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Use of transients in quasi-neutral regions for characterizing solar cells, diodes, and transistors

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Use of transients in quasi-neutral regions for characterizing solar cells, diodes, and transistors
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Jung, Tae-Won, 1953-
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English
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ix, 110 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Diodes ( jstor )
Electric current ( jstor )
Electric potential ( jstor )
Electrons ( jstor )
Equivalent circuits ( jstor )
Minority carriers ( jstor )
Photovoltaic cells ( jstor )
Propagation delay ( jstor )
Transistors ( jstor )
Velocity ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Semiconductors ( lcsh )
Solar cells ( lcsh )
Solid state electronics ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 107-109.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Tae-Won Jung.

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USE OF TRANSIENTS IN QUASI-NEUTRAL REGIONS FOR
CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS
















By

TAE-WONLJUNG


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

S-parameter 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. Hyung-Kyu Lim, Mr. Jong-Sik Park, Mr. J. J. Liou, Mr. M. K. Chen,

Dr. Soo-Young 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 parents-in-law 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-
SURFACE-FIELD SOLAR CELLS, WITH APPLICATION TO EXPERIMENTAL
SHORT-CIRCUIT-CURRENT DECAY.................................... 4

2.1 Introduction........................................... 4
2.2 Mathematical Framework................................5
2.3 Transient versus Steady-State Analysis via Two-Port
Techniques ...............................................9
2.4 Open-Circuit-Voltage Decay (OCVD).......................11
2.5 Reverse Step Recovery (RSR)..........................14
2.6 Electrical Short-Circuit-Current 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 SHORT-CIRCUIT-CURRENT
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 Short-Circuit-
Current Decay and Dark I-V Characteristics.......35









3.3 Experiments.............................................36
3.3.1 Improvements in the Circuit for Short-Circuit-
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 EQUIVALENT-CIRCUIT REPRESENTATION OF THE QUASI-NEUTRAL BASE,
WITH APPLICATION TO DIODES AND BIPOLAR TRANSISTORS............53

4.1 Introduction............................................53
4.2 Equivalent-Circuit for Large-Signal Transient...........54
4.2.1 Derivation by Two-Port Approach.................54
4.2.2 SPICE2 Simulation of the Equivalent Circuit for
ESCCD .......................... ............... 61
4.3 Equivalent Circuits for Low-Frequency Small-Signal
Analysis ................................................. 64
4.3.1 Derivation in Frequency Domain....................64
4.3.2 Derivation in the Time Domain for Short-Base
Case.............................................70
4.3.3 Calculation of the Delay Time....................77
4.3.4 Modification of the Conventional Hybrid-i Model
by Including the Minority-Carrier Current
Propagation Delay................................81
4.3.5 Minority-Carrier Delay Time with Built-In
Electric Field......................................87
4.3.6 Measurement of Minority-Carrier Delay Time Across
the Quasi-Neutral 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 SHORT-CIRCUIT-CURRENT 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




3.3 Experiments 36
3.3.1 Improvements in the Circuit for Short-Circuit-
Current Decay 36
3.3.2 Quality of the Short Circuit of the Switching
Ci rcuit 37
3.3.3 Measurement of the Dark I(V) Characteristics 42
3.4 Experimental Results and Discussions 43
FOUR EQUIVALENT-CIRCUIT REPRESENTATION OF THE QUASI-NEUTRAL BASE,
WITH APPLICATION TO DIODES AND BIPOLAR TRANSISTORS 53
4.1 Introduction 53
4.2 Equivalent-Circuit for Large-Signal Transient 54
4.2.1 Derivation by Two-Port Approach 54
4.2.2 SPICE2 Simulation of the Equivalent Circuit for
ESCCD 61
4.3 Equivalent Circuits for Low-Frequency Small-Signal
Analysis 64
4.3.1 Derivation in Frequency Domain 64
4.3.2 Derivation in the Time Domain for Short-Base
Case 70
4.3.3 Calculation of the Delay Time 77
4.3.4 Modification of the Conventional Hybrid-ir Model
by Including the Minority-Carrier Current
Propagation Delay 81
4.3.5 Minority-Carrier Delay Time with Built-In
Electric Field 87
4.3.6 Measurement of Minority-Carrier Delay Time Across
the Quasi-Neutral 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 SHORT-CIRCUIT-CURRENT 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







ii(t) incoming minority carrier current toward a subregion at
x = Xi

IQNBO pre-exponential factor of steady-state quasi-neutral-
base current

IQNEO pre-exponential factor of steady-state quasi-neutral-
emitter current

Ish current through the shunt resistor of a diode

I(x,s) Laplace transform of i(x,t)

Ki (-1-si )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) steady-state 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 turn-on resistance of MOS transistor

rs series resistance of a diode

rsh shunt resistance of a diode









si the ith natural-frequency

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
natural-frequency) current

T(delay) propagation delay time of minority carriers across the
quasi-neutral 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 p-n junction

XQNB quasi-neutral base width

Yi admittance-like elements for a quasi-neutral 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 QUASI-NEUTRAL REGIONS FOR
CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS



SBy



TAE-WON 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 quasi-neutral 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 quasi-neutral base region of p/n junction silicon solar

cells, including devices having conventional back-surface-field (BSF),

ion-implanted BSF and polysilicon BSF structures. This method, called

electrical-short-circuit-current-decay (ESSCD) avoids errors introduced

in other methods in common use, such as open-circuit-voltage-decay and

reverse step recovery, that arise from the capacitive effects of mobile


viii









holes and electrons in the volume of the p/n junction space-charge

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

quasi-neutral 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 transmission-line, model. The inclusion

of the inductor makes this possible. Models of this type are developed

for both small-signal and large-signal 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 hybrid-T model for bipolar transistors in the common-emitter

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

Si-device 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 quasi-neutral 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 in-process 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 quasi-neutral 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 quasi-neutral base of back-

surface-field (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 two-port approach [9]

to obtain a unifying framework for transient analysis and develop from

it a new method, experimental electrical-short-circuit-current-decay

(ESCCD) for the determination of the recombination lifetime and back-

surface recombination velocity of the quasi-neutral 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 4-mil BSF solar

cells). We derive various performance parameters for BSF solar cells by

exploring the ESCCD method. We propose a methodology to separate the

quasi-neutral 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 minority-carrier propagation delay

in a compact equivalent circuit. This enables the use of standard

circuit-analysis computer programs, without resorting to much more

sophisticated programs needed when transmission-line models [10] are

employed. These ideas are applied both to ESCCD and to bipolar





3





transistors, for which an improved hyprid-i 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
BACK-SURFACE-FIELD SOLAR CELLS,
WITH APPLICATION TO EXPERIMENTAL SHORT-CIRCUIT-CURRENT DECAYS



2.1 Introduction

This chapter has three purposes. First, we outline a mathematical

method that systematically and compactly describes the large-signal

transient and small-signal 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

quasi-neutral principal regions of the devices.

Second, exploiting this description, we survey the adequacy of

various experimental large-signal 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 quasi-neutral base from a

single treatment measurement for three different p+/n/n+ back-surface-

field solar cells.





5




2.2 Mathematical Framework

In this section, we develop a mathematical framework which could be

applicable to most of the large-signal transient measurement methods and

could include small-signal 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

minority-carrier density and the minority-carrier current in a quasi-

neutral base region in low injection. Focusing on the quasi-neutral

base, assumed to be n-type here (of x-independent donor density NDD)

with no loss in generality, will simplify the treatment; extensions to

the quasi-neutral emitter are straightforward, provided one inserts the

physics relevant to n+ or p+ regions.

Assume a p+/n diode in which the uniformly doped quasi-neutral 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 back-surface-field (BSF)

region. Assume also low-level 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:




5
2.2 Mathematical Framework
In this section, we develop a mathematical framework which could be
applicable to most of the large-signal transient measurement methods and
could include small-signal 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
minority-carrier density and the minority-carrier current in a quasi
neutral base region in low injection. Focusing on the quasi-neutral
base, assumed to be n-type here (of x-independent donor density Nqq)
with no loss in generality, will simplify the treatment; extensions to
the quasi-neutral emitter are straightforward, provided one inserts the
physics relevant to n+ or p+ regions.
Assume a p+/n diode in which the uniformly doped quasi-neutral base
starts at x = 0 and has a general contact defined by arbitrary effective
surface recombination velocity Seff at the far edge x = Xq^g. Such a
contact could result, for example, from a back-surface-field (BSF)
region. Assume also low-level injection and a uniform doping of the
base region. Then a linear continuity (partial differential) equation
describes the excess minority holes p(x,t)
3p(x ,t)/ at = Dp32p(x,t)/ax2 p(x,t)/T (2.1)
where Dp is the diffusion coefficient and Tp 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:


7
Regarding the minority carrier densities at the two edges as the
excitation terms for a system analogous to a linear two-port network of
circuit theory, we have the following two-port network matrix from (2.4)
and (2.5) for the two excitations (densities) and the two responses
(currents):
1(0,s) i(0,0")/s
"An
A12
_P(0,s) p(0,0")/s
I(xqNBs) 1(xqNB, )/s
_A21
A22_
P(XQNB *S) "p(XQNB
(2.6)
where p(0,0") and p(XgNg,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 quasi-neutral
base. In Eq. (2.6), A12=A21= -e(Dp/Lp*) cosech(XQNB/L *) and An= A22
e(Dp/L*)coth(Xq^/L*). Figure 2.1 displays the master equation, where
the initial values are included in 1(0,s), I(XqNB,s) ,P(0,S), and
P(XQNBs) *
Transient solutions can be derived from (2.6) by inserting proper
boundary conditions, initial values and constraints imposed by the
external circuit. For example, l(0,s)=0 in OCVQ open-circuit-voltage-
decay [1], l(0,s)=constant for reverse step recovery [2,3], and P(0,s) =
0 for short-circuit current decay, the latter of which is developed in
detail here. For small-signal methods [11-12], where dl, for example, is
an incremental change of current, I(0,s)=Iqq/s + dl(0,s) and
P(0,s)=Pqq/s + (edV/kT)PQQ. 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.









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 electrical-short-circuit-current-decay (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. Steady-State Analysis via Two-Port 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 quasi-neutral regions under study using the two-port

technique described in Sec. 2.2, the displacement current is

negligible. In the steady state, the two-port description leading to

the master equation simplifies because the hole current in our example

of Sec. 2.2 depends only on position x. Tnis x-dependence results from

volume recombination (relating to the minority-carrier lifetime) and

effective surface recombination (relating to the effective surface

recombination velocity). A two-port 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 quasi-neutral base;

and, using quasi-neutrality together with knowledge of the steady-state

currents in the junction space-charge region and in the p+ quasi-neutral

emitter region, one can thus find the steady-state current flowing in

the external circuit or the voltage at the terminals of the diode. If

the quasi-neutral 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 time-varying mode of operation leads to a

minority hole current in the n-type quasi-neutral 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 two-port 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.




10
leads to the same matrix description as that derived previously, in
which the matrix elements A^j(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 quasi-neutral base;
and, using quasi-neutrality together with knowledge of the steady-state
currents in the junction space-charge region and in the p+ quasi-neutral
emitter region, one can tnus find the steady-state current flowing in
the external circuit or the voltage at the terminals of the diode. If
the quasi-neutral 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 time-varying mode of operation leads to a
minority hole current in the n-type quasi-neutral 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 3p/3t 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 two-port 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.








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. = /-l-ST > 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 open-voltage

decay curve is usually bent up or bent down because of discharging and

recombination within the space-charge region.

As mentioned in [11], the mobile charge within the space-charge

region contributes significantly to the observed voltage transient for

Si, in which n 1010 cm3, but not in Ge, for which OCVD was first




12
p(0,t)
2i(0,0")L
eDpV
^ + 3
[cosec (xqnbK/Lp) + ^Dp/LpSeff^
(2.8)
where s.¡ is the ith singularity point (ith mode) which satisfies
coth(XQNB'/l+si Tp/Lp) + Dp /1+si Tp,/LpSeff ~ 0 (29^
and K. = /-1-s.t > 0.
i 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 s.¡ is called a mode, as
in the electromagnetic theory. Appendix A treats the details of
determining the Eigenvalues s^ from Eq. (2.9) (and from the similar
Eq. 2.11 derived below).
The decaying time constant -1/s^ 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 S0ff 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 = 30U K. In Si devices the open-voltage
decay curve is usually bent up or bent down because of discharging and
recombination within the space-charge region.
As mentioned in [11], the mobile charge within the space-charge
region contributes significantly to the observed voltage transient for
10 -3
Si, in which n^ =10 cm but not in Ge, for which OCVD was first









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 space-charge 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 first-mode 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.




14
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 1(0,s) = constant (reverse current) at 0 < t < ts, where ts
is the time needed for the excess hole density p(0,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 space-charge region
complicates the interpretation of the measured xs in terms of the
desired parameters, Xp and Seff.
In addition to this, during the recovery transient (xs < t < <*),
the reverse generation current is often large enough to saturate the
recovery current so quickly ttiat we have no sizable linear portion of
the first-mode curve on a plot of ln[i(t)] vs. t. This linear portion
provides interpretable data for Ge devices [3], but not often for Si
devices according to our experiments.









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 space-charge region within a transit

time of this region (about 10-11s 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 space-charge region in the ESCCD method occurs within a time of

the order of 10-11s, which is much less than any of the times associated

with discharge of the quasi-neutral regions. This absence in effect of

excess holes and electrons within the junction space-charge 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 space-charge region appears in Appendix B.

The discharge of the quasi-neutral emitter depends on the energy-

gap narrowing, the minority carrier mobility and diffusivity, the

minority-carrier 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 quasi-neutral base, and we shall assume

this is so in the discussion to follow.




16
its equilibrium value. (The physics governing this phenomenon comes
from Maxwell's Curl H_ = j_ + 3D_/3t; taking the divergence of both sides
yields 0 = div _i_ + d(div _D)/dt, which, when combined with _i_ = qO/e and
divD^ = p, yields a response of the order of e/a, the dielectric
relaxation time.)
Following this readjustment of the barrier height, the excess holes
and electrons exit the junction space-charge region within a transit
time of this region (about lCT^s 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 space-charge region in the ESCCD method occurs within a time of
the order of 10s, which is much less than any of the times associated
with discharge of the quasi-neutral regions. This absence in effect of
excess holes and electrons within the junction space-charge 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 space-charge region appears in Appendix B.
The discharge of the quasi-neutral emitter depends on the energy-
gap narrowing, the minority carrier mobility and diffusivity, the
minority-carrier 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 quasi-neutral base, and we shall assume
this is so in the discussion to follow.


17
Having established that the mobile carriers in the junction space-
charge region enter the electrical-short-circuit-decay 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
1(0,s) = i(0,U")/s
eDpp(0,0')
V
coth + VLp*Seff
1 + (pVW^VnbV
(2.10)
Cauchy's residue theorem yields the inverse transform of (2.10):
1(t) ? eDpP(-0')K1 "pWeff jjd _
1=1 1 p ^ Tp/si Ki ^ + ^XQN8/2Seff)cosec (KiXQNB/!V
(2.11)
where s^ is the ith singularity which satisfies the Eigenvalue equation,
D XnMR
1 + l-s2 /TTi7% coth( l = 0 (2.12)
p eff ^ p ^
and where = (-1-s.jTp)^2 > 0, with s^ < 0.
Truncating (2.11) and (2.12) to include only the first mode (s^),
we obtain
1/2
1 + (D /L S *-) /I + s, t coth[(XnKID/L )(1 + s.t ) ] = 0
v p p eff' 1 p v QNB p/v 1 p'
(2. 13)








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 quasi-neutral base charges

to store ultimately a steady-state 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

short-circuit constraint. The large capacitor maintains this constraint

nearly perfectly during the first-mode 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 short-circuit

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 1--NDD (substrate doping) = 6 x 1014 atoms/cm3; XQNB (base

thickness) = 348 jm, A = 4 cm2; DEVICE 2--NDD = 7 x 1014 atoms/cm3

XQNB = 320 um, A = .86 cm2, DEVICE 3--NDD = 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 first-mode 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 semi-logarithmic counterpart, illustrating the

straight-line 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

I-V characteristic is V = .03851n(I/Io+1). If the first-mode current is



Ifirst-mode(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 ifirst-mode (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.




22
much faster circuit. Here xd = -l/s^ is the first-mode 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 semi-logarithmic counterpart, illustrating the
straight-line portion of the transient obtained from the output of the
logarithmic amplifier in Fig. 2.2. From this Td 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
I-V characteristic is V = .03851n(I/IQ+1). If the first-mode current is
ifirst-mode^) = constant exp(-t/Td) td =-l/s1 (2.15)
then the slope of the output voltage of log amplifier is -38.5 mV/rd.
Extrapolation of the straight portion in Fig. 2.3(c) yields the initial
value ifirst-mode (+) as the intercept.
We measure the decay time constant and the initial amplitude of the
first model as follows: DEVICE 1, Td = -1/s^ = 29.3 us, i^1-rst(0+) =
2.73 mA for V(0") = 0.44 V and T = 303.1 K. For DEVICE 2, xd =
24.5 usee, if1-rst(0+) = 4.35 mA at v(0") = 0.5 V and T = 302.9 K. For
DEVICE 3, td = 28.5 usee, ifirst(0+) = -696 mA at v(0) = .47 V and
303.5 K. Here v(0) denotes the steady forward voltage applied across
the solar cell before the transient.


23
From the above development, these results give DEVICE 1, xp =
119 gs, Seff = 25 cm/sec; DEVICE 2, = xp = 119 gs, Seff = 60 cm/s;
DEVICE 3, Tp = 213 gs, Seff = 100 cm/s. These results agree favorably
with those obtained for the same devices by using the more time-
consuming methods detailed in [11-12].
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 space-charge-
region. These methods, among open-circuit 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 Eq 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 n^ 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 space-charge region play no role in determining the response of
the electrical-short-circuit-current 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 first-mode response,



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
SHORT-CIRCUIT CURRENT DECAY


3.1 Introduction

This chapter describes various improvements of the method of

electrical short-circuit 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 first-mode current Ifirst-mode(O+) together with the decay time

constant rd as the ESCCD parameters used to determine T and S. The

parameter ifirst-mode(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: ifirst-mode(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 Cifirst-mode(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 short-circuit 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 short-circuit

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 two-port 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.




Third, in this chapter we consider the sensitivity problem involved
in the method of electrical short-circuit 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, xm_jn and R^, the importance of which is
discussed in this chapter.
Fourth, we show quantitatively that the electrical short-circuit
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 Ipg in the ratio above brings the
emitter recombination current into our method for determining x and
S of the base region. This, however, is only apparently a problem.
Indeed, we illustrate that use of the S(x) locus enables a determination
of Iqnj:, 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 two-port network formulation introduced in
Chapter Two in (2.6), the representation for which is illustrated in
Fig. 3.1, where Y^s) = An(s) + A-i2(s), Y2(s) = -A12(s), P^s) =
p(0,0")/s, P2(s) = p(XgNB,0")/s, Ix(S) = i(0,0)/s, and I2(s) =
i UqnBO


Fig.3.1
Two-port network representation of a quasi-neutral region with boundary conditions
at x = XqNB and x = 0. The nodal variable is the excess minority carrier concentration.






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-




S-CU
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 low-injection
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 minority-carrier current at x = 0 for t < 0 is


i(0, 0-) = IQNBO (exp(qV(0")/kT 1)


(3.5)




31
*qnb,S,S^ ~ (3.1)
Solving the network of Fig. 3.2 for 1(0,s) under the low-injection
condition yields
1(0,s) = i(0,0)/s Yx p(0,0")/s
(Y1 + V Y2
~ ytTV 'Y~ (P( )/s) (3.2)
T1 t2 s
ni2
where p(0,0) = t¡ (exp(eV(0 )/kT) -1) (3.3)
WDD
If we use the Cauchy Residue Theorem to obtain the inverse
transform of 1(0,s), we get an infinite series for i(0,t). Truncating
this series after the first term, at t = 0+, yields
^first node^ ) = ^FMO (exP(eY(0 )/kT)-l)
where Ip^Q
AeDKjn.2
S1LN00
(2.14)
cot(K,XnNR/L) DK./LS
1 1 e (3.4)
(t/2Kx2) + (Xqnb/2S)csc(K1Xqnb/L)
The minority-carrier current at x = 0 for t < 0 is
i(0, 0) = I^NB0 (exp(qV(0")/kT 1)
(3.5)


32
y ADni sinh(XQNB/L^ + aCOShXQNg/1-) ^
e ¡QNB0 = LNdd cosh(XgNB/L) + asinn(XQNB/L) (3*6)
LS
and where a = g-. Here a is the ratio of the normalized surface
recombination velocity to the diffusion velocity [16]. Thus the ratio
R = IpM0
!QNB0
2K^ cotA^ + tanAj
-
Si (XqNB/A1)2/D + (XQNB/S) CSc2a1
cosh (Xq^g/L) + os i nh (Xg^jg/L)
si nh(XQNB/L) + etcosh(XqNB/L) (3-7)
where A^ = K^XqNB/L and where K1 and A^ are obtained by solving (A.3) of
Appendix A. The ratio R will be utilized for the determination of the
quasi-neutral 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 space-charge recombination current component is
negligible [17], the I(V) characteriStic of the solar cell is









I =D + Ish (3.8)




= IFO (exp(eV/kT)-1) + V/rsh (3.9)


Here IFO is the pre-exponential factor of the forward bias current and V
is the voltage across the space-charge region. The pre-exponential

factor IFO in (3.9) has two components:


FO = IQNBO + IQNEO (3.10)


where IQNBO is the quasi-neutral-base current component and IQNEO is the

quasi-neutral-emitter 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 I-V characteristics from the terminals of a solar

cell. The pre-exponential factor IFO will be utilized for the

determination of the base material parameters.


