier diffusion length, DpE and tp being, respectively, the minority carrier diffusion coefficient and lifetime in the emitter; SpE 10 is the surface recombination velocity for minority carriers at the emitter contact; n2 / pE = lE/AT is the equilibrium hole concentration in the emitter, rc and T V being, / M DE ' respectively, the intrinsic carrier concentration and the N-type doping concentration in the emitter; LpC = ^ jDpCxpC is the hole minority carrier diffusion length, DpC and r c being, respectively, the minority carrier diffusion coefficient and lifetime in the collector; SpC is the hole surface recombination velocity at the collector contact; pc = iC/AT is the / ^V DC equilibrium hole concentration in the collector, niC and NDC being, respectively, the intrinsic carrier concentration and N-type doping concentration in the collector; LnB = ^ DnBinB , is the electron minority carrier diffusion length, DnB and T B being, respectively, the minority carrier diffusion coefficient and lifetime in the base; mo and « B W are the electron equilibrium concentrations in the base at x = 0 and x = W, respectively. The approximation WEE -xE = WE is made since xE is generally small under forward-bias. WE is the thickness of the emitter region. Note that NDC is used for the entire collector-subcollector region, x = 0 is taken to be at the boundary of the quasi-neutral base adjacent to the emitter-base depletion region while x = W is taken to be at the boundary of the quasi-neutral base adjacent to the base-collector depletion region. xE and xc are taken to be at the boundaries of the quasi-neutral emitter and collector regions adjacent to the depletion regions, respectively. XB is the total base width. In the graded base case, AEg is the difference in E between the values at the two metallurgical junctions which define the 11 base. Figure 2.1 shows the energy-band diagram and the various current components present in an H B T in the active mode of operation without illumination. J„(0),dark • J,(x£),dark 4 J,(*c),dark Figure 2.1 Energy-band diagram and particle current components in an H B T in the dark. Three recombination processes are considered for the emitter-base space-charge region recombination current density JR and the base-collector space-charge region generation current density JG. They are the Shockley-Read-Hall (SRH), radiative, and Auger processes. Following the work of Searles and Pulfrey [20], the recombination processes can be expressed as SRH.B sink 2kT tan' fZ - Z ^ z Q p ^P \ Z O P Z P + 1 y JSRH,E ~ ^ sinn BE 2kT tan - i \ Z 0 n Z n + ly (2.5) 12 with while 0 q{Vbl-VBE)/ AT - 4 t pO,x^nO,x Z„ = exp ni,n 1 XnO,n qV} BE D > 0 , « J0n -exp 2kT 2Nrat{Vbl-VBE) + VBE -q- 2kT „ ni,p 1p0,p ZP=^T, — E XP 2kT ni,p jXpO,p -q-2Nrat% -VBE) + VBE + AEfr -2AEt 2kT Nrat = NA + ND WBE = xn+xp=. 2z(Vbi-VBE) 'qNA{l-Nrat) Vbi =—hi NAND V nip J + AEr kT 9 In NAND ^ ni,pni,n J + AE-JRad,B=qntpBpWBE{\-Nmt)\ jRad,E =qnlBnWBENmt exp exp f VBE-AE^ kT rqVBE ^ \ kT j - 1 (2.6) and 13 J Aug.B 2anlpWBE 0 Z Z n x p Op p exp kT sink VZE-AE,, 2kT {ZQP Z p ^ A n pxnQ PZ P Z 0 P + A p px p 0 p^j 2qnlWBE J Aug,E ®Z„Z0nx„ exp\ 'QVBE ' sink ~qvBE~ . kT _ . 2kT _ {Z„ Z0n )(Ani„xn0i„ZnZ0n + A p n x p 0 n ) (2.7) AEFN is the spHtting of the electron quasi-Fermi level that occurs in an abrupt emitter-base heterojunction of a n-p-n H B T [21,22]; B is the radiative recombination coefficient; An and Ap are the electron. and hole Auger coefficients, respectively. Hence, the total recombination current density in the emitter-base space-charge region is JR ~ JsRH 3 Rod + J Aug (2.8) In the reversed-biased collector-base junction, the generation current densities can be expressed as [19] jSRH „ a n i W B C ^nO + ^pO (2.9) J^=qBnfWBC (2.10) Aug qnfWBC(AnND+ApNA) In NAND + • BC\ (2.11) -i J kT 14 WBC is the base-collector depletion-layer width. Again, the total generation current density in the base-collector space-charge region is JG = JsGRH + JG[ad+JGAus (2-12) Figure 2.2 shows a schematic of all the main electron and hole current components with arrows indicating the direction of charge flow under normal operating conditions in the dark. E M I T T E R N B A S E P C O L L E C T O R N (dark) JXO) J M (dark) ( d a r k ) V K(0)-/„W| (dark) U w Figure 2.2 Schematic of charge flows in an H B T in the dark. 15 2.2 HBT Under Illumination The H B T under consideration has a wide-bandgap emitter, and narrow-bandgap base and collector. B y choosing an optical source of a certain wavelength such that electron-hole pair generation by the photons only occurs in the narrow-bandgap base and collector regions, the analysis can be greatly shnplified. The assumption that the emitter is transparent, together with the assumption of low-level optical generation in the base, means that J (xE) remains unchanged from the case where there is no iEmnination, and is identical to Eq . (2.1). 2.2.1 Collector Hole Current The base-collector junction, shown in Figure 2.3, is essentially a GaAs homojunction. The collector hole current is assumed to be governed by a simple diffusion process. With optical absorption, the continuity equation becomes -i^W-iw+aF(1_/?K«=0 (2.13) q dx XpC where a is the absorption coefficient, F is the photon flux density, and R is the reflectivity. The equation for the hole diffusion current in the collector is A Jp{x) = -qDpC-±± (2.14) Eqs. (2.13) and (2.14) combine to give the differential equation 16 VP{x) i > W + a F ( 1 _ ^ - „ = 0 dx 4 c (2.15) A where p(x) is the excess hole concentration at position x in the collector, and Lpc = •\JDpctpc is m e h ° l e minority carrier diffusion length, DpC and tpC being, respectively, the minority carrier diffusion coefficient and lifetime in the collector. Eq . (2.15) has the following general solution p(x) = &3 exp f x ^ V LPCJ ( x) aF{\-R)xpC _ + £ 4 expl | — e \ L P C J a2L2pC-l (2.16) which, when substituted into Eq . (2.14), yields the following equations for the hole diffusion current density evaluated at the collector boundaries x = xc and x = Wcc (see Figure 2.3): Jp(xc) = qD Pc L p c ^ c / L p c A ( ' \ A p(x c ) cosh — - p ( W c c ) \ L P C ) qDpC aF(l-R)x Pc L„r sinhf a2F(\-R)x pC -axr — e c e 0X0 cosh {Wc) yLpc) (2.17) 17 Figure 2.3 Energy-band diagram of the p-n base-collector junction under reverse-bias. Note that the substitution Wc =WCC -xc is made in obtaining Eqs. (2.17) and (2.18). The boundary conditions are (2.19) 18 Jp{Wcc) = qSpCp{Wcc) (2.20) where pc = 'C/AJ is the equikbrium hole concentration in the collector, n and / * DC ' being, respectively, the intrinsic carrier concentration and N-type doping concentration in the collector, and SpC is the hole surface recombination velocity at the collector contact. For the purpose of transport calculations, the doping densities in the collector and the subcollector regions are taken to be NDC. Equating Eqs. (2.17) to (2.20), the resulting collector hole current density evaluated at x = xc is JP(xc)= Jpc cosh \ L P C J /kT - 1 + qD2pC aF(l-R)xpC LpC sinhl ^ c x2 4 c - ! aLpC sinh Wc/ V / LPCJ -aWcc _ e - a X c + e - « > " c c c o s h {%/ + DpC cosh V / LPCJ -pC qDpc sinh v / aF(l-R)x Pc ^ a 2 4 c - l cosh KLpc) -aWr, (2.21) -qD a2F(l-R)x P C a 2 4 c - l PC -axr — e 0 where JPC = qDpcPc Lpc sinh v / LPCJ L„rS p C p C / n +tanh uPc Wr-, \ / L P C , LpcSpC/ + t a n h uPc Wc/ V /LPCJ 19 Eqs. (2.1) and (2.21) represent the hole current components of the total D C emitter and collector currents, respectively (in the case of no space-charge recombination and generation). To find the electron current components, one needs to solve for the electron diffusion current in the graded base. This is described in the next subsection. 2,2.2 Electron Diffusion Current in the Base Assuming that the hole current is small compared to the electron current, the following expression for the base electron current density is used [23]: qDnBnf(x) d (n(x)N NA dx nf(x) (2.22) Here, it is assumed that p = NA in the P-type heavily doped base, and that the nondegenerate Einstein relation DnB = (kTpin)/q holds. The electron diffusion coefficient in the base, DnB, is assumed to be constant at a value appropriate to the material of the A l mole fraction at the center of the base. Since base grading is taken to be linear, the bandgap can be expressed as EAx)-E^-^rx (2-23) where, E is the bandgap at x = 0, XB is the total base width, and AEg is the difference in Eg between the values at the two metallurgical junctions which define the base. The intrinsic carrier concentration is then expressed as [19] 20 nf (JC) = Nc (x)Nv (x)e~ / k i = aefx (2.24) where nf0Nc(x)Nv(x) a A Eg kTX„ Nc and Nv denote the effective densities of states in the conduction and valence bands, respectively. The subscript zero refers to conditions at x - 0. It is safe to assume that a changes much more slowly than /*, or more precisely, Ida a dx « \f\ [15], thus when E q . (2.24) is substituted into Eq. (2.22), the latter becomes Jn (x) = qDnBefx —(n(x)e'fx) = qD„B dn(x) dx (2.25) The electron carrier concentration can be written as the sum of the excess and equilibrium electron carrier concentrations, that is A nf(x) * a n[x) = n[x)-\ = n[x ) H e (2.26) Substituting Eq . (2.26) back into Eq . (2.25) yields J„(x) = qD, nB f A dn(x) dx - f n(x) (2.27) 21 which, when substituted into the following continuity equation for minority carrier electrons in the base ldJn(x) n(x) + a F { l _ R y „ = Q ( 2 2 g ) q dx XnB yields d2n(x) dn(x) n(x) aF(l-R)e~ax ox dx L„B DnB where the electron minority carrier diffusion length is given by LnB - ^DnBxnB , TnB being the minority carrier lifetime in the base evaluated at the center of the base region. The solution for y in the following second-order differential equation j^ + b^ + cy = R(x) (2.30) is y = Cxev + C2ev +— f e~riX R(x)dx + — f e-vR(x)dx (2.31) A Hence, the solution for n(x) in the second-order differential equation (2.29) is 22 where n(x) = C.e1 + C0e2 H ; r; re W M ' i - ' i X ' i + a ) — (XX rx = s + t r2=s-t t = C V / 2 2L„B A . , a F ( l - i ? ) , 1 r.W r,W e 1 - e 2 r,W -aW e 2 —e + r.W r2W e1 - e 2 C, = 2 r.W r,W el - e 2 « r ( i - * ) ( g - ^ - g ^ ) - ; ( w ) (2.32) A s shown in Figure 2.3, the boundaries of the quasi-neutral base are at x - 0 and x = W. Substituting Eq . (2.32) into Eq. (2.25) yields 23 Jn(x) = qD, nB -(a + / ) ^ DnB{h-r2){rx+a) aF(l-i?) (2.33) A s a result, the emitter electron density is / (0) and the collector electron current density is / (W). To evaluate these two current densities exactly, it is necessary to obtain A A expressions for n(0) and n(W). The latter is given by i(W) = n BW\ e 7kT (2.34) A where n Bw is the electron equilibrium concentration in the base at x = W. To obtain n(Q), one must match / n(0) with the electron thermionic-field-emission current density, which is written as [19] HBO\ e /kT 1 -n(0) (2.35) where z„ = qvTnEyne /kT When evaluated at* = 0, Eq . (2.33) reduces to 24 where J*(0) = y„ 2tn(W)-a„ n(0) +ym(K1+K1p1 -f)(e^-e-«") + (r2-f)(e-«"-e^j\ (2.36) -qDnB(a + f)(K1+K2) qD, nB r.W r,W el -e2 {r,-f)e^~{h-f)e' aF(l-R) VnBV-x-hXh+a) aF{\-R) DnB(r2 -h){r2 +a) A Equating the current densities of Eqs. (2.35) and (2.36), and solving for n(0) yields n (0) = A 2fy„ n(W) + znriBo e 1VBl /kT •1 + z„+a„y„ yn(K1+K2)\(ri-f)(e^-e-*") + {r2-f)(e^-e^)\ z„+a„yn qD„B(a + f)(K1+K2) z„ +any„ (2.37) Substituting Eqs. (2.34) and (2.37) into Eq . (2.36) yields the following expression for the emitter electron current density 25 '.