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USE OF TRANSIENTS IN QUASINEUTRAL REGIONS FOR CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS By TAEWONLJUNG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986 ACKNOWLEDGMENTS I wish to express my sincere appreciation to the chairman of my supervisory committee, Professor Fredrik A. Lindholm, for his guidance, encouragement, and support throughout the course of this work. I also thank Professor Arnost Neugroschel for his help in experiments, and Professors Peter T. Landsberg, Sheng S. Li, Dorothea E. Burk, and R. E. Hummel for their participation on my supervisory committee. I am grateful to Kevin S. Eshbaugh of Harris Semiconductor for Sparameter measurements, and to Dr. Taher Daud of the Jet Propulsion Laboratory and to Dr. Mark Spitzer of SPIRE Corp. and Mr. Peter Iles of Applied Solar Energy Corp. for discussions and for devices used in the experiments. Thanks are extended to my colleagues and friends, Dr. HyungKyu Lim, Mr. JongSik Park, Mr. J. J. Liou, Mr. M. K. Chen, Dr. SooYoung Lee, and Dr. Adelmo Ortiz Conde for helpful discussions and encouragement. I also thank Carole Boone for typing this dissertation. I am greatly indebted to my wife, Aerim, for her love and support during all the years of this study, my children, Jiyon, Dale, and Dane for their love, and my parent and parentsinlaw for their help and encouragement. The financial support of the Jet Propulsion Laboratory is gratefully acknowledged. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ...................................................... ii LIST OF SYMBOLS................................................ ...... v ABSTRACT...........................................................viii CHAPTER ONE INTRODUCTION. ............. ..... .............................. TWO UNIFYING VIEW OF TRANSIENT RESPONSES FOR DETERMINING LIFETIME AND SURFACE RECOMBINATION VELOCITY IN SILICON DIODES AND BACK SURFACEFIELD SOLAR CELLS, WITH APPLICATION TO EXPERIMENTAL SHORTCIRCUITCURRENT DECAY.................................... 4 2.1 Introduction........................................... 4 2.2 Mathematical Framework................................5 2.3 Transient versus SteadyState Analysis via TwoPort Techniques ...............................................9 2.4 OpenCircuitVoltage Decay (OCVD).......................11 2.5 Reverse Step Recovery (RSR)..........................14 2.6 Electrical ShortCircuitCurrent Decay (ESCCD)..........15 2.6.1 Brief Physics and Mathematics...................15 2.6.2 Experiments and Results.........................18 2.7 Discussion.. ................. .......... ....... ... 23 THREE EXTENSION OF THE METHOD OF ELECTRICAL SHORTCIRCUITCURRENT DECAY ......... ................................. ....... 26 3.1 Introduction. ...........................................26 3.2 Theory...................................... ........... 27 3.2.1 Theory of ESCCD Method.............. ...... 27 3.2.2 Dark I(V) Characteristic of a Solar Cell.........32 3.2.3 Combined Method of Electrical ShortCircuit Current Decay and Dark IV Characteristics.......35 3.3 Experiments.............................................36 3.3.1 Improvements in the Circuit for ShortCircuit Current Decay...................................36 3.3.2 Quality of the Short Circuit of the Switching Circuit ..........................................37 3.3.3 Measurement of the Dark I(V) Characteristics.....42 3.4 Experimental Results and Discussions....................43 FOUR EQUIVALENTCIRCUIT REPRESENTATION OF THE QUASINEUTRAL BASE, WITH APPLICATION TO DIODES AND BIPOLAR TRANSISTORS............53 4.1 Introduction............................................53 4.2 EquivalentCircuit for LargeSignal Transient...........54 4.2.1 Derivation by TwoPort Approach.................54 4.2.2 SPICE2 Simulation of the Equivalent Circuit for ESCCD .......................... ............... 61 4.3 Equivalent Circuits for LowFrequency SmallSignal Analysis ................................................. 64 4.3.1 Derivation in Frequency Domain....................64 4.3.2 Derivation in the Time Domain for ShortBase Case.............................................70 4.3.3 Calculation of the Delay Time....................77 4.3.4 Modification of the Conventional Hybridi Model by Including the MinorityCarrier Current Propagation Delay................................81 4.3.5 MinorityCarrier Delay Time with BuiltIn Electric Field......................................87 4.3.6 Measurement of MinorityCarrier Delay Time Across the QuasiNeutral Base Region of Bipolar Transistors...... ........ ..................... 88 FIVE SUMMARY AND RECOMMENDATIONS ............................. .93 APPENDICES A DETERMINATION OF THE EIGENVALUES FOR ESCCD AND OCVD...........95 B PHYSICS OF ELECTRICAL SHORTCIRCUITCURRENT DECAY ............98 C RELATION BETWEEN ASHAR'S AND ELMORE'S DEFINITIONS OF DELAY TIME ....................................... ................103 D EFFECTIVE BASEWIDTH ESTIMATION OF THE BIPOLAR TRANSISTORS MEASURED IN CHAPTER FOUR....................................105 REFERENCES ............................ ........................... 107 BIOGRAPHICAL SKETCH...................................................110 CSCR(V) CSCRO D Dij Dp Ax e i(x,O) ID IDC(O) i firstmode(t) IFMO IFO Ii(s) LIST OF SYMBOLS area of a device XQNBKi/L characteristic matrix elements of a quasineutral region for largesignal transient normalized surface recombination velocity characteristic matrix elements of a subregion for large signal transient spacecharge region capacitance of a pn junction forwardbiased with voltage V spacecharge region capacitance of a pn junction at V=O diffusion coefficient of minority carriers characteristic matrix elements of a quasineutral region for smallsignal lowfrequency analysis diffusion coefficient of minority holes thickness of a subregion magnitude of the electron charge minority carrier current at t = 0 dark current of an ideal diode dc steadystate current at x = 0 the firstnaturalfrequency (first transient mode) current at x = 0 preexponential factor of the first naturalfrequency current at t = 0" preexponential factor of a steadystate current of a diode with negligible spacecharge current Laplace transform of ii(t) ii(t) incoming minority carrier current toward a subregion at x = Xi IQNBO preexponential factor of steadystate quasineutral base current IQNEO preexponential factor of steadystate quasineutral emitter current Ish current through the shunt resistor of a diode I(x,s) Laplace transform of i(x,t) Ki (1si )1/2 L diffusion length of minority carriers Lp diffusion length of minority holes L* L/(1 + st)1/2 Lp* Lp/(l + STp)1/2 ni intrinsic carrier density NDD base doping density PDC(U) steadystate hole density at x = 0 p(x,O) p(x,t) at t = 0 P(x,s) Laplace transform of p(x,t) p(x,t) excess hole density Pi(s) Laplace transform of pi(t) Pi(t) excess minority carrier density at x = x R IFMO/'QNBO RI IQNEO/IQNBO RM IFMO/IFO rMOS turnon resistance of MOS transistor rs series resistance of a diode rsh shunt resistance of a diode si the ith naturalfrequency S, Seff effective surface recombination velocity Smax performance parameter of a solar cell derived from the ESCCD method Td decay time constant of the fundamental mode (first naturalfrequency) current T(delay) propagation delay time of minority carriers across the quasineutral base Tmin performance parameter of a solar cell derived from the ESCCD method TSCR discharge time constant associated with CSCR Vg gradient voltage of a pn junction XQNB quasineutral base width Yi admittancelike elements for a quasineutral base Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy USE OF TRANSIENTS IN QUASINEUTRAL REGIONS FOR CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS SBy TAEWON JUNG May 1986 Chairman: Fredrik A. Lindholm Major Department: Electrical Engineering This dissertation describes results of theoretical and experimental studies concerning the transient and frequency response of minority carriers within quasineutral regions of various semiconductor devices. The studies lead, in part, to the development of a new method for determining the recombination lifetime and surface recombination velocity of the quasineutral base region of p/n junction silicon solar cells, including devices having conventional backsurfacefield (BSF), ionimplanted BSF and polysilicon BSF structures. This method, called electricalshortcircuitcurrentdecay (ESSCD) avoids errors introduced in other methods in common use, such as opencircuitvoltagedecay and reverse step recovery, that arise from the capacitive effects of mobile viii holes and electrons in the volume of the p/n junction spacecharge region under forward voltage. Two circuit implementations of ESCCD are presented and evaluated. The ESCCD method derives from a theoretical development that provides a unifying view of various measurement methods for determining recombination lifetime and related parameters from the observation of transients following the sudden application or withdrawal of excitation. From this same theoretical framework we derive an equivalent circuit for quasineutral regions consisting of resistors, capacitors and inductors. This equivalent circuit approximates the effect of minority carrier propagation delay in a compact lumped circuit without the need to resort to a distributed, or transmissionline, model. The inclusion of the inductor makes this possible. Models of this type are developed for both smallsignal and largesignal variations. Their use enables the exploration of the effect of propagation delay in the ESCCD response through a standard circuit analysis computer program. Application of the same theoretical framework yields a modification of the hybridT model for bipolar transistors in the commonemitter configuration. This modified equivalent circuit is assessed experimentally. The experimental assessment demonstrates that it characterizes the effects of carrier propagation delay on phase shift with good accuracy. CHAPTER ONE INTRODUCTION Since the open circuit voltage decay method [1] and reverse step recovery method [2,3] were developed for the determination of the recombination lifetime of Ge diodes, other similar transient methods [4 8] have been also developed. These methods have been applied to Sidevice recombination characterization. Solar cells have received attention because recombination is a major physical mechanism governing solar cell performance. Transient methods for the determination of recombination parameters of the solar cell basically share a common origin: injecting minority carriers into the quasineutral region and electrically observing their vanishing that