3.2.3 Combined Method of Electrical Short-Circuit Current Decay and
Dark I-V 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 I-V 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 pre-exponential factor of the

first-mode current. Using the dark I-V measurement, we estimate the

pre-exponential 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

T-S 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 bipolar-transistor switching circuit in

Fig. 2.2 to measure the decay time constant and the initial amplitude of

the first natural-frequency 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




36
(3.7), since R is a function of both t and S. Also each point (t,S)
produces its own value for Iq^gg 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 Iq^q. Specifically
Td = VTS) (2.13)
ipo = Vt,SIQNE0^
and
RM = f3^xS,IQNE0^
Using (2.13), (3.10) and (3.14) and the measured variables, one can
solve for x, S and Iq^eq in a manner to be described later.
3.3 Experiments
3.3.1 Improvements in the Circuit for ESCCD
Previously we used a bipolar-transistor switching circuit in
Fig. 2.2 to measure the decay time constant and the initial amplitude of
the first natural-frequency current at t=0+. 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
(3.10)
(3.14)






























cuJ



b~I r



'V)i

Th


bI 0





o4


I0
0


& I


(A
I-


-il


_ _


"cM


,_ l
PCZi




Fig.3.4 Switching circuit used in measuring short-circuit-current decay parameters, x and S.








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 steady-state hole or electron density
through the volume of the space-charge 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 first-term or dominant natural-frequency current decay of the solar

cells.




41
where
Vi2U) = -(rHOS + rs + rd)i(t) .
(3.20)
Since the ratio (rMQ£ + 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] + ix(t) (3.21)
Solving (3.21) for i(t) yields
i(t) = E1exp(-t/xSCR) + E2exp(-t/xd)
(3.22)
where
tSCR0 r CSCR0^rM0S + r s + rd ^
(3.23)
As can be seen in (3.22), the first term of the right side can be
neglected and xd can be determined if the time constant x^qR is much
smaller than x. For the switching circuit of Fig. 3.4 x<^R 2 200 ns.
For the solar cells described in this chapter, 0.5 us < xd < 30 us. Thus
the RC time constant of the measurement circuit negligibly influences
the first-term or dominant natural-frequency current decay of the solar
cells.









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 ohm-cm p-type 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 cm-3 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 = s-1 + T-1 (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 quasi-neutral 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 thin-base 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 long-base 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 p-type 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




47
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(x)
exhibits dS/di 0 over a large range of t. This is the mathematical
statement, for our procedure, that S is more accurately determined than
is t for a thin-base solar cell. (If Xq^b/L 1, dS/dx = over a
large range of S, which means that t is more accurately determined than
is S for a long-base 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 p-type substrates. The top n+ layer is about 0.3 um deep.
The thickness of these cells is about 100 un.
Using xd, we generated S( t) 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/dx approaches zero,
defines Smax. If for an extreme case for which negligible volume
recombination occurs during the ESCCD transient, S = S(T)ax.
Similarly, the limit dS/dx > defines a minimum value of the
lifetime xmin, as illustrated in Fig. 3.7. For the all corresponding to
the lower locus of Fig. 3.7, Tjn^n 40 us. This value xj^ occurs for









the extreme case of negligible surface recombination at the back contact

during the ESCCD transient. For the upper-locus 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 open-circuit 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 poly-Si layers on the back surface of the base. The value

of S is estimated to be about 2000 cm/s for n+/p/p+-poly-Si cell

(poly 1) and about 400 cm/s for p+/n/n+-poly-Si cell (poly 2). Thin

cells are characterized in terms of mnin and Smax.




49
the extreme case of negligible surface recombination at the back contact
during the ESCCD transient. For the upper-locus cell in Fig. 3.7,
Tm^n = 0. These two parameters, Smax and Tml-n, can be used as
performance parameters for thin solar cells; small Smax and large i^in
is desirable for thin BSF solar cells for a given base thickness and
doping concentration.
We also measured the values of R^: = 20 for the lower locus and
Rm 3 for the upper locus. But we cannot use the measured R^ 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 Iq^eq and small S for a thin solar cell. These
conditions imply a large open-circuit voltage for a given base thickness
and doping. Small RM usually implies either a poor BSF contract at the
back surface or a large Iq|\|£q For example, for the better BSF solar
cell (the lower locus), we have Rr1 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 poly-Si layers on the back surface of the base. The value
of S is estimated to be about 2000 cm/s for n+/p/p+-poly-Si cell
(poly 1) and about 400 cm/s for p+/n/n+-poly-Si cell (poly 2). Thin
cells are characterized in terms of -i^n and Smax.


50
Table
3.1 ESCCD MEASUREMENTS
FOR VARIOUS SOLAR
CELLS.
NAME
RESISTIVITY
(ohm-cm)
THICKNESS
(ym)
Td(ys)
t (y s)
S(cm/s)
L(ym)
SPIRE
.31
374
6.5
15
1300
185
ASEC1
.15
301
3.6
7
ohmic
93
ASEC2
.15
267
4.0
13
ohmic
126
BSF#1
10
240
5.3
35+
100-400*
350+
BSF#2
10
260
6.3
45+
100-350*
400+
BSF#3
10
284
7.85
75+
100-225*
512+
BSF#4
10
96
.98
20+
100-380*
265+
BSF#5
10
91
.9
25+
100-290*
295+
BSF#6
10
107
.7
4500*
BSF#7
10
102
.73
3200*
LEU#1++
8
328
28.0
145+
40-80*
417+
LEU#2++
1.5
325
25.7
105+
40-150*
347+
P0LY1
2
203
2.7
2000*
P0LY2++
2
208
8.8
25+
100-400*
168+
++ denotes p+/n/n+ BSF solar cell. + denotes Tmjn or Lmin.
* denotes Smax. P0LY1-2 have poly-Si layers at the back surfaces.












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(C-3-5).
Locus 2: BSF solar cell(2-53).

















CHAPTER FOUR
EQUIVALENT-CIRCUIT REPRESENTATION OF THE
QUASI-NEUTRAL BASE,
WITH APPLICATIONS TO DIODES AND TRANSISTORS



4.1 Introduction

In the previous chapters, we treated the quasi-neutral base region

using the two-port 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

large-signal 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 base-region behavior. This avoids

difficulties associated with the infinite number of natural frequencies

characterizing a distributed system. It makes possible use of a

circuit-analysis 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.




CHAPTER FOUR
EQUIVALENT-CIRCUIT REPRESENTATION OF THE
QUASI-NEUTRAL BASE,
WITH APPLICATIONS TO DIODES AND TRANSISTORS
4.1 Introduction
In the previous chapters, we treated the quasi-neutral base region
using the two-port 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
large-signal 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 base-region behavior. This avoids
difficulties associated with the infinite number of natural frequencies
characterizing a distributed system. It makes possible use of a
circuit-analysis 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.
53



























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 equivalent-circuit representation of a quasi-
neutral region, which enables use of circuit-analysis 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







4-3
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

admittance-like 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




58
Yi(s) = B^(s) + B^2(s) AeAx/st + sAeAx/2
(4.7)
and
Y2(s) = -B-^2(s) = AeD/Ax sAeAx/6 (4.8)
= (AeD/Ax)(1 sAx2/6D) = (AeD/Ax)(l + sAx2/6D)_1 (4.9)
for |s| 6D/Ax2.
From (4. 7), Bn(s) + B12(s) in Fig. 4.2 is realized in terms of
admittance-like elements [9,21]:
BH (s) + B12(s) = G + sC (4.10)
where
G = AeAx/2 t (4.11)
and
C = AeAx/2
(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











O .
0 IB0


+
cn






































oc
-I




















0
W


+

II
x


C
0

L
.C

0


-x


A

0.
Y

aT




61
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 quasi-neutral
region. The thickness of the quasi-neutral region is 96 urn, the
lifetime of the minority carriers is IDO us, the diffusivity is
35 cm^/s, and the surface recombination velocity of the back low-high
junction is 200 cm/s. The results of the 2- and 3- and 15-subregion
equivalent circuits are shown in the linear-linear graph of Fig. 4.4.
The short-circuit current of the 3-subregion equivalent circuit nearly
coincides with that of the 15-subregion equivalent circuit for the time
range where the first-mode current dominates the short-circuit current.
Recall that the first-mode component contains the information about
the parameters t and S. Our earlier work in Chapters Two and Three,
focussed on determining these parameters.
Figure 4.5 displays the short-circuit current decay for the same
solar cell on a semi logarithmic graph. The results for having two or
more subregions nearly coincide for t > 0.4 gs. Thus from Figs. 4.4 and
4.5, a 3-subregion equivalent-circuit suffices for the determination
of t and S for this solar cell. The decay time constant and the
current ratio R are determined from Fig. 4.4 and Fig. 4.5:
= 0.98 ys and R = 31
These values coincide with the exact solutions obtained by solving (3.7)
and (A.3) of Appendix A.











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 electrical-short-circuit-current decay
responses displayed semilogarithmically for the quasi-neutral
base sliced into (a) one, (b) two, and (c) fifteen equally
thick subregions.




NORMALIZED CURRENT
63
SPICE2 simulation of electrical-short-circuit-current decay
responses displayed semilogarithmically for the quasi-neutral
base sliced into (a) one, (b) two, and (c) fifteen equally
thick subregions.
Fig.4.5











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 two-subregion
equivalent circuit is illustrated in (a) and (b), (a) without
inductors, (b) with inductors. Response (c) corresponds to a
fifteen-subregion model, with or without inductors.




NORMALIZED CURRENT
65
Fig.4.6 The role of inductors in the response of a two-subregion
equivalent circuit is illustrated in (a) and (b), (a) without
inductors, (b) with inductors. Response (c) corresponds to a
fifteen-subregion model, with or without inductors.








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 quasi-neutral 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 quasi-neutral region
under low-frequency small-signal 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




'(XqNBs)
+ o-
- o-
Li ni
1 .
j *
1
1
1
1
o,s) ¡ Q
1
I
=ci :
G G <
> 1 2 <
>
2
1
1
' ¡ V(XQNB
1
1
1
-O +
-o -
x = O
x = X
QNB
Fig.4.7 Low-frequency equivalent circuit for the quasi-neutral base in low injection for small-
signal variations in the applied voltage and quasi-Fermi potentials.
Here i^s) = G(s)v(XQNB>s).








established this convention, we then identify the incremental voltages

v(0,s) and v(XQNB,s) with the incremental quasi-Fermi 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 quasi-neutral region with a general contact at x =

XQNB, we can derive the low-frequency small-signal equivalent-circuit

directly by truncating the DC components of minority-carrier density and

current from Fig. 3.2 instead of starting from the matrix equation.

Figure 4.8 displays this equivalent circuit of a quasi-neutral region

with a general contact for low-frequency small-signal 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 Short-Base Case

The foregoing derivation in the frequency domain yields an

equivalent circuit involving an inductor L, which does not commonly

appear in small-signal or incremental models for a quasi-neutral base




70
established this convention, we then identify the incremental voltages
v(0,s) and v(Xq^g,s) with the incremental quasi-Fermi 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 quasi-neutral region with a general contact at x =
XqNB, we can derive the low-frequency small-signal equivalent-circuit
directly by truncating the DC components of minority-carrier density and
current from Fig. 3.2 instead of starting from the matrix equation.
Figure 4.8 displays this equivalent circuit of a quasi-neutral region
with a general contact for low-frequency small-signal excitation. The
values of the circuit elements are G = eAK^XQNg/2x, C = eAK^Xq^g/2, R =
XqNs/eAKiD, and L = XqNg3/6D2eAK1. The element G$ represents the
general contact at the back surface, the value of which follows from
Fig. 3.2:
G$ = [eAKxS] (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 Short-Base Case
The foregoing derivation in the frequency domain yields an
equivalent circuit involving an inductor L, which does not commonly
appear in small-signal or incremental models for a quasi-neutral base








i(O,s)


IVVV vv


v(,s) I C G



-n___K ,____,__ _________


x=-


(a)


i(O,s) L R
I


v(Os)


x-O (b)


Fig.4.9 Equivalent circuit of
surface recombination
ohmic contact and (b)


a quasi-neutral base when the back
velocity is (a) infinite, as in an
zero, as in a blocking contact.




72
F i g. 4.9
Equivalent circuit of a quasi-neutral base when the back
surface recombination velocity is (a) infinite, as in an
ohmic contact and (b) zero, as in a blocking contact.








To derive an equivalent-circuit containing an inductor L from the

time domain, we start from the quasi-static equivalent circuit for the

common-emitter configuration commonly called the hybrid-i 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 minority-carrier 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 (quasi-static)

charge storage within the base region. The conductance in Fig. 4.10(a)

is the derivative of the (quasi-static) input current with respect to

the input voltage divided by the dc common-emitter current gain B [14].

We formally include the effect of propagation delay by inserting a

time-delay 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 low-frequency expansion

of delay in the complex-frequency or Laplace domain:


exp[-sr(delay)] = 1 -sT(delay) = 1/[1 + sr(delay)]


(4.32)











































(a) The input stage of the conventional hybrid-r model of
bipolar transistors. (b) Modified hybrid-w model with the
inclusion of minority carrier delay across the quasi-
neutral base. (c) Modified hybrid-w 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 common-emitter 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 common-base rather than the common-emitter

configuration, then L' = L which is the L in Fig. 4.9.




4.3.3. Calculation of the Delay Time

We now determine the minority-current propagation delay of a short

base by calculating the Elmore delay time [15], also used later by Ashar

[25]:




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 ^ + sx(delay)]-1 [1 + (sL'/R'gsA) + (^'/R'QSA)]_1 .
(4.33)
This equation yields R1 = 0 and L' so that
i(delay) = L'/R'qsa (4.34)
where 6 is the incremental common-emitter 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 x(delay). This
determination will demonstrate that (6+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 common-base rather than the common-emitter
configuration, then L1 = L which is the L in Fig. 4.9.
4.3.3. Calculation of the Delay Time
We now determine the minority-current propagation delay of a short
base by calculating the Elmore delay time [15], also used later by Ashar
[25]:











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-

A-S


Ca

(3nllA G3ZIlUWNQN)



AiISN30 3Y0H SS33X3








80













SI0







A u
S-
O'
iII



-







.>

S-
4-)

( Z )








00
1 ir




> a



V 0.







U--o
y**4

I -
I I I I Il
















iN3afnl 31OH SS33X3




h-
z
u
X
q:
id
u
u
_j
o
x
m
CD
U
U
X
LiJ
Fig.4.11(b)
TIME (pis )
Impulse response of minority-carrier current at x = X.-^g.
CO
o



























-lAAA


.0)
E
-o













0,
I




CO
ro
Fig.4.12 The conventional hybrid-ir model for bipolar transistors.



























t =


emitter SCR base SCR collector


Fig.4.13(b)


Actual response of the minority-carrier profile for
a sudden change in the emitter-base voltage.











hybrid-r 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 hybrid-i 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 common-emitter configuration only slightly

whereas for a transistor in the common-base configuration the correction

reduces the input capacitance by one third. For the common-emitter

configuration, however, inclusion of delay is important to improving the

accuracy of the response of the incremental collector current to

incremental base-emitter 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 quasi-neutral base

region. An additional component of delay comes from the propagation of

these same carriers across the emitter-base junction space-charge

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 Minority-Carrier Delay Time with Built-In Electric Field
In this section, we estimate the delay time of the minority carrier

delay time across the quasi-neutral base region when the electric field
exists due to the non-uniformity of the base doping concentration
profile.
Assume a uniform built-in electric field within the quasi-neutral
base. Then the minority-carrier 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):




87
4.3.5 Minority-Carrier Delay Time with Built-In Electric Field
In this section, we estimate the delay time of the minority carrier
delay time across the quasi-neutral base region when the electric field
exists due to the non-uniformity of the base doping concentration
profile.
Assume a uniform built-in electric field within the quasi-neutral
base. Then the minority-carrier current at x = Xq^b is [22]
I(XQNB,s) = -AeDnpoeeV/kT^ G csch(GXQNB)P(0,s)
(4.40)
= -A21(s)P(0,s)
where K is the ratio of the base doping concentrations at the two edges,
x=0 and x=XqNB and where
(4.41)
Using Ashar's definition [25] of delay time, we have
T(delay) = 1 i nr
s-0
For the simple case when the base doping concentration is uniform, K=l,
we have the following from (4.42):






























































C,,

U,





U,


0
0o




Silicon dioxide
poly-silicon substrate
Fig.4.15 Crosssectional view of the measured transistors.































- > 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


__




Full Text

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PAGE 122

81,9(56,7< 2) )/25,'$ ,,,


44
To determine R^, 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 ohm-cm p-type substrate. The top n+ layer is
about 0.3 ym 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 10^ cm"^ and the junction depth is
1 urn. The thickness of the base is 374 un.
The measured values of Td, RM and IpQ are 6.5 ys, 0.23 and 2 pA
respectively. Using = 6.5 ys, we yenerate the S(x) locus shown in
Fig. 3.6. From this locus, we determine the values, 15 ys and
1300 cm/s, for the lifetime and the surface recombination velocity. The
other parameters of this cell are also determined: 1QNB0 PA
^NEO 1 pA, IFM0 ~ 0,46 pA and * = lil111* The ratio the cel1
thickness to the diffusion length is =2 for this particular cell.


SURFACE RECOMBINATION VELOCITY (cm/s)
48
Fig.3.7 S(x) loci for two different solar cells with thin base.


74
To derive an equivalent-circuit containing an inductor L from the
time domain, we start from the quasi-static equivalent circuit for the
common-emitter configuration commonly called the hybrid-ir 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
9mvbe 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 minority-carrier 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 (quasi-static)
charge storage within the base region. The conductance in Fig. 4.10(a)
is the derivative of the (quasi-static) input current with respect to
the input voltage divided by the dc common-emitter current gain $ [14].
We formally include the effect of propagation delay by inserting a
time-delay circuit element, producing Fig. 4.10(b). We seek a
conventional circuit element that will produce a delay x(delay) in the
output current, or equivalently in the voltage Vbe' relative to Vbe.
The simplest such element results from use of a low-frequency expansion
of delay in the complex-frequency or Laplace domain:
exp[-sx(delay)] 1 -sx(delay) 1/[1 + sx(delay)]
(4.32)


USE OF TRANSIENTS IN QUASI-NEUTRAL REGIONS FOR
CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS
By
TAE-WON,JUNG
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
S-parameter measurements, and to Dr. Taher Daud of the Jet Propulsion
Laboratory and to Dr. Mark Spitzer of SPIRE Corp. and Mr. Peter lies of
Applied Solar Energy Corp. for discussions and for devices used in the
experiments. Thanks are extended to my colleagues and friends,
Dr. Hyung-Kyu Lim, Mr. Jong-Sik Park, Mr. J. J. Liou, Mr. M. K. Chen,
Dr. Soo-Young Lee, and Dr. Adelrno 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 parents-in-law for their help and
encouragement.
The financial support of the Jet Propulsion Laboratory is
gratefully acknowledged.

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS Ü
LIST OF SYMBOLS v
ABSTRACT viii
CHAPTER
ONE INTRODUCTION 1
TWO UNIFYING VIEW OF TRANSIENT RESPONSES FOR DETERMINING LIFETIME
AND SURFACE RECOMBINATION VELOCITY IN SILICON DIODES AND BACK-
SURFACE-FIELD SOLAR CELLS, WITH APPLICATION TO EXPERIMENTAL
SHORT-CIRCUIT-CURRENT DECAY 4
2.1 Introduction 4
2.2 Mathematical Framework 5
2.3 Transient versus Steady-State Analysis via Two-Port
Techniques 9
2.4 Open-Circuit-Voltage Decay (OCVD) 11
2.5 Reverse Step Recovery (RSR) 14
2.6 Electrical Short-Circuit-Current 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 SHORT-CIRCUIT-CURRENT
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 Short-Circuit-
Current Decay and Dark I-V Characteristics 35

3.3 Experiments 36
3.3.1 Improvements in the Circuit for Short-Circuit-
Current Decay 36
3.3.2 Quality of the Short Circuit of the Switching
Ci rcuit 37
3.3.3 Measurement of the Dark I(V) Characteristics 42
3.4 Experimental Results and Discussions 43
FOUR EQUIVALENT-CIRCUIT REPRESENTATION OF THE QUASI-NEUTRAL BASE,
WITH APPLICATION TO DIODES AND BIPOLAR TRANSISTORS 53
4.1 Introduction 53
4.2 Equivalent-Circuit for Large-Signal Transient 54
4.2.1 Derivation by Two-Port Approach 54
4.2.2 SPICE2 Simulation of the Equivalent Circuit for
ESCCD 61
4.3 Equivalent Circuits for Low-Frequency Small-Signal
Analysis 64
4.3.1 Derivation in Frequency Domain 64
4.3.2 Derivation in the Time Domain for Short-Base
Case 70
4.3.3 Calculation of the Delay Time 77
4.3.4 Modification of the Conventional Hybrid-ir Model
by Including the Minority-Carrier Current
Propagation Delay 81
4.3.5 Minority-Carrier Delay Time with Built-In
Electric Field 87
4.3.6 Measurement of Minority-Carrier Delay Time Across
the Quasi-Neutral 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 SHORT-CIRCUIT-CURRENT 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