(o) = - a„y„ HBO] e K i - 1 riBW\ e / I a - 1 n J anyn z„+a„y„ -qDnB(a + f){Kx+K2) +yM +K2)[(ri -f)(e^-e-«w)+(r2 -e^)] -qDnB(a + f){K1+K2) (2.38) Similarly, the collector electron current density can be found by evaluating Eq . (2.33) at x = W, giving J„(w) = yn bnn(W)-2te2sW n(0) +y„(Kl +K2)[(ri -f)(e'>w-e-«wyw +(r2 - f ) ( e ^ - e ^ w ] -?A*(a + / ) ( * i + * 2 K (2.39) where r,W A A Finally, replacing «(W) and n(0) in Eq.(2.39) by their known equivalents in Eqs. (2.34) and (2.37) leads to 26 l + any '-nj nso\ e bH+(aJbn-4t*e™)%-2te 2sW riBw\ e 'kT - 1 - I . law 2te y„ Zn+a«y« K(K1+K2)[(ri-f)(e^-e^)+(r2 -/)(«-"-e^)}} -qDrtia + fX^+Ki) +yM+K2)[(ri -f)(e*-e-*»yw + ( r 2 - / ) ( * - " - e ^ ] -qDnB(a + f)(Kl+K2)e-aW (2.40) 2.2.3 Optical Generation in the Depletion Regions The current density generated optically in the depletion region can be expressed as JOPT = q£^F(\ - R)e~axdx (2.41) For the case of the base side of the emitter-base depletion region, which has a width of dp, the optically generated current density is J0PTl=qF{\-R){\-e-^) (2.42) In a similar fashion, by taking the limits of integration from x - W to x = x c (see Figure 2.1), the current density generated optically in the base-collector depletion region is obtained as follows 27 (2.43) 2.2.4 Charge Flows in an HBT J„(0),dark • J„(^ ,dark - • -J/x^.dark Figure 2.4 Energy-band diagram and particle current components in an H B T under illumination. 28 Figure 2.5 E M I T T E R N B A S E P C O L L E C T O R N Wcc Schematic of charge flows in an H B T under illumination. Note: in the emitter-base space-charge region, JR arises principally from recombination in the emitter side, whereas Jopn arises from optical generation on the base side. Figure 2.4 shows the various electron and hole current components in an H B T under illumination. The emitter and collector electron current densities, / (0) and Jn(W), are the electron current densities entering the quasi-neutral base and the base-collector depletion region, respectively. / (xE) is the hole current density entering the quasi-neutral emitter from the base, and -J (xc) is the analogous hole current density entering the base-collector depletion region from the collector. JR, in Eq . (2.8), is the recombination current density in the emitter-base depletion region for the forward-biased base-emitter junction and JG, in Eq . (2.12), is the space-charge region generation current density for the 29 reversed-biased collector-base junction; Jopn, in Eq . (2.42), is the current density generated optically in the emitter-base depletion region while Jom, in Eq . (2.43), is the current density generated optically in the base-collector depletion region; Jom is the current density generated optically in the quasi-neutral base and back-injected into the emitter; Jom is the current density generated optically in the quasi-neutral base that collects at the collector; and Jom is the current density generated optically in the quasi-neutral collector that collects at the collector. The total D C emitter and collector current densities, JE and Jc, are drawn with arrows in Figure 2.5 to indicate the direction of positive charge flow under normal operating conditions. The term | / ( 0 ) - / (W) | represents the part of the base current density due to recombination in the quasi-neutral base. From Figure 2.5 we have JE=-Jn(0) + Jp{xE)+JR- Jam (2-44) Jc=~Jn(W)- JP(xc)+JG + JOPTA (2.45) Substituting Eqs. (2.1), (2.21), (2.38), and (2.40) into Eqs. (2.44) and (2.45) leads to the following Ebers-Moll expressions: qVBE/ \ ( qVBC/ JE=An\e / k T - \ \ + AJe / k T - l | + JR - J o m - J0PT2 (2.46) qVBE/ \ / qVBC/ Jc=All\e /kT-l\ + A22\e / k T -1 I + JG + J0PT3 + J0PTA + J0PT5 (2.47) 30 where a„y„nB0 JpE H cosh / z„ yLpEJ An -2ty„riBW 1 + a„yn/ J, OPT2 *21 i 2 2 J OPT3 1 + a„yn/ -(K1+K2} yn +(r2-f)(e-aW-e^) -qD^{a + f)\ A 0 , — 2sW 2te y„n.Bo 1 + 2 „2^W\ }'« * . + ( a A - 4 * V " ) 1 + anyn/ ynnBw - JpC cosh 2 J W 2 ^ y„ z„ +anyn (K1+K2) y„ {K1+K2) yn (ri-fX<2"-e-*w) +(r2-f)(e-aW-er>w) k - f ) ^ -e-aW)er>w +(r2-f)(e-«w-er'wy> -qDnB(a + f)\ -qDnB(a + f)e -aW J and J p were given in Eqs. (2.42) and (2.43), respectively, and 31 aLpC sinh \ / LPCJ -aWrr -axr , -aWrr „ „ „ u e 00 — e 0 + e c c cosh V / L P C J 'Pc cosh qDpC aF(l-R)x Pc LpC sinh V / LPCJ alUpC-\ e °" c cosh yLpcj -aWri +qDp i L£-o-°*c P C a 2 L 2 p C - l If no optical power is present, that is F = 0, Eqs. (2.46) and (2.47) reduce to « W "\ ( ivBC/ JE=An\e ]+An ie /kT-l\+JR (2.49) Jc = A ^ j ? ^ - l } + AJ^^ - l ] + JG (2.50) Eqs. (2.49) and (2.50) are identical to those derived by Ho [19]. Ho also showed that, with the proper assumptions made, the Ebers-Moll equations (2.49) and (2.50) do reduce to those predicted by the conventional diffusion model for the case of the simple homojunction transistor, yielding 32 Jv = \ q D » * ~ n B coth { W) + qDpEpE KLnB ) LpE /kT qDnBnB (2.51) /kT 1 + 7 . Jc = qDnBnB LnB sinh( W/L 'nB 9VBl '/kT - 1 (2.52) qDnBnB coth r W \ . qDpcPc "nB \ L n B j + - - 1 \+Jr. Eqs. (2.51) and (2.52) are the normal diffusion-model Ebers-Moll current density equations [24, p. 260]. 2.3 High-Frequency Modeling In this section, two widely used figures-of-merit that characterize the high-frequency performance of an O C H B T : i) the cut-off frequency, fr and ii) the maximum frequency of oscillation, fmax, wi l l be presented. A l l formulations are based on the H B T structure shown in Figure 2.6, whose geometrical and doping density parameters are given in Table 2.1. 33 2.3.1 Cut-off Frequency The cut-off frequency fT, also known as the gain-bandwidth product, is defined as the frequency at which the common-emitter short-circuit current gain is unity. The cut-off frequency is usually expressed as f r = ^ ~ (2-53) where [25] Xec ^^B ^^SCR ^C xE is the emitter charging time, xB is the base transit time, xSCR is the collector space-charge region signal delay time, and zc is the collector charging time. 34 Figure 2.6 The OCHBT structure. Table 2.1 Parameters for the pyramidal heterojunction bipolar transistor. Layer # Material Thickness (A) Doping (cm3) A l or In composition X Emitter cap 1 n+-In,Gall(As 300 1X10" 0.6 2 n-In,Ga,,As 300 1X10" 0.6-0 linear 3 n-GaAs 1500 2X10" 0 Emitter grading 4 n-AlxGa,.As 300 5X10" 0-0.3 linear Emitter 5 n-ALGa,.,As 1600 5X10" 0.3 Emitter grading 6 n-ALGa,,As 300 5X1017 0.3-0 linear Base 7 p+-AlxGa,.^s 1000 1X10" 0 Collector 8 n-GaAs 4000 3X10" 0 Collector buffer 9 n+-GaAs 4000 2X10" 0 35 The emitter charging time, tE, is a time constant representing the delay in the input of a common-emitter circuit. It is expressed as [26] where iB=re (CEj + CCj) + (REC + REX + REI )CCj (2.54) REC ~ 'cE SE^E r> ( P c o p l ^ c a p l + Pcap\2^cap\l + P cap2^cap2 ) KEX = V „ _ [ p « W „ + p M ( ^ - X B ) ] SE^E SEEE 3 / , 1 dV, BE $E LE dJE Q _ SELE&E£B 'Ej ^•E^E ~*~ ^B^BE XC + %-BC P capl 'capU ' capl 1 p£ 2 = 1 Cap\l\^ncap\l 1 ' capl\^ncap2 1 1 «E2 36 Referring to Figure 2.7, r is the emitter differential resistance, REC is the emitter contact resistance, REX is the extrinsic emitter resistance, and REI is the intrinsic emitter resistance, pcE is the specific contact resistivity of the emitter; li v M-ncapU, and fj.ncap2 are the electron mobilities of I n 0 S G a 0 4 A s , I n 0 3 G a 0 7 A s , and GaAs, respectively, jinEl and finE2 are the electron mobilities of A l 0 1 5 G a 0 8 5 A s and A l 0 3 G a 0 7 A s , respectively; CEj and CCj are the emitter and collector junction capacitances; XE is the depletion-layer width in the emitter, Xc is the depletion-layer width in the collector, and XBE and XBC are the base depletion-layer widths next to the emitter-base and base-collector interfaces, respectively. Figure 2.7 Equivalent circuit resistances for the emitter layers and emitter-base junction. 37 The base transit time is the time for electrons to cross the quasi-neutral base region. It can be calculated from the known distribution of base excess electron density and the collector electron current density by the following equation — j — r [ -qnixfix Jn(w)]o D, nB (n-fyw-(r2-f)Ce^-^-(K1 + K2)e -aW (2.55) where C = —CJCl ~ e'w. Assuming no base grading, no optical power, and WlLnB « 1, Eq . (2.55) reduces to the usual expression W^HD^. Note that Eq . (2.55) is only an estimate for xB. It becomes inaccurate under high-current conditions and high field conditions. Neither does it account for ballistic effects [19]. However, none of these effects are important in the present work. tSCR is the time delay for transport through the base-collector space-charge layer, and is given by [27, p. 35] ^SCR ~ BC 2-0, (2.56) where WBC is the width of the base-collector space-charge region and MS is the saturation velocity for electrons in GaAs. The collector charging time is the time delay caused by the charging of the collector junction capacitance through the collector series resistance, and is given by 38 - (Rcc + R-CB + Ra )Cq (2.57) where [26] Sbuf CC Lr coth R-Sbuf 'cC V -J? 4. £>' _ RSbufScD RsbufSBC ^CB ~ KCB + KCB ~ " 3Lr R pc{WCI-Xc) ci Rsbuf ~ 1 qNhuf\inbufwbuf l qNa\x„c € N H n Rr n vV vV Figure 2.8 Equivalent circuit resistances for the intrinsic and buffer regions of the collector. 39 With reference to Figure 2.8, Rcc, RCB, and RCI are the resistances of the collector contact, n + buffer layer, and intrinsic n layer, respectively; Rsbuf is the sheet resistance of the n + collector buffer layer, \i is the electron mobility of the buffer layer, and pcC is the collector specific contact resistivity; }inC and Xc are the electron mobility and depletion width in the intrinsic collector layer, and NCI and WCI are the doping density and thickness of the intrinsic collector layer. 