follows the withdrawal of excitation. The rapidity of measurement by transient response makes it attractive in general and in particular for inprocess control at key steps in manufacturing solar cells. The general purpose of this study is to explore theoretically and experimentally the transient responses of excess minority carriers within the quasineutral base. This is done in part to develop reliable methods mainly for the determination of the lifetime and the back surface recombination velocity of the quasineutral base of back surfacefield (BSF) silicon solar cells. The results of this study are directly applicable to any bipolar device including diodes, solar cells, and transistors. As will be seen, they have use beyond that of determining lifetime and surface recombination velocity. In Chapter Two, we illustrate the use of the twoport approach [9] to obtain a unifying framework for transient analysis and develop from it a new method, experimental electricalshortcircuitcurrentdecay (ESCCD) for the determination of the recombination lifetime and back surface recombination velocity of the quasineutral base of BSF silicon solar cells. In the implementation of this method in Chapter Two, we use a bipolar switching circuit to provide a short circuit between the two terminals of solar cells. In Chapter Three we present an improved switching circuit for ESCCD measurement applicable to submicrosecond response (for 4mil BSF solar cells). We derive various performance parameters for BSF solar cells by exploring the ESCCD method. We propose a methodology to separate the quasineutral emitter current component of BSF solar cells by using ESCCD. We develop also various other improvements in the underlying theory and in the interpretation of the experimental results. We apply the improved ESCCD circuit to determine the back surface recombination velocity of the first polysilicon BSF solar cells. The same theoretical framework used in developing ESCCD leads to the incorporation of the effects of minoritycarrier propagation delay in a compact equivalent circuit. This enables the use of standard circuitanalysis computer programs, without resorting to much more sophisticated programs needed when transmissionline models [10] are employed. These ideas are applied both to ESCCD and to bipolar 3 transistors, for which an improved hypridi model is derived. This model is assessed experimentally for bipolar transistors of known geometry and diffusion profiles. Chapter Five summarizes the contributions of this dissertation and presents recommendations for extension of the present study. CHAPTER TWO UNIFYING VIEW OF TRANSIENT RESPONSES FOR DETERMINING LIFETIME AND SURFACE RECOMBINATION VELOCITY IN SILICON BACKSURFACEFIELD SOLAR CELLS, WITH APPLICATION TO EXPERIMENTAL SHORTCIRCUITCURRENT DECAYS 2.1 Introduction This chapter has three purposes. First, we outline a mathematical method that systematically and compactly describes the largesignal transient and smallsignal frequency responses of diodes and the related devices such as transistors, diodes, and solar cells. This mathematical framework enables a comparison among available methods for determining carrier recombination lifetime and surface recombination velocity of quasineutral principal regions of the devices. Second, exploiting this description, we survey the adequacy of various experimental largesignal transient methods for deducing these parameters. The survey is indicative, not exhaustive. Third, we examine in detail, both theoretically and experimentally, a method that apparently has not been much explored previously. We demonstrate that this method yields both the surface recombination velocity and the recombination lifetime of the quasineutral base from a single treatment measurement for three different p+/n/n+ backsurface field solar cells. 5 2.2 Mathematical Framework In this section, we develop a mathematical framework which could be applicable to most of the largesignal transient measurement methods and could include smallsignal admittance methods for the determination of the lifetime and the back surface recombination velocity of the base region of a diode or a solar cell. This analysis will treat the minoritycarrier density and the minoritycarrier current in a quasi neutral base region in low injection. Focusing on the quasineutral base, assumed to be ntype here (of xindependent donor density NDD) with no loss in generality, will simplify the treatment; extensions to the quasineutral emitter are straightforward, provided one inserts the physics relevant to n+ or p+ regions. Assume a p+/n diode in which the uniformly doped quasineutral base starts at x = 0 and has a general contact defined by arbitrary effective surface recombination velocity Seff at the far edge x = XQNB. Such a contact could result, for example, from a backsurfacefield (BSF) region. Assume also lowlevel injection and a uniform doping of the base region. Then a linear continuity (partial differential) equation describes the excess minority holes p(x,t) p(x,t)/at = D 2p(x,t)/ax2 p(x,t)/T (2.1) where Dp is the diffusion coefficient and yp is the lifetime of holes. If we take the Laplace transform of Eq. (2.1) with respect to time, we get an ordinary differential equation in x with parameter s: p(x,0) + sP(x,s) = Dp d2P(x,s)/dx2 P(x,s)/Tp ,(2.2) where t=co P(x,s) = f etp(x,t)dt s = + jw, j = (1)1/2 (2.3) t=O Solving Eq. (2.2) yields P(x,s) = p(x,0)/s + M1 exp(x/L ) + M2 exp(x/L ) (2.4) where L = [(D T )/(1 + ST )]12 and where M1 and M2, given below, are to be determined by the boundary values at the two edges of the quasi neutral base region: P(O,s) at x=0, and P(XQNB,s) at x=XQNB. Substitution of (2.4) into (2.2) yields the steadystate continuity equation for p(x,O), verifying that (2.4) is the solution of (2.2). Because of quasineutrality and low injection, the minority hole diffusion current dominates in determining the response from the quasi neutral base. The following matrix describes this current at x = 0 and x = XQNB: I(0,s) i(0,0")/s 3 L eD e i (2.5) :_ 1 1 1 (2.5) I(XQNBs) i(XQNB,0 )/s p e QNB/p e QNBp M2 where I(0,0) and I(XQNB,0~) are the initial values of the minority hole diffusion current at x=0 and x=XQNB. In (2.5), hole current entering the quasineutral base is positive, by definition. Regarding the minority carrier densities at the two edges as the excitation terms for a system analogous to a linear twoport network of circuit theory, we have the following twoport network matrix from (2.4) and (2.5) for the two excitations (densities) and the two responses (currents): I(0,s) i(0,0")/s A 11 A12 P(0,s) p(0,0)/s I(XQNBs) i(XQNB,0)/s A21 A22J P(XQNBS) p(XQNBO)/s (2.6) where p(O,O0) and p(XQNB,0) are the initial values of the excess hole densities. Equation (2.6) extends a similar earlier development [9] by including initial conditions so that transients may be directly studied. We call Eq. (2.6) the master equation for the quasineutral base. In Eq. (2.6), A12=A21= e(D /L *) cosech(XQNB/L *) and A11= A22 e(Dp/L*)coth(XQNB/L*). Figure 2.1 displays the master equation, where the initial values are included in I(O,s), I(XQNB,s),P(O,S), and P(XQNB,S). Transient solutions can be derived from (2.6) by inserting proper boundary conditions, initial values and constraints imposed by the external circuit. For example, I(O,s)=O in OCVD opencircuitvoltage decay [1], I(O,s)=constant for reverse step recovery [2,3], and P(O,s) = 0 for shortcircuit current decay, the latter of which is developed in detail here. For smallsignal methods [1112], where dl, for example, is an incremental change of current, I(0,s)=IDC/s + dI(O,s) and P(O,s)=PDC/s + (edV/kT)PDC. Here the suffix DC denotes a dc steady state variable. In later sections, we will show briefly how to get solutions from the master equation for various of these methods. + z C Ix _ + o n_ S zb a x S 0 C, r 0 0 *r co u 0 4 c.I = 4.) J 0 0 .: C1 0) S. i . r V). L 0 In a solar cell, the back contact is generally characterized in terms of effective recombination velocity, Seff. The boundary condition at the back contact is I(XQNB,s) = eSeffP(XQNBs). From a circuit viewpoint, this relation is equivalent to terminating Fig. 2.1 by a resistor of appropriate value dependent partly on Seff. Because Seff in part determines the transient in the various methods named above, we can determine Seff from the transient response, as will be shown. In Secs. 2.4 to 2.6 we consider the utility of the master equation in characterizing selected measurement methods. The main emphasis will be placed on the electricalshortcircuitcurrentdecay (Sec. 2.6). Before doing this, however, we shall remark on the simplicity provided by the master equation (Eq. 2.6) by comparing it with its counterpart in the steady state. 2.3. Transient vs. SteadyState Analysis via TwoPort Techniques In general, the current (current density for a unit area) is the sum of the hole current, the electron current and the displacement current. For the quasineutral regions under study using the twoport technique described in Sec. 2.2, the displacement current is negligible. In the steady state, the twoport description leading to the master equation simplifies because the hole current in our example of Sec. 2.2 depends only on position x. Tnis xdependence results from volume recombination (relating to the minoritycarrier lifetime) and effective surface recombination (relating to the effective surface recombination velocity). A twoport formulation for the steady state leads to the same matrix description as that derived previously, in which the matrix elements Aij(s) of Eq. (2.6) still hold but with the simplification that s = 0. From such a master equation, one can determine the hole current at the two edges of the quasineutral base; and, using quasineutrality together with knowledge of the steadystate currents in the junction spacecharge region and in the p+ quasineutral emitter region, one can thus find the steadystate current flowing in the external circuit or the voltage at the terminals of the diode. If the quasineutral base is the principal region of the device, in the sense that it contributes dominantly to the current or voltage at the diode terminals, then one has no need to consider the current components from the other two regions. In contrast the general timevarying mode of operation leads to a minority hole current in the ntype quasineutral base of our example that depends on two independent variables, x and t. The time dependence results because the holes not only recombine within the region and at its surface, but also their number stored within the base varies with time. This may be regarded as resulting from the charging or discharging hole current associated with ap/9t in the hole continuity equation. This charging or discharging current complicates the variation of the hole current in space and time. But the use of the Laplace transform of the twoport technique in effect reduces the complexity of the differential equation to the level of that describing the steady state; the dependence on variable t vanishes, reducing the partial differential equation to an ordinary differential equation in x, just as in the steady state. This comparison also brings out another point. Just as in the steady state, one must interpret the transient voltage and current at the diode terminals as resulting not only from the quasineutral base but also from the junction spacecharge region and the quasineutral emitter. In the interpretation of experiments to follow, we shall account for this multiregional dependence. 