A
LIST OF SYMBOLS
area of a device
Ai
xQNBKi/L
Aij
characteristic matrix elements of a quasi-neutral region
for large-signal transient normalized surface
recombination velocity
Bij
characteristic matrix elements of a subregion for large-
signal transient
CSCr(V)
space-charge region capacitance of a p-n junction
forward-biased with voltage V
CSCRO
space-charge region capacitance of a p-n junction at
V = 0
D
diffusion coefficient of minority carriers
characteristic matrix elements of a quasi-neutral region
for small-signal low-frequency analysis
°P
diffusion coefficient of minority holes
Ax
thickness of a subregion
e
magnitude of the electron charge
i(x,0“)
minority carrier current at t = O’
!D
dark current of an ideal diode
ÍQct0)
dc steady-state current at x = 0
^ first-mode^^
the first-natural-frequency (first transient mode)
current at x = 0
^MO
pre-exponential factor of the first natural-frequency
current at t = 0"
!fo
pre-exponential factor of a steady-state current of a
diode with negligible space-charge current
Ii (S)
Laplace transform of i^(t)
V

i -j (t)
!qnbo
¡QNEO
*Sh
I(x,s)
Ki
L
Lp
L*
V
N
DD
Pdc^
P(x.ü-)
P(x,s)
p(x,t)
pi (s)
Pi(t)
R
KI
R,
M
MÃœS
sh
incoming minority carrier current toward a subregion at
x = xi
pre-exponential factor of steady-state quasi-neutral -
base current
pre-exponential factor of steady-state quasi-neutral-
emitter current
current through the shunt resistor of a diode
Laplace transform of i(x,t)
(—1 — Sit)1/2
diffusion length of minority carriers
diffusion length of minority holes
L/(l + st)1/2
y(1 + ®y1/2
intrinsic carrier density
base doping density
steady-state hole density at x = 0
p(x,t) at t = 0“
Laplace transform of p(x,t)
excess hole density
Laplace transform of p^(t)
excess minority carrier density at x = x^
^MO^QNBO
^NEO^QNBO
^MO^FO
turn-on resistance of MOS transistor
series resistance of a diode
shunt resistance of a diode
vi

si
the natural-frequency
s> Seff
effective surface recombination velocity
^nax
performance parameter of a solar cell derived from the
ESCCD method
Td
decay time constant of the fundamental mode (first
natural-frequency) current
t(delay)
propagation delay time of minority carriers across the
quasi-neutral base
Tmi n
performance parameter of a solar cell derived from the
ESCCD method
tSCR
discharge time constant associated with C$qR
V9
gradient voltage of a p-n junction
XQNB
quasi-neutral base width
Yi
admittance-like elements for a quasi-neutral 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 QUASI-NEUTRAL REGIONS FOR
CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS
By
TAE-WON 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 quasi-neutral 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 quasi-neutral base region of p/n junction silicon solar
cells, including devices having conventional back-surface-field (BSF),
ion-implanted BSF and polysilicon BSF structures. This method, called
electrical-short-circuit-current-decay (ESSCD) avoids errors introduced
in other methods in common use, such as open-circuit-voltage-decay and
reverse step recovery, that arise from the capacitive effects of mobile
v i i i

holes and electrons in the volume of the p/n junction space-charge
region under forward voltage. Two circuit implementations of ESCCD are
presented and evaluated.
The ESCCD method derives from a tneoretical 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
quasi-neutral 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 transmission-line, model. The inclusion
of the inductor makes this possible. Models of this type are developed
for both small-signal and large-signal 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 hybrid-iT model for bipolar transistors in the common-emitter
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
Si-device 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 quasi-neutral 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 in-process 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 quasi-neutral 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 quasi-neutral base of back-
surface-field (8SF) silicon solar cells. The results of this study are
directly applicable to any bipolar device including diodes, solar cells,
1

2
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 two-port approach [9]
to obtain a unifying framework for transient analysis and develop from
it a new method, experimental electrical-short-circuit-current-decay
(ESCCD) for the determination of the recombination lifetime and back-
surface recombination velocity of the quasi-neutral 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 4-mil BSF solar
cells). We derive various performance parameters for BSF solar cells by
exploring the ESCCD method. We propose a methodology to separate the
quasi-neutral 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 minority-carrier propagation delay
in a compact equivalent circuit. This enables the use of standard
circuit-analysis computer programs, without resorting to much more
sophisticated programs needed when transmission-line models [10] are
employed. These ideas are applied both to ESCCD and to bipolar

3
transistors, for which an improved hyprid-tr 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
BACK-SURFACE-FIELD SOLAR CELLS,
WITH APPLICATION TO EXPERIMENTAL SHORT-CIRCUIT-CURRENT DECAYS
2.1 Introduction
This chapter has three purposes. First, we outline a mathematical
method that systematically and compactly describes the large-signal
transient and smal1-signal 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
quasi-neutral principal regions of the devices.
Second, exploiting this description, we survey the adequacy of
various experimental large-signal 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 quasi-neutral base from a
single treatment measurement for three different p+/n/n+ back-surface-
field solar cells.
4

5
2.2 Mathematical Framework
In this section, we develop a mathematical framework which could be
applicable to most of the large-signal transient measurement methods and
could include small-signal 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
minority-carrier density and the minority-carrier current in a quasi¬
neutral base region in low injection. Focusing on the quasi-neutral
base, assumed to be n-type here (of x-independent donor density Nqq)
with no loss in generality, will simplify the treatment; extensions to
the quasi-neutral emitter are straightforward, provided one inserts the
physics relevant to n+ or p+ regions.
Assume a p+/n diode in which the uniformly doped quasi-neutral base
starts at x = 0 and has a general contact defined by arbitrary effective
surface recombination velocity Seff at the far edge x = Xq^g. Such a
contact could result, for example, from a back-surface-field (BSF)
region. Assume also low-level injection and a uniform doping of the
base region. Then a linear continuity (partial differential) equation
describes the excess minority holes p(x,t)
3p(x ,t)/ at = Dp82p(x,t)/ax2 - p(x,t)/T , (2.1)
where Dp is the diffusion coefficient and Tp 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:

6
-p(x,0~) + sP(x,s) = Dp d2P(x,s)/dx2 - P(x,s)/t
(2.2)
where
P(x,s) = / e“stp(x,t)dt
t=0
Solving Eq. (2.2) yields
, s = a + ju>, j = (-1)1/2 . (2.3)
P(x,s) = p(x,0~)/s + exp(-x/Lp*) + M2 exp(x/Lp*)
(2.4)
* 1/2
where L = [(D x )/(1 + st )] and where and M2, given below, are
r r r r
to be determined by the boundary values at the two edges of the quasi -
neutral base region: P(0,s) at x=0, and P(XqNB,s) at x=XqNb.
Substitution of (2.4) into (2.2) yields the steady-state continuity
equation for p(x,0"), verifying that (2.4) is the solution of (2.2).
Because of quasi-neutrality 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:
1(0,s) - 1(0,0')/s
i(XQNB»s) ” i(XQNB,°
where 1(0,0") and I(XgNg,0~) are the initial values of the minority hole
diffusion current at x=0 and x=xqNB. In (2.5), hole current entering
the quasi-neutral base is positive, by definition.
A
L *
P
-X /i *
-e QNB/Lp
x /i *
e QNB7 p
_Mr
.M2.
(2.5)

7
Regarding the minority carrier densities at the two edges as the
excitation terms for a system analogous to a linear two-port network of
circuit theory, we have the following two-port network matrix from (2.4)
and (2.5) for the two excitations (densities) and the two responses
(currents):
1(0,s) - i(0,0")/s
"An
A12
_P(0,s) - p(0,0")/s
I(xqNB»s) - 1(xqNB,° )/s
_A21
A22_
P(XQNB *S) "p(XQNB’°
(2.6)
where p(0,0") and p(XgNg,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 quasi-neutral
base. In Eq. (2.6), A12=A21= -e(Dp/Lp*) cosech(XgNB/L *) and An= A22
e(Dp/L*)coth(Xq^/L*). Figure 2.1 displays the master equation, where
the initial values are included in 1(0, s), I(XqNB,s) ,P(0,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, l(0,s)=0 in OCVQ open-circuit-voltage-
decay [1], l(0,s)=constant for reverse step recovery [2,3], and P(0,s) =
0 for short-circuit current decay, the latter of which is developed in
detail here. For small-signal methods [11-12], where dl, for example, is
an incremental change of current, I(0,s)=Iqq/s + dl(0,s) and
P(0,s)=Pqq/s + (edV/kT)PQQ. 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.

I (0,s)
+ -
P(0,s)
Fig.2.1
^*QNB’8^
>
A^-j (s)
A12(s)
A21 (s)
*V)
eg
<
< +
^*QNB’8^
Two-port representation for excess hole density and hole current density
at the two edges of the n-type quasi-neutral base region.

9
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(XgNB,s) = -eSeffP(XgNB,s). 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 electrical-short-circuit-current-decay (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. Steady-State Analysis via Two-Port 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 quasi-neutral regions under study using the two-port
technique described in Sec. 2.2, the displacement current is
negligible. In the steady state, the two-port description leading to
the master equation simplifies because the hole current in our example
of Sec. 2.2 depends only on position x. Tnis x-dependence results from
volume recombination (relating to the minority-carrier lifetime) and
effective surface recombination (relating to the effective surface
recombination velocity). A two-port formulation for the steady state

10
leads to the same matrix description as that derived previously, in
which the matrix elements A^j(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 quasi-neutral base;
and, using quasi-neutrality together with knowledge of the steady-state
currents in the junction space-charge region and in the p+ quasi-neutral
emitter region, one can tnus find the steady-state current flowing in
the external circuit or the voltage at the terminals of the diode. If
the quasi-neutral 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 time-varying mode of operation leads to a
minority hole current in the n-type quasi-neutral 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 3p/3t 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 two-port 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.

11
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 quasi-neutral base
but also from the junction space-charge region and the quasi-neutral
emitter. In the interpretation of experiments to follow, we shall
account for this multi-regional dependence.
2.4 Open Circuit Voltage Decay (QCVD)
In this widely used method [1], the free carriers in the junction
space-charge region enter to contribute to the transient. But,
consistently with Sec. 2.2, and with most common usage, we concentrate
on the n-type quasi-neutral base.
From the master equation [Eq. (2.6)], the transient solution for
the junction voltage is obtained from open-circuit constraint that
I(0,s)=0:
P(0,s) = p(0,0")/s -
i (0,0~)L*
eD s
P
1 + Dp[c°t*(XWB/Lp«)]/lp«S
c°th(XqNB/Lp*) + Dp/Lp*S
(2.7)
Here we have assumed that the quasi-neutral 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):

12
p(0,t)
2i(0,0")L
“eDpV
^ + 3
[cosec CXqNBki/Lp) + (Dp/LpSeff)]
(2.8)
where s.¡ is the ith singularity point (ith mode) which satisfies
coth(XQNB'/l+si Tp/Lp) + Dp /1+si Tp,/LpSeff ~ 0 (2*9^
and K. = /-1-s.t > 0.
i 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 s.¡ is called a mode, as
in the electromagnetic theory. Appendix A treats the details of
determining the Eigenvalues s^ from Eq. (2.9) (and from the similar
Eq. 2.11 derived below).
The decaying time constant -1/s^ 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 S0ff 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 = 30U K. In Si devices the open-voltage
decay curve is usually bent up or bent down because of discharging and
recombination within the space-charge region.
As mentioned in [11], the mobile charge within the space-charge
region contributes significantly to the observed voltage transient for
10 -3
Si, in which n^ =10 cm , but not in Ge, for which OCVD was first

13
13-3 . .
developed, and for which =10 cm . Here is the intrinsic
density and is also the ratio of the pre-exponential factors that govern
contributions from the quasi-neutral regions relative to those from the
junction space-charge region.
Thus we identify the transient decay of mobile electrons and holes
within the p/n junction space-charge region, which persists throughout
the open-circuit voltage decay (OCVD), as a mechanism that distorts OCVD
so significantly that the conventional treatment of OCVD will not
reliably determine ip 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-
charge-region contribution to the observed transient voltage [13],
quasi-static approximations and a description of the forward-voltage
capacitance of the space-charge 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 ip and Seff.

14
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 1(0,s) = constant (reverse current) at 0 < t < ts, where ts
is the time needed for the excess hole density p(0,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 xs, 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 < x$, the decay of mobile hole and
electron concentrations in the p/n junction space-charge region
complicates the interpretation of the measured xs in terms of the
desired parameters, Xp and Seff.
In addition to this, during the recovery transient (xs -< t < <*»),
the reverse generation current is often large enough to saturate the
recovery current so quickly ttiat we have no sizable linear portion of
the first-mode curve on a plot of ln[i(t)] vs. t. This linear portion
provides interpretable data for Ge devices [3], but not often for Si
devices according to our experiments.

15
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
steady-state condition and then suddenly applies zero bias through a
small resistance. This causes the mobile charges stored within the
junction space-charge and quasi-neutral 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 quasi-neutral emitter and the junction space-charge
region are much smaller than from the quasi-neutral base, one can
separate the first mode of the quasi-neutral-base current and determine
Seff and tp.
We first consider the time of response of the junction space-charge
region. Upon the removal of the forward voltage, the constraint at the
terminals becomes essentially that of a short circuit. The majority-
carrier quasi-Fermi 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
quasi-neutral regions, times that are of the order of no greater than
10 iCs. 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 quasi-neutral base. The resulting Coulomb forces cause
majority carriers to rush from the edges of the junction barrier
reyions, thus causing the nearly sudden rise of the barrier height to

16
its equilibrium value. (The physics governing this phenomenon comes
from Maxwell's Curl H_ = j_ + 3D_/3t; taking the divergence of both sides
yields 0 = div _i_ + d(div _D)/dt, which, when combined with _i_ = qO/e and
divD^ = p, yields a response of the order of e/a, the dielectric
relaxation time.)
Following this readjustment of the barrier height, the excess holes
and electrons exit the junction space-charge region within a transit
time of this region (about lCT^s 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 space-charge region in the ESCCD method occurs within a time of
the order of 10““s, which is much less than any of the times associated
with discharge of the quasi-neutral regions. This absence in effect of
excess holes and electrons within the junction space-charge 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 space-charge region appears in Appendix B.
The discharge of the quasi-neutral emitter depends on the energy-
gap narrowing, the minority carrier mobility and diffusivity, the
minority-carrier 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 quasi-neutral base, and we shall assume
this is so in the discussion to follow.

17
Having established that the mobile carriers in the junction space-
charge region enter the electrical-short-circuit-decay 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
1(0,s) = i(0,U")/s
eDpp(0,0')
V
coth(X0NB/Lp*) ♦ yLD*Seff
1 + (“pVw^VnbV
(2.10)
Cauchy's residue theorem yields the inverse transform of (2.10):
1(t) . . ? eV<0-°~>K1 cot(KiXQNB/Lp) - DpVLpSeff jji* _
1=1 1 p ^ Tp/si Ki ^ + ^XQN8/2Seff )cosec (Ki XQNB/!V
(2.11)
where s.¡ is the ith singularity which satisfies the Eigenvalue equation,
D XnMR
1 + l-s2— coth( "l - = 0 , (2.12)
p eff ^ p ^
and where = (-1-s.jTp)^2 > 0, with s^ < 0.
Truncating (2.11) and (2.12) to include only the first mode (s^),
we obtain
1/2
1 + (D /L S --) /I + s, t coth[(XnKID/L )(1 + s.t ) ] = 0
v p p eff' 1 p v QNB p/v 1 p'
(2. 13)

18
and
Virst mode(O)
eDpp(0,0")K1
s, L
1 P
cot(kixqNB/lp) - (DpKi/LpSeff)
(Tp/2Kl) + ^QNB^eff) C0SeC ^KlXONB/Lp^
(2.14)
Equations (2.13) and (2.14) contain four unknowns: if-¡rst mode^» sl*
Tp, and Seff. The parameters, s^ and ifq-rst mode are determined
from the straight-line portion of the observed decay (in Fig. 2.3(c) to
be discussed below) p(0,0“) = (n2/NDpj)[exp(eV(0")/kT)-l]. Here v(0“) is
known and the doping concentration Nqq of the base is measured by usual
methods; Dp(NDD) is known, and XgN8 is measured. Combining (2.14) and
(2.13) then yields the desired parameters: xp and Seff.
2.6.2 Experiments and Results
To explore the utility of the ESCCO 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 V^(t) is high, switching
transistor T1 turns on, which charges the large capacitor in parallel
with it and divides the high voltage about equally between the
solar cell and the emitter-collector terminals of the transistor. Thus

'high
V^t)
low
Fig.2.2 Electronic switching circuit used in the ESCCD method. The circuit elements are
bipolar transistors(2N3906), pulse generator(HP8004), operational amp(Burr-Brown
3500c), and diodes(lN914).

20
the voltage across the solar cell becomes about 0.6 V, which one may
control by altering (Vh-¡yri), and the variable resistor connected to the
transistor base, or both. In this mode, the quasi-neutral base charges
to store ultimately a steady-state charge of excess holes and electrons,
and p(0,0-) of Eqs. (2.10), (2.12) and (2.13) is established.
Now assume that V^(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
short-circuit constraint. The large capacitor maintains this constraint
nearly perfectly during the first-mode transient of the solar cell; that
is, during this transient, this capacitor and the input voltage source,
which has a small resistance of 50 $2 (in parallel with 10 n), act as
nearly incremental short circuits. Thus the desired short-circuit
constraint is maintained to a good approximation during the ESCC0
transient of interest.
We use three different BSF solar cells for which the parameters
are: DEVICE 1—Nqq (substrate doping) = 6 x 1014 atoms/cm3; XqNB (base
thickness) = 348 ym, A = 4 cm^; DEVICE 2--Nqq = 7 x 10^4 atoms/cm3,
*QNB = 3^° A = *86 cm^» DEVICE 3--Nqq = 3.5 x 1035 atoms/cm3,
XqnB = ym, A = 4 cm^.
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 ys. The speed is circuit limited. One could design a

21
(a)
(b)
(c)
Fig.2.3 (a) Voltage across BSF #1 solar cell (vertical:0.2V/div),
(b) Current through BSF #1 solar cell (vertical:ImA/div),
(c) Log scale representation of (b) (vertical:0.lV/div),
where v(t) = (mkT/e)ln(i(t)/I0+l).

22
much faster circuit. Here xd = -l/s^ is the first-mode 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 semi-logarithmic counterpart, illustrating the
straight-line portion of the transient obtained from the output of the
logarithmic amplifier in Fig. 2.2. From this xd 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
I-V characteristic is V = .03851n(I/IQ+1). If the first-mode current is
ifirst-mode^) = constant * exp(-t/xd) , xrf =-l/s1 , (2.15)
then the slope of the output voltage of log amplifier is -38.5 mV/xd.
Extrapolation of the straight portion in Fig. 2.3(c) yields the initial
value ifirst-mode (°+) as the intercept.
We measure the decay time constant and the initial amplitude of the
first model as follows: DEVICE 1, xd = -1/s^ = 29.3 us, i^1-rst(0+) =
2.73 mA for V(0") = 0.44 V and T = 303.1 K. For DEVICE 2, xd =
24.5 usee, i|r1-rst(0+) = 4.35 mA at v(0") = 0.5 V and T = 302.9 K. For
DEVICE 3, xd = 28.5 usee, ifirst(0+) = -696 mA at v(0“) = .47 V and
303.5 K. Here v(0”) denotes the steady forward voltage applied across
the solar cell before the transient.

23
From the above development, these results give DEVICE 1, xp =
119 gs, Seff = 25 cm/sec; DEVICE 2, = xp = 119 gs, Seff = 60 cm/s;
DEVICE 3, Tp = 213 gs, Seff = 100 cm/s. These results agree favorably
with those obtained for the same devices by using the more time-
consuming methods detailed in [11-12].
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 space-charge-
region. These methods, among open-circuit 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 Eq 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 n^ 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 space-charge region play no role in determining the response of
the electrical-short-circuit-current 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 first-mode response,

24
which accounts for vanishing of minority holes both by volume
recombination within the quasi-neutral base and effectively by surface
recombination, greatly exceeded the time required for the excess hole
density at the base edge of the space-charge 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 electrical-short-circuit-current decay (ESCCD)
method. In general, the current response derives from vanishing of
minority carriers not only in the quasi-neutral base but also in the
quasi-neutral 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 low-injection
conditions for which the response was measured. But for other solar
cells or for higher levels of excitation, the recombination current of
the quasi-neutral 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 in-process control in solar-cell
manufacturing.
This chapter began with a mathematical formulation of the relevant
boundary-value 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
SHORT-CIRCUIT CURRENT DECAY
3.1 Introduction
This chapter describes various improvements of the method of
electrical short-circuit 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 MÃœS 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 first-mode current Ifirst-mode(0+) together with the decay time
constant as the ESCCD parameters used to determine t and S. The
parameter i ^ rst_moc|e(0+) 1S Proportional to exp[ev(0")/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, x and S are
determined by three measurable parameters: ifirst-mode^+^» Td and
v(0“). 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(0“)
by treating [i fi rst-niode^+^/^F^”) as t^ie 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 short-circuit 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, xm_jn and R^, the importance of which is
discussed in this chapter.
Fourth, we show quantitatively that the electrical short-circuit
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 Ipg in the ratio above brings the
emitter recombination current into our method for determining x and
S of the base region. This, however, is only apparently a problem.
Indeed, we illustrate that use of the S(x) locus enables a determination
of Iqnj:, 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 two-port network formulation introduced in
Chapter Two in (2.6), the representation for which is illustrated in
Fig. 3.1, where Y^s) = An(s) + A-i2(s), Y2(s) = -A12(s), P^s) =
p(0,0")/s, P2(s) = p(XqNB,0")/s, Ix(S) = i(0,0“)/s, and I2(s) =
i UqnB’O •

Fig.3.1
Two-port network representation of a quasi-neutral region with boundary conditions
at x = XqNB and x = 0. The nodal variable is the excess minority carrier concentration.

29
In this figure we have used the y-parameter 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 two-port network
representation.
Other advantages include
(a) systematic determination of the natural frequencies [14];
(b) systematic conversion to the case of steady-state excitations,
attained by setting the complex frequency variable s to zero;
(c) systematic connections to the underlying physics, as we shall
i 11ustrate;
(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 = XgNB to Fig. 3.1. The boundary
condition at x = Xg^g, I(XgNB,s) = -AeSP(XgNB,s), removes Igis) and
Pg(s) in Fig. 3.1 from consideration. Figure 3.2 displays the resulting
two-port network representation of the quasi-neutral base region of a
solar cell. Here

x - O x - XQNB
Fig.3.2 Equivalent two-port network representation of a quasi-neutral base of a solar cell
with a general contact at x = Xg^g under electrical-short-circuit-current-decay
condition. The excess minority carrier concentration at x = 0 vanishes in this case.