2.3.2 Maximum Frequency of Oscillation The maximum oscillation frequency, fmax, is the frequency at which the unilateral gain becomes unity. It can be expressed as [28, p. 164] where fT is the cut-off frequency, Rb is the base resistance, and C is the collector capacitance. A more accurate result is obtained by using the following effective RbCc (2.58) [2,29,30] {RbCc) = CCI(Rm + RBX +RBC) + CCX RBC + (2.59) where [26] 40 R BI R BX H-SB^E 3LB _ RsB$EB RBC ~ 4Pc~BR~SB coth V 1 ' PcB J C, ci XBc + %c Q _ 2SEBLBzc r = ^cc RsB 2SBLBEC %BC + Xc 1 RSB ~ qNB\ipB(WB-XBC-XBE) 1 <lNB\LpB(WB-XBC) Referring to Figure 2.9, RBI, RBX, and RBC, are, respectively, the intrinsic, extrinsic, and contact resistances of the base, and CCI, Ccx, and Ccc are the base-collector junction capacitances underneath the intrinsic, extrinsic, and contact regions of the base. RSB and RSB are the sheet resistance of the intrinsic and extrinsic bases, NB is the base doping density, WB is the thickness of the base layer, and fx is the base hole mobility. 41 E M I T T E R B A S E AA SUB i*~ h— R.. AA n - C O L L E C T O R R. Figure 2.9 Equivalent circuit resistances and capacitances for the base and base-collector junction. 2.4 Material Parameters A l l the material parameters used in the model are obtained from D A P H N E , which is an H B T analysis program developed from Ho's work by researchers at U B C , principally: Ang [31], Laser [25] and St. Denis [32]. The source code for the version of D A P H N E used in this work is listed in the appendix. The more important ones include bandgap, electron affinity, electron and hole effective masses, dielectric constant, mobility and diffusivity, and minority carrier lifetimes. A l l these parameters depend on the A l mole fraction. In addition, the bandgap, the minority carrier lifetimes, and the mobility and 42 diffusion coefficients are directly dependent on the doping densities. The parameters are listed as follows: Bandgap: Eg (x, N) = m i n ( £ [ (x), E? (x)) -1.6 x 10"8 N K (2.60) 1.424+ 1.247x 1.424 + 1.247x + 1.147(x - 0.45) 0 < x < 0 . 4 5 2 0 . 4 5 < x < L 0 (2.61) Ef (x) = 1.900 + 0.125x + 0.143x2 (2.62) where N is the net doping concentration in cm"3, min() means "the minimum o f , x is the A l mole fraction, and E is in eV. Electron Mfin i ty : X ( x ) = 4 . 0 7 - A £ c ( x ) (2.63) where A £ c ( x ) = 0.798 l x 0 < x < 0 . 4 5 AEc(x) = 0.392 + 0 .048x-0.27x 2 0.45 < x < 1 0 43 Effective Mass: m* =(0.48+ 0.3 l x ) ^ (2.64) m„ K r ) % +K)^exp f A £ „ L " r ^ kT iK + ( m „ x ) / 2 e x p x < 0 . 4 5 kT (2.65) J (m„ ) 2 exp kT ^2A m„ + ( m „ L ) ^ e x p AE L-X \ x > 0 . 4 5 (2.66) kT + K ) ' where r m„ = (0.067+0.083x)mo m„L = (0.55 +0.12x)m 0 m„x = :(0.85-0.07x)m 0 AEJ- r = ^ L ( x ) - < ( x ) AEf •r=£f(x)-£[(x) AK x = < ( x ) - £ f ( x ) AEY x=£j-W-£f(x) 44 Dielectric Constant: 1+2 E1 +2 + ( l - x ) y£2+2j 1 - x v e 1 + 2 . + ( l - x ) V £ 2 + 2 . (2.67) where x is the A l mole fraction, and 1 and 2 refer to A l A s and GaAs, respectively. Mobility: where \ip(NT,x) = mp(x = 0) e"1 - £ _ 1 V-p,GaAs(NT) (2.68) (2.69) 1 + V V 2 ( A £ " L _ r ^  expi AT + m„ X A X ra, exp n J kT m„ :(x=o) ra, F " 1 — P _ 1 fcfc2 W 2 K(x = i)f + K ( x = i)f e - ;- E - . M-i 3 / , 3 / -1 / \ - 1 / \ r-n,AlAs [ntMY*£a(x)_£' (X) erw-^ w 1 / \ r%,GoAs 45 Again, 1 and 2 refer to ALAs and GaAs, respectively, eh2 and £ l 2 are the high-frequency and the low-frequency dielectric constants of GaAs, respectively, and NT is the total doping concentration. Diffusivity: where D = — \l n (2.70) F - > 2 ^ ~ Vrc ^ l + e x p ( x - r ) ) 3 x i V c = 2 Ny=2 2nmpkT kT ^v 46 Minority Carrier Lifetimes: 1 1 . 1 1 1 • + — + — + (2.71) ^SRH XR ^ A ^ INT where rl + ft: D + tlj <T = T - + T -lSRH ^ po - — ^ Lno - — n + p n + p 1 1R = B{n + p) l AN d ^ INT ~ eff SlNT T„B„, T„, and i. are the carrier lifetimes associated with the S R H , radiative, and Auger recombination processes, respectively, and xim is the carrier lifetime due to interface traps; B is the radiative constant which is dependent on A l mole fraction and doping density; A is the Auger coefficient which is dependent on A l mole fraction and temperature; \ is the effective active layer width and SINT is the interface recombination velocity. In this work, S1NT was taken as infinite, as is probably reasonable for such a highly lattice-matched combination of materials such as A lGaAs /GaAs . 47 A minor change was incorporated into Eq . (2.60) for the effective bandgap energy. Harmon et al. [33] found that bandgap shrinkage for p-GaAs can be expressed as AEg = 2 . 5 5 x l 0 " 8 7 V ^ (2.72) where N is the doping density and AE is in eV. Assuming that this bandgap shrinkage also applies to / 7 - A L G a ^ A s , Eq. (2.60) can be rewritten as Eg (x, N) = m i n « (x), £ , x ( x ) ) - 2.55 x 10' 8 N% (2-73) We assume that this bandgap shrinkage is taken up entirely by a shift in the valence band edge. This means that bandgap narrowing influences the current only through changes in intrinsic carrier concentration, and not through changes in the barrier profiles "seen" by electrons. 2.5 Incorporation of New Equations into DAPHNE To allow for optical injection, the optical generation terms in Eqs. (2.21), (2.38), (2.40), (2.42), (2.43), and (2.44) to (2.47) were incorporated into D A P H N E . 48 CHAPTER 3: SIMULATION RESULTS For simplicity, a one-dimensional model is used in the thesis'for the derivation of the H B T current equations, and a quasi-two-dimensional model is used for the formulation of the high-frequency figures-of-merit. The reasons for using a simple model include the desire to obtain some reasonable estimates of the device performance without resorting to two-dimensional models which involve extensive computations, and to investigate qualitatively the effects of optical injection on the device performance of H B T s . As with Ho 's work, the present model has two major limitations. First, the model does not include any of the effects that occur at high injection levels. This includes base push-out due to the Ki rk effect [34], and the emitter and base resistance drops. To avoid complications due to the Ki rk effect, Jc is restricted to values below 10 s A / c m 2 [19]. In addition, the model only takes intrinsic recombination mechanisms into account. Recombination at the surface of the emitter periphery and in the external base are neglected. These surface recombination effects are very difficult to model because of their two-dimensional nature. However, in newer H B T s with better passivated surfaces and graded bases, surface recombination is often suppressed [19]. Unless otherwise specified, all the calculations presented in this chapter are based on the pyramidal H B T structure shown in Figure 2.6, whose geometrical and doping parameters are given in Table 2.1. The surface recombination velocities at all contacts are taken to be infinite and the actual base-emitter junction is taken to be ungraded. The base-collector reverse-bias voltage is set to - 3 V . The fiber had a diameter of 50pm, and light 49 from it was assumed to illuminate the entire base region (see Figure 4.3). While the emitter, base, and collector current densities can be converted to the respective currents by multiplying the various current densities by the emitter area, in reality, however, light absorption takes place only in the base region which is not under the emitter and base contacts. This means that the optically generated currents in the base and collector wi l l be under-estimated. Hence, in calculating F, a scaling factor is included to take into account the difference between the area of absorption in the base region and the emitter area. This factor is the ratio of the base area not covered by the emitter and base contacts to the emitter area. 3.1 DC Characteristics 3.1.1 Emitter and Collector Current Density Components Recalling from Eqs. (2.46) and (2.47), the emitter and collector current densities for the H B T under illumination are JE = An e k T - \ +J„-J, OPTl OPTl (3.1) 0PT4 + J, OPTS (3.2) 50 The dependence of the emitter current density and its various current density components on the base-emitter voltage are shown in Figure 3.1. Note that A^e^^-Y) and A12(g*v*^*r-1) have been replaced by A l l and A12, respectively. 0.2 0.4 0.6 0.8 1 Base-Emitter Voltage (V) Figure 3.1 Plot of emitter current density and its various components with respect to BE' From Figure 3.1, it can be seen that JE is dominated by Jopn at low-bias. As the base-emitter voltage starts to increase, JOPn becomes dominant as Jom starts to decrease. Finally, A l l becomes the dominating factor when VBE is about 1.25V. Hence, JE starts with a negative value and makes the transition to a positive value as VBE increases from OV 51 to 1.6V, i.e., when A l l becomes dominant. Note that the magnitude of JE is plotted in Figure 3.1. J0Pn decreases as VBE increases because the emitter-base depletion region shrinks under increased forward-bias. This leads to a decrease in the number of electron-hole pairs generated in the emitter-base depletion region. The reason for the decrease in the magnitude of Jom as VBE is increased lies in the energy barrier present in the conduction band at the emitter-base junction. With increasing bias, this energy barrier increases and further blocks the flow of electrons from the base into the emitter. Base-Emitter Voltage (V) Figure 3.2 Plot of emitter current density and its various components with respect to V (zoomed in). 52 JR increases with VBE. A l l increases exponentially with VBE, with an ideality factor of about 1.09. This is a typical value for a transport current which is mainly due to tunneling [35]. A12 decreases with VBE and remains insignificant compared to A l l and JR. As with the case of 7 o m , the reason for the decrease in A12 as VBE increases is due to the presence of an energy barrier in the conduction band at the emitter-base junction. With increasing bias, this energy barrier increases and makes it more difficult for electrons to back-inject from the base into the emitter, thereby causing A12 to decrease. Figure 3.1 is redrawn in Figure 3.2 with a smaller emitter current density range to enhance resolution. 53 The dependence of the collector current density and its various components on the base-emitter voltage are shown in Figure 3.3. Note that A21(e,v***r-1) and A22(e,v*^*r-1) have been replaced by A21 and A22, respectively. From Figure 3.3, it can be seen that Jc is dominated by Jom at low-bias. As the base-emitter voltage starts to increase, A21 becomes the dominating factor. Unlike Jopn, which decreases as V^. increases, 7 o m , being independent of V^, remains constant. 