2.4 Open Circuit Voltage Decay (OCVD) In this widely used method [1], the free carriers in the junction spacecharge region enter to contribute to the transient. But, consistently with Sec. 2.2, and with most common usage, we concentrate on the ntype quasineutral base. From the master equation [Eq. (2.6)], the transient solution for the junction voltage is obtained from opencircuit constraint that I(O,s)=O: i(0,0)L* 1 + D [coth(X NB/L *)]/L *S P(O,s) = p(O,0 )/s e P p eff eD s coth(XQB/L *) + Dp/L *Seff (2.7) Here we have assumed that the quasineutral base is the principal region in the sense described in Sec. 2.2; that is, we neglect contributions from all the other regions of the device. Using the Cauchy residue theorem, we find the inverse transform of Eq. (2.7): p(0,t) 2i(0,0 )L [1 + ( K/L Sff)cot(XQNBKi/L)] p(Ot)= 2 e i=1 eDpSi p [cosec2(XQNBKi/Lp) + (Dp/LpSeff)] (2.8) where si is the ith singularity point (ith mode) which satisfies coth(XQNB li+sip/L ) + Dp 1+s p/LpSeff = 0 (2.9) and K. = /lST > 0. 1 i p As can be seen in Eq. (2.8), the decay of the excess hole density at x=0 is a sum of exponentials; each Eigenvalue si is called a mode, as in the electromagnetic theory. Appendix A treats the details of determining the Eigenvalues si from Eq. (2.9) (and from the similar Eq. 2.11 derived below). The decaying time constant 1/sI of the first mode is much the largest of the modes. Both s, and the initial amplitude of the first model are functions of Seff and Tp. Thus separating the first mode from the observed junction voltage decay curve will enable, in principle, determination of Seff and Tp simultaneously. But our recent experience, coupled with that cited in [13], suggests that this is seldom possible in practice for Si devices at T = 300 K. In Si devices the openvoltage decay curve is usually bent up or bent down because of discharging and recombination within the spacecharge region. As mentioned in [11], the mobile charge within the spacecharge region contributes significantly to the observed voltage transient for Si, in which n 1010 cm3, but not in Ge, for which OCVD was first 13 3 developed, and for which ni 10 cm Here ni is the intrinsic density and is also the ratio of the preexponential factors that govern contributions from the quasineutral regions relative to those from the junction spacecharge region. Thus we identify the transient decay of mobile electrons and holes within the p/n junction spacecharge region, which persists throughout the opencircuit voltage decay (OCVD), as a mechanism that distorts OCVD so significantly that the conventional treatment of OCVD will not reliably determine Tp or Seff. The conventional treatment is consistent with that proceeding from the master equation, as described in this section. The interested reader may consult Ref. 11 for experimental comparisons that lead to this conclusion. We shall not pause here to present these. Rather we shall turn briefly to possible methods to remove the effects of this distortion. In an attempt to characterize the space chargeregion contribution to the observed transient voltage [13], quasistatic approximations and a description of the forwardvoltage capacitance of the spacecharge region based on the depletion approximation were combined to give rough estimates of this contribution. We plan to refine the approximations and the estimates in a future publication, leading possibly to a variant of OCVD useful for determining Tp and Seff. 2.5. Reverse Step Recovery (RSR) For this method [2,3], in which again the diode is subjected to steady forward voltage for t > 0, we have two constraints (for t > 0). The first is I(0,s) = constant (reverse current) at 0 < t < TS, where 's is the time needed for the excess hole density p(O,t) to vanish. This is the primary constraint. (The second constraint is p(0,t) = 0 for Ts < t < , a result of the applied reverse bias through a resistor. The primary observable, storage time Ts, is estimated by following a procedure similar to that described in Sec. 2.2, proceeding from the master equation. This method suffers difficulties similar to that of the OCVD method. Because p(0,t) > 0 for 0 < t < Ts, the decay of mobile hole and electron concentrations in the p/n junction spacecharge region complicates the interpretation of the measured Ts in terms of the desired parameters, Tp and Seff* In addition to this, during the recovery transient (Ts < t < c), the reverse generation current is often large enough to saturate the recovery current so quickly that we have no sizable linear portion of the firstmode curve on a plot of In[i(t)] vs. t. This linear portion provides interpretable data for Ge devices [3], but not often for Si devices according to our experiments. 2.6 Electrical Short Circuit Current Decay (ESCCD) 2.6.1 Brief Physics and Mathematics In this method, one first applies a forward bias to set up a steadystate condition and then suddenly applies zero bias through a small resistance. This causes the mobile charges stored within the junction spacecharge and quasineutral regions to discharge rapidly. One then measures the transient current by measuring voltage across the small resistor. If the discharging time constants related to the charge stored within the quasineutral emitter and the junction spacecharge region are much smaller than from the quasineutral base, one can separate the first mode of the quasineutralbase current and determine Seff and Tp. We first consider the time of response of the junction spacecharge region. Upon the removal of the forward voltage, the constraint at the terminals becomes essentially that of a short circuit. The majority carrier quasiFermi levels at the two ohmic contacts immediately become coincident, and the junction barrier voltage rises to its height at equilibrium within the order of the dielectric relaxation time of the quasineutral regions, times that are of the order of no greater than 1012s. This occurs because the negative change in the applied forward voltage introduces a deficit of majority holes near the ohmic contact of the p+ emitter and a deficit of majority electrons near the ohmic contact in the quasineutral base. The resulting Coulomb forces cause majority carriers to rush from the edges of the junction barrier regions, thus causing the nearly sudden rise of the barrier height to its equilibrium value. (The physics governing this phenomenon comes from Maxwell's Curl H = i + aD/at; taking the divergence of both sides yields 0 = div i + d(div D)/dt, which, when combined with i = aD/e and divD = p, yields a response of the order of c/a, the dielectric relaxation time.) Following this readjustment of the barrier height, the excess holes and electrons exit the junction spacecharge region within a transit time of this region (about 1011s typically), where they become majority carriers in the quasineutral region and thus exit the device within the order of a dielectric relaxation time. Thus the discharging of excess holes and electrons within the junction spacecharge region in the ESCCD method occurs within a time of the order of 1011s, which is much less than any of the times associated with discharge of the quasineutral regions. This absence in effect of excess holes and electrons within the junction spacecharge region greatly simplifies the interpretation of the observed transient. It is one of the main advantages of this method of measurement. A more detailed discussion of the vanishing of excess holes and electrons within the junction spacecharge region appears in Appendix B. The discharge of the quasineutral emitter depends on the energy gap narrowing, the minority carrier mobility and diffusivity, the minoritycarrier lifetime, and the effective surface recombination velocity of this region. For many solar cells, this discharge time will be much faster than that of the quasineutral base, and we shall assume this is so in the discussion to follow. Having established that the mobile carriers in the junction space charge region enter the electricalshortcircuitdecay transient during an interval of time too short to be observed, and noting also now that negligible generation or recombination of electrons or holes within this region will occur during the transient, we now turn to the observable transient current. Inserting the constraint, P(0,s) = 0, into the master equation, Eq. 2.6, leads to eD p(0,0) coth(X /L *) + D /L *S I(O,s) = i(0,O)/s QN P p eff sL 1 + (D/L *Seff)cotn(XQNB/L *) (2.10) Cauchy's residue theorem yields the inverse transform of (2.10): () eD p(0,0)K cot(KiXQNB/L ) DpK /LSeff i.t ii(t) Spp p poef2 e) i= p ( s ) + (XQNB/2Sff)cosec (KiXNB/Lp) (2.11) where si is the ith singularity which satisfies the Eigenvalue equation, D XN 1 + LSeP V1+si T coth( L 1+sJ ) = 0 (2.12) p eff L p and where Ki = (1siTp)1/2 > 0, with si < 0. Truncating (2.11) and (2.12) to include only the first mode (sl), we obtain 1/2 1 + (Dp/LpSff) V1 + 1Tp coth[(XQNB/Lp)(1 + S1 p) ] = 0 (2. 