31
*qnb,S,S^ ~ ’ (3.1)
Solving the network of Fig. 3.2 for 1(0,s) under the low-injection
condition yields
1(0,s) = i(0,0“)/s - Yx p(0,0")/s
(Y1 + V Y2
- rTTVV~ (p(0>0 )/s) » (3-2)
T1 t2 ts
ni2
where p(0,0“) = t¡— (exp(eV(0 )/kT) -1) . (3.3)
mDD
If we use the Cauchy Residue Theorem to obtain the inverse
transform of 1(0,s), we get an infinite series for i(0,t). Truncating
this series after the first term, at t = 0+, yields
ifirst node^ ) = ^FMO (exP(e^(0 )/kT)-l)
where Ip^Q
AeD^n.2
S1LN00
(2.14)
cot(K,XnNR/L) - DK./LS
1 1 e (3.4)
(t/2Kx2) + (Xq^jB/2S)csc(K1Xq^b/L)
The minority-carrier current at x = 0 for t < 0 is
i(0, 0“) = I^NB0 (exp(qV(0")/kT - 1)
(3.5)

32
y _ ADni sinh(XQNB/L^ + aCOShíXQNg/1-) ^
e I(3NB0 = LNdd * cosh(XgNB/L) + asinn(XQNB/L) * (3*6)
LS
and where a = —g-. Here a is the ratio of the normalized surface
recombination velocity to the diffusion velocity [16]. Thus the ratio
R = IfM0
!QNB0
2K^ cotA^ + tanAj
— • — - - - — —— — ■ - •
Si (XqNB/A1)2/D + (XQNB/S) CSc2a1
cosh (Xq^g/L) + oís i nh (Xg^jg/L)
sinh(xqNB/L) + «cosh(XqNB/L) * (3*7)
where A^ = K^XqNB/L and where K1 and A^ are obtained by solving (A.3) of
Appendix A. The ratio R will be utilized for the determination of the
quasi-neutral 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 space-charge recombination current component is
negligible [17], the I(V) characteriStic of the solar cell is

Fig.3.
A simple diode model with series resistance and
shunt resistance in the dark condition.

34
(3.8)
= IF0 (exp(eV/kT)-l) + V/r$h .
(3.9)
Here Ipg is the pre-exponential factor of the forward bias current and V
is the voltage across the space-charge region. The pre-exponential
factor Ipg in (3.9) has two components:
FO “ ^NBO + !QNE0
(3.10)
where Ig^gg is the quasi-neutral-base current component and IQNEO
quasi-neutral-emitter current component.
The voltage across the two terminals of the solar cell Vout is
As the forward bias increases, the current I in (3.8) becomes more
dominated by the component Ig and the effect of I$n becomes negligible
for the solar cell . Thus
FQ exp(eV/kT) .
(3.12)
Combining (3.11) and (3.12), we obtain an expression for VQut in terms
of I, rs and Ipg:

35
“out ■ V ♦T,OSe • <3-‘3>
There are two unknowns, rs and Ipg, in (3.13). We estimate rs and Ipg
by measuring the dark I-V characteristics from the terminals of a solar
cell. The pre-exponential factor Ipg will be utilized for the
determination of the base material parameters.
3.2.3 Combined Method of Electrical Short-Circuit Current Decay and
Dark I-V 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 I-V characteristic methods. Using the ESCCD method, we measure
the decaying time constant of the first mode and the ratio R^ of pre-
exponential factors from Fig. 2.3:
!FM0
in which the subscript FMO means the pre-exponential factor of the
first-mode current. Using the dark I-V measurement, we estimate the
pre-exponential factor Ipg by eliminating the series resistance effect
as described in Section 3.2.2.
From the measured value of tg, one can generate a s(t) locus on the
i-S plane; each point on this locus must produce the measured value of
Td. Each point (x,S) also has its own value of the ratio R, defined in

36
(3.7), since R is a function of both t and S. Also each point (t,S)
produces its own value for Iq^gg 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 Iq^q. Specifically
td = f^t.S) , (2.13)
ipo = Vt,S»IQNE0^
and
RM = f3^x’S,IQNE0^
Using (2.13), (3.10) and (3.14) and the measured variables, one can
solve for x, S and Iq^eq in a manner to be described later.
3.3 Experiments
3.3.1 Improvements in the Circuit for ESCCD
Previously we used a bipolar-transistor switching circuit in
Fig. 2.2 to measure the decay time constant and the initial amplitude of
the first natural-frequency current at t=0+. 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
(3.10)
(3.14)

37
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 turn-on 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 turn-on switching time of this measurement
circuit is 12.5 ns (250 pF times 50 onm). Thus the speed of the
measurement circuit is adequate for any bipolar devices having xd 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 shorted-path. The voltage
across the junction space-charge-region does not vanish as long as the
current flows through the series resistance. Fig. 3.5 displays the
equivalent circuit during discharge when the first-term natural-
frequency current dominates the discharging current. Higher-term
natural-frequency 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 xd of the first-term
natural-frequency current. In Fig. 3.5, rs and r$d are the series and
shunt resistance of a solar cell, rd detects the discharging current,
and i^(t) is the first-term natural-frequency current.

Fig.3.4 Switching circuit used in measuring short-circuit-current decay parameters, t and S.

39
rMOS
i(t)
Fig.3.5 Equivalent-circuit representation of the measurement
circuit of Fig.3.4 when the first mode dominates
electrical-short-circuit-current decay.

40
The current i(t) flowing through rd in Fig. 3.5 is
i(t) = CSCR^dv12^t^dt^ + v12^t)/rsh + hU) (3.15)
where
d^SCR
CSCR E 17^ = f(v12) ’ (3-16)
in which Q^/e is the integrated steady-state hole or electron density
through the volume of the space-charge region. From Chawla and Gummel
[18]
CSCR/CSCR0 " t1 " (Vi2/Vg)]_m (3.17)
where C^q is the for variations in v^ about bias voltage v^ = 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 = CSCR0 * f^v12)
(3.18)
Thus
i(t) = CSCR0dv12(t)/dt + v12^t)/rsh + 1l(t)
(3.19)

41
where
vi2(t) = -(rHOS + rs + rd)i(t) .
(3.20)
Since the ratio (rMQ£ + 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] + ix(t) . (3.21)
Solving (3.21) for i(t) yields
i(t) = E1exp(-t/xSCR) + E2exp(-t/xd)
(3.22)
where
tSCR0 r CSCR0^rM0S + r s + rd ^
(3.23)
As can be seen in (3.22), the first term of the right side can be
neglected and xd can be determined if the time constant x^R is much
smaller than x. For the switching circuit of Fig. 3.4 x<^R - 200 ns.
For the solar cells described in this chapter, 0.5 ys < xd < 30 us. Thus
the RC time constant of the measurement circuit negligibly influences
the first-term or dominant natural-frequency current decay of the solar
cells.

42
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(V0(Jt)
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(VQUt) characteristic with idealized
diode theory, we obtain
(Vout)i = Vs + (kT/q) 1 °9eC xi/IF0) (3.24)
where rs is the series resistance, IFq is the idealized pre-exponential
current (corresponding to unity slope), and subscript i denotes
different data points. Applied to two such data points, Eq. (3.24)
yields
AVout = rsM + (kT/e)1°9e(I2/Ii) (3.25)
upon subtraction. This determines rs, which we may thus ignore in the
subsequent discussion. To determine IpQ, we use the procedure of
Ref. [19].

43
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 xd. 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 = ^MO^QNBO T,S)
(3.26)
RI = ^NEO^QNBO " ^1FO " ^NBO^QNBO
(3.27)
RM = ^MO^FO
(3.14)
The relation among these parameters is
R = (1 + Rj)Rm. (3.28)
The ratio R is determined by theory for any assumed values of S and
t lying on the S(x) locus corresponding to the measured value of the
decay time constant xd. The ratio Rj is determined by the measured
value of IpQ and by the value of Iq^BO wh''c^ "'s obtained from (3.6) for
any assumed values of S and x. The ratio RM is determined by
measurement. Thus (3.28) enables a determination of S and x by an
iterative procedure.

44
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 x 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 ohm-cm p-type substrate. The top n+ layer is
about 0.3 ym 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 1088 cm"8 and the junction depth is
1 urn. The thickness of the base is 374 un.
The measured values of x^, R^ and Ipg are 6.5 ys, 0.23 and = 2 pA
respectively. Using x^ = 6.5 ys, we generate the S(x) locus shown in
Fig. 3.6. From this locus, we determine the values, 15 ys 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» ^FMO ~ 0,46 PA and L " 185 vjm. The ratio of the cell
thickness to the diffusion length is =2 for this particular cell.

SURFACE RECOMBINATION VELOCITY (cm/s)
45
Fig.3.6 S(t) locus for a BSF solar cell. The locus is generated from
= 6.5 ys.

46
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 in terms of the following two time
constants by solving (A.3) of Appendix A:
Td_1 = "S1 = Ts_1 + T_1 (3.29)
where x5 = Uq^g/Aj )^/D. Equal rates occur if
ts = t (3.30)
Here A^ is obtained from (A.3) of Appendix A:
A1 = it/2 for S(back) = 0 (3.31)
and
A^ = it 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 quasi-neutral region,
whereas the parameter t^/t is the probability that a minority carrier
vanishes by volume recombination.

47
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(x)
exhibits dS/di * 0 over a large range of t. This is the mathematical
statement, for our procedure, that S is more accurately determined than
is t for a thin-base solar cell. (If Xq^b/L » 1, dS/dx = « over a
large range of S, which means that t is more accurately determined than
is S for a long-base 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 p-type substrates. The top n+ layer is about 0.3 um deep.
The thickness of these cells is about 100 un.
Using xd, we generated S( t) 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/dx approaches zero,
defines Smax. If for an extreme case for which negligible volume
recombination occurs during the ESCCD transient, S = S|T)ax.
Similarly, the limit dS/dx > « defines a minimum value of the
lifetime xmin, as illustrated in Fig. 3.7. For the all corresponding to
the lower locus of Fig. 3.7, Tjn^n = 40 us. This value xm^n occurs for

SURFACE RECOMBINATION VELOCITY (cm/s)
48
Fig.3.7
S(x) loci for two different solar cells with thin base.

49
the extreme case of negligible surface recombination at the back contact
during the ESCCD transient. For the upper-locus cell in Fig. 3.7,
Tm^n = 0. These two parameters, Smax and Tml-n, can be used as
performance parameters for thin solar cells; small Smax and large T^in
is desirable for thin BSF solar cells for a given base thickness and
doping concentration.
We also measured the values of R^: = 20 for the lower locus and
Rm « 3 for the upper locus. But we cannot use the measured R^ 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 Iq^eq and small S for a thin solar cell. These
conditions imply a large open-circuit voltage for a given base thickness
and doping. Small RM usually implies either a poor BSF contract at the
back surface or a large Iqjvjeq• For example, for the better BSF solar
cell (the lower locus), we have Rf1 = 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 poly-Si layers on the back surface of the base. The value
of S is estimated to be about 2000 cm/s for n+/p/p+-poly-Si cell
(poly 1) and about 400 cm/s for p+/n/n+-poly-Si cell (poly 2). Thin
cells are characterized in terms of -i^n and Smax.

50
Table
3.1 ESCCD MEASUREMENTS
FOR VARIOUS SOLAR
CELLS.
NAME
RESISTIVITY
(ohm-cm)
THICKNESS
(pm)
t^Íps)
T (p S )
S(cm/s)
L(pm)
SPIRE
.31
374
6.5
15
1300
185
ASEC1
.15
301
3.6
7
ohmic
93
ASEC2
.15
267
4.0
13
ohmic
126
BSF#1
10
240
5.3
35+
100-400*
350+
BSF#2
10
260
6.3
45+
100-350*
400+
BSF#3
10
284
7.85
75+
100-225*
512+
BSF#4
10
96
.98
20+
100-380*
265+
BSF#5
10
91
.9
25+
100-290*
295+
BSF#6
10
107
.7
4500*
BSF#7
10
102
.73
3200*
LEU#1++
8
328
28.0
145+
40-80*
417+
LEU#2++
1.5
325
25.7
105+
40-150*
347+
P0LY1
2
203
2.7
2000*
P0LY2++
2
208
8.8
25+
100-400*
168+
++ denotes p+/n/n+ BSF solar cell. + denotes Tmjn or Lmin.
* denotes Smax. P0LY1-2 have poly-Si layers at the back surfaces.

51
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 id of the ohmic-contact solar cell and by using (3.31):
Td-1 = C(Xqnb/tt)2/D]-1 + t_1 (3.33)
Second, one measures xd 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(i)
locus.
An illustrative example is shown in Fig. 3.8. In this example, we
used a wafer which is 10 ohm-cm and p-type. The upper locus corresponds
to the ohmic-contact 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 Xg^g/L decreases. For example, for a 10 ohm-cm
n+p/p+ solar cell (ohmic contact) with a thickness of 350 ± 3 un, the
error in the lifetime is estimated to be about 20%. In doing this
calculation, we assumed that the lifetime is 50 gs (Xg^g/L = 0.84) and
that 0 = 35 cm2/s. For a 0.3 ohm-cm n+/p/p+ solar cell (ohmic contact)
with a thickness of 350 ± 3 urn, 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.

SURFACE RECOMBINATION VELOCITY (cm/s)
52
Fig.3.8 Illustration of the determination procedure of S and x using
one ohmic contact solar cell and one BSF solar cell from the
same material. Locus 1: ohmic contact solar cell(C-3-5).
Locus 2: BSF solar cell(2-53).

CHAPTER FOUR
EQUIVALENT-CIRCUIT REPRESENTATION OF THE
QUASI-NEUTRAL BASE,
WITH APPLICATIONS TO DIODES AND TRANSISTORS
4.1 Introduction
In the previous chapters, we treated the quasi-neutral base region
using the two-port 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
large-signal 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 base-region behavior. This avoids
difficulties associated with the infinite number of natural frequencies
characterizing a distributed system. It makes possible use of a
circuit-analysis 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.
53

54
Along with the equivalent circuit for transient analysis, we
develop equivalent circuits for low-frequency smal1-signal excitation.
We correct the quasi-static input capacitance of the hybrid-tt 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 hybrid-ir
model for a junction transistor, which is advanced near the end of the
chapter.
4.2 Equivalent-Circuit for Large-Signal Transients
4.2.1 Derivation by Two-Port Approach
We slice the quasi-neutral base, assumed to be in low injection,
into many subregions. A typical subregion is shown in Fig. 4.1. From
the two-port approach, we have the following linear matrix equation
relating the excitations (minority-carrier densities) and the responses
(minority carrier currents) of the ith subregion:
I-j(s) - i-j (0")/s
B11 B12
P-j (s) - p-j (0")/s
Ii+1(s) - ii+1(0")/s
B21 822
pi+l " Pi+i(° )/s
.
(4.1)

en
en
Fig.4.1 One-dimensional schematic diagram of a BSF solar cell with a quasi-neutral
base sliced into subregions.

56
where Bn(s) = B22(S) = (AeD/L*)coth(Ax/L*)
(4.2)
and B^2(s) = B2i(S) = -(AeD/L*)csch(Ax/L*)
(4.3)
Figure 4.2 displays the matrix equation (4.1), where p^(0")/s and
Pi+1(0-)/s are the minority carrier densities and i^(0“)/s and
ii+l(0-)/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 Y^ and Y2 in Fig. 4.2 are related to B^ and B2^ as follows:
(4.4)
Y1 = B11 + B12
(4.5)
To realize an RLC equivalent-circuit representation of a quasi-
neutral region, which enables use of circuit-analysis software, such as
SPICE2, one has to algebraically approximate the transcendental
functions Y^(s) and Y2(s) in Fig. 4.2. The condition which makes
possible a series expansion of (4.2) and (4.3) is
| Ax/L* | = | Ax(1 + st)1/^/(Dt)1/^| < it
(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 Y^s) and Y2(S) by

Fig.4.2 Two-port network description for the ith subregion.

58
Y^s) = B^(s) + B^2(s) « AeAx/st + sAeAx/2
(4.7)
and
Y2(s) = -B-^2(s) = AeD/Ax - sAeAx/6 (4.8)
= (AeD/Ax)(1 - sAx2/6D) = (AeD/Ax)(l + sAx2/6D)_1 (4.9)
for |s| « 6D/Ax2.
From (4. 7), Bn(s) + B12(s) in Fig. 4.2 is realized in terms of
admittance-like elements [9,21]:
BH (s) + B12(s) = G + sC (4.10)
where
G = AeAx/2 t (4.11)
and
C = AeAx/2
(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

59
voltage. Thus G in (4.11) has dimensions of [A/cm"3] and is associated
with volume recombination, whereas C has dimensions of [C/cm“3] and is
associated with minority carrier storage.
For small |s|, a simple network realization of -B^s) in (4.9) is
-B12(s) = 1/(R + sL) , (4.13)
where
R = Ax/AeD , (cm“3/A)
(4.14)
and
L = Ax3/6AeD2 , (cm_3s/A) . (4.15)
We associate R with minority-carrier 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 quasi-neutral base for large-signal transient excitation
of the minority-carrier densities (Fig. 4.3).
Previous uses of the two-port 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.

en
o
Fig.4.3
RLC equivalent-circuit representation for the i
.th
subregion.

61
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 quasi-neutral
region. The thickness of the quasi-neutral region is 96 urn, the
lifetime of the minority carriers is IDO us, the diffusivity is
35 cm^/s, and the surface recombination velocity of the back low-high
junction is 200 cm/s. The results of the 2- and 3- and 15-subregion
equivalent circuits are shown in the linear-linear graph of Fig. 4.4.
The short-circuit current of the 3-subregion equivalent circuit nearly
coincides with that of the 15-subregion equivalent circuit for the time
range where the first-mode current dominates the short-circuit current.
Recall that the first-mode component contains the information about
the parameters t and S. Our earlier work in Chapters Two and Three,
focussed on determining these parameters.
Figure 4.5 displays the short-circuit 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 gs. Thus from Figs. 4.4 and
4.5, a 3-subregion equivalent-circuit suffices for the determination
of t and S for this solar cell. The decay time constant and the
current ratio R are determined from Fig. 4.4 and Fig. 4.5:
= 0.98 ys and R = 31
These values coincide with the exact solutions obtained by solving (3.7)
and (A.3) of Appendix A.

62
Fig.4.4 SPICE2 simulation of electrical-short-circuit-current decay
as a function of the number of the subregions, indicated
parametrically, used in the equivalent circuit for the
quasi-neutral base.

NORMALIZED CURRENT
63
SPICE2 simulation of electrical-short-circuit-current decay
responses displayed semilogarithmically for the quasi-neutral
base sliced into (a) one, (b) two, and (c) fifteen equally
thick subregions.
Fig.4.5

64
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 2-subregion 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 x through the slope and the intercept
of the straight-line portion of the transient. Thus, inclusion of the
inductor L in a 2-subregion equivalent circuit are needed for accurate
modeling.
In the time domain, we see that the inductors contribute delay,
designated by Tq in Fig. 4.6. In the frequency domain, the inductors
filter out the high-frequency components of the response. The same
effects--delay in the time domain and filtering in the frequency
domain--result from using a inany-section 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 Low-Frequency Small-Signal Analysis
4.3.1 Derivation in Frequency Domain
For low-frequency small-signal excitation, the matrix equation
(2.6) becomes the following:

NORMALIZED CURRENT
65
Fig.4.6 The role of inductors in the response of a two-subregion
equivalent circuit is illustrated in (a) and (b), (a) without
inductors, (b) with inductors. Response (c) corresponds to a
fifteen-subregion model, with or without inductors.

66
i (0, s)
^11(S) 4i2(s)
ev(0,s) (Q)
kT kDC^u;
1(XQNB »s)
Agi(s) Ag2(s)
eV ^ ^ONB ,S ^
giNB p (V )
kT r0C'AQNB _
(4.16)
where |v(0,s)| < and Iv (XQNB»s) I < ^ and v is the hole
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[l + v/(kT/e)]
- exp[v0C/(kT/e)][l + v/(kT/e)]
Rewriting (4.16), we have
i(0,s)
Di1(s)
v(0,s)
1(XQNB »s)
D2i(s) 022^s^
v(xqnb>s)
In (4.17),

67
^11(s)
= ♦
(4.18)
= ^2^12^s^ *
(4.19)
^2i(s)
= »
(4.20)
D22(s)
= K2A22(s) ,
(4.21)
where
= (e/kT)(ni2/NDD)exp(eV(0)/kT (4.22)
and
K2 = (e/kT)(ni2/NDD)exp(eV(XQNB)/kT) . (4.23)
If we assume a thin quasi-neutral region (Xq^b << 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 quasi-neutral region
under low-frequency small-signal excitation is shown in Fig. 4.7, in
which:
G1 - eAK1XgNB/2T
(4.24)

'(XqNB»s)
+ o-
- o-
Li ni
1 —— .
j . *
1
1
1
1
o,s) ¡ Q
1
I
:
â–º G . G <
> 1 2 <
>
2
1
1
' ¡ V(XQNB
1
1
1
-O +
-o -
x = O
x = X
QNB
Fig.4.7 Low-frequency equivalent circuit for the quasi-neutral base in low injection for small-
signal variations in the applied voltage and quasi-Fermi potentials.
Here i^s) = G(s)v(XQNB>s).

69
C1 =
eAKlxQNB/2
(4.25)
Rl =
Xqnb/eAK2D ,
(4.26)
L1 =
XQNB^/6D2eAK2 »
(4.27)
g2 =
eAKlXQNB/2t 5
(4.28)
C2 =
eAKlXQNB/2
(4.29)
G(s) = -eA(K^ - K2)(Xq^g/D “ ^XQNB^°) * (4.30)
In Fig. 4.7, the nodal variable has the dimensions of voltage.
This contrasts with our earlier models, for large-signals, for which the
nodal variable is minority-carrier density (see Fig. 4.3, for example).
For small-signal 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 quasi-Fermi 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

70
established this convention, we then identify the incremental voltages
v(0,s) and v(XqNB,s) with the incremental quasi-Fermi 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 quasi-neutral region with a general contact at x =
^QNB’ we can derive t*1e low-frequency small-signal equivalent-circuit
directly by truncating the DC components of minority-carrier density and
current from Fig. 3.2 instead of starting from the matrix equation.
Figure 4.8 displays this equivalent circuit of a quasi-neutral region
with a general contact for low-frequency small-signal excitation. The
values of the circuit elements are G = eAK^XQNg/2x, C = eAK^Xq^g/2, R =
XqNs/eAKiD, and L = XqNg3/6D2eAK1. The element G$ represents the
general contact at the back surface, the value of which follows from
Fig. 3.2:
G$ = [eAKxS] (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 Short-Base Case
The foregoing derivation in the frequency domain yields an
equivalent circuit involving an inductor L, which does not commonly
appear in small-signal or incremental models for a quasi-neutral base

Fig.4.8 Equivalent circuit of Fig.4.7 augmented by a circuit element that accounts for an arbitrary
value of minority-carrier recombination velocity at the back surface.