28 26 24 CM < §22 £ 20 CD D g 18 n O 16 14 12 Vbc=-3V Vbc=-2V Vbc=-1V Vbc=0V Jopt5 Jopt5 Jopt5 Jopt3 _ Jopt5 I 1 - -Popt=1mW i i i i 1 0.2 0.4 0.6 0.8 1 Base-Emitter Voltage (V) 1.2 1.4 1.6 Figure 3.4 Plot of J0PTi and Joprs with respect to VBE. Figure 3.4 shows the dependence of J0PTi and J0PTS on for different values of VBC. With a fixed VBC, J0PT5 remains constant and is independent of VBE because the quasi-neutral 54 collector width is independent of VBE. Jopn, however, is seen to increase with VBE. As mentioned earlier, Jopn decreases as VBE increases due to the rise in the energy barrier in the conduction band at the base-emitter junction. These electrons, which find it increasingly more difficult to overcome the energy barrier as the bias is increased, wi l l start to accumulate in the neutral base and diffuse across the base toward the collector. Finally, when they reach the base-collector depletion region, they wil l be sucked into the collector region. This results in an increase in J0PTi, which is observed in Figure 3.4. The increase in J0PT3 as VBE is increased is reflected in the increase in J o m + J0PT5 with increasing bias as shown in Figure 3.3. As the base-collector junction becomes more reverse-biased, the quasi-neutral collector region shrinks and causes J0PT5 to decrease. However, since the change in depletion region width with VBC occurs predominantly on the collector side, J o m changes only marginally with VBC as observed in Figure 3.4. From Figure 3.3, A21 is observed to increase exponentially with VBE, with an ideality factor of about 1.09. This is a typical value for a transport current which is mainly due to tunneling [35]. A 2 2 and JG, however, being independent on VBE, remain constant. From both Figures 3.2 and 3.3, it can be concluded that optical generation occurs predominantly in the base-collector depletion region. This is not unexpected, given that efficient separation of photogenerated electrons and holes takes place in the field region, and that the depletion region width (at VBC = - 3 V ) is so large (4570A). 55 3.1.2 Collector and Base Current Densities 10" 10' 10' I 10° I .£10" 2 CO c co CO 5 1 0 -10 10 Popt=0mW Popt=1mW Popt=2mW Popt=3mW 0.2 0.4 0.6 0.8 1 Base-Emitter Voltage (V) 1.2 1.4 1.6 Figure 3.5 Dependence of collector current density on VBE and optical power. The collector current density and its relationship with VBE, with and without optical excitation, is shown in Figure 3.5. The solid line represents the collector current density in the dark. The other lines represent the collector current density with injected optical powers of lmW, 2mW, and 3mW. From Figure 3.5, it can be seen that, at low-bias, the collector is flooded by the optically generated carriers. As the bias is increased, the carriers generated electrically start to increase exponentially. Finally, when VBE 56 approaches 1.45V, the carrier population generated electrically becomes comparable to that generated optically. Beyond this, electrical generation dominates over optical generation. Figure 3.5 is redrawn in Figure 3.6 with a linear scale for the collector current density. Note that, from Figure 3.6, as the optical power increases from lmW to 2mW to 3mW, the difference between the optically generated collector current density from each case to the next is the same, thereby displaying a linear relationship. This is the expected relationship with optical flux for Jopn (see Eq. 2.43). 300 250 w 200 E £ 150 <x> O c CD 100 50 1 1 T 1 I Popt=3mW Popt=2mW " I " /'••; / : / : • I • I ;' 1 1 1 i 1 -/ / / / -/ - Popt=1mW / • ' 1 -i i i Popt=0mW / i ' i 0.2 0.4 0.6 0.8 1 Base-Emitter Voltage (V) 1.2 1.4 1.6 Figure 3.6 Dependence of collector current density on (linear scale). 57 The dependence of the collector current density on the base-emitter voltage for different amounts of base grading and under illumination (lmW) is shown in Figure 3.7. 10 F 10' | i o 4 to c 0) o gio 3 o 10' 10 : 1 1 : 1 1 1 : r xbe=0 / / / / xbe=0.1 / / xbe=0.2 / / r — xbe=0.3 / / " '• Popt=1mW / ' '• / ' / ' / •' ' r / / ' -/ / / / ' ' , 1 1 1 1 1.1 1.2 1.3 1.4 Base-Emitter Voltage (V) 1.5 1.6 Figure 3.7 Dependence of collector current density on VBE for different amounts of base grading with optical power of lmW. As with the pure electrical case, the magnitude of the collector current density decreases as the amount of base grading increases in the region where electrical generation is dominant, i.e., when VBE in larger than 1.3V. This can be directly attributed to a smaller saturation current density (A21) [19]. As Ho stated, the major contribution to the decrease of An with increasing \ b e is the exponential decline of the intrinsic carrier concentration, 58 nt0, which is about 1000 times smaller for Al 0 3Ga 0_ 7As than that for GaAs, due to the differences in bandgap. The base current density can be obtained by subtracting Jc from JE using the expressions given in Eqs. (3.1) and (3.2), i.e., JB = JE - Jc. Figure 3.8 illustrates the dependence of the base current density on the base-emitter voltage for different optical powers. Note that the magnitude of JB is used in Figure 3.8. 1.2 1.3 1.4 Base-Emitter Voltage (V) Figure 3.8 Dependence of the magnitude of the base current density on VBE for varying amounts of optical power. 59 A t low-bias and under illumination, JB can be approximated by —Jc because the magnitude of Jc is much larger than the magnitude of JE, and hence, has a large negative value. As VBE increases beyond 1.2V, JE starts to increase positively at a faster rate compared to Jc. A t this point, JB is observed to increase positively as well. Finally, when JE becomes larger than Jc, JB changes to a positive value. As optical power increases, Jc increases while JE becomes more negative at low bias. As a result, VBE has to be raised to a higher value before JB can make the transition from a negative to a positive value. This explains the observed shift to the right in Figure 3.8 of the dips where JB changes from a negative to a positive value. Figure 3.9 illustrates the dependence of the base current density on the base-emitter voltage for varying amounts of base grading. Note that the magnitude of JB is used in Figure 3.9. As mentioned before, the magnitude of JB is approximately equal to the magnitude of Jc at low VBE. Because J o m (photogeneration in the base-collector space-charge region) dominates the photocurrent and is independent of xbe, the JB curves do not alter at all with xbe at this bias. This is in contrast to the pure electrical case, where S R H recombination is the dominant space-charge recombination process for all degrees of base grading [19]. As VBE increases, JE starts to increase (in the positive sense). Finally, when enough carriers are generated electrically through the forward-bias at the emitter-base junction, JB makes a transition from being a negative current to being a positive current (into the base). This explains the dips observed in Figure 3.9 when VBE ~ 1.55V. A t high-bias, the electrically generated carriers dominate. In this case, JB is dominated by 60 JRMg, JRrad, and I/„(0)-/(W) | (neutral base recombination) at low xbe, and by Jp{xE) (back-injected hole current from base into the emitter) at high xbe [19], as the base-emitter junction becomes more like a homojunction. Base-Emitter Voltage (V) Figure 3.9 Dependence of the magnitude of the base current density on VBE for varying amounts of xbe. 3.1.3 Optical Gain The optical gain can be expressed as G = Jc(oTpt)/(qF)/(\-R), where /c(opt) = JOPTi + J +J (the optically generated collector current density) and F(l-R) is that fraction 61 of the photon flux density actually injected into the device. As can be anticipated from Figure 3.3, which shows / c(opt) to be essentially independent of bias, there wil l not be much dependence of optical gain on the total collector current. As well, the optical gain wil l be independent of base grading because / c(opt), being dominated by Jom, is independent of base grading. Figure 3.7 shows that Jc is independent of base grading when optical generation is dominant, i.e., when Jopn is dominant. The optical gain calculated for the H B T with parameters specified in Figure 2.6 and Table 2.1, and with VBC = - 3 V is 0.39. 3.2 High-Frequency Characteristics Using the equations in Section 2.3.1 and the physical parameters listed in Table 2.1, the parasitic resistances are calculated as follows for VBC = —3V: REC = 0 .09^ REX = 0.04ft RR = 28.15ft R„ = 132.5ft Rm = 50.60ft JSC o A HI Rcc = 3.27ft RCB = 64.14ft Ra = 0.98ft These resistances are calculated for the case of no base grading. A l l but the three base resistances are bias-independent. The base resistances change by a few tens of milliohms as VBE changes from 0.1 to 1.6V and can be treated as bias-independent [19]. The collector capacitances, computed for VBC = - 3 V in the case of no base grading, are C =11.68fF C = 4 5 . 1 8 f F C = 25.70fF 62 The remaining hybrid-7t model components are dependent on VBE, and are calculated separately. Figure 3.10 shows the hybrid-7t equivalent circuit used in the analysis. Figure 3.10 Hybrid-Tr, equivalent circuit for a transistor. Note that r is the input resistance, and can be expressed as " l - a „ where r is the emitter differential resistance and a n is the D C common-base short-circuit e 0 gain. RE, RB, and Rc are the emitter, base, and collector resistances, respectively, while Cc. and CE. are the collector and emitter junction capacitances, respectively. In calculating the cut-off frequency and the maximum frequency of oscillation, the circuit in Figure 3.10 is 63 used for both the dark case and the iUurninated case. This is possible because D A P H N E takes into account the changes in current density (at the same VBE as in the dark case) due to optical excitation when calculating the emitter differential resistance, the output resistance, the reverse feedback resistance, and the transconductance ( r , r, r, and g , 7 7 x e7 o7 p/ O respectively) of the hybrid-Tt equivalent circuit. Optical injection enhances the rate of increase of JE with respect to VBE, leading to a decrease in re. As a result, an iUuminated H B T wi l l have higher fT mdfmax in the region dominated by optical generation compared to an H B T in the dark even i f both have the same biases (voltage). S21 is defined as the wave ratio of the transmitted wave at the output to the incident wave at the input when the output is attached to a matched load, i.e., the forward-transmission gain with output terminated in a matched load. To calculate 5 2 1 , the components of the hybrid-rc equivalent circuit for the transistor (shown in Figure 3.10) are first evaluated at a chosen bias using D A P H N E . Using these values in a more complete representation of the measurement circuitry (see Section 3.2.3), Sn is then calculated using a software package available at Telecommunications Research Laboratory (TRLabs), Edmonton called Microwave Design System (MDS). 