13) and first mode(0) eDp(0,0)K1 cot(KlXQNB/Lp) (DpK1/LpSeff) lLp (T /2K ) + (XQNB/2S ) cosec2(KX /L) (2.14) Equations (2.13) and (2.14) contain four unknowns: first mode(0)' Sl* Tp, and Seff. The parameters, s, and first mode (0) are determined from the straightline portion of the observed decay (in Fig. 2.3(c) to be discussed below) p(0,0O) = (n 2/ND)[exp(eV(0O)/kT)1]. Here v(O) is known and the doping concentration NDD of the base is measured by usual methods; Dp(NDD) is known, and XQNB is measured. Combining (2.14) and (2.13) then yields the desired parameters: Tp and Seff* 2.6.2 Experiments and Results To explore the utility of the ESCCD method, we connect the solar cell under study to node B of the electronic switching circuit illustrated in Fig. 2.2. The circuit works as follows. When V1(t) is high, switching transistor T1 turns on, which charges the large capacitor in parallel with it and divides the high voltage Vhigh about equally between the solar cell and the emittercollector terminals of the transistor. Thus 19 rC a)3 EO o o " 0 C .0 S1C C) a E 0 co CC +3>  C cu 0 e l"0 . u C a 0 uo 4)  CE e a o *o Su OJa) Q1) cc C UCC C OL Sgg 00 E 4'O 0 (T V) 2 *~ fL *X u s e: r *rC~ 4>r "SUA 1 the voltage across the solar cell becomes about 0.6 V, which one may control by altering (Vhigh), and the variable resistor connected to the transistor base, or both. In this mode, the quasineutral base charges to store ultimately a steadystate charge of excess holes and electrons, and p(0,0") of Eqs. (2.10), (2.12) and (2.13) is established. Now assume that V1(t) drops to its low value, an incremental change of about 0.6 V. The capacitor across the transistor acts as an incremental short circuit and the voltage across the solar cell suddenly vanishes to a good approximation, thereby establishing the desired shortcircuit constraint. The large capacitor maintains this constraint nearly perfectly during the firstmode transient of the solar cell; that is, during this transient, this capacitor and the input voltage source, which has a small resistance of 50 n (in parallel with 10 Q), act as nearly incremental short circuits. Thus the desired shortcircuit constraint is maintained to a good approximation during the ESCCO transient of interest. We use three different BSF solar cells for which the parameters are: DEVICE 1NDD (substrate doping) = 6 x 1014 atoms/cm3; XQNB (base thickness) = 348 jm, A = 4 cm2; DEVICE 2NDD = 7 x 1014 atoms/cm3 XQNB = 320 um, A = .86 cm2, DEVICE 3NDD = 3.5 x 1015 atoms/cm3, XQNB = 348 um, A = 4 cm2 We measure the voltage across the solar cell under study. As illustrated in Fig. 2.3(a), in which the voltage of the emitter drops by 0.1 V within 1 us. The speed is circuit limited. One could design a Sv(t) 50Ps Si(t) 1mA 0  1mA ~ . I I I I I I * 50us v(t) SI 50ps Fig.2.3 (a) (b) (c) Voltage across BSF #1 solar cell (vertical:0.2V/div), Current through BSF #1 solar cell (vertical:lmA/div), Log scale representation of (b) (vertical:0.1V/div), where v(t) = (mkT/e)ln(i(t)/Io+l). (b) t a l l I I ! I I I I I I I I I . I I I much faster circuit. Here Td = 1/sI is the firstmode decay time, influenced by both volume and surface recombination in the base. But the circuit used suffices because Tp >> 1 us for the solar cells studied. Figure 2.3(b) shows the current during the transient. Fig. 2.3(c) is its semilogarithmic counterpart, illustrating the straightline portion of the transient obtained from the output of the logarithmic amplifier in Fig. 2.2. From this rd is determined. Since the voltage at node B is purely exponential for a time, the corresponding output voltage at node C is linear in time, as Fig. 2.3(c) illustrates. We use switching diodes in the log amplifier of which the IV characteristic is V = .03851n(I/Io+1). If the firstmode current is Ifirstmode(t) = constant exp(t/rd) Td =l/s, (2.15) then the slope of the output voltage of log amplifier is 38.5 mV/Td. Extrapolation of the straight portion in Fig. 2.3(c) yields the initial value ifirstmode (0+) as the intercept. We measure the decay time constant and the initial amplitude of the first model as follows: DEVICE 1, Td E 1/sI = 29.3 us, ifirst(O) = 2.73 mA for V(O0) = 0.44 V and T = 303.1 K. For DEVICE 2, rd = 24.5 usec, ifirst(0+) = 4.35 mA at v(O0) = 0.5 V and T = 302.9 K. For DEVICE 3, rd = 28.5 lisec, ifirst(0+) = .696 mA at v(0") = .47 V and 303.5 K. Here v(O0) denotes the steady forward voltage applied across the solar cell before the transient. From the above development, these results give DEVICE 1, rp = 119 us, Seff = 25 cm/sec; DEVICE 2, = Tp = 119 us, Seff = 60 cm/s; DEVICE 3, Tp = 213 us, Seff = 100 cm/s. These results agree favorably with those obtained for the same devices by using the more time consuming methods detailed in [1112]. 2.7 Discussion Most measurement methods for the determination of the minority carrier lifetime and the surface recombination velocity of the base region of Si solar cells share a common problem caused by the existence of the sizable number of the mobile carriers within the spacecharge region. These methods, among opencircuit voltage decay (Secs. 2.4) and reverse step recovery (Sec. 2.5), were originally developed for Ge devices. Si has a much larger energy gap EG than does Ge. Thus the distortion of the measured response by carriers stored in the space charge region is more pronounced in Si, mathematically because of the role of the intrinsic density ni discussed in Sec. 2.4. If the electronic switch providing the short circuit closes fast enough, the mobile holes and electrons stored for negative time in the junction spacecharge region play no role in determining the response of the electricalshortcircuitcurrent decay described in Sec. 2.6. In our experiments, the simple circuit of Fig. 2.2 had speed limitations, but these limitations did not markedly influence the accuracy of the determined base lifetime and surface recombination velocity. This lack of influence results because the decay time of the firstmode response, which accounts for vanishing of minority holes both by volume recombination within the quasineutral base and effectively by surface recombination, greatly exceeded the time required for the excess hole density at the base edge of the spacecharge region to decrease by two orders of magnitude. Details concerning this issue appear in Sec. 2.6. Apart from this potential circuit limitation, which one can overcome by improved circuit design, a more basic consideration can limit the accuracy of the electricalshortcircuitcurrent decay (ESCCD) method. In general, the current response derives from vanishing of minority carriers not only in the quasineutral base but also in the quasineutral emitter. For the solar cells explored in this study, the emitter contributes negligibly to the observed response because of the low doping concentration of the base and because of the lowinjection conditions for which the response was measured. But for other solar cells or for higher levels of excitation, the recombination current of the quasineutral emitter can contribute significantly. Note that the ESCCD method determines the base lifetime and the effective surface recombination velocity of a BSF solar cell by a single transient measurement. One can easily automate the determination of these parameters from parameters directly measured from the transient by a computer program, and the measurement itself may be automated. This suggests that ESCCD may be useful for inprocess control in solarcell manufacturing. This chapter began with a mathematical formulation of the relevant boundaryvalue problem that led to a description similar to that of two 25 port network theory. The advantages of this formulation were touched upon in Sec. 2.1 and only the bare elements of its relation to open circuit voltage decay and step reverse recovery were developed. Further exploitation to enable systematic development and comparison of small signal and transient methods for the determination of material parameters of solar cells and other junction devices is recommended as a subject for further study. CHAPTER THREE EXTENSION OF THE METHOD OF ELECTRICAL SHORTCIRCUIT CURRENT DECAY 3.1 Introduction This chapter describes various improvements of the method of electrical shortcircuit current decay. First, the switching circuit in Fig. 2.2 has been improved to accommodate decay time constants down to the submicrosecond range. We used MOS transistors to provide a voltage controlled switch between the two terminals of a solar cell. The use of the MOS transistors yields a much faster switching time and a simpler circuit in comparison with the bipolar transistor in Chapter Two. Second, in the previous chapter, we used the initial amplitude of the firstmode current Ifirstmode(O+) together with the decay time constant rd as the ESCCD parameters used to determine T and S. The parameter ifirstmode(O+) is proportional to exp[ev(O)/kT] where v(0") is the voltage at the terminals at t=0~ minus the voltage drop in the series resistance. Thus, in the method of Chapter Two, T and S are determined by three measurable parameters: ifirstmode(0+), rd and v(O0). The last of these is the least accurately determined of the three because of possible contact and cell series resistances. In the improved approach of this chapter we eliminate the need to measure v(O) by treating Cifirstmode(O+)/IF(O) as the measurable parameter. In the ratio the factor exp[ev(0)/kT] cancels out. 26 Third, in this chapter we consider the sensitivity problem involved in the method of electrical shortcircuit current decay for thin or thick solar cells. By a thin solar cell, for example, we mean that the thickness of its base region is much less than the diffusion length. We analyze this problem by using S(T) locus for a given measured decay time constant. For a thin solar cell, we introduce new performance parameters, such as Smax, Tmin and RM, the importance of which is discussed in this chapter. Fourth, we show quantitatively that the electrical shortcircuit current decay curve is not affected by either the series resistance or shunt resistance of the usual solar cell. Finally, we note that the use of IFO in the ratio above brings the emitter recombination current IQNE into our method for determining T and S of the base region. This, however, is only apparently a problem. Indeed, we illustrate that use of the S(T) locus enables a determination of IQNE, thus adding to the utility of the method to be described. 3.2 Theory 3.2.1 Theory of ESCCD Method A general description of the theory and the underlying physics for the ESCCD method appeared in Chapter Two. In this section we exploit advantages of the twoport network formulation introduced in Chapter Two in (2.6), the representation for which is illustrated in Fig. 3.1, where Y1(s) = A11(s) + A12(s), Y2(s) = A12(s), Pl(s) = p(O,0')/s, P2(s) = p(XQNB,O')/s, I1(S) = i(O,0)/s, and 12(s) = i(XQNB,O)/s. z x 0. 