72
F i g. 4.9
Equivalent circuit of a quasi-neutral base when the back
surface recombination velocity is (a) infinite, as in an
ohmic contact and (b) zero, as in a blocking contact.

73
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 = Xg^g to lag (in phase angle) the voltage phasor, v(0,ju>).
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 short-base case, for
which the minority-carrier 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 short-base 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 short-base devices,
such as junction transistors.
To simplify the discussion, we assume that the quasi-neutral 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 reverse-biased 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.

74
To derive an equivalent-circuit containing an inductor L from the
time domain, we start from the quasi-static equivalent circuit for the
common-emitter configuration commonly called the hybrid-ir model [24].
We focus on the input stage of this equivalent circuit, it being
understood tnat 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 minority-carrier 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 (quasi-static)
charge storage within the base region. The conductance in Fig. 4.10(a)
is the derivative of the (quasi-static) input current with respect to
the input voltage divided by the dc common-emitter current gain 3 [14].
We formally include the effect of propagation delay by inserting a
time-delay 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 low-frequency expansion
of delay in the complex-frequency or Laplace domain:
exp[-sx( delay)] = 1 -sr(delay) = 1/[1 + st(delay)]
(4.32)

Fig.4.10 (a) The input stage of the conventional hybrid-ir model of
bipolar transistors, (b) Modified hybrid-ir model with the
inclusion of minority carrier delay across the quasi¬
neutral base, (c) Modified hybrid-ir model with the time
delay element realized with inductor.

76
RQSA
be
â– N
J
^ CQSA '
» 1
* K
>rqsa
i (
TIME DELAY
ELEMENT
) <
-o +
be
-0 +
(b)
(c)

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 “ ^ + sx(delay)]-1 » [1 + (sL'/R'^) + (R'/R1 qsA)]_1 •
(4.33)
This equation yields R' = 0 and L1 so that
i(delay) = L'/R'qsa (4.34)
where 6 is the incremental common-emitter 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 x(delay). This
determination will demonstrate that (6+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 common-base rather than the common-emitter
configuration, then L' = L which is the L in Fig. 4.9.
4.3.3. Calculation of the Delay Time
We now determine the minority-current propagation delay of a short
base by calculating the Elmore delay time [15], also used later by Ashar
[25]:

78
oo
00
t(delay) = / te(t)dt/ / e(t)dt
(4.35)
o
o
where e(t) is the impulse response of the current at x = Xg^g to v^(t) =
A5(t). In the complex frequency domain, this becomes [26]
x(delay) = 1 im{-[dF(s)/ds]/F(s)}
s>0
(4.36)
F(s) is the system function
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 minority-carrier delay time of
the thin quasi-neutral base region with an ohmic contact at x = Xg^g, we
apply impulse excitation at x = 0 and solve for the impulse response
using the two-port approach. The quasi-neutral base region used in this
calculation is thin compared with the diffusion length and we assign the
values for D and XgNg arbitrarily. Figure 4.11(a) displays an example
of the impulse response of the minority-carrier density profile in
position as time passes. Figure 4.11(b) displays the current impulse
response at x = XgNg. We derive the delay time x(delay) from this
simulation by using (4.35) and numerical integration:
t(delay) = Xqnb2/6Q
(4.37)

X
I—
I—I
in
z
u
a
u
_j
o
i
cn
ai
u
u
x
u
0 .5 1.0
POSITION
(NORMRLIZED VHLUE)
Impulse response of minority carrier profile in position within the quasi-neutral base.
Fig.4.11(a)

h-
z
u
x
q:
id
u
u
_j
o
x
m
CD
U
U
X
LiJ
Fig.4.11(b)
TIME (pis )
Impulse response of minority-carrier current at x = X.-^g.
CO
o

81
We substitute (4.37) into (4.34) to get the inductance. This
inductance in (4.40) is larger, by a factor (p + 1), than the inductance
in Fig. 4.9 as we indicated at the end of Sec. 4.3.2:
L' = (6 + 1)(RXqnb2/6D) = (0 + 1)L (4.38)
where L1 is the inductor for the common-emitter mode, L is the inductor
for the common-base mode, and R is the R in Fig. 4.9.
4.3.4 Modification of the Conventional Hybrid-Pi Transistor Model by
Including the Minority-Carrier Current Propagation Delay
The conventional hybrid-pi transistor model [24] in Fig. 4.12 is
based on the quasi-static approximation which does not include the
effects coming from the propagation delay of the minority-carrier
current across the base region. To include the propagation delay of the
minority-carrier current across the base in the conventional hybrid-pi
transistor model, we consider the collector current delay when we change
the emitter-base voltage.
Figure 4.13(a) shows the minority-carrier density corresponding to
a step change in v^ based on the conventional quasi-static
approximation. Note the instantaneous readjustment implied by this
model. Contrast this with the transient reponse 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

CO
ro
Fig.4.12 The conventional hybrid-ir model for bipolar transistors.

83
Fig.4.13(a) Quasi-static response of minority-carrier profile
for a sudden change in the emitter-base voltage.

34
Fig.4.13(b) Actual response of the minority-carrier profile for
a sudden change in the emitter-base voltage.

85
hybrid-ir 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, ^be • Thus if basewidth
modulation is neglected, the input admittance is
Y(oj) = 1/r + ju>C -jwL'/r^l/r + ja£ -jwC/3(3+l)
IT IT TT TT TT1T
(4.39)
Equation (4.39) yields a more accurate approximation for low
frequencies, oi << l/x(delay), than does the conventional hybrid-* model ,
which overestimates the input capacitance by a factor (33 + 2)/
(33 + 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 common-emitter configuration only slightly
whereas for a transistor in the common-base configuration the correction
reduces the input capacitance by one third. For the common-emitter
configuration, however, inclusion of delay is important to improving the
accuracy of the response of the incremental collector current to
incremental base-emitter 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 quasi-neutral base
region. An additional component of delay comes from the propagation of
these same carriers across the emitter-base junction space-charge
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.

I
L
co
CTl
Fig.4.14 Modified hybrid-ir model including minority-carrier delay across the quasi-neutral
base. Here L' = xidelay)^, r^ = R(g + 1), and = C.

87
4.3.5 Minority-Carrier Delay Time with Built-In Electric Field
In this section, we estimate the delay time of the minority carrier
delay time across the quasi-neutral base region when the electric field
exists due to the non-uniformity of the base doping concentration
profile.
Assume a uniform built-in electric field within the quasi-neutral
base. Then the minority-carrier current at x = Xq^b is [22]
I(XQNB,s) = -AeDnpoeeV/kT^ G csch(GXQNB)P(0,s)
(4.40)
= -A21(s)P(0,s)
where K is the ratio of the base doping concentrations at the two edges,
x=0 and x=XqNB and where
(4.41)
Using Ashar's definition [25] of delay time, we have
T(delay) = 1 i rr
s-»0
For the simple case when the base doping concentration is uniform, K=l,
we have the following from (4.42):

88
x(delay) = . (4.43)
Since the expression of delay time in (4.42) includes the uniform-doping
case and uniform-electric-field case, (4.42) holds more generally for
the delay time of minority-carrier across the quasi-neutral base.
4.3.6 Measurement of Minority-Carrier 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 urn 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
h2]L of transistors at Iq = 10 pA, and 100 pA. 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
quasi-neutral 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.

Silicon dioxide
poly-silicon substrate
Fig.4.15 Crosssectional view of the measured transistors.

IMPURITY CONCENTRATION PROFILE(atoms/cm"
90
Fig.4.16 Impurity profile of the measured transistors. The dashed line
is for the exponential fucntion approximation.

FREQUENCY(Mhz)
Measurement of phase of current gain of transistor 127.
Fig.4.17(a)

PHRSE OF ,(degree)
FREQUENCY(Mhz )
Fig.4.17(b) Measurement of phase of current gain of transistor 132.

CHAPTER FIVE
SUMMARY AND RECOMMENDATIONS
The major accomplishments of this study are
1) Development and illustration of the ESCCD measurement method.
The accuracy of ESCCD is not affected by the existence of
mobile electrons and holes in the p/n junction space-charge
region in contrast to the accuracy of the other methods
commonly used for silicon bipolar devices.
2) Derivation of a unifying view of excess minority-carrier
behavior within quasi-neutral regions in regard to transients
and their use in determining recombination lifetime and surface
recombination velocity.
3) Improvement of the speed of ESCCD measurement system by using
MOSFET transistors as switching devices rather than the
bipolar-circuit realization first used.
4) Derivation of a compact equivalent-circuit representation of
quasi-neutral regions which includes the minority-carrier
propagation delay.
5) Demonstrations that application of existing circuit analysis
programs to this equivalent circuit enables detailed and
accurate device analysis for both large- and small-signal
excitation.
6) Use of this equivalent-circuit to improve existing small-signal
models for bipolar transistors, with experimental
i 11ustrations.
93

94
We suggest the following as useful future efforts based on the
methods and approaches of the present study:
1) Application of ESCCD to determine lifetime t and surface
recombination velocity S to various devices used in power
electronics where open-circuit-voltage decay and reverse step
recovery are routinely employed.
2) Use of ESCCD for determining S and t in solar cells by a
variant of the method in which light or other radiation
replaces the electrical excitation employed in the present
version of ESCCD.
3) Improve the circuit for ESCCD. No claim is made that the
improved configuration of ESCCD is optimal.
4) Explore extending ESCCD to the determination of t and S for
quasi-neutral regions in moderate or high injection.
5) Apply the modeling approach of Chapter Four to the heavily
doped emitter of silicon bipolar transistors and to
heterojunction bipolar transistors, where energy-gap and other
parameter changes with position complicates the physics.
6) Assess more fully and improve the large-signal and small-signal
equivalent circuits for diodes, solar cells and transistors
developed in Chapter Four.

APPENDIX A
DETERMINATION OF THE EIGENVALUES FOR ESCCD AND OCVD
In this paper, we have two eigenvalue equations (2.9) and (2.12),
that determine s.¡ of each mode for OCVD and ESCCD. These are
coth(XqNB(l + s.¡ ip) 1/^/Lp) + Dp(1 + Sj ip)^/^/LpSgff - 0 (2.9)
and
1 + (Dp(1 + siTp)1/2/LpSeff)coth(XQNB(l + siTp)1/2/Lp) = 0 . (2.12)
In (2.9) and (2.12), eigenvalues exist only if
1 + s.j tp < 0 (or s.¡ < -1/Tp or < xp)
(A.l)
where x.¡ = -1/s.j.
Granting (A.l), we have
(1 + sixp)1/2 = j(-l - si xp)1/2
where (-1 - S-jXp)^2 > 0. Replacing (1 + s^ xp)^2 in (2.9) and (2.12)
with j(-l - S-jXp)*/2 yields
95

96
cot (Ai) - (Dp/SXQNB) A-j = 0
(A.2)
and
tan(Ai) + (Dp/SXg^g)Aj = 0
where
(A.3)
VP)
P
1/2
Equations (2.9) and (A.2) are identical and so are (2.12) and (A.3)
under the condition of (A.l), (A.2) and (A.3) imply an infinite number
of eigenvalues as shown in Figs. A.l and A.2.

97
Fig.A.1
The eigenvalues for OCVD where Y-
Ai = XQNsVLp’ a"d Yi = DpAi/SXQNB*
cot(Ai),
Fig.A.2
The eigenvalues for ESCCD where Y. = tan(A.),
Ai = XQ.NBKi>/Lp’ and Yi = "DpAi//SXQNB-

APPENDIX B
PHYSICS OF SHORT-CIRCUIT-CURRENT DECAY
Although there are several ways to treat the sudden application of
a short circuit replacing forward bias V, perhaps the simplest way is to
think of voltage -V being applied in series with V at t = 0. This
treatment emphasizes the change in voltage that starts the ensuing
transient. See Fig. B.l(a), below.
Thus at t = 0, this change in voltage raises the right ohmic
contact by magnitude eV relative to the left ohmic contact because the
ohmic contacts are in equilibrium with the adjoining semiconductor in
the sense that the distance between the quasi-Fermi level of majority
carriers and the majority-carrier band edge remains the same as in
equilibrium. They are in nonequilibrium in the sense that charge
carriers can pass through the contacts. At t = 0+, some arbitrarily
small time after the application of the short circuit, the change in
applied voltage has caused electrons to exit the n-type material
adjacent to the contact, leaving behind bared donor atoms and the
positive charge shown in Fig. 8.1(b). Similarly, holes exit the p-type
material (electrons enter the valence band from the metal), giving rise
to the negative charge shown in Fig. B.l(b). A near-delta function of
current i(t), flowing in the direction shown in Fig. B.l(a), establishes
this charge configuration at t = 0+. Note that I(t) during the entire
98

Fig.B.l (a) For t < 0, switch Si is closed, S2 is open; conversely
for t > 0, SI is open and S2 is closed. The junction space-
charge region is defined by xp < x < xn. (b) charge density
at t = 0+. (c) charge density for t of the order of a
dielectric relaxation time, (d) the total current is x-
independent but is essentially majority-carrier convection
current in the quasi-neutral regions and is displacement
current in the space-charge region for t of the order of
the dielectric relaxation time, (e) electrons and holes
drift out of the space-charge region in a transit time.
(f) the resulting excess hole density in the space-charge
region after a transit time has lapsed.

100

101
transient for t > 0 flows in a direction opposite to that occuring for
negative time because the transient results in removing the electrons
and holes present under forward V.
Having established the existence of this negative charge, we now
consider what happens subsequently. Here enters a result developed
earlier from operating on Maxwell's equation by the divergence operator
0 = div curl H = div(j^) + div(jp) + div[3( eE)/3t] . (B. 1)
From this result, two consequences emerge: 1) the charges Fig. B.l(b)
redistribute to the positions shown in Fig. B.l(c) within the order of a
dielectric relaxation time t = e/a; and 2) the total current is
solenoidal , that is, its divergence is zero, where here the total
current includes the displacement current.
The consequence of (2) is illustrated in Fig. B.l(d) for a
particular time of order of t. Notice the large time-rate of change of
electric field E within the junction space-charge region, Xp < x < x^.
Here we have employed a one-dimensional model so that the operator div
becomes the operator 3/3x. Thus we see that the electric field in the
space-charge region grows rapidly so that within t of the order of x the
barrier height has returned to its near-equilibrium value and the
electric field is several times larger than it was in negative time. In
negative time, however, the drift and diffusion tendencies of the
junction space-charge region were perturbed only by perhaps one part in
104 in the forward voltage steady state; that is, the space-charge

102
region was in quasi-equilibrium. For t of the order of x, the drift
tendency now overwhelms the diffusion tendency, and holes and electrons
drift out of the space charge region in a transit time t' determined by
U|\j " *p)/velocity where the velocity approaches the scatter limited
velocity because of the high field (Fig. B.l(e)). For typical devices,
t' will be of the order of 10"^ s. After this time has passed, the
hole and electron concentrations will have returned nearly to their
equilibrium values.
Because 10"^ s is a time not observed by typical measurement
equipment, we think of the initial condition established by shorting the
terminals suddenly as that of quasi-neutral regions still storing
approximately the same excess charge as was present in the steady state
of negative time, of a space-charge region at the equilibrium barrier
height and devoid of excess holes and electrons, and as an excess
minority-carrier density in the quasi-neutral regions that drops sharply
to zero at the space-charge region edges (Fig. B.l(f)).

APPENDIX C
RELATION BETWEEN ASHAR'S DEFINITION AND
ELMORE'S DEFINITION OF DELAY TIME
Ashar's definition of delay time is equivalent to Elmore's. We
will show this equality by deriving Elmore's definition from Ashar's.
The system function F(s) of a system under impulse excitation is
expressed in terms of this impulse response, e(t):
oo
F(s) = / e(t)e"st dt . (C.l)
o
If we substitute this expression, (C.l), into Ashar's definition (4.36),
we have the following expression for delay time:
t(delay)
1 im
s-K)
- it fe(t)e-5tdt]
0
00
/ e(t)e stdt
o
(C .2)
Since the Laplace variable s is orthogonal to time variable t, we
manipulate equation (C.2) in the following manner:
/ te(t)e"stdt
x(delay) = 1 im — (C.3)
s*° f e(t)e~stdt
103

104
00
/ 1im[te(t)e~st]dt
o s-K)
00
/ 1im[e(t)e”st]dt
o s+0
oo
/ te(t)dt
CO
/ e(t)dt
(C.4)
(4.35)
Thus, we derived Elmore's definition of delay time from Ashar's by using
the relation between system function and impulse response.

APPENDIX D
EFFECTIVE BASE WIDTH ESTIMATION OF
THE TRANSISTORS MEASURED IN CHAPTER FOUR
We measure the transistors for S-parameters in Chapter Four for
Vq^ = 10 V. Hence we expect shrinkage of tne basewidth due to the
expansion of the space-charge region of the collector-base junction. We
estimate the resultant quasi-neutral basewidth by approximating the
impurity concentration N(x) of the base with an exponential function of
x/Ld, as shown by the dashed line in Fig. 4.16:
N(x) = Nd exp(-x/Ld) , (D.l)
where Ld = 4.3 x 10"4 cm.
Here x originates from the position where the impurity
concentration is Nd (5 x 1014 cnT^) and directs toward the buried layer
side. If we assume that the space-charge region is extended to x=-X$cR,
then the electric field E(x) is expressed by using Poisson's equation:
x
E(x) = I N(x)dx, -X$CR < x < 0 . (D.2)
Si “XSCR
The potential difference from the base to the collector by using one¬
sided step junction approximation is
105

106
VCB = / E(x)dx
-X
(0.3)
SCR
eLdNd XSCR/Ld , . eLd Nd , XSCR/Ld .
e (XSCR } + ~ (e ' l)
Si
si
(0.4)
By substituting VCB = 9.3V in (D.3), we estimate XSCR and XqNB like as
follows: X QNB
XSCR = 4 ^ 1
XQNB " 10

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for solid-state devices," Solid-State Electron., vol. 7, pp. 771-
783, 1964.
10. C. T. Sah, "Tne equivalent circuit model in solid-state
electronics, III: Conduction and displacement currents," Solid-
State Electron., vol. 13, pp. 1547-1575, 1970.
107

108
11. A. Neugroschel , P.J. Chen, S. C. Pao, and F. A. Lindholm,
"Diffusion length and lifetime determination in p-n junction solar
cells and diodes by forward biased capacitance measurements," IEEE
Trans. Electron Devices, vol. ED-25, pp. 485-490, 1978.
12. A. Neugroschel, "Determination of lifetime and recombination
currents in p-n junction solar cells, diodes, and transistors,"
IEEE Trans. Electron Devices, vol. ED-28, pp. 108-115, 1981.
13. J. E. Mahan and D. L. Barns, "Depletion layer effects in the open
circuit voltage decay lifetime measurements," Solid-State
Electronics, vol. 24, pp. 989-994, 1981.
14. J. G. Linvill and J. F. Gibbons, Transistors and Active Circuits,
New York: McGraw-Hill Book Company, Inc., 1961.
15. W. C. Elmore, "Tne transient response of dampted linear networks
with particular regard to wideband amplifiers," J. Appl. Phys.,
vol. 19, pp. 55-63, 1948.
16. R. W. Dutton and R. J. Whittier, "Forward current-voltage and
switching characterisecs of p+-n-n+ (epitaxial) diodes, IEEE
Trans. Electron Devices, vol. ED-16, pp. 458-467, 1969.
17. S. M. Sze, Physics of Semiconductor Devices, New York: John Wiley
and Sons, Inc., 1981.
18. B. R. Chawla and H. K. Gummel, "Transition region capacitance of
diffused p-n junctions," IEEE Trans. Electron Devices, vol. ED-18,
pp. 178-195, 1971.
19. T. H. Ning and D. D. Tang, "Method for determining the emitter and
base series resistance of bipolar transistors," IEEE Trans.
Electron Devices, vol. ED-31, pp. 409-412, 1984.
20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions, New York: Dover Publications, Inc., 1970.
21. D. J. Hamilton, F. A. Lindholm, and J. A. Narud, "Comparison of
large signal models for junction transistors," Proc. IEEE,
vol . 52, pp. 239-248, 1964.
22. D. J. Hamilton, F. A. Lindholm, and A. H. Marshak, Principles and
Applications of Semiconductor Device Modeling, New York: Holt,
Rinehart and Winston, Inc., 1971.
23. J. L. Moll, "Large-signal transient response of junction
transistors," Proc. IRE, vol. 42, pp. 1773-1784, 1954.

109
24. P. E. Gray, D. De Witt, A. R. Boothroyd, and G. F. Gibbons,
Electronics and Circuit Models of Transistors, S.E.E.C. Notes,
vol. 2, New York: John Wiley & Sons, Inc., 1965.
25. K. G. Ashar, "The method of estimating delay in switching circuits
and the figure of merit of a switching transistor," IEEE Trans.
Electron Devices, vol. ED-11, pp. 497-506, 1964.
26. W. T. Thomson, Laplace Transformation, Englewood Cliffs, NJ:
Prentice-Hall, Inc., I960.
27. D. K. Lynn, C. S. Heyer, and D. J. Hamilton, Analysis and Design
of Integrated Circuits, New York: McGraw-Hill Book Company, 1968.