3.2.1 Cut-off Frequency The dependence of the cut-off frequency on Jc is illustrated in Figure 3.11. The solid line represents the cut-off frequency for the dark case while the other lines represent the cut-off frequencies for optical powers of l m W , 2mW, and 3mW. From Figure 3.11, it is shown that cut-off frequency decreases with increasing optical power at a given 64 collector current density. At first glance, this seems to contradict the general assumption that optical injection enhances device performance. On further examination, this is found not to be the case. Figure 3.11 Dependence of cut-off frequency on Jc. A more meaningful comparison is the variation of fT with respect to VBE, which is shown in Figure 3.12. Notice in Figure 3.12 that/ r decreases as VBE increases. The reason for the drop in fT as VBE increases is due to the increase in the emitter differential resistance. This in turn is due to the fact that the rate of increase of JE with respect to VBE diminishes as VKK increases for the region where Jopn and Jopn are still the dominating 65 components in JE. Finally, when A l 1 becomes significantly large, the emitter differential resistance starts to drop again, resulting in the observed rise in fT as VBE increases further. Figure 3.12 also shows that, for a fixed bias,/r increases as the optical power is increased. This is a direct result of the decrease in the emitter differential resistance as optical power is increased. Base-Emitter Voltage (V) Figure 3.12 Dependence of cut-off frequency on VBE. The dependence of the cut-off frequency on Jc with varying amounts of base grading and under lmW of optical injection is illustrated in Figure 3.13. As with the pure electrical case [19], increasing the Al mole fraction causes the cut-off frequency to 66 increase in the region dominated by electrical generation. This is due to the reduction of base transit time with base grading. This, in turn, is due to the aiding field in the base and the associated decrease in neutral base recombination [19]. In the region where optical generation dominates, fT is independent of xbe. - • ' — ' — 1 ' ' ' ' 1 ' 1 — • • • • •"-n 1 1—1— •< — • /.y /•/ /;/ '•/ '/ 7 xbe=0 xbe=0.1 xbe=0.2 - - xbe=0.3 Popt=1mW _L_L-J 1 • L _ 103 104 io5 Collector Current Density (A/cmA2) Figure 3.13 Dependence of cut-off frequency on Jc with varing xbe. 3.2.2 Maximum Frequency of Oscillation The dependence of the maximum frequency of oscillation on Jc is illustrated in Figure 3.14. The solid line represents the maximum frequency of oscillation for the dark 67 case while the other lines represent the maximum frequency of oscillation for optical powers of lmW, 2mW, and 3mW. Again, the maximum frequency of oscillation is seen to decrease with increasing optical power for a fixed value of Jc. Because RB and Cc are fixed under these circumstances, fmax follows fv so the results of Figure 3.14 are similar in form to those for/ r shown in Figure 3.11. Collector Current Density (A/cmA2) Figure 3.14 Dependence of maximum frequency of oscillation on Jc. The variation of/mai with respect to VBE is shown in Figure 3.15. Over the bias range OV < VBE < 1.6V, RBCC changes by approximately 0.35%. Thus, once again, fmax is dominated by the bias dependence of/ r and so Figure 3.15 is of similar form to Figure 3.12. 68 Base-Emitter Voltage (V) Figure 3.15 Dependence of maximum frequency of oscillation on VBE. The dependence of the maximum frequency of oscillation on Jc with varying amounts of base grading and under lmW of optical power is illustrated in Figure 3.16. As with the pure electrical case [19], increasing the Al mole fraction causes the maximum frequency of oscillation to increase in the region dominated by electrical generation. By comparing Figures 3.16 and 3.13, it is noted that/max is less dependent than/r on the variation of Jc This is due to the square root dependence of fmax on fr In the region where optical generation dominates, is independent of xbe. 69 3.2.3 521 Gain Response As mentioned earlier, to calculate 521, the components of the hybrid-7t equivalent circuit are first evaluated at a chosen bias using DAPHNE. Using these values, 521 is calculated using a software package available at TRLabs called MDS. Using the equations in Section 2.3.1 and the physical parameters listed in Table 2.1, the hybrid-7T component values for applied biases of VBE = 1.6V and VBC = -3V are calculated as follows: 70 RE = 0.3144ft RB = 211.23ft Rc = 68.39ft r = 132.83ft r = 273.3Kft r = 276.5Kft' n o n g =0.112S C =82.56fF C = 442.2fF Om Cj Ej RE, RB, and Rc are the emitter, base, and collector resistances, respectively; r^, ro, and are the input, output, and reverse feedback resistances, respectively; gm is the transconductance; Cc. and CE. are the collector and emitter junction capacitances, respectively. The plots of Sn, with and without illumination, as a function of frequency for VBE -1.6V are shown in Figures 3.17 and 3.18, respectively. The optical power used is l m W . The 3dB bandwidths for the dark case and the illuininated case are 950MHz and 1GHz, respectively. The difference in the 3dB bandwidths between the dark case and the illuminated case arises from the slight difference in the equivalent circuit used. The equivalent circuits used to calculate Sn of the H B T in the dark and under illumination are displayed in Figures 3.19 and 3.20, respectively. Notice that, in Figure 3.19, Port 1 is connected to the base, while in Figure 3.20, Port 1 is connected to the base-collector space-charge region via a voltage controlled current source. The subcircuit that controls this current source represents the laser diode that produces the optical signal. The transconductance is chosen arbitrarily to model the actual optical link. Figures 3.19 and 3.20 are based on the hybrid-n; model shown in Figure 3.10, but with inductors included to account for the inductive effects of the wires used as a result of the wire bonding process. The inductors values are L £ = 2nH LB = I n H L c = 2nH 71 5 2 ] is calculated using a program available in TRLabs called MDS. This is accomplished simply by inserting the S-Parameter Ports 1 and 2 at the input and output connections, respectively, of the equivalent circuit using MDS. The S-Parameter Ports are built-in functions of MDS. Notice in Figure 3.20 that there is an extra current source between the base and collector. This current source represents the optical generation that occurs in the base-collector depletion region. Note that optical generation is ignored in the neutral base and collector regions since optical generation occurs predominantly in the base-collector depletion region.. The emitter is connected to a 50Q termination in the uluminated case. \? 1 \ \ r \ \ \ \ V *-45 .0 MHz f r e q 5.0 GHz Figure 3.17 Plot of S21 in the dark. 72 73 CM BB B5 M O O 5 & 6-W v — • • o « o <: r-•a CD •5 a CO OB 1 3 o o .O I > co o •a T 3 C 3 C Q S OA 13 o -a o l-l .o 3 .8 0 1 w o (S P OA PL, CHAPTER 4: EXPERIMENTAL RESULTS In this chapter, the process of preparing the H B T device for measurement is presented. As well, the experimental setups used to obtain the D C and high-frequency measurements are described. Finally, the measured D C and high-frequency results are presented. 4.1 Device Preparation Ideally, the experimental H B T s would have a transparent emitter contact through which the illumination could be directed. Some devices with such a contact, using indium-tin-oxide metallization have been reported [36]. The University of Michigan was contacted to see i f indium-tin-oxide H B T s could be obtained for the present study, but none were available. Instead devices from C R C were used. These had conventional gold metallization but a relatively large spacing between the emitter stack and collector contacts, which could be used for the illumination aperture. The H B T s were components of a much larger circuitry which forms a reticle. The layout of such a reticle is shown in Figure 4.1. Each reticle contained only six devices which wil l work. This is due to the fabrication process involved. A l l the six devices were located at the top right hand corner of the reticle. In order to process the individual devices in preparation for packaging, the wafer containing the reticles must first be diced to obtain the individual H B T s . This was accomplished by using the micro dicing saw at 76 AMC. Figure 4.2 shows the plan view of one of the device blocks. The dimensions of the emitter, base, and collector metallization pads are also shown. Figure 4.1 Layout of circuitry containing HBT devices from CRC. 77 EMITTER I EMITTER j 340um Figure 4.2 Plan view of a device block. Schematics of the H B T s are shown in Figure 4.3. Figure 4.3a shows the plan view of the structure of the actual device. Figure 4.3b shows the cross-sectional view of the device. While it may appear in Figure 4.2 that the H B T has two emitter contacts, it has in fact only one. The rectangles and squares in black shown in Figure 4.2 are actually metallization pads that are connected to the respective contacts. As mentioned earlier, the reason for choosing these C R C devices is because these devices have a large gap between the base contact and the collector contact, see Figure 4.3b. While it is assumed in the model that light penetrates the entire device, in reality, the contacts are opaque and hence, there is no optical injection in the areas under the contacts. With the majority of the carriers being generated optically in the base-collector space-charge region, it seems reasonable that a device which has a large SBC wi l l capture a sufficient amount of the incident light in order to demonstrate a significant response. The tradeoff, however, is that f_ and f w i l l deteriorate as a result. J T J max 78 | Collector 1 contact . . . . . . . . . . . . . . . . . . | Py t fM 11 Base conta j tiesa H / I T ctpil IIIIIlT • ] 1 | Hast- pedestal SI SUBSTRATE j a) SC SI SUBSTRATE! b) Figure 4.3 Schematic of the H B T : a) Plan view b) Cross sectional view. 79 Having diced the wafer to obtain the H B T s , the next step was to design the alumina substrate upon which to mount the device. The design was done using a drawing package available at A M C called L-edit [37]. When the design was completed, it was transferred to a G D S file, which was then sent to P P M Photomask, a company in St.-Laurent, Quebec, where the chrome-on-glass mask was fabricated. After the completion of the mask, a chrome-gold photolithography run was performed at A M C to transfer the pattern from the mask to a 25mil thick, 2inch by 2inch alumina substrate. The substrate was then sent to Laser Processing Technology (LPT) in Portland, Oregon, where via holes were drilled using a carbon dioxide laser. The design of the alumina substrate is shown in Figure 4.4. Gold Trace Grounding Pad H B T Spot Alumina Substrate Figure 4.4 Design of the alumina substrate. 