04 '. ( Au *1 SI C co 04) *C L C 0 0 U u S.r *0 S E 0 0 LU 'O *r E a ." x 0" Si 4 4 C 3 I 0). Lr 0 4L) r C0. *r_ 0 34J 0S  oil C: Z cyrr In this figure we have used the yparameter set [14]. This choice is arbitrary. Instead we could have chosen any of the four parameter sets. Mapping into the other three sets is straightforward and may be desirable, for example, for certain input excitations and output terminations. That is one advantage of a twoport network representation. Other advantages include (a) systematic determination of the natural frequencies [14]; (b) systematic conversion to the case of steadystate excitations, attained by setting the complex frequency variable s to zero; (c) systematic connections to the underlying physics, as we shall illustrate; (d) systematic treatment of various terminations and excitations; (e) systematic derivation of the system function in the complex frequency domain, which maps into the impulse response (Green's function) in the time domain, an advantage we will illustrate later by use of the Elmore definition of delay [15]. For the analysis of ESCCD method of a solar cell using Fig. 3.1, one must provide a shorted path at x = 0 and a back contact having recombination velocity S at x = XQNB to Fig. 3.1. The boundary condition at x = XQNB, I(XQNB,s) = AeSP(XQNB,s), removes 12(s) and P2(s) in Fig. 3.1 from consideration. Figure 3.2 displays the resulting twoport network representation of the quasineutral base region of a solar cell. Here 30 0 . VI U 0 I z S o 0 r 0 1 II ') C. r s 0 Sr 0 0 I .U II C O 1a 1 4"" X *r 0 = (f U4 C tv r S S *r I+ u r. 4 0) Iw UC 0 0 , mut 0 ii S r 0 O .C U S 0 0 *r LS U C *r SCU 0 .U II o 0 S  *4  > E > .4. 3 Cu S3 m c ,s .LL Ys I(XQNBs)/P(XQN,'s) = AeS (3.1) Solving the network of Fig. 3.2 for I(O,s) under the lowinjection condition yields I(O,s) = i(O,0)/s Y1 p(O,0)/s (Y1 + Ys) Y2 Y1 + Y Y (p(O,0)/s) (3.2) ni2 where p(O,0) = (exp(eV(O)/kT) 1) (3.3) DD If we use the Cauchy Residue Theorem to obtain the inverse transform of I(O,s), we get an infinite series for i(O,t). Truncating this series after the first term, at t = 0+, yields first mode(0+) = IFMO (exp(eV(O)/kT)1) (2.14) AeDKlni2 where IFMO = AeDK S1LNDD cot(KlXQNB/L) DK1/LS (T/2K1 ) + (XQNB/2S)csc(KlXQNB/L) The minoritycarrier current at x = 0 for t < 0 is i(0, 0) = IQNBO (exp(qV(0")/kT 1) (3.5) ADni sinh(X NB/L) + acosh(XQNB/L) whe NBO LNDD cosh(XNB/L) + asinh(XQNB/L) 36) LS and where a = . Here a is the ratio of the normalized surface recombination velocity to the diffusion velocity [16]. Thus the ratio IFMO R  QNBO 2K1 cotA1 + tanA1 21 12 Sl (XQNB/A) /D + (XQNB/S) csc2A1 cosh(XQNB/L) + tsinh(XQNB/L) sinh(XQNB/L) + acosh(XQNB/L) (3.7) where Al = K1XQNB/L and where K1 and AI are obtained by solving (A.3) of Appendix A. The ratio R will be utilized for the determination of the quasineutral base parameters. 3.2.2 Dark I(V) Characteristic of a Solar Cell The equivalent circuit of a solar cell in the dark condition, including series and shunt resistances, is shown in Fig. 3.3. If we assume that the spacecharge recombination current component is negligible [17], the I(V) characteristic of the solar cell is *b~ S o r_ 0 0 c *r O  *r ,) 0 C I  gO 0 c0 ON c.. E( .w U *0 . E) 0 U o+ *1 i *r ,c I =D + Ish (3.8) = IFO (exp(eV/kT)1) + V/rsh (3.9) Here IFO is the preexponential factor of the forward bias current and V is the voltage across the spacecharge region. The preexponential factor IFO in (3.9) has two components: FO = IQNBO + IQNEO (3.10) where IQNBO is the quasineutralbase current component and IQNEO is the quasineutralemitter current component. The voltage across the two terminals of the solar cell Vout is Vout = I rs + V (3.11) As the forward bias increases, the current I in (3.8) becomes more dominated by the component ID and the effect of Ish becomes negligible for the solar cell. Thus I IFO exp(eV/kT) (3.12) Combining (3.11) and (3.12), we obtain an expression for Vout in terms of I, rs and IFO: Vout rs I + loge(/IFO (3.13) There are two unknowns, rs and IF0, in (3.13). We estimate rs and IFO by measuring the dark IV characteristics from the terminals of a solar cell. The preexponential factor IFO will be utilized for the determination of the base material parameters. 3.2.3 Combined Method of Electrical ShortCircuit Current Decay and Dark IV Characteristic In this section, we present a method for the determination of the parameters of a solar cell. This method involves combining the ESCCD and dark IV characteristic methods. Using the ESCCD method, we measure the decaying time constant of the first mode Td and the ratio RM of pre exponential factors from Fig. 2.3: FMO RM 'F (3.14) FO in which the subscript FMO means the preexponential factor of the firstmode current. Using the dark IV measurement, we estimate the preexponential factor IFO by eliminating the series resistance effect as described in Section 3.2.2. From the measured value of Td, one can generate a s(T) locus on the TS plane; each point on this locus must produce the measured value of T.. Each point (T,S) also has its own value of the ratio R, defined in (3.7), since R is a function of both T and S. Also each point (T,S) produces its own value for IQNBO in (3.6). Now we have three equations for three unknowns: The three equations are (2.13), (3.10) and (3.14) and the three unknown are T, S and IQNEO. Specifically d = f (T,S) (2.13) IF = 2(T,S,IQNEO) (3.10) and RM = f3(T,S,IQNEO) (3.14) Using (2.13), (3.10) and (3.14) and the measured variables, one can solve for T, S and IQNEO in a manner to be described later. 3.3 Experiments 3.3.1 Improvements in the Circuit for ESCCD Previously we used a bipolartransistor switching circuit in Fig. 2.2 to measure the decay time constant and the initial amplitude of the first naturalfrequency current at t=O+. We have made this switching circuit faster and simpler by replacing bipolar transistors by power MOSFET switches. To increase speed further, we reduced the parasitic effects existing in the measurement circuit. To decrease the parasitic inductance, we shortened the discharge path of the stored carriers and also shortened the length of the probes of the oscilloscope. The improved circuit is illustrated in Fig. 3.4. In this circuit the power MOSFET switch has a turnon resistance of 0.6 ohm. The input capacitance of the MOSFET is 250 pF. The output impedance of the pulse generator is 50 ohm. The turnon switching time of this measurement circuit is 12.5 ns (250 pF times 50 ohm). Thus the speed of the measurement circuit is adequate for any bipolar devices having Td larger than 100 ns. This switching circuit provides a sudden shorted path across the two terminals of a solar cell in a manner similar to that of the bipolar switching circuit described in Chapter Two. 3.3.2 Quality of the Short Circuit of the Switching Circuit We now consider the quality of short circuit provided by the switching circuit of Fig. 3.4. The discharging path has a series resistance of a few ohms instead of a perfect shortedpath. The voltage across the junction spacechargeregion does not vanish as long as the current flows through the series resistance. Fig. 3.5 displays the equivalent circuit during discharge when the firstterm natural frequency current dominates the discharging current. Higherterm naturalfrequency current components have vanished previously from the equivalent circuit representation of Fig. 3.5 since they have shorter decay time constants than the time constant Td of the firstterm naturalfrequency current. In Fig. 3.5, rs and rsh are the series and shunt resistance of a solar cell, rd detects the discharging current, and il(t) is the firstterm naturalfrequency current. cuJ b~I r 'V)i Th bI 0 o4 I0 0 & I (A I il _ _ "cM ,_ l PCZi i(t) 0+ C (t) rMOS SCR i (t) rsh 12 0 2 < rd Fig.3.5 Equivalentcircuit representation of the measurement circuit of Fig.3.4 when the first mode dominates electricalshortcircuitcurrent decay. The current i(t) flowing through rd in Fig. 3.5 is i(t) = CSCR(dvl2(t)/dt) + vl2(t)/rsh + il(t) (3.15) where dQsc CSCR CCR dv1 f12) (3.16) in which QSCR/e is the integrated steadystate hole or electron density through the volume of the spacecharge region. From Chawla and Gummel [18] CSCR/CSCRO = [1 (v12/Vg)m (3.17) where CSCRO is the CSCR for variations in v12 about bias voltage v12 = 0 and where 1/3 < m < 1/2 and Vg is the gradient voltage, which includes the contribution of mobile holes and electrons within the SCR. Since Vg > v12(t), CSCR CSCRO f(v12) (3.18) Thus i(t) CSCROdv12(t)/dt + v12(t)/rsh + il(t) (3.19) where v12(t) = (rMOS + rs + rd)i(t) (3.20) Since the ratio (rMOS + rd + rs)/rsh is usually very small for practical solar cells, we obtain from (3.19) i(t) = CSCRO(rMOS + rs + rd)[di(t)/dt] + il(t) Solving (3.21) for i(t) yields i(t) = Elexp(t/TSCR) + E2exp(t/Td) where TSCRO CSCR(rMOS + r s + rd) (3.21) (3.22) (3.23) As can be seen in (3.22), the first term of the right side can be neglected and Td can be determined if the time constant TSCR is much smaller than T. For the switching circuit of Fig. 3.4 TSCR = 200 ns. For the solar cells described in this chapter, 0.5 us < Td < 30 us. Thus the RC time constant of the measurement circuit negligibly influences the firstterm or dominant naturalfrequency current decay of the solar cells. 3.3.3 Measurement of the Dark I(V) Characteristics The measurement of the dark I(V) characteristics of a solar cell is straightforward. One first measures the terminal I(Vout) characteristics in the dark condition and then corrects for the effects coming from the existence of the series resistance. This method is based on the assumption that the main deviation of the diode current from the ideal exp(qV/kT) behavior at high currents can be attributed solely and relatively simply to series resistances [19]. From combining the measured I(Vout) characteristic with idealized diode theory, we obtain (Vout)i = Iirs + (kT/q)loge(Ii/IFo) (3.24) where rs is the series resistance, IFO is the idealized preexponential current (corresponding to unity slope), and subscript i denotes different data points. Applied to two such data points, Eq. (3.24) yields AVout = r1AI + (kT/e)loge (12/1) (3.25) upon subtraction. This determines rs, which we may thus ignore in the subsequent discussion. To determine IFO, we use the procedure of Ref. [19]. 