BIOGRAPHICAL SKETCH
Tae-Won Jung was born in Seoul, Korea, on May 21, 1953. He
received the B.S. degree in electronic engineering from the Seoul
National University in 1976 and the M.S. degree in electrical
engineering from the Korea Advanced Institute of Science, Seoul, Korea,
in 1978. Since August 1981, he has been working for the Ph.D. degree at
the University of Florida, Gainesville.
In July 1985, he joined the Linear Product Design group at Harris
Semiconductor, Melbourne, Florida, where he has been designing analog
devices such as operational amplifiers and analog switches. From 1978
to 1981, he was with Kyung-Pook National University where he was engaged
in teaching semiconductor device physics.
His area of interest includes bipolar and MOSFET transistor
modeling and integrated circuit design.
He is a member of the honor society of Phi Kappa Phi.
110

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy. ^ ( ( /
[ /"} l iJiLfrt LH-
F. A. Lindholm, Chairman
Professor of Electrical Engineering
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
// ^ <<7
Peter T. Landsberg *
Graduate Research Professor
of Electrical Engineering
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
Sheng S. Li
Professor of
Electrical Engineering
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
A
"t (.. L i A
D. E. Burk
Associate Professor
Engineering
of Electrical
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
Rolf E. Hummel
Professor of Materials Science
and Engineering

This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
May 1986
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08554 1505



Fig.4.10 (a) The input stage of the conventional hybrid-ir model of
bipolar transistors, (b) Modified hybrid-^ model with the
inclusion of minority carrier delay across the quasi
neutral base, (c) Modified hybrid-ir model with the time
delay element realized with inductor.


40
The current i(t) flowing through rd in Fig. 3.5 is
i(t) = CSCR(dv12(t)/dt) + v12(t)/rsh + ix(t) (3.15)
where
CSCR
f(vi2)
(3.16)
in which Q$QR/e 1S the integrated steady-state hole or electron density
through the volume of the space-charge region. From Chawla and Gummel
[18]
CSCR/CSCR0 s t1 (vi2/Vg)]m (3.17)
where C$CRq is the C$qR for variations in v12 about bias voltage v^2 = 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 2 cSCR0 f(v12)
(3.18)
Thus
i(t) CSCR0dv12(t)/dt + v12(t)/rsh + 1j(t)
(3.19)


h-
z:
Ld
CK
CK
ID
o
u
o
I
m
01
u
u
x
LJ
Fig.4.11(b)
TIME ( |js )
Impulse response of minority-carrier current at x = Xq^b


Fig.B.l (a) For t < 0, switch S\ is closed, S2 is open; conversely
for t > 0, SI is open and S2 is closed. The junction space-
charge region is defined by xp < x < xn. (b) charge density
at t = 0+. (c) charge density for t of the order of a
dielectric relaxation time, (d) the total current is x-
independent but is essentially majority-carrier convection
current in the quasi-neutral regions and is displacement
current in the space-charge region for t of the order of
the dielectric relaxation time, (e) electrons and holes
drift out of the space-charge region in a transit time.
(f) the resulting excess hole density in the space-charge
region after a transit time has lapsed.


2
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 two-port approach [9]
to obtain a unifying framework for transient analysis and develop from
it a new method, experimental electrical-short-circuit-current-decay
(ESCCD) for the determination of the recombination lifetime and back-
surface recombination velocity of the quasi-neutral 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 4-mil BSF solar
cells). We derive various performance parameters for BSF solar cells by
exploring the ESCCD method. We propose a methodology to separate the
quasi-neutral 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 minority-carrier propagation delay
in a compact equivalent circuit. This enables the use of standard
circuit-analysis computer programs, without resorting to much more
sophisticated programs needed when transmission-line models [10] are
employed. These ideas are applied both to ESCCD and to bipolar


51
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 ohmic-contact solar cell and by using (3.31):
Td_1 = C(XqNB/^)2/D]_1 + t_1 (3.33)
Second, one measures td 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(t)
locus.
An illustrative example is shown in Fig. 3.8. In this example, we
used a wafer which is 10 ohm-cm and p-type. The upper locus corresponds
to the ohmic-contact cell. The lower one corresponds to the BSF cell.
The lifetimes of these cells are estimated about 200 ys 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 ohm-cm
n+p/p+ solar cell (ohmic contact) with a thickness of 350 3 un, the
error in the lifetime is estimated to be about 20%. In doing this
calculation, we assumed that the lifetime is 50 ys (Xg^g/L = 0.84) and
that 0 = 35 cm^/s. For a 0.3 ohm-cm n+/p/p+ solar cell (ohmic contact)
with a thickness of 350 i 3 nn, this error is estimated to be about
5%. Here for this calculation, we assumed that the lifetime is 20 ys
(XqN8/L = 1.64) for diffusivity D = 23 cm2/s.


73
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,j(u).
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 short-base case, for
which the minority-carrier 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 short-base 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 short-base devices,
such as junction transistors.
To simplify the discussion, we assume that the quasi-neutral 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 reverse-biased 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.


V^t)
Fig.2.2 Electronic switching circuit used in the ESCCD method. The circuit elements are
bipolar transistors(2N3906), pulse generator(HP8004), operational amp(Burr-Brown
3500c), and diodes(lN914).


16
its equilibrium value. (The physics governing this phenomenon comes
from Maxwell's Curl H_ = j_ + 80/at; taking the divergence of both sides
yields 0 = div j_ + d(div J))/dt, which, when combined with _i_ = qO/e and
divD^ = p, yields a response of the order of e/o, the dielectric
relaxation time.)
Following this readjustment of the barrier height, the excess holes
and electrons exit the junction space-charge region within a transit
time of this region (about lCT^s 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 space-charge region in the ESCCD method occurs within a time of
the order of 10 ^s, which is much less than any of the times associated
with discharge of the quasi-neutral regions. This absence in effect of
excess holes and electrons within the junction space-charge 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 space-charge region appears in Appendix B.
The discharge of the quasi-neutral emitter depends on the energy-
gap narrowing, the minority carrier mobility and diffusivity, the
minority-carrier 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 quasi-neutral base, and we shall assume
this is so in the discussion to follow.


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 F TRANSIENTS IN QUASI-NEUTRAL REGIONS FOR
CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS
- By
TAE-WON 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 quasi-neutral 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 quasi-neutral base region of p/n junction silicon solar
cells, including devices having conventional back-surface-field (BSF),
ion-implanted BSF and polysilicon BSF structures. This method, called
electrical-short-circuit-current-decay (ESSCD) avoids errors introduced
in other methods in common use, such as open-circuit-voltage-decay and
reverse step recovery, that arise from the capacitive effects of mobile
v i i i


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.


96
cot(Ai) (Dp/SXQfsjB) Ai = 0
(A.2)
and
tan^) + (Dp/SXQNB)Ai = 0
where
(A.3)
1/2
Equations (2.9) and (A.2) are identical and so are (2.12) and (A.3)
under the condition of (A.l), (A.2) and (A.3) imply an infinite number
of eigenvalues as shown in Figs. A.l and A.2.


APPENDIX D
EFFECTIVE BASE WIDTH ESTIMATION OF
THE TRANSISTORS MEASURED IN CHAPTER FOUR
We measure the transistors for S-parameters in Chapter Four for
Vq£ = 10 V. Hence we expect shrinkage of tne basewidth due to the
expansion of the space-charge region of the collector-base junction. We
estimate the resultant quasi-neutral basewidth by approximating the
impurity concentration N(x) of the base with an exponential function of
x/Ld, as shown by the dashed line in Fig. 4.16:
N(x) = Nd exp(-x/Ld) (D.l)
where Ld = 4.3 x 10"4 cm.
Here x originates from the position where the impurity
concentration is Nd (5 x 10*4 cm"^) and directs toward the buried layer
side. If we assume that the space-charge region is extended to x=-X<;cp,
then the electric field E(x) is expressed by using Poisson's equation:
x
E(x) = I N(x)dx, -X$CR < x < 0 (D.2)
Si ~XSCR
The potential difference from the base to the collector by using one
sided step junction approximation is
105


UNIVERSITY OF FLORIDA
III
3 1262 08554 1505


APPENDIX A
DETERMINATION OF THE EIGENVALUES FOR ESCCD AND OCVD
In this paper, we have two eigenvalue equations (2.9) and (2.12),
that determine s^ of each mode for OCVD and ESCCD. These are
coth(XQNB(l + s^p^/Lp) + Dp (1 + Sitp^/Z/LpSeff = 0 (2.9)
and
1 + (Dp(1 + Si tp)1/2/LpSeff)coth(XQ^B( 1 + s-j Tp)1/2/Lp) = 0 (2.12)
In (2.9) and (2.12), eigenvalues exist only if
1 + Si Tp <0 (or Si < 1 /Tp or ^ < tp) (A.l)
where Tj = -1/Si.
Granting (A.l), we have
(1 + SiTp)1/2 = j(-l S i Tp) ^ ^2
where (-1 s^p)^2 > 0. Replacing (1 + s^p)1/2 in (2.9) and (2.12)
with j(-l Sitp)1/2 yields
95


39
i(t)
Fig.3.5 Equivalent-circuit representation of the measurement
circuit of Fig.3.4 when the first mode dominates
electrical-short-circuit-current decay.


37
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 turn-on 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 turn-on 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 xd 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 shorted-path. The voltage
across the junction space-charge-region does not vanish as long as the
current flows through the series resistance. Fig. 3.5 displays the
equivalent circuit during discharge when the first-term natural-
frequency current dominates the discharging current. Higher-term
natural-frequency 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 xd of the first-term
natural-frequency current. In Fig. 3.5, rs and r$h are the series and
shunt resistance of a solar cell, rd detects the discharging current,
and i^(t) is the first-term natural-frequency current.


56
where Bu(s) = B22(S) = (AeD/L*)coth(Ax/L*)
(4.2)
and B^2(s) = B2^(S) = -(AeD/L*)csch(Ax/L*)
(4.3)
Figure 4.2 displays the matrix equation (4.1), where p^(0")/s and
pi+l(0-)/s are the minority carrier densities and i-¡(0")/s and
ii+1(0-)/s are the minority carrier currents, respectively, at the two
edges of the i^*1 subregion. The designation, t = 0, means the instant
before we apply the excitation to start the transient. The circuit
elements Y^ and Y2 in Fig. 4.2 are related to B^ and B2^ as follows:
(4.4)
Y1 = B11 + b12
(4.5)
To realize an RLC equivalent-circuit representation of a quasi
neutral region, which enables use of circuit-analysis software, such as
SPICE2, one has to algebraically approximate the transcendental
functions Y^(s) and Y2(s) in Fig. 4.2. The condition which makes
possible a series expansion of (4.2) and (4.3) is
| Ax/L* I = | Ax( 1 + St)1/,2/(Dt)1/,2| < it
(4.6)
where x is the recombination lifetime [20]. For thickness Ax and a
natural frequency s which satisfy (4.6), one can truncate the series to
approximate Yj(s) and Y2(S) by


This approximation corresponds to a passive network having impedance,
Z(s) = R' + jX(s), and, thus from Fig. 4.10(b),
Vbe(s)/Vbe = [1 + sx(delay)]-1 [1 + (sL7R'QSA) + .
(4.33)
This equation yields R1 = 0 and L so that
t(delay) = L7R'QSA (4.34)
where 6 is the incremental common-emitter 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 x(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 common-base rather than the common-emitter
configuration, then L' = L which is the L in Fig. 4.9.
4.3.3. Calculation of the Delay Time
We now determine the minority-current propagation delay of a short
base by calculating the Elmore delay time [15], also used later by Ashar
[25]:


47
Although one will not know Xq^/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(x)
exhibits dS/dx 0 over a large range of x. This is the mathematical
statement, for our procedure, that S is more accurately determined than
is x for a thin-base solar cell. (If Xq^b/L 1, dS/dx = over a
large range of S, which means that x is more accurately determined than
is S for a long-base 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/crn p-type substrates. The top n+ layer is about 0.3 urn deep.
The thickness of these cells is about 100 un.
Using xd, we generated S(x) 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/dx approaches zero,
defines Smax. If for an extreme case for which negligible volume
recombination occurs during the ESCCD transient, S = S)T,ax.
Similarly, the limit dS/dx + > defines a minimum value of the
lifetime xml-n, as illustrated in Fig. 3.7. For the all corresponding to
the lower locus of Fig. 3.7, Tjnl-n = 40 us. This value xm^n occurs for


IMPURITY CONCENTRATION PROFILE(atoms/cm'
90
Fig.4.16 Impurity profile of the measured transistors. The dashed line
is for the exponential fucntion approximation.


APPENDIX B
PHYSICS OF SHORT-CIRCUIT-CURRENT DECAY
Although there are several ways to treat the sudden application of
a short circuit replacing forward bias V, perhaps the simplest way is to
think of voltage -V being applied in series with V at t = 0. This
treatment emphasizes the change in voltage that starts the ensuing
transient. See Fig. 8.1(a), below.
Thus at t = 0, this change in voltage raises the right ohmic
contact by magnitude eV relative to the left ohmic contact because the
ohmic contacts are in equilibrium with the adjoining semiconductor in
the sense that the distance between the quasi-Fermi level of majority
carriers and the majority-carrier band edge remains the same as in
equilibrium. They are in nonequilibrium in the sense that charge
carriers can pass through the contacts. At t = 0+, some arbritrarily
small time after the application of the short circuit, the change in
applied voltage has caused electrons to exit the n-type material
adjacent to the contact, leaving behind bared donor atoms and the
positive charge shown in Fig. 8.1(b). Similarly, holes exit the p-type
material (electrons enter the valence band from the metal), giving rise
to the negative charge shown in Fig. 8.1(b). A near-delta function of
current i(t), flowing in the direction shown in Fig. B.l(a), establishes
this charge configuration at t = 0+. Note that I(t) during the entire
98


USE OF TRANSIENTS IN QUASI-NEUTRAL REGIONS FOR
CHARACTERIZING SOLAR CELLS, DIODES, AND TRANSISTORS
By
TAE-WONjJUNG
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


I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
I certify that I have read
conforms to acceptable standards
adequate, in scope and quality,
Doctor of Philosophy.
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
F. A. Lindholm, Chairman
Professor of Electrical Engineering
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
Peter T. Landsberg
Graduate Research Professor
of Electrical Engineering
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
Professor of Electrical Engineering
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
A. w 't (. i :
D. E. Burk
Associate Professor of Electrical
Engineering
this study and that in my opinion it
of scholarly presentation and is fully
as a dissertation for the degree of
Rolf E. Hummer
Professor of Materials Science
and Engineering


43
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 t(1. 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 ^FMO^QNBO Ts)
(3.26)
RI = ^NEO^QNBO ^FO ^NBO^QNBO
(3.27)
RM ^MO^FO
(3.14)
The relation among these parameters is
R = (1 + Rj)Rm. (3.28)
The ratio R is determined by theory for any assumed values of S and
t lying on the S(x) locus corresponding to the measured value of the
decay time constant xg. The ratio Rj is determined by the measured
value of Ipg and by the value of Iq^g which is obtained from (3.6) for
any assumed values of S and x. The ratio RM is determined by
measurement. Thus (3.28) enables a determination of S and x by an
iterative procedure.


21
(a)
(b)
Fig.2.3 (a) Voltage across BSF #1 solar cell (vertical:!
(b) Current through BSF #1 solar cell (vertical
(c) Log scale representation of (b) (vertical:0
where v(t) = (mkT/e)ln(i(t)/I0+l).
2V/div),
:lmA/div),
.lV/div),


CHAPTER TWO
UNIFYING VIEW OF TRANSIENT RESPONSES FOR DETERMINING
LIFETIME AND SURFACE RECOMBINATION VELOCITY IN SILICON
BACK-SURFACE-FIELD SOLAR CELLS,
WITH APPLICATION TO EXPERIMENTAL SHORT-CIRCUIT-CURRENT DECAYS
2.1 Introduction
This chapter has three purposes. First, we outline a mathematical
method that systematically and compactly describes the large-signal
transient and small-signal 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
quasi-neutral principal regions of the devices.
Second, exploiting this description, we survey the adequacy of
various experimental large-signal 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 quasi-neutral base from a
single treatment measurement for three different p+/n/n+ back-surface-
field solar cells.
4


i H ni
L-* t
1
1
(0,s) ¡ Q
1
1
1
D ¡,(s)'
-Cj <
> G i G <
1 2<
>
2
1
1
1
1
" | V(XQNB
1
1
> /
-o +
o -
QNB
Fig.4.7 Low-frequency equivalent circuit for the quasi-neutral base in low injection for small
signal variations in the applied voltage and quasi-Fermi potentials.
Here ijis) =
G(S)v(XqnbS)'


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23
From the above development, these results give DEVICE 1, Xp =
119 ys, Seff = 25 cm/sec; DEVICE 2, = ip = 119 ys, Sgff = 60 cm/s;
DEVICE 3, Xp = 213 ys, Seff = 100 cm/s- These results agree favorably
with those obtained for the same devices by using the more time-
consuming methods detailed in [11-12],
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 space-charge-
region. These methods, among open-circuit 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 n^ 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 space-charge region play no role in determining the response of
the electrical-short-circuit-current 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 first-mode response,


72
i (O, s)
L R
+ o
smwvwr
v(0,s)
- o
x O
(b)
C
Fig.4.9 Equivalent circuit of a quasi-neutral base when the back
surface recombination velocity is (a) infinite, as in an
ohmic contact and (b) zero, as in a blocking contact.


106
0
VCB / E(x)dx
XSCR
sL. N X o/^ o / L j eL 2N. XcrD/l_.
d a ^ SCK d \ d d SCR d
e (XSCR } + ~ (e 1}
Si
By substituting VCB = 9.3V in (D.3), we estimate XSqR and X
'QNB
follows: X
SCR a 4 Mm and XqNB 10 nn.
lSCR
= 4 um
Xqnb = 10 um
(0.3)
(0.4)
like as


42
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
i Vs + (kT/q)loge(I-j/1F0) (3.24)
where r$ is the series resistance, IpQ is the idealized pre-exponential
current (corresponding to unity slope), and subscript i denotes
different data points. Applied to two such data points, Eq. (3.24)
yields
AVout = rsAI + (kT/e)19e(I2/Il^ (3.25)
upon subtraction. This determines rs, which we may thus ignore in the
subsequent discussion. To determine IpQ, we use the procedure of
Ref. [19].


14
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 l(0,s) = constant (reverse current) at 0 < t < xs, where ts
is the time needed for the excess hole density p(0,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 space-charge region
complicates the interpretation of the measured x$ in terms of the
desired parameters, xp and Seff.
In addition to this, during the recovery transient (x$ < t < ),
the reverse generation current is often large enough to saturate the
recovery current so quickly that we have no sizable linear portion of
the first-mode curve on a plot of ln[i(t)] vs. t. This linear portion
provides interpretable data for Ge devices [3], but not often for Si
devices according to our experiments.


35
(3.13)
There are two unknowns, r$ and Ip0, in (3.13). We estimate rs and IFq
by measuring the dark I-V characteristics from the terminals of a solar
cell. The pre-exponential factor Ipg will be utilized for the
determination of the base material parameters.
3.2.3 Combined Method of Electrical Short-Circuit Current Decay and
Dark I-V 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 I-V characteristic methods. Using the ESCCO method, we measure
the decaying time constant of the first mode x^ and the ratio R^ of pre
exponential factors from Fig. 2.3:
FMO
R,
(3.14)
M
FO
in which the subscript FMO means the pre-exponential factor of the
first-mode current. Using the dark I-V measurement, we estimate the
pre-exponential factor IFq by eliminating the series resistance effect
as described in Section 3.2.2.
From the measured value of x^, one can generate a s(x) locus on the
t-S plane; each point on this locus must produce the measured value of
Td. Each point (x,S) also has its own value of the ratio R, defined in


CHAPTER THREE
EXTENSION OF THE METHOD OF ELECTRICAL
SHORT-CIRCUIT CURRENT DECAY
3.1 Introduction
This chapter describes various improvements of the method of
electrical short-circuit 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 first-mode current If^rst-mode^+) together with the decay time
constant Td as the ESCCD parameters used to determine t and S. The
parameter ifirst-mode^+^ proportional to exp[ev(0")/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: ifi rst-mode^+^ Td anc*
v(0). 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(0)
by treating [i f.j rst_moCfe(0+)]/Ip(0'') as the measurable parameter. In
the ratio the factor exp[ev(0)/kT] cancels out.
26


Silicon dioxide
CO
v£>
poly-silicon substrate
Fig.4.15 Crosssectional view of the measured transistors.


10
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 quasi-neutral base;
and, using quasi-neutrality together with knowledge of the steady-state
currents in the junction space-charge region and in the p+ quasi-neutral
emitter region, one can thus find the steady-state current flowing in
the external circuit or the voltage at the terminals of the diode. If
the quasi-neutral 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 time-varying mode of operation leads to a
minority hole current in the n-type quasi-neutral 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 3p/3t 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 two-port technique in effect reduces the
complexity of tne 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.


A
LIST OF SYMBOLS
area of a device
Ai
XQNBKi/L
Aij
characteristic matrix elements of a quasi-neutral region
for large-signal transient normalized surface
recombination velocity
Bij
characteristic matrix elements of a subregion for large-
signal transient
CSCR^V^
space-charge region capacitance of a p-n junction
forward-biased with voltage V
CSCRO
space-charge region capacitance of a p-n junction at
V = 0
D
diffusion coefficient of minority carriers
ij
characteristic matrix elements of a quasi-neutral region
for small-signal low-frequency analysis
DP
diffusion coefficient of minority holes
AX
thickness of a subregion
e
magnitude of the electron charge
i(x,0")
minority carrier current at t = 0
!d
dark current of an ideal diode
ioc()
dc steady-state current at x = 0
^ fi rst-mode^)
the first-natural-frequency (first transient mode)
current at x = 0
tfmo
pre-exponential factor of the first natural-frequency
current at t = 0
tfo
pre-exponential factor of a steady-state current of a
diode with negligible space-charge current
Ii (s)
Laplace transform of ij(t)
V


12
P(0 t) = I 2i(M')L [1 + cot3 esi
t=l eDps1 TP [cosec2(Xp^/Lp) (Op/LpSeff)]
(2.8)
where s.¡ is the ith singularity point (ith mode) which satisfies
coth(,lqNB/rrVp/Lp> + Dp /rfV^/LpSeff 0 <2-9>
and K. = /-1-s.t > 0.
i 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 s^ is called a mode, as
in the electromagnetic theory. Appendix A treats the details of
determining the Eigenvalues s^ from Eq. (2.9) (and from the similar
Eq. 2.11 derived below).
The decaying time constant -1/s^ 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 xp 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 open-voltage
decay curve is usually bent up or bent down because of discharging and
recombination within the space-charge region.
As mentioned in [11], the mobile charge within the space-charge
region contributes significantly to the observed voltage transient for
10 -3
Si, in which n^ =10 cm but not in Ge, for which OCVD was first


holes and electrons in the volume of the p/n junction space-charge
region under forward voltage. Two circuit implementations of ESCCD are
presented and evaluated.
The ESCCD method derives from a tneoretical 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
quasi-neutral 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 transmission-line, model. The inclusion
of the inductor makes this possible. Models of this type are developed
for both small-signal and large-signal 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 hybrid-n model for bipolar transistors in the coiranon-emitter
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.