80 After preparation of both the device and the alumina substrate, the next step was to mount the H B T onto the substrate. This was done by using a non-conductive crystalline gel. A small piece of the gel was placed at the centre of the substrate. Upon heating, it melted. The H B T was then placed on the gel and the substrate allowed to cool down to room temperature. This process attached securely the H B T to the alumina substrate. The next step in the packaging process was to connect the metallization pads on the H B T to the traces on the alumina substrate. This was done at A M C using the K & S manual wedge bonder. 1.25mil diameter aluminum wire, was used. Considerable difficulty was encountered during the learning process to perfect a bonding technique. The wire wi l l not adhere to the surface of the metallization i f too little force is applied on the wedge. On the other hand, i f too much force is applied, the metallization pad wil l be torn off completely. To complicate matters further, the device was thick. This created a big step between the top of the device and the surface of the alumina substrate, making the bonding process more challenging. Eventually, success was achieved by improvising on a procedure suggested by Dr. Jackson of U B C . Dr. Jackson suggested that the bonding should be done in reverse, making a bond on the substrate first and drawing the wire to the metallization pad. Instead of completing the loop by making the second bond, the wire should be broken off, leaving a tail of considerable length, enough such that the tail can be used to make the second bond by squishing it with the tip of the wedge. Based on this idea, the author used the wedge to squish the wire first (to make it easier to bond and requiring less force) before making the first bond on the metallization pad, and completed the second bond on the substrate in the usual way. 81 — Middle Piece • End Piece Figure 4.5 Plan view of the aluminum casing. The aluminum casing for the alumina substrate is shown in Figure 4.5. It comprises three pieces. It was designed by the author and produced by the machine shop located in the Electrical Engineering Department at the University of Alberta. The alumina substrate fits on the floor of the large middle piece. The alumina substrate was' secured to the aluminum block by four plastic screws at each of its corners. The two end pieces are each fitted with a gold-plated female S M A socket. The centre conductors of the S M A connectors are connected to the input and output traces on the substrate with EpoTek's H 2 0 E two-part, heat-cured conductive silver epoxy. The input and output traces refer to the traces on the substrate connected to the base and collector of the H B T , respectively. Each of the two end pieces is held to the middle piece by two screws. Connection between the ground pad and the aluminum block is made through the 82 conductive silver epoxy deposited on the walls of the via holes. To ensure good grounding, a layer of gold is deposited on the under side of the alumina substrate, and the conductive silver epoxy is extended to the hps of the via holes on both sides of the alumina substrate. 4.2 Experimental Setup The experimental setup for the D C measurements is illustrated in Figure 4.6. The H P 4145A semiconductor parameter analyzer (from A M C ) was used to provide the stimulus and to measure the D C response of the device. The packaged device was placed on a probe station with the probes making contacts with the traces on the substrate to provide the bias as well as to monitor the response through the parameter analyzer. The computer controls the parameter analyzer and stores the exported data. The optical source used was a 780nm, variable-power laser source with the laser driver current controlling the optical power. The optical power is coupled by a multimode fiber which ends right over the surface of the H B T . To minimize optical loss, the fiber end was cleaved and the spacing between the fiber and the H B T was minimized. 83 Laser Diode r. P C Parameter Analyzer 780nm Fiber llllllll Probe JUKI Figure 4.6 Experimental setup for D C measurements. The experimental setup for the high-frequency measurements in the case of optical injection is illustrated in Figure 4.7. The H P 8510B network analyzer at TRLabs was used to provide the stimulus and to measure the high-frequency response of the device. The laser used was the same as the one used in the D C measurements, and has a bandwidth of about 1GHz. D C biases from power supplies were fed to the base and the collector of the H B T through an external bias tee and Port 2 of the network analyzer, respectively. Each port of the network analyzer has a built-in bias tee to allow a small signal to be applied on top of a D C bias. Port 1 of the network analyzer was connected to the laser driver to provide the small signal input while Port 2 of the network analyzer was connected to the collector to measure the response. The bias tee attached to the base of the H B T was connected to a 50ft termination. 84 Laser Diode Network Analyzer Portl Port 2 780nm Fiber T E R M Bias T R F Output (Collector) Probe Figure 4.7 Experimental setup for high-frequency measurements. To measure the electrical high-frequency characteristics (in the dark), the laser was switched off and Port 1 of the network analyzer was connected directly to the base of the H B T . The D C bias to the base was then applied to Port 1 of the network analyzer with the external bias tee i n Figure 4.7 removed: 4.3 Experimental Measurements The experimental results, unless otherwise specified, were measured using the experimental setups described in the previous section. The packaged device has an emitter size of 2/im x 15jUm. The dimensions of the complete device are shown in Figure 2.6. 85 4.3.1 D C Characteristics Base-Emitter Voltage (V) Figure 4.8 Measured dependence of Ic on VBE (log scale). Using the experimental setup shown in Figure 4.6, measurement of the DC characteristics of the packaged HBT was performed (the packaging process did not alter the measured data). The dependence of collector current on base-emitter voltage for varying amounts of optical power is illustrated in Figure 4.8. Optical power was varied by changing the drive current though the laser diode. From Figure 4.8, it is difficult to distinguish the individual curves for optical power ranging from 0.2mW to 5.5mW. Hence, a linear scale is used in Figure 4.9 for the collector current instead. 86 Base-Emitter Voltage (V) Figure 4.9 Measured dependence of Ic on VBE (linear scale). Note that, from Figure 4.9, the measured collector current (optically generated) increases quite linearly when the injected optical power increases. The dependence of the measured base current on the base-emitter voltage for varying amounts of optical excitation is shown in Figure 4.10. Note that the magnitude of the base current is shown in Figure 4.10. The magnitude of the base current increases quite linearly with the injected optical power for low VBE because JB = Jr As mentioned earlier, JE becomes larger than Jc at higher values of VBE when the optical power injected is 87 increased. This causes the observed shift of the dips (where JB changes from negative to positive) to the right in Figure 4.10 when optical power is increased. 10' •10 - - Popt=0W, Vbc=0V 0.2mW<Popt<5.5mW 0 2 0.4 0.6 0.8 1 1.2 Base-Emitter Voltage (V) 1.4 1.6 Figure 4.10 Measured dependence of IB on VBE. Figure 4.11 shows the measured common-emitter characteristics of the packaged device. Optical injection is seen to increase the collector current. The sudden drop in collector current as the collector-emitter voltage increases is possibly due to the switching of the device from the transistor mode to a resistive mode. The device behaves like a transistor when VCE is low. At higher VCE, the base-collector junction becomes reverse-biased and the device behaves like a resistor. The reason for the switching behavior is 8 8 probably due to the poor metallization on the experimental device since, as mentioned earlier, packaging did not alter the measured data. Figure 4.11 Measured common-emitter characteristics of the packaged device. JB is varied from OmA to 1.2mA with increments of 0.2mA. 4.3.2 High-Frequency Characteristics The high-frequency characteristics of the packaged HBT were measured using the experiment setup shown in Figure 4.7. The measured gain response (521) of the device in 89 the dark is shown in Figure 4.12. The applied biases were 1.6V and 2.2V for VBE and VCE respectively. 5 2 i Iog MAG R E F 0 . 0 dB 1 1 0 . 0 d B / V 2 . 6 4 9 8 dB hp M A R < E R 6 5 . : 1 2 7 5 M H z L S T A R T 0 . 0 4 5 0 0 0 0 0 0 GHz STOP 5 . 0 0 0 0 0 0 0 0 0 GHz Figure 4.12 Measured gain response of the packaged HBT in the dark. 90 S 2 1 log MAG REF -50.0 dB 1 10.0 dB/ -34.355 dB hp MAR <ER 36 . 1 ^75 MHz - . _ i I START 0.045000000 GHz STOP 2.00000000® GHz Figure 4.13 Measured gain response of the packaged HBT under iUumination. The 3dB bandwidth, as indicated by the vertical arrow marker in Figure 4.12, is about 565MHz. The low S2l (5.6dB) at 45MHz, the lowest frequency measured, of the HBT is due to the fabrication process involved. As mentioned earlier, the HBT used is an experimental device with inconsistencies in the quality of the gold metallization. As a result, the gain of a device can differ significantly from that of the other devices, even 91 i f they belonged to the same batch. The reason for using this device, even though it did not have the highest measured S21, is because it had the most complete set of D C and A C measurements. A n earlier batch of devices, which the author failed to package successfully, had a CRC-measured 5 2 1 of 25dB. The measured gain response (5 2 ]) of the device under iUurnination is shown in Figure 4.13. The 3dB bandwidth, as shown in Figure 4.13, is about 837MHz, and might be limited by the 3dB bandwidth of the laser used. The laser has a 3dB bandwidth of about 1GHz and is very close to the measured 3dB gain bandwidth of the device. The applied biases were 1.6V and 2.2V for VBE and VCE, respectively. The optical power used was 1.8mW. To see i f the 3dB bandwidth was limited by the laser, the packaged device was sent for measurement to the Communication Laboratory at the University of Saskatchewan in Saskatoon, where an externally modulated laser diode was available. The setup is identical to the one shown in Figure 4.7, except that, in place of the laser diode which is modulated directly by Port 1 of the network analyzer, the laser diode is now biased at a fixed point. It produces light output at 830nm and is fed into a Mach-Zehnder optical modulator, which is driven by Port 1 of the network analyzer through a wideband amplifier, thereby producing the necessary R F power to drive the modulator. One of the complementary outputs of the modulator is monitored by an optical power meter. The optical setup is shown in Figure 4.14. With this setup, modulation up to a frequency of 18GHz is possible. 