3.4 Experimental Results and Discussions In the most general case, the ratio XQNB/L is arbitrary. For this case, we generated the S(T) locus corresponding to the measured value of the decay time constant Td. This locus is generated by solving the transcendental equation of (A.3) of Appendix A. We consider the following ratios, for reasons that will become apparent: R = IFMO/QNBO = f(r,S) (3.26) R = IQNEO /QNBO = FO IQNBO)/IQNBO (3.27) RM = IFMO/IFO (3.14) The relation among these parameters is R = (1 + RI)RM. (3.28) The ratio R is determined by theory for any assumed values of S and T lying on the S(T) locus corresponding to the measured value of the decay time constant Td. The ratio Ry is determined by the measured value of IFO and by the value of IQNBO which is obtained from (3.6) for any assumed values of S and T. The ratio RM is determined by measurement. Thus (3.28) enables a determination of S and T by an iterative procedure. To determine RM, we use the ESCCD method as discussed in connection with (3.14) to determine the ratio (not the individual components of the ratio). Having formed the three ratios above by a combination of experiment and theory, we search for the values of S and T that satisfy (3.29). Completion of this search yields the actual values of S and T for the solar cell under study. It also yields the ratio of the emitter to the base components of the total current, and hence these components separately if the base doping concentration is determined in the usual manner. As an illustrative example of the above, we consider a particular cell fabricated on a 0.3 ohmcm ptype substrate. The top n+ layer is about 0.3 um deep. The front surface is texturized and covered with AR coating. The back surface has been implanted with boron. The concentration of boron is about 1020 cm3 and the junction depth is 1 pm. The thickness of the base is 374 un. The measured values of Td, RM and IFO are 6.5 ps, 0.23 and = 2 pA respectively. Using Td = 6.5 us, we generate the S(T) locus shown in Fig. 3.6. From this locus, we determine the values, 15 us and 1300 cm/s, for the lifetime and the surface recombination velocity. The other parameters of this cell are also determined: IQNBO = 1 pA, IQNEO = 1 pA, IFMO 0.46 pA and L 185 lm. The ratio of the cell thickness to the diffusion length is = 2 for this particular cell. MINORITY CARRIER LIFETIME Fig.3.6 S(r) Td = locus for a BSF solar cell. 6.5 's. The locus is generated from 104 103 102 100 (Ms) Such a solar cell has moderate thickness in the sense that, in the ESCCD transient, the minority carriers vanish from volume recombination and from exiting the surface at comparable rates. To sharpen this definition of a moderately thick solar cell, we note that one can express the decay time constant Td in terms of the following two time constants by solving (A.3) of Appendix A: d1 = 1 = s1 + T1 (3.29) where Ts = (XQNB/AI)2/D. Equal rates occur if Ts = T (3.30) Here Al is obtained from (A.3) of Appendix A: A1 = w/2 for S(back) = 0 (3.31) and A1 = i for S(back) = (3.32) Here in (3.29), the parameter Td/Ts is the probability that a minority carrier vanishes through the surfaces bounding the quasineutral region, whereas the parameter Td/T is the probability that a minority carrier vanishes by volume recombination. Although one will not know XQNB/L for any given solar cell at the outset, XQNB can be easily measured, and one can make an initial estimate of L as a function of the base doping concentration from past experience. If XQNB/L << 1, the procedure simplifies because the locus S(T) exhibits dS/dr = 0 over a large range of T. This is the mathematical statement, for our procedure, that S is more accurately determined than is r for a thinbase solar cell. (If XQNB/L >> 1, dS/dr = over a large range of S, which means that T is more accurately determined than is S for a longbase solar cell.) To illustrate the procedure for thin solar cells, we consider two different n+/p/p+ BSF solar cells. These cells are fabricated on 10 ohm/cm ptype substrates. The top n+ layer is about 0.3 im deep. The thickness of these cells is about 100 in. Using Td, we generated S(r) loci of the two cells as shown in Fig. 3.7. For the cells corresponding to the lower and the upper loci, the actual values of S are estimated to be less than 190 cm/s and less than 3000 cm/s, respectively. These maximum values (190 and 3000 cm/s), obtained from the region of the loci for which dS/dT approaches zero, defines Smax. If for an extreme case for which negligible volume recombination occurs during the ESCCD transient, S = Smax. Similarly, the limit dS/dT + = defines a minimum value of the lifetime Tmin, as illustrated in Fig. 3.7. For the all corresponding to the lower locus of Fig. 3.7, Tnin = 40 us. This value Tmin occurs for 48 104 S R=3.5 R=3.56 R=3.58 locus 2: + S10 ohmcm n+/p/p, Sff, max O d = 0.74 us, XQNB = 101 Um. 0 1103 O C Z R=3 5 R=3 6 R=32  locus 1: S ' 0 2 10 ohmcm n+/p/p+, Seff, max 0 100 200 300 400 MINORITY CARRIER LIFETIME (92s) Fig.3.7 S(T) loci for two different solar cells with thin base. 0  0 100 200 300 400 MINORITY CARRIER LIFETIME (ps) Fig.3.7 S(T) loci for two different solar cells with thin base. the extreme case of negligible surface recombination at the back contact during the ESCCD transient. For the upperlocus cell in Fig. 3.7, Tmin = 0. These two parameters, Smax and Tmin, can be used as performance parameters for thin solar cells; small Smax and large Tmin is desirable for thin BSF solar cells for a given base thickness and doping concentration. We also measured the values of RM: RM = 20 for the lower locus and RM = 3 for the upper locus. But we cannot use the measured RM directly to determine T, because R does not change much as T increases as illustrated by marks on the loci Fig. 3.7. Instead, the measured RM can be used as another performance parameter for thin BSF solar cells, since large RM means small IQNEO and small S for a thin solar cell. These conditions imply a large opencircuit voltage for a given base thickness and doping. Small RM usually implies either a poor BSF contract at the back surface or a large IQNEO* For example, for the better BSF solar cell (the lower locus), we have RM = 20, whereas RM = 3 for a poorer BSF solar cell (the upper locus). We measured various kinds of solar cells and characterized them as shown in Table 3.1. Among the cells in Table 1, poly 1 and poly 2 have highly doped polySi layers on the back surface of the base. The value of S is estimated to be about 2000 cm/s for n+/p/p+polySi cell (poly 1) and about 400 cm/s for p+/n/n+polySi cell (poly 2). Thin cells are characterized in terms of mnin and Smax. Table 3.1 ESCCD MEASUREMENTS NAME RESISTIVITY THICKNESS (ohmcm) (pm) SPIRE .31 374 ASEC1 .15 301 ASEC2 .15 267 BSF#1 10 240 BSF#2 10 260 BSF#3 10 284 BSF#4 10 96 BSF#5 10 91 BSF#6 10 107 BSF#7 10 102 LEU#1++ 8 328 LEU#2++ 1.5 325 POLY1 2 203 POLY2++ 2 208 ++ denotes p+/n/n+ BSF solar ce * denotes Smax. POLY12 have po FOR VARIOUS SOLAR Td(vs) T(Ps) 15 7 13 35+ 45+ 75+ 20+ 25+ 6.5 3.6 4.0 5.3 6.3 7.85 .98 .9 .7 .73 28.0 25.7 2.7 8.8 11. + denotes Tmin lySi layers at the CELLS. S(cm/s) L(pm) 1 5 3 6 0+ 0+ 2+ 5+ 5+ I 1300 18! ohmic 9 ohmic 12( 100400* 35( 100350 40( 100225 51 100380 26! 100290 29! 4500* 3200 4080* 41 40150 34 2000* 100400* 161 or Lmin. back surfaces. ~ 8+ 7+ 7+ 145+ 105+ Finally, we present one more method to determine the recombination parameters of solar cells. In this method, one fabricates two different solar cells out of the same wafer, one BSF solar cell and one ohmic contact solar cell. Then one first estimates the lifetime of the cells by measuring Td of the ohmiccontact solar cell and by using (3.31): Td' = [(XQNB/r)2/D]1 + T1 (3.33) Second, one measures rd of the BSF solar cell and generates the S(T) locus on the same plot. Since the lifetimes of the two cells are the same, S of the BSF cell can be obtained from the corresponding S(r) locus. An illustrative example is shown in Fig. 3.8. In this example, we used a wafer which is 10 ohmcm and ptype. The upper locus corresponds to the ohmiccontact cell. The lower one corresponds to the BSF cell. The lifetimes of these cells are estimated about 200 us and the recombination velocity of the BSF cell is estimated to be 2000 cm/s. The error in T introduced by error bounds on the measured thickness increases when the ratio XQNB/L decreases. For example, for a 10 ohmcm n+p/p+ solar cell ohmicc contact) with a thickness of 350 3 mn, the error in the lifetime is estimated to be about 20%. In doing this calculation, we assumed that the lifetime is 50 us (XQNB/L = 0.84) and that 0 = 35 cm2/s. For a 0.3 ohmcm n+/p/p+ solar cell ohmicc contact) with a thickness of 350 3 in, this error is estimated to be about 5%. Here for this calculation, we assumed that the lifetime is 20 us (XQNB/L = 1.64) for diffusivity D = 23 cm2/s. 107  10 O 0j locus 1 10 UO  I 104 0 U locus 2 102 1 10 0 100 200 300 400 MINORITY CARRIER LIFETIME (ps) Fig.3.8 Illustration of the determination procedure of S and T using one ohmic contact solar cell and one BSF solar cell from the same material. Locus 1: ohmic contact solar cell(C35). Locus 2: BSF solar cell(253). CHAPTER FOUR EQUIVALENTCIRCUIT REPRESENTATION OF THE QUASINEUTRAL BASE, WITH APPLICATIONS TO DIODES AND TRANSISTORS 4.1 Introduction In the previous chapters, we treated the quasineutral base region using the twoport approach. This approach provides solutions of the distributed system (independent variables, x and t or x and s) without approximations. Thus it is accurate. This chapter describes an alternative approach for modeling the largesignal transient response. In this approach, one considers thin subregions to constitute the whole base region. This enables algebraic approximations of the transcendental functions of s associated with each subregion, yielding thereby a lumped circuit representation made of capacitors, resistors, etc. Thus, circuit analysis software, such as SPICE2, becomes available to predict baseregion behavior. This avoids difficulties associated with the infinite number of natural frequencies characterizing a distributed system. It makes possible use of a circuitanalysis computer program such as SPICE for device analysis. This chapter also deals with a problem that arises in any lumped circuit approximation: the selection of the size (thickness) and the number of the small subregions (or lumps [14]). A criterion for this selection will be considered with the help of SPICE2 simulations. Along with the equivalent circuit for transient analysis, we develop equivalent circuits for lowfrequency smallsignal excitation. We correct the quasistatic input capacitance of the hybridw model for a transistor and include an inductance in the equivalent circuit. These changes arise systematically from the approach employed. The physical meaning of the inductance traces to the propagation delay of the minority carriers. These developments lead to an improved hybridir model for a junction transistor, which is advanced near the end of the chapter. 