85
hybrid-ir 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( (4.39)
Equation (4.39) yields a more accurate approximation for low
frequencies, w << l/x(delay), than does the conventional hybrid-u model,
which overestimates the input capacitance by a factor (3$ + 2)/
(33 + 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 common-emitter configuration only slightly
whereas for a transistor in the common-base configuration the correction
reduces the input capacitance by one third. For the common-emitter
configuration, however, inclusion of delay is important to improving the
accuracy of the response of the incremental collector current to
incremental base-emitter 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 quasi-neutral base
region. An additional component of delay comes from the propagation of
these same carriers across the ernitter-base junction space-charge
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.


APPENDIX C
RELATION BETWEEN ASHAR'S DEFINITION AND
ELMORE'S DEFINITION OF DELAY TIME
Ashar's definition of delay time is equivalent to Elmore's. We
will show this equality by deriving Elmore's definition from Ashar's.
The system function F(s) of a system under impulse excitation is
expressed in terms of this impulse response, e(t):
oo
F(s) = / e(t)est dt (C.l)
o
If we substitute this expression, (C.l), into Ashar's definition (4.36),
we have the following expression for delay time:
t(delay)
- gjt fe(t)e'stdt]
0
00
/ e(t)e-stdt
(C.2)
Since the Laplace variable s is orthogonal to time variable t, we
manipulate equation (C.2) in the following manner:
/ te(t)e~stdt
t(delay) = lim (C.3)
S_K) / e(t)e"stdt
103


18
and
1 fi rst mode^)
eDpp(0,0 )K1 CQt(KjXqNB/lp) ~ (PpK]/LpSeff\
1 p V2IV + ^XQNB/2Seff^ C0Sec ^K1XQNB/Lp^ ^ ^
Equations (2.13) and (2.14) contain four unknowns: if-jrst mode^ sl
ip, and Seff. The parameters, s-^ and if-jrst mode are determined
from the straight-line portion of the observed decay (in Fig. 2.3(c) to
be discussed below) p(0,0) = (n2/N^)[exp(eV(0~)/kT)-l]. Here v(0) is
known and the doping concentration Ngg of the base is measured by usual
methods; Dp(NQg) is known, and Xg^g is measured. Combining (2.14) and
(2.13) then yields the desired parameters: and Seff.
2.6.2 Experiments and Results
To explore the utility of the ESCCO 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 (t) is high, switching
transistor T1 turns on, which charges the large capacitor in parallel
with it and divides the high voltage about equally between the
solar cell and the emitter-collector terminals of the transistor. Thus


84
emitter SCR
Fig.4.13(b)
Actual response of the minority-carrier profile for
a sudden change in the emitter-base voltage.


81
We substitute (4.37) into (4.34) to get the inductance. This
inductance in (4.40) is larger, by a factor (p+ 1), than the inductance
in Fig. 4.9 as we indicated at the end of Sec. 4.3.2:
L' (B + 1)(RXqnb2/6D) = (B + 1)L (4.38)
where L' is the inductor for the common-emitter mode, L is the inductor
for the common-base mode, and R is the R in Fig. 4.9.
4.3.4 Modification of the Conventional Hybrid-Pi Transistor Model by
Including the Minority-Carrier Current Propagation Delay
The conventional hybrid-pi transistor model [24] in Fig. 4.12 is
based on the quasi-static approximation which does not include the
effects coming from the propagation delay of the minority-carrier
current across the base region. To include the propagation delay of the
minority-carrier current across the base in the conventional hybrid-pi
transistor model, we consider the collector current delay when we change
the emitter-base voltage.
Figure 4.13(a) shows the minority-carrier density corresponding to
a step change in v^ based on the conventional quasi-static
approximation. Note the instantaneous readjustment implied by this
model. Contrast this with the transient reponse 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


67
^11 (s) = KiAn(s) ,
(4.18)
0l2(s) = ^A^is) ,
(4.19)
^2i(s) = KjA2^(s) ,
(4.20)
^22^= ^2^22^*
(4.21)
where
Kx = (e/kT)(n12/NDD)exp(eV(0)/kT (4.22)
and
K2 = (e/kT)(n^/N00)exp(eV(XQNB)/kT) (4.23)
If we assume a thin quasi-neutral 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 quasi-neutral region
under low-frequency small-signal excitation is shown in Fig. 4.7, in
which:
G1 = eAK1XgNB/2x
(4.24)


64
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 2-subregion 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 straight-line portion of the transient. Thus, inclusion of the
inductor L in a 2-subregion equivalent circuit are needed for accurate
modeling.
In the time domain, we see that the inductors contribute delay,
designated by Tq in Fig. 4.6. In the frequency domain, the inductors
filter out the high-frequency components of the response. The same
effects--delay in the time domain and filtering in the frequency
domain--result from using a inany-section 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 Low-Frequency Small-Signal Analysis
4.3.1 Derivation in Frequency Domain
For low-frequency small-signal excitation, the matrix equation
(2.6) becomes the following:


50
Table 3.1 ESCCD MEASUREMENTS FOR VARIOUS SOLAR CELLS.
NAME RESISTIVITY
(ohm-cm)
THICKNESS tj(ys)
(ym)
t(s)
S(cm/s)
L(ym)
SPIRE
.31
374
6.5
15
1300
185
ASEC1
.15
301
3.6
7
ohmic
93
ASEC2
.15
267
4.0
13
ohmic
126
BSF#1
10
240
5.3
35+
100-400*
350+
BSF#2
10
260
6.3
45+
100-350*
400+
BSF#3
10
284
7.85
75+
100-225*
512+
BSF#4
10
96
.98
20+
100-380*
265+
BSF#5
10
91
.9
25+
100-290*
ro
KO
v
BSF#6
10
107
.7
4500*
BSF#7
10
102
.73
3200*
LEU#1++
8
328
28.0
145+
40-80*
417+
LEU#2++
1.5
325
25.7
105+
40-150*
347+
P0LY1
2
203
2.7
2000*
POLY2++
2
208
8.8
25+
100-400*
168+
++ denotes p+/n/n+ BSF solar cell. + denotes Tmin or Mnin*
* denotes S^*. P0LY1-2 have poly-Si layers at the back surfaces.


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
S-parameter measurements, and to Dr. Taher Daud of the Jet Propulsion
Laboratory and to Dr. Mark Spitzer of SPIRE Corp. and Mr. Peter lies of
Applied Solar Energy Corp. for discussions and for devices used in the
experiments. Thanks are extended to my colleagues and friends,
Dr. Hyung-Kyu Lim, Mr. Jong-Sik Park, Mr. J. J. Liou, Mr. M. K. Chen,
Dr. Soo-Young 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 parents-in-law for their help and
encouragement.
The financial support of the Jet Propulsion Laboratory is
gratefully acknowledged.
ii


88
x(delay)
(4.43)
Since the expression of delay time in (4.42) includes the uniform-doping
case and uniform-electric-field case, (4.42) holds more generally for
the delay time of minority-carrier across the quasi-neutral base.
4.3.6 Measurement of Minority-Carrier 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 urn 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
1^! of transistors at Iq = 10 uA, and 100 uA. 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
quasi-neutral 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.


SURFACE RECOMBINATION VELOCITY (cm/s)
52
Fig.3.8 Illustration of the determination procedure of S and x using
one ohmic contact solar cell and one BSF solar cell from the
same material. Locus 1: ohmic contact solar cell(C-3-5).
Locus 2: BSF solar cell(2-53).


22
much faster circuit. Here xd = -l/s1 is the first-mode decay time,
influenced by both volume and surface recombination in the base. But
the circuit used suffices because Xp >> 1 gs for the solar cells
studied. Figure 2.3(b) shows the current during the transient.
Fig. 2.3(c) is its semi-logarithmic counterpart, illustrating the
straight-line portion of the transient obtained from the output of the
logarithmic amplifier in Fig. 2.2. From this xd 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
I-V characteristic is V = .03851n(I/IQ+1). If the first-mode current is
1 fjrst-mode^) constant exp(-t/Td) xd -1/s^ (2.15)
then the slope of the output voltage of log amplifier is -38.5 mV/xd.
Extrapolation of the straight portion in Fig. 2.3(c) yields the initial
value ifirst-mode (+) as the intercept.
We measure the decay time constant and the initial amplitude of the
first model as follows: DEVICE 1, xd = -1/s^ = 29.3 gs, if-¡rst(0+) =
2.73 mA for V(O') = 0.44 V and T = 303.1 K. For DEVICE 2, xd =
24.5 gsec, i^^rs^.(0+) = 4.35 mA at v(0") = 0.5 V and T = 302.9 K. For
DEVICE 3, xd = 28.5 gsec, ifirst(0+) = .696 mA at v(0) = .47 V and
303.5 K. Here v(0") denotes the steady forward voltage applied across
the solar cell before the transient.


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
Si-device 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 quasi-neutral 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 in-process 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 quasi-neutral 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 quasi-neutral base of back-
surface-field (8SF) silicon solar cells. The results of this study are
directly applicable to any bipolar device including diodes, solar cells,
1


I
L
Fig.4.14 Modified hybrid-ir model including minority-carrier delay across the quasi-neutral
base. Here L' = xidelay)^, r^ = R($ + 1), and = C.


59
voltage. Thus G in (4.11) has dimensions of [A/cm"3] and is associated
with volume recombination, whereas C has dimensions of [C/cm-3] and is
associated with minority carrier storage.
For small |s|, a simple network realization of -B12(s) in (4.9) is
-B12(s) = 1/(R + sL) (4.13)
where
R = Ax/AeD (cm"3/A)
(4.14)
and
L = Ax3/6AeD2 (cm"3s/A) (4.15)
We associate R with minority-carrier 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 quasi-neutral base for large-signal transient excitation
of the minority-carrier densities (Fig. 4.3).
Previous uses of the two-port 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.


s seff
$max
Td
x(delay)
Tmin
tSCR
V9
XQNB
Yi
the ith natural-frequency
effective surface recombination velocity
performance parameter of a solar cell derived from the
ESCCD method
decay time constant of the fundamental mode (first
natural-frequency) current
propagation delay time of minority carriers across the
quasi-neutral base
performance parameter of a solar cell derived from the
ESCCD method
discharge time constant associated with C$CR
gradient voltage of a p-n junction
quasi-neutral base width
admittance-like elements for a quasi-neutral base


76
QSA
(a)
+ o-
be
-o-
..... g
^ a
*
J
^ CQSA
> K
* \
Cd 1
kqsa
i c
TIME DELAY
ELEMENT
)
-o +
v1
vbe
-0 +
(b)
(c)


11
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 quasi-neutral base
but also from the junction space-charge region and the quasi-neutral
emitter. In the interpretation of experiments to follow, we shall
account for this multi-regional dependence.
2.4 Open Circuit Voltage Decay (QCVD)
In this widely used method [1], the free carriers in the junction
space-charge region enter to contribute to the transient. But,
consistently with Sec. 2.2, and with most common usage, we concentrate
on the n-type quasi-neutral base.
From the master equation [Eq. (2.6)], the transient solution for
the junction voltage is obtained from open-circuit constraint that
l(0,s)=0:
P(0,s) = p(0,0~)/s
1 Dp[cth( XQNB/l_p*)]/Lp*Seff
eD s
P
ctn(XQNB/Lp*)+Dp7Lp*Seff
(2.7)
Here we have assumed that the quasi-neutral 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):


41
where
vl2(t) = (rMOS + rs +
(3.20)
Since the ratio (rMg$ + rd + rs )/rsh is usually very small for practical
solar cells, we obtain from (3.19)
i(t) = -CscR0(rM0S + rs + rd)[di(t)/dt] + ix(t) (3.21)
Solving (3.21) for i(t) yields
i(t) = E^xpi-t/tjgp) + E2exp(-t/xd)
(3.22)
where
tSCR0 = CSCR0^rM0S + r s + rd^
(3.23)
As can be seen in (3.22), the first term of the right side can be
neglected and xd can be determined if the time constant x$gR is much
smaller than x. For the switching circuit of Fig. 3.4 x^CR 200 ns.
For the solar cells described in this chapter, 0.5 us < xd < 30 us. Thus
the RC time constant of the measurement circuit negligibly influences
the first-term or dominant natural-frequency current decay of the solar
cells.


>-
I
II
cn
z:
LlI
a
Ld
_i
o
X
U)
U)
u
u
X
u
POSITION
Fig.4.11(a)
(NORMALIZED VALUE)
Impulse response of minority carrier profile in position within the quasi-neutral base.


87
4.3.5 Minority-Carrier Delay Time with Built-In Electric Field
In this section, we estimate the delay time of the minority carrier
delay time across the quasi-neutral base region when the electric field
exists due to the non-uniformity of the base doping concentration
profile.
Assume a uniform built-in electric field within the quasi-neutral
base. Then the minority-carrier current at x = XqNB is [22]
I(XQNB,s} = -AeDnpoeeV/kTvK G csch(GXQNB)P(0,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=XgNg and where
G
l9eK 2 1/2
iljr2-) + ^
QN8
(4.41)
Using Ashar's definition [25] of delay time, we have
x(delay) = lim
dsl
(s)
s-0 'A21^s)
'QNB
6D

log K 2 ;
!(5s-) + (-
QNB
) ]}
(4.42)
For the simple case when the base doping concentration is uniform, K=l,
we have the following from (4.42):


I (0,s)
^XQNB8^
+ -
P(0,s)
Fig.2.1
A-j-j (s)
A-J2 (s)
A21(s)
*
04
<
+
^XQNB,S^
Two-port representation for excess hole density and hole current density
at the two edges of the n-type quasi-neutral base region.


66
i(0,s)
A^(S) A^2(s)
ev(Qs) p (Q)
kT *DClu'
i(XqNBs)
A21(s) A£2(s)
)
Hi.?,.., p (Y )
kT rDC'AQNB _
where |v(0,s)| < ^ and Iv(XQNBs) I K e and v the ho^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[l + v/(kT/e)]
exp[vDC/(kT/e)][l + v/(kT/e)]
Rewriting (4.16), we have
i(0,s)
Dll(s) 0^2(s)
v(0,s)
1(XQNBS)
^21(s)
v(XqNB,s)
(4.17)
In (4.17),


54
Alony with the equivalent circuit for transient analysis, we
develop equivalent circuits for low-frequency small-signal excitation.
We correct the quasi-static input capacitance of the hybrid-ir 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 hybrid-ir
model for a junction transistor, which is advanced near the end of the
chapter.
4.2 Equivalent-Circuit for Large-Signal Transients
4.2.1 Derivation by Two-Port Approach
We slice the quasi-neutral base, assumed to be in low injection,
into many subregions. A typical subregion is shown in Fig. 4.1. From
the two-port approach, we have the following linear matrix equation
relating the excitations (minority-carrier densities) and the responses
(minority carrier currents) of the ith subregion:
I -j (s) ij (0)/s
B11 b12
P^s) pi(0")/s
Ii+1(s) ii+1(0")/s
B21 B22
pi+l Pi+in/s
J
(4.1)


29
In this figure we have used the y-parameter 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 two-port network
representation.
Other advantages include
(a) systematic determination of the natural frequencies [14];
(b) systematic conversion to the case of steady-state 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 = XgNB, I(XgNB,s) = -AeSP(XgNB,s), removes I2(s) and
P2(s) in Fig. 3.1 from consideration. Figure 3.2 displays the resulting
two-port network representation of the quasi-neutral base region of a
solar cell. Here


13
13 -3
developed, and for which =10 cm Here n^ is the intrinsic
density and is also the ratio of the pre-exponential factors that govern
contributions from the quasi-neutral regions relative to those from the
junction space-charge region.
Thus we identify the transient decay of mobile electrons and holes
within the p/n junction space-charge region, which persists throughout
the open-circuit 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-
charge-region contribution to the observed transient voltage [13],
quasi-static approximations and a description of the forward-voltage
capacitance of the space-charge 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.


100


36
(3.7), since R is a function of both t and S. Also each point (t,S)
produces its own value for Iqnbq 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 Iqneq* Specifically
Td = fi(Ts) (2.13)
ipo = f2^r,S,IQNE0^
and
RM = f3^T,S,IQNE0^
Using (2.13), (3.10) and (3.14) and the measured variables, one can
solve for t, S and Iq^eq in a manner to be described later.
3.3 Experiments
3.3.1 Improvements in the Circuit for ESCCD
Previously we used a bipolar-transistor switching circuit in
Fig. 2.2 to measure the decay time constant and the initial amplitude of
the first natural-frequency current at t=0+. 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
(3.10)
(3.14)


15
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
steady-state condition and then suddenly applies zero bias through a
small resistance. This causes the mobile charges stored within the
junction space-charge and quasi-neutral 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 quasi-neutral emitter and the junction space-charge
region are much smaller than from the quasi-neutral base, one can
separate the first mode of the quasi-neutral-base current and determine
Seff and tp.
We first consider the time of response of the junction space-charge
region. Upon the removal of the forward voltage, the constraint at the
terminals becomes essentially that of a short circuit. The majority-
carrier quasi-Fermi 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
quasi-neutral regions, times that are of the order of no greater than
lCT^s. 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 quasi-neutral base. The resulting Coulomb forces cause
majority carriers to rush from the edges of the junction barrier
reyions, thus causing the nearly sudden rise of the barrier height to


49
the extreme case of negligible surface recombination at the back contact
during the ESCCD transient. For the upper-locus cell in Fig. 3.7,
Tm1-n 0. These two parameters, Smax and xmin, can be used as
performance parameters for thin solar cells; small Smax and large
is desirable for thin BSF solar cells for a given base thickness and
doping concentration.
We also measured the values of R^: = 20 for the lower locus and
Rm 3 for the upper locus. But we cannot use the measured RM directly
to determine x, because R does not change much as x increases as
illustrated by marks on the loci Fig. 3.7. Instead, the measured R^ can
be used as another performance parameter for thin BSF solar cells, since
large RM means small Iqnjtq and small S for a thin solar cell. These
conditions imply a large open-circuit voltage for a given base thickness
and doping. Small Rf/| usually implies either a poor BSF contract at the
back surface or a large Iqneq. 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 poly-Si layers on the back surface of the base. The value
of S is estimated to be about 2000 cm/s for n+/p/p+-poly-Si cell
(poly 1) and about 400 cm/s for p+/n/n+-poly-Si cell (poly 2). Thin
cells are characterized in terms of Tjn^n and Smax.


SURFACE RECOMBINATION VELOCITY (cm/s)
45
Fig.3.6 S(t) locus for a BSF solar cell. The locus is generated from
xd = 6.5 us.


34
1 *D + !sh
(3.8)
= IFQ (exp(eV/kT)-l) + V/r$h (3.9)
Here Ipg is the pre-exponential factor of the forward bias current and V
is the voltage across the space-charge region. The pre-exponential
factor Ipg in (3.9) has two components:
!F0 = !QNB0 + *QNE0 (3*10)
where IqnbO is the <1uasT-neutral-base current component and Iqneq is the
quasi-neutral-emitter current component.
The voltage across the two terminals of the solar cell vout
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 = Ipo exp(eV/kT) (3.12)
Combining (3.11) and (3.12), we obtain an expression for VQut in terms
of I, rs and Ipg:


17
Having established that the mobile carriers in the junction space-
charge region enter the electrical-short-circuit-decay 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
1(0,s) = i(0,0")/s -
eDpp(0,0")
V
coth(XQNB/Lp*) Dp/Lp*Seff
1 + (Dp/Lp*Seff>cot"(X/Lp*)
(2.10)
Cauchy's residue theorem yields the inverse transform of (2.10):
i(t)
; eytO.Q-X) cot(KiXQWB/Lp) WLpSeff
1=1 1 p ^ Tpy/si Ki ^ + ^XQNB^2Seff )cosec ^KiXQNB'/Lp^
(2.11)
where s.¡ is the ith singularity which satisfies the Eigenvalue equation,
1 +
LpSeff
/T+sTt^ coth(
QNB
= 0
(2.12)
and where = (-1-s.jTp)1^ > 0, with s^ < 0.
Truncating (2.11) and (2.12) to include only the first mode (s^),
we obtain
1/2
1 + (Dp/LpSeff} + slTp cothC(XQNB/Lp)(1 + slTp) ] = 0 <2- 13>


PHRSE OF hQ. (degree )
FREQUENCY(Mhz )
Measurement of phase of current gain of transistor 127.
Fig.4.17(a)


CHAPTER FOUR
EQUIVALENT-CIRCUIT REPRESENTATION OF THE
QUASI-NEUTRAL BASE,
WITH APPLICATIONS TO DIODES AND TRANSISTORS
4.1 Introduction
In the previous chapters, we treated the quasi-neutral base region
using the two-port 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
large-signal 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 base-region behavior. This avoids
difficulties associated with the infinite number of natural frequencies
characterizing a distributed system. It makes possible use of a
circuit-analysis 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.
53


Fig.4.2 Two-port network description for the ith subregion.


i i (t)
lqnbo
LqNEO
^sh
I(X,S)
Ki
L
LP
L*
lp*
NDD
pc(0)
P(x.-)
P(x,s)
p(x,t)
P-i (s)
Pi(0
R
Ri
I
R
M
MOS
sh
incoming minority carrier current toward a subregion at
x = xi
pre-exponential factor of steady-state quasi-neutral-
base current
pre-exponential factor of steady-state quasi-neutral-
emitter current
current through the shunt resistor of a diode
Laplace transform of i(x,t)
(-l-si t)1/2
diffusion length of minority carriers
diffusion length of minority holes
L/(l + st)-*-/2
Lp/(1 + sxp)1/2
intrinsic carrier density
base doping density
steady-state hole density at x = 0
p(x,t) at t = 0"
Laplace transform of p(x,t)
excess hole density
Laplace transform of p^(t)
excess minority carrier density at x = Xj
^MO^QNBO
^NEO^QNBO
^MO^FO
turn-on resistance of MOS transistor
series resistance of a diode
shunt resistance of a diode
VI


102
region was in quasi-equilibrium. For t of the order of t, the drift
tendency now overwhelms the diffusion tendency, and holes and electrons
drift out of the space charge region in a transit time t' determined by
(xN Xp)/velocity where the velocity approaches the scatter limited
velocity because of the high field (Fig. B.l(e)). For typical devices,
t1 will be of the order of 10"^ s. After this time has passed, the
hole and electron concentrations will have returned nearly to their
equilibrium values.
Because 10-1* s is a time not observed by typical measurement
equipment, we think of the initial condition established by shorting the
terminals suddenly as that of quasi-neutral regions still storing
approximately the same excess charge as was present in the steady state
of negative time, of a space-charge region at the equilibrium barrier
height and devoid of excess holes and electrons, and as an excess
minority-carrier density in the quasi-neutral regions that drops sharply
to zero at the space-charge region edges (Fig. B.l(f)).