92 +v o Laser Diode Mach-Zehnder Modulator \ / / \ Fiber V Optical Power Meter Miteq Wideband Amplifier 18dB, 18GHz Port 1 of Network Analyzer Figure 4.14 Optical setup used in Saskatoon. Using the setup shown in Figure 4.14, the gain response (52 1) of the packaged H B T was re-measured in Saskatoon and the result shown in Figure 4.15. Again, the applied biases were 1.6V and 2.2V for VBE and VCE, respectively, and the optical power used was 1.8mW. From Figure 4.15, the 3dB bandwidth is 610MHz, which is lower than that originally measured. It is not possible, therefore, to draw any conclusion about whether the H B T or the laser in the earlier measurement (Figure 4.13) was limiting the performance. The measured 5 2 1 in Figure 4.15 is also much higher due to the additional amplification achieved as a result of the amplifier used in between Port 1 of the network 93 analyzer and the Mach-Zehnder modulator. The reference level (OdB) shown in Figure 4.15 corresponds to a responsivity of 0.065A/W. S 2 1 Iog MAG REF 0 . 0 dB 2 5 . 0 d B / -2.984-4- dB hp 1 . A \ G KER ;09 . 2 - 1 7 437 5 Mr Hz START 0.04-5000000 GHz STOP 1 0 . 0 0 0 0 0 0 0 0 0 GHz Figure 4.15 Re-measured gain response of the packaged HBT under illumination. 94 C H A P T E R 5: C O M P A R I S O N O F M O D E L A N D M E A S U R E M E N T S In this chapter, the measured data from Chapter 4 is compared with the simulation results from Chapter 3. The current density used in the simulation results must first be converted to current before any comparison can be made. This is done by multiplying the current density by SjLE. Because of the one-dimensional nature of the model, the emitter, base and collector currents are obtained by multiplying the respective current densities by the emitter area. This is done for both the dark and the iUuminated case. Comparison of the modeled and the measured D C characteristics wi l l first be presented. This is then followed by the A C characteristics. 5.1 DC Analysis The device parameters used in the simulations are taken from Figure 2.6 and Table 2.1. The base-collector voltage used was OV instead of - 3 V as in Chapter 3. This was to enable a comparison to be made between theory and experiment under bias conditions for which the transistor was not influenced by contact problems (see Figure 4.11). The plot of collector current vs base-emitter voltage for the simulated case and the measured case is shown in Figure 5.1 for the device in the dark. The simulated line with no fitting attempted uses the device parameters as indicated in Figure 2.6 and Table 2.1. The following changes were made to obtain an optimal fit between the simulated data and the measured data. SE = 3.3um LE = 16.3um 95 WGE = 100A (emitter junction grading) 10" 10" -2 10' 10" 10 § 10"5 F 10' -7 10 10" -8 10 -10 0.8 Popt=OmW Measurement Simulation (no fitting) — Simulation (fitting) 1.1 1.2 1.3 Base-Emitter Voltage (V) 1.6 Figure 5.1 Plot of collector current vs base-emitter voltage (dark case). The measured collector current is larger than the simulated collector current (no fitting) because the actual emitter size may be larger than that designed. This can probably be attributed to the diffraction which occurs in the photolithographic process during fabrication. Hence, a larger emitter size was used to fit the measured and simulated currents. Emitter junction grading was utilized because an abrupt change in aluminum mole fraction across the emitter-base junction is very difficult to achieve in practice; it is quite reasonable that the actual transistor used would behave more like a graded-junction 96 device than an abrupt junction device. It has been found necessary to" invoke a small amount of junction grading in achieving good fits of simulated results from DAPHNE to experimental data from some of Bell-Northern Research's high-performance devices [35]. From Figure 5.1, there is close agreement between the simulated (fitted) and the measured collector current for VBE < 1.2V. At higher values of VBE, the measured collector current falls below the model predictions. This can be attributed directly to the contact problems as illustrated in Figure 4.11, which affect the device not only when the base-collector region is re versed-biased, but also when VBE is larger than 1.2 V. 10° n!;;;!!:::~::::::::::::;:::::::::: 10"1 lO"2 Measurement Simulation (no fitting) — Simulation (fitting) j rrent (A) o Jill j!M::::::5:::::::::::i:::::::::::: .^ J^ <rT?T"T"~/ ': -O 10* 10"5 :::::::::::::::::::::::::::::::::::::::: .:pppt^ :1,6TOVy: .„-6 i I i '. i i 1 ' ' _ i \ I I 1 1 1 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Base-Emitter Voltage (V) Figure 5.2 Plot of collector current vs base-emitter voltage (muminated case). 97 The plot of collector current vs base-emitter voltage for the simulated case and the measured case under iUumination (1.8mW) is shown in Figure 5.2. Again, the simulated line with no fitting attempted uses the device parameters as presented in Figure 2.6 and Table 2.1. Fitting requires the changes mentioned earlier for achieving optimal fit in the dark case. From Figure 5.2, there is close agreement between the simulated (fitted) and the measured collector current for VBE < 1.2V. At higher values of VBE, the measured collector current falls below the model predictions. As with the dark case, this can be attributed direcdy to the contact problems as illustrated in Figure 4.11. Base-Emitter Voltage (V) Figure 5.3 Plot of base current vs base-emitter voltage. 98 The dependencies of the base current on the base-emitter voltage for the dark case and the illuminated case are shown in Figure 5.3. Note that the magnitude of the base current is displayed for the iUuminated case. The base current is under-estimated for the dark case when VBE is less than 1.55V. This is very likely due to large contributions from surface recombination, which is not treated in the model. As a result, no attempt was made to fit the measured and simulated base currents in the dark. There is, however, close agreement between the measured data and simulated data for the magnitude of the base current under iUurnination when V„„ is less than 1.1V. This is because the base current is BE dependent on the collector current (JB = -Jc) at this bias range, the range over which the model is able to accurately predict Jc, as shown in Figure 5.2. 5.2 A C Analysis • Simulation Measurement 45 200 400 600 800 Frequency (MHz) 1000 Figure 5.4 Plot of 5 2 1 in the dark. 99 The gain response of the device is illustrated in Figures 5.4 and 5.6. Figure 5.4 compares the gain response between the simulated result and the measured result for the device in the dark. Figure 5.6 compares the gain response between the simulated data and the measured data for the device under optical injection. From Figure 5.4, it is observed that the simulated data and the measured data both have a similar shape. As mentioned earlier, the low value of the S2: measured is not necessarily a true indication of the performance of the device as designed. It was mentioned before that, because of inconsistent metallization, the experimental devices had significantly different gains even though they were fabricated in the same way. The device presented here had one of the lower gains measured. LEGEND: 18 N=6B 8 S21 dB 6 4 2 8 -2 -4 -6 -8 - IB Figure 5.5 Plot of CRC-measured S21 on the second batch of devices. 100 Figure 5.5 illustrates the range of the CRC-measured S 2 1 obtained from measurement on the second batch of devices. The measured S 2 1 ranged from OdB to lOdB at 200MHz, and from - 5 d B to lOdB at 2GHz. The 3dB bandwidths for the simulated data and the measured data are 950MHz and 565MHz, respectively. From Figure 5.6, the 1st measurement refers to the data measured by the author at TRLabs and the 2nd measurement refers to the data measured at Saskatoon using the E O M . Again, the simulated data and the measured data have similar shapes, except for the difference in the 5 2 1 values. The reason for the large difference between the S2l of the 2nd measurement and the other data (simulation and 1st measurement) is because the setup for the optical source was different. The laser diode used was different, and there was an additional amplifier hooked up between port 1 of the network analyzer and the E O M . Notice that in Figure 5.4, the simulated data is larger than the measured data. In Figure 5.6, however, the simulated data is less than the first measurement. The reason for the difference between the S 2 1 of the first measurement and the simulation is that the transconductance of the voltage controlled current source used in Figure 3.20 to model the optical signal was chosen arbitrarily to model the optical excitation process; a higher value of transconductance would bring the curves (simulation and first measurement) into better agreement. To make a fairer comparison between simulation and measurement, i.e., to allow for the additional amplifier gain in measurement 2, and for a different transconductance in measurement 1, it is reasonable to normalize the results to the same S2l at the lowest frequency. This is done in Figure 5.7, from which it can be seen there is reasonable agreement between both sets of measured data and the simulated data. The 101 3dB bandwidths for the simulated data, the 1st measured data, and the 2nd measured data are 1GHz, 837MHz, and 610MHz, respectively. T J Frequency ( M H z ) Figure 5.6 Plot of S2l with optical injection. \ 1 1 1 h 600 800 1060 Frequency ( M H z ) Figure 5.7 Plot of normalized S21 with optical injection. • Simulation 2nd Measurement 1st Measurement • Simulation 2nd Measurement 1st Measurement 102 6. Summary 6.1 Conclusions In this thesis, the D C characteristics of the optically controlled AlGaAs /GaAs n-p-n Heterojunction Bipolar Transistor, such as emitter current, base current, and collector current have been examined in detail using a comprehensive one-dimensional analytical model. The H B T ' s high-frequency characteristics have also been studied through a quasi-two-dimensional model for pyramid-structured devices (shown in Figure 2.6). The following conclusions can be drawn from this work: 1. Optical generation occurs predominantly in the base-collector space-charge region. The optically generated carriers dominate when the base-emitter voltage is less than 1.2V. A t higher bias (greater than 1.4V), the electrically generated carriers become the dominating factor in determining the D C current. 