4.2 EquivalentCircuit for LargeSignal Transients 4.2.1 Derivation by TwoPort Approach We slice the quasineutral base, assumed to be in low injection, into many subregions. A typical subregion is shown in Fig. 4.1. From the twoport approach, we have the following linear matrix equation relating the excitations (minoritycarrier densities) and the responses (minority carrier currents) of the ith subregion: Ii(s) ii(O)/s Bi B12 Pi(s) Pi(O)/s (4.1) Ii+1(s) ii+1(0)/s B21 B22 Pi+ Pi+1(0)/s Y m z 0 x I 4 C *r 4 U) 0 J 0 S I U) U C 4 S.0 0 CO *r "U *0..1 IC) U 0 *r 0. Sa t+ (0 C *i 0r~ *  vi a> C: uvl where B11(s) = B22(S) = (AeD/L*)coth(Ax/L*) (4.2) and B12(s) = B21(S) = (AeD/L*)csch(Ax/L*) (4.3) Figure 4.2 displays the matrix equation (4.1), where pi(0)/s and Pi+1(0~)/s are the minority carrier densities and ii(0")/s and ii+1(O)/s are the minority carrier currents, respectively, at the two edges of the ith subregion. The designation, t = 0, means the instant before we apply the excitation to start the transient. The circuit elements Y1 and Y2 in Fig. 4.2 are related to B11 and B21 as follows: Y1 = B11 + B12 (4.4) Y2 = B12 (4.5) To realize an RLC equivalentcircuit representation of a quasi neutral region, which enables use of circuitanalysis software, such as SPICE2, one has to algebraically approximate the transcendental functions Y1(s) and Y2(s) in Fig. 4.2. The condition which makes possible a series expansion of (4.2) and (4.3) is IAx/L*I = IAx(1 + ST)1/2/(DT)1/21 < i (4.6) where T is the recombination lifetime [20]. For thickness Ax and a natural frequency s which satisfy (4.6), one can truncate the series to approximate Y1(s) and Y2(S) by (A ~1  S" X  I C 0 a CD 0 S U, aJ 0 L 43 S C s 0 0 3 I r I _ I a4 I coT CN r >3 e (4.7) Y1(s) = B11(s) + B12(s) = AeAx/sT + sAeAx/2 Y2(s) = B12(s) = AeD/Ax sAeAx/6 = (AeD/Ax)(1 SAX2/6D) = (AeD/Ax)(1 + sAx2/6D)"1 (4.8) (4.9) for Is << 6D/Ax2. From (4.7), B11(s) + B12(s) in Fig. 4.2 is realized in terms of admittancelike elements [9,21]: B11(s) + B12(s) = G + sC (4.10) where G = AeAx/2T C = AeAx/2 (4.11) (4.12) These circuit elements have unconventional dimensions because they describe, in (4.1), the linear relation between current and minority carrier density, rather than the usual relations between current and voltage. Thus G in (4.11) has dimensions of [A/cm3] and is associated with volume recombination, whereas C has dimensions of [C/cm3] and is associated with minority carrier storage. For small Isl, a simple network realization of B12(s) in (4.9) is B12(s) = 1/(R + sL) (4.13) where R = Ax/AeD (cm3/A) (4.14) and L = Ax3/6AeD2 (cm3s/A) (4.15) We associate R with minoritycarrier transport; for reasons to be discussed, L relates to minority carrier propagation delay. By combining (4.1), (4.10) and (4.13), we derive the equivalent circuit for the quasineutral base for largesignal transient excitation of the minoritycarrier densities (Fig. 4.3). Previous uses of the twoport approach explicitly [9,10,21,22] or implicitly [23] have neglected the factor multiplying s in (4.9). Hence the corresponding inductor appears here apparently for the first time. O . 0 IB0 + cn oc I 0 W + II x C 0 L .C 0 x A 0. Y aT 4.2.2 SPICE Simulation of the Equivalent Circuit for ESSCD We have carried out the SPICE simulations of RLC equivalent circuits with different numbers of subregions for a given quasineutral region. The thickness of the quasineutral region is 96 um, the lifetime of the minority carriers is 100 us, the diffusivity is 35 cm2/s, and the surface recombination velocity of the back lowhigh junction is 200 cm/s. The results of the 2 and 3 and 15subregion equivalent circuits are shown in the linearlinear graph of Fig. 4.4. The shortcircuit current of the 3subregion equivalent circuit nearly coincides with that of the 15subregion equivalent circuit for the time range where the firstmode current dominates the shortcircuit current. Recall that the firstmode component contains the information about the parameters T and S. Our earlier work in Chapters Two and Three, focused on determining these parameters. Figure 4.5 displays the shortcircuit current decay for the same solar cell on a semilogarithmic graph. The results for having two or more subregions nearly coincide for t > 0.4 us. Thus from Figs. 4.4 and 4.5, a 3subregion equivalentcircuit suffices for the determination of T and S for this solar cell. The decay time constant Td and the current ratio R are determined from Fig. 4.4 and Fig. 4.5: Td = 0.98 Ps and R = 31 These values coincide with the exact solutions obtained by solving (3.7) and (A.3) of Appendix A. TIME (p.s) +10 40 Fig.4.4 SPICE2 simulation of electricalshortcircuitcurrent decay as a function of the number of the subregions, indicated parametrically, used in the equivalent circuit for the quasineutral base. +3 1 (b) (c) I rd 0.98 ps Z0.5 W (a) 2  0 z 0.1 I I 0 0.5 1 1.5 TIME (vs) Fig.4.5 SPICE2 simulation of electricalshortcircuitcurrent decay responses displayed semilogarithmically for the quasineutral base sliced into (a) one, (b) two, and (c) fifteen equally thick subregions. The equivalent circuit used for this SPICE2 simulation includes inductors L, which come from the expansion of (4.11). Previous work has employed an expansion in which the factor multiplying s has been neglected [10,22]. For a 2subregion equivalent circuit, Fig. 4.6 displays the current decay with and without L. Recall that the current decay of Fig. 4.6 determines S and T through the slope and the intercept of the straightline portion of the transient. Thus, inclusion of the inductor L in a 2subregion equivalent circuit are needed for accurate modeling. In the time domain, we see that the inductors contribute delay, designated by TD in Fig. 4.6. In the frequency domain, the inductors filter out the highfrequency components of the response. The same effectsdelay in the time domain and filtering in the frequency domainresult from using a manysection equivalent circuit without inductors. The advantage in including inductors is that accuracy in the response is achieved while retaining a simple equivalent circuit. The same advantages are emphasized in the circuit simulation of bipolar integrated circuits. This subject is treated in the next section of this chapter. 4.3 Equivalent Circuits for LowFrequency SmallSignal Analysis 4.3.1 Derivation in Frequency Domain For lowfrequency smallsignal excitation, the matrix equation (2.6) becomes the following: 1 (C) (b) T z D . 0.5 (a) O N z 0.1III 0" 0.5 1 1.5 TIME (ps) Fig.4.6 The role of inductors in the response of a twosubregion equivalent circuit is illustrated in (a) and (b), (a) without inductors, (b) with inductors. Response (c) corresponds to a fifteensubregion model, with or without inductors. r All(S) A12(s) A21(s) A22(s) ev(O,s)p ( kT PDC(0) ev(XQNB ,s) kT PDC(XQNB) where Iv(O,s) < and V(XQN,S) < kand v is the hole e e electrochemical potential and P, as before, is excess hole density. This result is derived by using the approximation, exp[(VDC + v)/(kT/e)] = exp[VDC/(kT/e)]exp[1 + v/(kT/e)] = exp[vDC/(kT/e)][1 + v/(kT/e)] Rewriting (4.16), we have i(0,s) i(XQNB,S) D11(s) D12(s) D21(s) 022(s) v(0,s) V(XQNB.S) (4.17) In (4.17), i(0,s) i(XQNB,s) (4.16) D11(s) = KIA11(s) 012(s) = K2A12(s) D21(s) = K1A21(s) D22(s) = K2A22(s) where K1 = (e/kT)(ni2/NDD)exp(eV(O)/kT K2 = (e/kT)(ni2/NDD)exp(eV(XQNB)/kT) If we assume a thin quasineutral region (XQNB << L), we can realize (4.17) with RLC elements in a manner similar to that of the previous section. The equivalent circuit of a thin quasineutral region under lowfrequency smallsignal excitation is shown in Fig. 4.7, in which: G1 = eAK1XQNB/2T (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) a x %0. Sn 4 0 *r e cc r *r 4 > CC *r ( ) 0 0 3 *r 3 0) 1 0 (D .3 C cr0 4 > *r 0 U . UC I V *r C" * 0' ) I * > 3 C) U *r 0 C. * 0 'a I CU ) 3 0 S 0 UT ) _1 (>3: c.'J C)~ 0 uan 0o 01 0  I x C1 = eAK1XQNB/2 (4.25) R= XQNB/eAK2D (4.26) L= XQNB3/62eAK2 (4.27) G2= eAKXQNB/2T (4.28) C2 = eAKlXQNB/2 (4.29) and G(s) = eA(K1 K2)(XQNB/D SXQNB/6) (4.30) In Fig. 4.7, the nodal variable has the dimensions of voltage. This contrasts with our earlier models, for largesignals, for which the nodal variable is minoritycarrier density (see Fig. 4.3, for example). For smallsignal excitation, such as that assumed in Fig. 4.7, the incremental voltage across the p/n junction, and the incremental quasi Fermi potential for minority carriers, become linear in the minority carrier current. Thus, because the quasiFermi potential for majority carriers is essentially independent of position, if one subtracts any variations arising from ohmic drops in the base, one may regard this potential as the reference potential and set it to zero. Having established this convention, we then identify the incremental voltages v(0,s) and v(XQNB,s) with the incremental quasiFermi potential for the minority carriers. The equivalent circuit in Fig. 4.7 corresponds to that of Sah [10] except for the inductance present in Fig. 4.7. When we have a quasineutral region with a general contact at x = XQNB, we can derive the lowfrequency smallsignal equivalentcircuit directly by truncating the DC components of minoritycarrier density and current from Fig. 3.2 instead of starting from the matrix equation. Figure 4.8 displays this equivalent circuit of a quasineutral region with a general contact for lowfrequency smallsignal excitation. The values of the circuit elements are G = eAKlXQNB/2T, C = eAKlXQNB/2, R = XQNB/eAKlD, and L = XQNB3/6D2eAK1. The element Gs represents the general contact at the back surface, the value of which follows from Fig. 3.2: Gs = [eAK1S] (4.31) where S is the surface recombination velocity of the back surface. Figures 4.9.a and 4.9.b display the equivalent circuits for an ohmic contact and a blocking contact respectively. 4.3.2 Derivation in the Time Domain for the ShortBase Case The foregoing derivation in the frequency domain yields an equivalent circuit involving an inductor L, which does not commonly appear in smallsignal or incremental models for a quasineutral base a 0 0 0 P1 II0 (A 61 Ij L. r 4) Co 4. 0 c, 0 4 U Co CI J4J 4 c4, E cr, CO 0 O u r) 3 4 (0 U  X N U L4 C) 0) 5 or, >0) r0 0 LJJ> 0C, I U L. i(O,s) IVVV vv v(,s) I C G n___K ,____,__ _________ x= (a) i(O,s) L R I v(Os) xO (b) Fig.4.9 Equivalent circuit of surface recombination ohmic contact and (b) a quasineutral base when the back velocity is (a) infinite, as in an zero, as in a blocking contact. region. Thus we need to comment on the physical mechanisms occurring within the base region that give rise to this inductance. From the viewpoint of frequency response, the inductance causes the current phasor at x = XQNB to lag (in phase angle) the voltage phasor, v(0,jw). In the time domain, this lagging phase angle corresponds to a delay in the current at the back contact following the sudden application of voltage at x = 0. This delay is emphasized in the shortbase case, for which the minoritycarrier diffusion length greatly exceeds the quasi neutral base thickness. Hence, to interpret the inductance in Figs. 4.8 and 4.9, we now fix attention on the shortbase case, which enables a detailed consideration of physical mechanisms ongoing in the time domain. This yields the added advantage of making possible a derivation of a new equivalent circuit especially suited to shortbase devices, such as junction transistors. To simplify the discussion, we assume that the quasineutral base region is terminated by either an ohmic contact or by some other mechanisms preventing the accumulation of minority carriers at the back surface. One example of such a mechanism is a reversebiased or zero biased p/n collector junction. Before proceeding to details, we note, on qualitative grounds, that the delay under study arises because part of the particle current yields storage of particles; the remainder, which is the convection current (particle charge density times net particle velocity) acts to propagate the particles. If the convection current were absent, only capacitors, that is, no inductors, would appear in the equivalent circuit model. To derive an equivalentcircuit containing an inductor L from the time domain, we start from the quasistatic equivalent circuit for the commonemitter configuration commonly called the hybridi model [24]. We focus on the input stage of this equivalent circuit, it being understood that the voltage across the parallel combination of capacitance and conductanc in Fig. 4.10(a) controls a current source gmvbe in the output or collector circuit. We ignore for the present extrinsic elements such as base resistance. In this figure, the capacitance is the derivative of the minoritycarrier charge in the base region with respect to the input voltage, under the assumption that this charge retains for time variations the functional dependence that it has in the dc steady state. This capacitance corresponds to (quasistatic) charge storage within the base region. The conductance in Fig. 4.10(a) is the derivative of the (quasistatic) input current with respect to the input voltage divided by the dc commonemitter current gain B [14]. We formally include the effect of propagation delay by inserting a timedelay circuit element, producing Fig. 4.10(b). We seek a conventional circuit element that will produce a delay r(delay) in the output current, or equivalently in the voltage Vbe' relative to Vbe. The simplest such element results from use of a lowfrequency expansion of delay in the complexfrequency or Laplace domain: exp[sr(delay)] = 1 sT(delay) = 1/[1 + sr(delay)] (4.32) (a) The input stage of the conventional hybridr model of bipolar transistors. (b) Modified hybridw model with the inclusion of minority carrier delay across the quasi neutral base. (c) Modified hybridw model with the time delay element realized with inductor. Fig.4.10 + Vbe + be +C _L CQSA c * :1 I RQSA + V' be + 0+ Vbe SRQSA 6 'SA 3 d ^ r ~ r v This approximation corresponds to a passive network having impedance, Z(s) = R' + jX(s), and, thus from Fig. 4.10(b), V'be(s)/Vbe = [1 + st(delay)]1 = [1 + (sL'/R'QA) + (R'/R'QSA)]1 (4.33) This equation yields R' = 0 and L' so that r(delay) = L'/R'QSA (4.34) where a is the incremental commonemitter current gain. Thus Fig. 4.10(b), at this level of approximation, becomes the simple circuit of Fig. 4.10(c). It remains to determine the delay time T(delay). This determination will demonstrate that (B + 1) times L in Fig. 4.9 has the same value as the inductor L' in Fig. 4.10(c). If at the outset of this section we had dealt with the commonbase rather than the commonemitter configuration, then L' = L which is the L in Fig. 4.9. 4.3.3. Calculation of the Delay Time We now determine the minoritycurrent propagation delay of a short base by calculating the Elmore delay time [15], also used later by Ashar [25]: T(delay) = f te(t)dt/ f e(t)dt (4.35) 0 o where e(t) is the impulse response of the current at x = XQNB to vl(t) A6(t). In the complex frequency domain, this becomes [26] T(delay) = lim{[dF(s)/ds]/F(s)} .(4.36) s+O F(s) is the system function Vbe(s)/Vbe(s) the inverse transform of which is the impulse response of the system. The equality of (4.35) and (4.36) is shown in Appendix C. To determine in the time domain the minoritycarrier delay time of the thin quasineutral base region with an ohmic contact at x = XQNB, we apply impulse excitation at x = 0 and solve for the impulse response using the twoport approach. The quasineutral base region used in this calculation is thin compared with the diffusion length and we assign the values for D and XQNB arbitrarily. Figure 4.11(a) displays an example of the impulse response of the minoritycarrier density profile in position as time passes. Figure 4.11(b) displays the current impulse response at x = XQNB. We derive the delay time r(delay) from this simulation by using (4.35) and numerical integration: T(delay) = XQNB2/60 (4.37) (4,37 a; Sr 4) I c E EI c u E o o i o lr oo . C CL Z 0: S> 0 nn 4 C CL. . U, ^^ n 4. < I R m S 0 4.Q 0 . 0 a r_ v AS Ca (3nllA G3ZIlUWNQN) AiISN30 3Y0H SS33X3 80 SI0 A u S O' iII  .> S 4) ( Z ) 00 1 ir > a V 0. Uo y**4 I  I I I I Il iN3afnl 31OH SS33X3 We substitute (4.37) into (4.34) to get the inductance. This inductance in (4.40) is larger, by a factor (6 + 1), than the inductance in Fig. 4.9 as we indicated at the end of Sec. 4.3.2: L' = (B+ 1)(RXQNB2/6D) = ( + 1)L (4.38) where L' is the inductor for the commonemitter mode, L is the inductor for the commonbase mode, and R is the R in Fig. 4.9. 4.3.4 Modification of the Conventional HybridPi Transistor Model by Including the MinorityCarrier Current Propagation Delay The conventional hybridpi transistor model [24] in Fig. 4.12 is based on the quasistatic approximation which does not include the effects coming from the propagation delay of the minoritycarrier current across the base region. To include the propagation delay of the minoritycarrier current across the base in the conventional hybridpi transistor model, we consider the collector current delay when we change the emitterbase voltage. Figure 4.13(a) shows the minoritycarrier density corresponding to a step change in vbe based on the conventional quasistatic approximation. Note the instantaneous readjustment implied by this model. Contrast this with the transient response of the minority carriers determined by solving the continuity equation displayed in Fig. 4.13(b). This figure illustrates the physical origin of the delay. From the results of the previous section, the input port of the lAAA .0) E o 0, I t = 0+ emitter SCR base SCR collector Fig.4.13(a) Quasistatic response of minoritycarrier profile for a sudden change in the emitterbase voltage. t = emitter SCR base SCR collector Fig.4.13(b) Actual response of the minoritycarrier profile for a sudden change in the emitterbase voltage. hybridr model becomes as shown in Fig. 4.14. This inductance at the input terminal delays the voltage across the resistor r,. Thus the collector current lags the input voltage, Vbe. Thus if basewidth modulation is neglected, the input admittance is Y()) = 1/r + joiC j L'/r 2 = 1/r + jwC juwC/3(B + 1) (4.39) Equation (4.39) yields a more accurate approximation for low frequencies, w << 1/T(delay), than does the conventional hybridi model, which overestimates the input capacitance by a factor (38 + 2)/ (38 + 3). This factor differs only slightly from unity for many transistors. Thus the inclusion of delay corrects the input admittance of a transistor in the commonemitter configuration only slightly whereas for a transistor in the commonbase configuration the correction reduces the input capacitance by one third. For the commonemitter configuration, however, inclusion of delay is important to improving the accuracy of the response of the incremental collector current to incremental baseemitter voltage. The delay in the time domain, and the corresponding phase shift in the frequency domain, as derived in the foregoing treatment, comes only from the propagation of minority carriers across the quasineutral base region. An additional component of delay comes from the propagation of these same carriers across the emitterbase junction spacecharge region. This consideration lies beyond the scope of the present study; it constitutes part of the ongoing research of J. J. Liou at the Department of Electrical Engineering of the University of Florida. SI A^&  a) .0 S  I= 'H ..J "  f0 C, S 3 4) U U) r ) 0 "CII Ln *r 0 01 0 (U 0 OT II *r 5 I C *r Ec I I II *I  *. j U to E. Q) ^2? *o n 4.3.5 MinorityCarrier Delay Time with BuiltIn Electric Field In this section, we estimate the delay time of the minority carrier delay time across the quasineutral base region when the electric field exists due to the nonuniformity of the base doping concentration profile. Assume a uniform builtin electric field within the quasineutral base. Then the minoritycarrier current at x = XQNB is [22] I(XQNB,s) = AeDnpo eeV/kT G csch(GXQNB)P(O,s) = A21(s)P(0,s) (4.40) where K is the ratio of the base doping concentrations at the two edges, x=0 and X=XQNB and where logK 2 1+s 1/2 G = [(2X ) + ] (4.41) QNB L Using Ashar's definition [25] of delay time, we have da(s) X 2 1 log K 2 X 2 T(delay) = lim 21 s X leK2 X+ N 2]} (4.42) s6 21(s) 6 {i 1 e( ) + ( L (4.42) s*O 21 For the simple case when the base doping concentration is uniform, K=1, we have the following from (4.42): 2 T(delay) = XNB (4.43) Since the expression of delay time in (4.42) includes the uniformdoping case and uniformelectricfield case, (4.42) holds more generally for the delay time of minoritycarrier across the quasineutral base. 4.3.6 Measurement of MinorityCarrier Delay Time Across the Base Region of Transistors We estimated the delay time by measuring the excess phase shift [27] of two transistors of the same type. Figure 4.15 displays the geometry of transistors measured. Figure 4.16 displays the impurity doping profile of the transistor. The base width of the transistors is estimated to be about 10 wm if we consider the influence of the space charge region width and of the overlapping of the buried layer and the base region. The estimation of the base width is in Appendix D. The parameter K is about 6. We put these parameter values into (4.42). The delay time is estimated to be about 5 ns. We measured the phase angle of the common emitter hybrid parameter h21 of transistors at IC = 10 pA, and 100 iA. The results of these measurements are shown in Fig. 4.17(a)(b). The delay times estimated from these figures are 5.3 ns and 5.6 ns. (The frequency range used to estimate the delay time should be much less than the reciprocal value of the delay time; only then can we regard the actual delay across the quasineutral base as originating from the time required for carriers to propagate across the base.) The measurement results are in good agreement with the theoretical value. C,, U, U, 0 0o 20 10 emitter n+ r^ 18 E 10 buried layer n E 0 LU I LL C 16 S10 Base p 5 1 zS 0 10 20 30(pm) DISTANCE FROM THE SURFACE Fig.4.16 Impurity profile of the measured transistors. The dashed line is for the exponential fucntion approximation.  > u ' I :: '4 'N. N 4. 0 LJ , S.. YC aJ m 4 Z c5 S S 0 4 ) (apI I (2sj62P)^L .O 3SWHd __ 