2. Optical injection causes the collector current to increase significantly at low bias. However, at low bias, it causes the emitter and base currents to decrease significantly and flow in the opposite direction compared to the dark case. 3. Base grading has no effect on the base and collector currents when these currents are dominated by optically generated carriers at low bias. This is because the predominant optical generation process (generation in the space-charge region) is independent of base grading. Base grading starts to have an effect on the base and collector currents when the electrically generated carriers become significant. 103 4. Optical gain does not have much dependence on the base-emitter voltage because the optically generated currents change only slightly with bias. Optical gain is independent of base grading because the current generated in the base-collector space-charge region, which is the dominant optically generated current, is independent of base grading. 5. Optical injection improves the / 7 and/ r a m r of the device at low bias. This is due to the enhanced rate of increase of emitter current with respect to base-emitter voltage under fflumination, which then causes the emitter differential resistance to decrease. 6. Base grading has no effect o n / r and/ m a j . when the optically generated carriers are dominant. This is because the dominant optical generation process (in the base-collector space-charge region) is independent of base grading. When the electrically generated carriers become dominant, base grading improves the values o f / r mdfmax by reducing the neutral base recombination and hence, the base transit time, just like in the dark case. 7. Optical injection improves the S2l 3dB bandwidth of the device. 6.2 Considerations for Future Work There are two areas in which improvements are warranted. First, the present model should be improved to produce simulation results which wil l be in better agreement with the measured results. Second, better devices should be used in the measurement. 104 To obtain better agreement between the simulated and the measured results, the present model should be improved by including in the model the effects of the surface recombination process that occurs around the emitter periphery and the bulk recombination process that occurs in the external base region. These improvements, however, would require the present model to be extended to a two-dimensional model. The model should also be improved to yield simulated S21 which wil l be in better agreement with the measured S2V Instead of arbitrarily choosing the transconductance value for the current source between the base and the collector, more work should be done to better understand the process of optical excitation so as to be able to relate the transconductance value to the optical excitation process. H B T s with transparent emitter contacts should be used in the measurement because they allow maximum optical injection. Moreover, such devices do not need a large separation between the emitter stack and the collector contact to allow sufficient optical power to penetrate the device. This wi l l produce better fT and/ m a x . If such devices are still not available in the future, devices with better metallization should be used. 105 References [1] H. Kroemer, "Theory of a wide-gap emitter for transistors," Proc. IRE , vol . 45, pp. 1535-1537, Nov. 1957. [2] G . O. Ladd, D . L . Feucht, "Performance potential of high frequency heterojunction transistors," IEEE Trans. Electron Devices, vol . ED-17, pp. 413-420, May 1970. [3] H . Kroemer, "Heterostructure bipolar transistors and integrated circuits," Proc. IEEE, vol . 70, pp. 13-25, Jan. 1982. [4] J. C. Campbell and K . Ogawa, "Heterojunction phototransistors for long wavelength optical receivers," / . Appl. Phys., vol . 53, pp. 1203-1208, Feb. 1982. [5] S. Chandrasekhar, M . K . Hoppe, A . G . Dentai, C . H . Joyner, and G . J. Qua, "Demonstration of enhanced performance of an InP/InGaAs heterojunction phototransistor with a base terminal," IEEE Electron Device Lett., vol. 12, pp. 550-552, Oct. 1991. [6] H . Wang, D . Ankri , "Monokthic integrated photoreceiver implemented with GaAs /GaAlAs heterojunction bipolar phototransistor and transistors," Electron. Lett, vo l . 22, pp. 391-393, Mar. 1986. [7] S. Chandrasekhar, A . H . Gnauck, R. A . Hamm, and G . J. Qua, "The phototransistor revisited: All-bipolar monokthic photoreceiver at 2 Gb/s with high sensitivity," IEEE Trans. Electron Devices, vo l . 39, pp. 2677-2678, Nov. 1992. 106 [8] D . Fritzsche, E . Kuphal, and R. Aulbach, "Fast response InP/InGaAs heterojunctionphototransistors," Electron. Lett, vol . 17, pp. 178-180, 1981. [9] E . Suematsu, H . Ogawa, "Frequency response of HBT's as photodetectors," I E E E Microwave Guided Wave Let t , vol . 3, pp. 217-218, July 1993. [10] T. Mori izumi and K . Takahashi, "Theoretical analysis of heterojunction phototransistors," IEEE Trans. Electron Devices, vol. ED-19, pp. 152-159, Feb. 1972. [11] N . Chand, P. A . Houston, and P . . N . Robson, "Gain of a heterojunction bipolar phototransistor," IEEE Trans. Electron Devices, vol. ED-32 , pp. 622-627, Mar. 1985. [12] B . C. Roy and N . B . Chakrabarti, "Gain and frequency response of a graded-base heterojunction bipolar phototransistor," IEEE Trans. Electron Devices, vol. E D -34, pp. 1482-1490, July 1987. [13] A . A . Grinberg, M . S. Shur, R. J. Fischer, and H . Morkoc, " A n investigation of the effect of graded layers and tunneling on the performance of A lGaAs /GaAs heterojunction bipolar transistors," IEEE Trans. Electron Devices, vol. ED-31 , pp. 1758-1765, Dec. 1984. [14] M . S. Lundstrom, " A n Ebers-Moll model for the heterostructure bipolar transistor," Solid-State Electron., vol . 29, pp. 1173-1179, 1986. [15] S. C. M . Ho and D . L . Pulfrey, "The effect of base grading on the gain and high-frequency performance of AlGaAs/GaAs heterojunction bipolar transistors," IEEE Trans. Electron Devices, vol . 36, pp. 2173-2182, Oct. 1989. 107 [16] M . C. Brian, and D . R. Smith, "Phototransistors, A P D - F E T and P I N F E T optical receivers for 1-1.6 /urn wavelength," IEEE Trans. Electron Devices, vol. ED-30 , p. 390, 1983. [17] S. D . Personick, "Receiver design for optical fiber systems," Proc. IEEE, vol. 65, p. 1670, 1977. [18] Z . Y u and R. W . Dutton, " S E D A N III — A generalized electronic material device analysis program," Technical Rept., Integrated Circuits Laboratory, Stanford University, July 1985. [19] S. Ho , "The effect of base grading on the gain and high-frequency performance of AlGaAs /GaAs heterojunction bipolar transistors," M . A . S c . Thesis, University of British Columbia, 1989. [20] S. Searles and D . L . Pulfrey, " A n analysis of space-charge-region recombination in H B T ' s , " IEEE Trans. Electron Devices, vol . 41, pp. 476-483, Apr . 1994. [21] S. S. Perlman and D . L . Feucht, "p-n Heterojunctions," Solid-State Electron., vol. 7, pp. 911-923, 1964. [22] D . L . Pulfrey and S. Searles, "Electron quasi-ferrni level splitting at the base-emitter junction of AlGaAs/GaAs H B T ' s , " IEEE Trans. Electron Devices, vol. 40, pp.1183-1185, June 1993. [23] H . Kroemer, "Two integral relations pertaining to the electron transport through a bipolar transistor with a nonuniform energy gap in the base region," Solid-State Electron., vo l . 28, pp. 1101-1103, 1985. 108 [24] A . Bar-Lev, Semiconductor and Electronic Devices. 2nd Ed . , Englewood Cliffs: Prentice-Hall Inc., 1984. [25] H . F. Cooke, "Microwave transistors: theory and design," Proc. IEEE, vol. 59, pp. 1163-1181, Aug . 1971. [26] A . P. Laser, "Calculation of the maximum frequency of oscillation for microwave heterojunction bipolar transistors," M . A . S c . Thesis, University of British Columbia, 1990. [27] I. E . Getreu, Modeling the Bipolar Transistor. Amsterdam: Elsevier Scientific Publishing Co. , 1978. [28] S. M . Sze, Physics of Semiconductor Devices. 2nd Ed . , New York : John Wiley & Sons, 1981. [29] S. S. Tan and A . G . Milnes, "Consideration of the frequency performance potential of GaAs homojunction and heterojunction n-p-n transistors," IEEE Trans. Electron Devices, vol . ED-30, pp. 1289-1294, Oct. 1983. [30] D . A . Sutherland and P. D . Dapkus, "Optimizing n-p-n and p-n-p heterojunction bipolar transistors for speed," IEEE Trans. Electron Devices, vol. ED-34, pp. 367-377, Feb. 1987, [31] O. S. Ang , Source code for D A P H N E , University of British Columbia, unpublished, 1990. [32] A . St. Denis, Source code for D A P H N E , University of British Columbia, unpublished, 1994. 109 [33] E . S. Harmon, M . R. Melloch, and M . S. Lundstrom, "Effective band-gap shrinkage i n GaAs ," Appl. Phys. Lett. vol . 64, pp. 502-504, Jan. 1994. [34] C. T. Kirk , " A theory of transistor cut-off frequncy (/j.) fall-off at high current density," IEEE Trans. Electron Devices, vol . ED-34, pp. 367-377, Feb. 1987. [35] J. J. X . Feng, D . L . Pulfrey, J. Sitch, and R. Surridge, " A physics-based H B T S P I C E model for large-signal applications," IEEE Trans. Electron Devices, vol. 42, pp. 8-14, Jan. 1995. [36] M . Karakucuk, W. L i , P. Freeman, and J. East, "Transparent emitter contact H B T ' s for direct optical injection locking of oscillators," I E E E M T T - S Digest, pp. 1391-1393, Apr . 1994. [37] M . Graham, Alberta Microelectronics Centre, personal communication. [38] A . P. Laser, "Program documentation for H B T device simulation and fmax calculation," Department of Electrical Engineering, U B C , unpublished report, July 1990. 110 APPENDIX A: SOURCE CODE FOR DAPHNE The simulation package comprises five programs, namely: Makefile, Dhbt.inp, Dhbt.c, Gip.c, and Lib.c [38]. However, Stdin.c, Parse.c, Parse.y, Token.c, and Token.l have been replaced by Gip.c, which was written by A . St. Denis of U B C . The programs can be run on any Sun workstation. After compilation of the source code, a total of nine files exist. The device modeling program is incorporated into two sets of source code: Dhbt.c - main program Lib.c - library of subroutines The main program contains variable declarations, subroutine declarations, and controls the calculation of all the H B T parameters. The library consists of all subroutines called by the main program. The program requires that initial conditions be supplied via the input file called Dhbt.inp. This file lists the H B T parameters and initial bias conditions. Gip.c is the program that actually reads the values stored in Dhbt.inp and exports the values to Dhbt.c. Makefile allows the source codes to be compiled successfully. It generates the object files and links them up. 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