5
2.2 Mathematical Framework
In this section, we develop a mathematical framework which could be
applicable to most of the large-signal transient measurement methods and
could include small-signal 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
minority-carrier density and the minority-carrier current in a quasi
neutral base region in low injection. Focusing on the quasi-neutral
base, assumed to be n-type here (of x-independent donor density Nqq)
with no loss in generality, will simplify the treatment; extensions to
the quasi-neutral emitter are straightforward, provided one inserts the
physics relevant to n+ or p+ regions.
Assume a p+/n diode in which the uniformly doped quasi-neutral base
starts at x = 0 and has a general contact defined by arbitrary effective
surface recombination velocity Seff at the far edge x = Xq^. Such a
contact could result, for example, from a back-surface-field (BSF)
region. Assume also low-level injection and a uniform doping of the
base region. Then a linear continuity (partial differential) equation
describes the excess minority holes p(x,t)
3p(x,t)/3t = Dp32p(x,t)/3x2 p(x,t)/r (2.1)
where Dp is the diffusion coefficient and tp 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:


Fig.4.12 The conventional hybrid-ir model for bipolar transistors.


PHRSE OF h-. (degree )
FREQUENCY(Mhz)
Fig.4.17(b) Measurement of phase of current gain of transistor 132.


Fig.3.
A simple diode model with series resistance and
shunt resistance in the dark condition.


108
11. A. Neugroschel, P.J. Chen, S. C. Pao, and F. A. Lindholm,
"Diffusion length and lifetime determination in p-n junction solar
cells and diodes by forward biased capacitance measurements," IEEE
Trans. Electron Devices, vol. ED-25, pp. 485-490, 1978.
12. A. Neugroschel, "Determination of lifetime and recombination
currents in p-n junction solar cells, diodes, and transistors,"
IEEE Trans. Electron Devices, vol. ED-28, pp. 108-115, 1981.
13. J. E. Mahan and D. L. Barns, "Depletion layer effects in the open
circuit voltage decay lifetime measurements," Solid-State
Electronics, vol. 24, pp. 989-994, 1981.
14. J. G. Linvill and J. F. Gibbons, Transistors and Active Circuits,
New York: McGraw-Hill Book Company, Inc., 1961.
15. W. C. Elmore, "Tne transient response of dampted linear networks
with particular regard to wideband amplifiers," J. Appl. Phys.,
vol. 19, pp. 55-63, 1948.
16. R. W. Dutton and R. J. Whittier, "Forward current-voltage and
switching characteristics of p+-n-n+ (epitaxial) diodes, IEEE
Trans. Electron Devices, vol. ED-16, pp. 458-467, 1969.
17. S. M. Sze, Physics of Semiconductor Devices, New York: John Wiley
and Sons, Inc., 1981.
18. B. R. Chawla and H. K. Gummel, "Transition region capacitance of
diffused p-n junctions," IEEE Trans. Electron Devices, vol. ED-18,
pp. 178-195, 1971.
19. T. H. Ning and D. D. Tang, "Method for determining the emitter and
base series resistance of bipolar transistors," IEEE Trans.
Electron Devices, vol. ED-31, pp. 409-412, 1984.
20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions, New York: Dover Publications, Inc., 1970.
21. D. J. Hamilton, F. A. Lindholm, and J. A. Narud, "Comparison of
large signal models for junction transistors," Proc. IEEE,
vol. 52, pp. 239-248, 1964.
22. D. J. Hamilton, F. A. Lindholm, and A. H. Marshak, Principles and
Applications of Semiconductor Device Modeling, New York: Holt,
Rinehart and Winston, Inc., 1971.
23. J. L. Moll, "Large-signal transient response of junction
transistors," Proc. IRE, vol. 42, pp. 1773-1784, 1954.


32
"i* s1nh(XQNB/L) + ocosh(XqN6/L)
where QNBO LNdd cosn(XqNB/L) + <6inh(XqNB/L) (3'6)
is
and where a = g-. Here a is the ratio of the normalized surface
recombination velocity to the diffusion velocity [16], Thus the ratio
R = IpM0
!QNB0
2K^ cotA^ + tanA^
Si iXQNB/Al)2/D + (XQNB/S) CSc2a1
cosh(Xq^g/L) + otsi nh( Xq^g/L)
sinh(XQNB/D + ctcosh(XQNB/L) (3-7)
where A^ = K^Xg^g/L and where and Aj are obtained by solving (A.3) of
Appendix A. The ratio R will be utilized for the determination of the
quasi-neutral 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 space-charge recombination current component is
negligible [17], the I(V) characteristic of the solar cell is


46
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 in terms of the following two time
constants by solving (A.3) of Appendix A:
Td_1 = "S1 = Ts"1 + T_1 (3.29)
where t$ = Ug^/Aj )^/D. Equal rates occur if
xs = t (3.30)
Here Aj is obtained from (A.3) of Appendix A:
Aj = tt/2 for S(back) = 0 (3.31)
and
A^ = rr for S(back) = (3.32)
Here in (3.29), the parameter x^/ig is the probability that a minority
carrier vanishes through the surfaces bounding the quasi-neutral region,
whereas the parameter t^/t is the probability that a minority carrier
vanishes by volume recombination.


Third, in this chapter we consider the sensitivity problem involved
in the method of electrical short-circuit 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, xmin and RM, the importance of which is
discussed in this chapter.
Fourth, we show quantitatively that the electrical short-circuit
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 IFq in the ratio above brings the
emitter recombination current Iq^ 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 Iqn£5 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 two-port network formulation introduced in
Chapter Two in (2.6), the representation for which is illustrated in
Fig. 3.1, where Yx(s) = An(s) + A12(s), Y2(s) = -A12(s), P^s) =
p(0,0')/s, P2(s) = p(XqNB,0")/s, I^S) = i(0,0")/s, and I2(s) =
1(XQNB,0~)/S*


3
transistors, for which an improved hyprid-ir 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.


Fig.3.1 Two-port network representation of a quasi-neutral region with boundary conditions
at x = Xqnb and x = 0. The nodal variable is the excess minority carrier concentration.


DC
Power
Supply
GO
00
Fig.3.4 Switching circuit used in measuring short-circuit-current decay parameters, t and S.


83
Fig.4.13(a) Quasi-static response of minority-carrier profile
for a sudden change in the emitter-base voltage.


109
24. P. E. Gray, D. Oe Witt, A. R. Boothroyd, and G. F. Gibbons,
Electronics and Circuit Models of Transistors, S.E.E.C. Notes,
vol, 2, New York: John Wiley & Sons, Inc., 1965.
25. K. G. Ashar, "The method of estimating delay in switching circuits
and the figure of merit of a switching transistor," IEEE Trans.
Electron Devices, vol. ED-11, pp. 497-56, 1964.
26. W. T. Thomson, Laplace Transformation, Englewood Cliffs, NJ:
Prentice-Hall, Inc., 1960.
27. D. K. Lynn, C. S. Meyer, and D. J. Hamilton, Analysis and Design
of Integrated Circuits, New York: McGraw-Hill Book Company, 1968.


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF SYMBOLS v
ABSTRACT viii
CHAPTER
ONE INTRODUCTION 1
TWO UNIFYING VIEW OF TRANSIENT RESPONSES FOR DETERMINING LIFETIME
AND SURFACE RECOMBINATION VELOCITY IN SILICON DIODES AND BACK-
SURFACE-FIELD SOLAR CELLS, WITH APPLICATION TO EXPERIMENTAL
SHORT-CIRCUIT-CURRENT DECAY 4
2.1 Introduction 4
2.2 Mathematical Framework 5
2.3 Transient versus Steady-State Analysis via Two-Port
Techniques 9
2.4 Open-Circuit-Voltage Decay (OCVD) 11
2.5 Reverse Step Recovery (RSR) 14
2.6 Electrical Short-Circuit-Current 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 SHORT-CIRCUIT-CURRENT
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 Short-Circuit-
Current Decay and Dark I-V Characteristics 35


9
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 H^q^g>s) = "e^eff^^QNBs^ From a c''rcu1t
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 electrical-short-circuit-current-decay (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. Steady-State Analysis via Two-Port 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 quasi-neutral regions under study using the two-port
technique described in Sec. 2.2, the displacement current is
negligible. In the steady state, the two-port description leading to
the master equation simplifies because the hole current in our example
of Sec. 2.2 depends only on position x. Tnis x-dependence results from
volume recombination (relating to the minority-carrier lifetime) and
effective surface recombination (relating to the effective surface
recombination velocity). A two-port formulation for the steady state


20
the voltage across the solar cell becomes about 0.6 V, which one may
control by altering (Vh-¡gh), and the variable resistor connected to the
transistor base, or both. In this mode, the quasi-neutral base charges
to store ultimately a steady-state charge of excess holes and electrons,
and p(0,0") of Eqs. (2.10), (2.12) and (2.13) is established.
Now assume that V^(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
short-circuit constraint. The large capacitor maintains this constraint
nearly perfectly during the first-mode transient of the solar cell; that
is, during this transient, this capacitor and the input voltage source,
which has a small resistance of 50 ft (in parallel with 10 ft), act as
nearly incremental short circuits. Thus the desired short-circuit
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 1--Nqq (substrate doping) = 6 x 10^ atoms/cm^; Xg^g (base
thickness) = 348 ym, A = 4 cm^; DEVICE 2--Nqq = 7 x 10^ atoms/cm^,
XqNB = 320 gm, A = .86 cm^, DEVICE 3--NqD = 3.5 x 1015 atoms/cm^,
Xqnb = 348 ym, A = 4 cm^.
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 ys. The speed is circuit limited. One could design a


104
/ 1im[te(t)est]dt
o s+O
/ 1im[e(t)est]dt
o s-0
00
/ te(t)dt
o
00
/ e(t)dt
(C.4)
(4.35)
Thus, we derived Elmore's definition of delay time from Ashar's by using
the relation between system function and impulse response.


78
oo
00
x(delay) = / te(t)dt/ / e(t)dt
(4.35)
o
o
where e(t) is the impulse response of the current at x = Xg^g to v^(t) =
A5(t). In the complex frequency domain, this becomes [26]
r(delay) = lim{-[dF(s)/ds]/F(s)}
s-K)
(4.36)
F(s) is the system function
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 minority-carrier delay time of
the thin quasi-neutral base region with an ohmic contact at x = Xg^g, we
apply impulse excitation at x = 0 and solve for the impulse response
using the two-port approach. The quasi-neutral base region used in this
calculation is thin compared with the diffusion length and we assign the
values for D and Xg^g arbitrarily. Figure 4.11(a) displays an example
of the impulse response of the minority-carrier density profile in
position as time passes. Figure 4.11(b) displays the current impulse
response at x = Xg^g. We derive the delay time x(delay) from this
simulation by using (4.35) and numerical integration:
x(delay) = XQNB2/6D
(4.37)


7
Regarding the minority carrier densities at the two edges as the
excitation terms for a system analogous to a linear two-port network of
circuit theory, we have the following two-port network matrix from (2,4)
and (2.5) for the two excitations (densities) and the two responses
(currents):
1(0,s) i(0,0)/s
A12
~P(0,s) p(0,0')/s
1(XQNB,S) i^XQNB )/S
_A21
A22_
P(XqNBs) -P(xqnB0 )/s
(2.6)
where p(0,0") and P(XQNg0) 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 quasi-neutral
base. In Eq. (2.6), A12=A21= -e(Dp/Lp*) cosech(XgNB/Lp*) and An= A22
e(Dp/L*)coth(XQNt}/L*). Figure 2.1 displays the master equation, where
the initial values are included in 1(0,s), I(XqNB,s),P(0,S), and
p(xQNBs)
Transient solutions can be derived from (2.6) by inserting proper
boundary conditions, initial values and constraints imposed by the
external circuit. For example, l(0,s)=0 in OCVD open-circuit-voltage-
decay [1], l(0,s)=constant for reverse step recovery [2,3], and P(0,s) =
0 for short-circuit current decay, the latter of which is developed in
detail here. For small-signal methods [11-12], where dl, for example, is
an incremental change of current, I(0,s)=Iqq/s + dl(0,s) and
P(0,s)=Pqq/s + (edV/kT)P0C. 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.


62
Fig.4.4
SPICE2 simulation of
as a function of the
parametrically, used
quasi-neutral base.
electrical-short-circuit-current decay
number of the subregions, indicated
in the equivalent circuit for the


97
Fig.A.l The eigenvalues for OCVD where Y- = cot(A^),
A1 XqnbVV and V1 = DpAi/SXQNB'
Fig.A.2 The eigenvalues for ESCCD where Y. = tan(A.),
Ai = XQNBK1/Lp- and Yi -Vf/SXqNB-


NORMALIZED CURRENT
65
Fig.4.6 The role of inductors in the response of a two-subregion
equivalent circuit is illustrated in (a) and (b), (a) without
inductors, (b) with inductors. Response (c) corresponds to a
fifteen-subregion model, with or without inductors.


the ith subregion
777
7
Ax

ONE
SCR
Q
N
'll
i
i
Xi 1 XQNB
Fig.4.1 One-dimensional schematic diagram of a BSF solar cell with a quasi-neutral
base sliced into subregions.


70
established this convention, we then identify the incremental voltages
v(0,s) and v(XqNB,s) with the incremental quasi-Fermi 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 quasi-neutral region with a general contact at x =
Xqnb> we can derive the low-frequency small-signal equivalent-circuit
directly by truncating the DC components of minority-carrier density and
current from Fig. 3.2 instead of starting from the matrix equation.
Figure 4.8 displays this equivalent circuit of a quasi-neutral region
with a general contact for low-frequency small-signal excitation. The
values of the circuit elements are G = eAK^XqNg/2r, C = eAK^XqNB/2, R =
XqNB/eAKiD, and L = Xq^g^/D^eAK^ The element G$ represents the
general contact at the back surface, the value of which follows from
Fig. 3.2:
G$ = [eAKxS] (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 Short-Base Case
The foregoing derivation in the frequency domain yields an
equivalent circuit involving an inductor L, which does not commonly
appear in small-signal or incremental models for a quasi-neutral base


94
We suggest the following as useful future efforts based on the
methods and approaches of the present study:
1) Application of ESCCD to determine lifetime t and surface
recombination velocity S to various devices used in power
electronics where open-circuit-voltage decay and reverse step
recovery are routinely employed.
2) Use of ESCCO for determining S and t in solar cells by a
variant of the method in which light or other radiation
replaces the electrical excitation employed in the present
version of ESCCD.
3) Improve the circuit for ESCCD. No claim is made that the
improved configuration of ESCCD is optimal.
4) Explore extending ESCCD to the determination of t and S for
quasi-neutral regions in moderate or high injection.
5) Apply the modeling approach of Chapter Four to the heavily
doped emitter of silicon bipolar transistors and to
heterojunction bipolar transistors, where energy-gap and other
parameter changes with position complicates the physics.
6) Assess more fully and improve the large-signal and small-signal
equivalent circuits for diodes, solar cells and transistors
developed in Chapter Four.


6
-p(x,(f) + sP(x,s) = Dp d2P(x,s)/dx2 P(x,s)/t
(2.2)
where
P(x,s) = / e'stp(x,t)dt s = o+ jw, j = (-1)1/2 (2.3)
t=0
Solving Eq. (2.2) yields
P(x,s) = p(x,0")/s + M1 exp(-x/Lp ) + M2 exp(x/Lp )
(2.4)
1/2
where Lp = t(0pTp)/(l + sip)] and where and M2, given below, are
to be determined by the boundary values at the two edges of the quasi -
neutral base region: P(0,s) at x=0, and P(XgNB,s) at x=XgNB.
Substitution of (2.4) into (2.2) yields the steady-state continuity
equation for p(x,0"), verifying that (2.4) is the solution of (2.2).
Because of quasi-neutrality 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 = XgNB:
I" KO,s) i(0,0")/s
-(XqnBS) i^XQNB,
= A
L *
P
1
-X /i *
-e AQNB/Lp
X /I i
e QNB/Lp
V
.V
(2.5)
where 1(0,0) and I(Xg^g,0~) are the initial values of the minority hole
diffusion current at x=0 and *=XqNB. In (2.5), hole current entering
the quasi-neutral base is positive, by definition.


NORMALIZED CURRENT
63
SPICE2 simulation of electrical-short-circuit-current decay
responses displayed semilogarithmically for the quasi-neutral
base sliced into (a) one, (b) two, and (c) fifteen equally
thick subregions.
Fig.4.5


31
YS -I^XQNBS^P^XQN8*S^ = AeS t3*1)
Solving the network of Fig. 3.2 for 1(0,s) under the low-injection
condition yields
1(0,s) = i(0,0)/s Yx p(0,0")/s
(Y1 + V Y2
Y-Vy % Y (p(Q>0 )/s) (3.2)
ni2
where p(0,0) = jf (exp(eV(0 )/kT) -1) (3.3)
nDD
If we use the Cauchy Residue Theorem to obtain the inverse
transform of 1(0,s), we yet an infinite series for i(0,t). Truncating
this series after the first term, at t = 0+, yields
1 first mode(+) = ^MO (exp(eV(0)/kT)-l)
(2.14)
AeDK.n.2
where Ipfvjo = 12
s LN
51LIN00
COt(KlXQNB/L) DK1/LS
(t/2K12) + (XQNB/2S)csc(K1XQNB/L)
(3.4)
The minority-carrier current at x = 0 for t < 0 is
1(0, O') = IQNB0 (exp(qV(0")/kT 1)
(3.5)


101
transient for t >0 flows in a direction opposite to that occuring for
negative time because the transient results in removing the electrons
and holes present under forward V.
Having established the existence of this negative charge, we now
consider what happens subsequently. Here enters a result developed
earlier from operating on Maxwell's equation by the divergence operator
0 = div curl H = div(jN) + div(jp) + div[9(eE)/9t] (B. 1)
From this result, two consequences emerge: 1) the charges Fig. B.l(b)
redistribute to the positions shown in Fig. B.l(c) within the order of a
dielectric relaxation time x = e/o; and 2) the total current is
solenoidal, that is, its divergence is zero, where here the total
current includes the displacement current.
The consequence of (2) is illustrated in Fig. B.l(d) for a
particular time of order of x. Notice the large time-rate of change of
electric field E within the junction space-charge region, xp < x < xN.
Here we have employed a one-dimensional model so that the operator div
becomes the operator 9/9x. Thus we see that the electric field in the
space-charge region grows rapidly so that within t of the order of x the
barrier height has returned to its near-equilibrium value and the
electric field is several times larger than it was in negative time. In
negative time, however, the drift and diffusion tendencies of the
junction space-charge region were perturbed only by perhaps one part in
10*^ in the forward voltage steady state; that is, the space-charge


CHAPTER FIVE
SUMMARY AND RECOMMENDATIONS
The major accomplishments of this study are
1) Development and illustration of the ESCCD measurement method.
The accuracy of ESCCD is not affected by the existence of
mobile electrons and holes in the p/n junction space-charge
region in contrast to the accuracy of the other methods
commonly used for silicon bipolar devices.
2) Derivation of a unifying view of excess minority-carrier
behavior within quasi-neutral regions in regard to transients
and their use in determining recombination lifetime and surface
recombination velocity.
3) Improvement of the speed of ESCCD measurement system by using
MOSFET transistors as switching devices rather than the
bipolar-circuit realization first used.
4) Derivation of a compact equivalent-circuit representation of
quasi-neutral regions which includes the minority-carrier
propagation delay.
5) Demonstrations that application of existing circuit analysis
programs to this equivalent circuit enables detailed and
accurate device analysis for both large- and small-signal
excitation.
6) Use of this equivalent-circuit to improve existing small-signal
models for bipolar transistors, with experimental
illustrations.
93


UNIVERSITY OF FLORIDA
3 1262 08554 1505


X
x XQNB
CO
o
Fig.3.2 Equivalent two-port network representation of a quasi-neutral base of a solar cell
with a general contact at x = Xg^g under electrical-short-circuit-current-decay
condition. The excess minority carrier concentration at x = 0 vanishes in this case.


69
- *
(4.25)
= xQNB/eAK2D
(4.26)
(4.27)
= eAK^Xq^B/2T
(4.28)
= eAK^Qi^/2 ,
(4.29)
G(s) = -eAO^ 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 large-signals, for which the
nodal variable is minority-carrier density (see Fig. 4.3, for example).
For small-signal 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 quasi-Fermi 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


58
Y^(s) = B^s) + B^2 (s) = AeAx/sx + sAeAx/2
(4.7)
and
Y2(s) = -B12(s) = AeD/Ax sAeAx/6 (4.8)
= (AeD/Ax)(1 sAx2/6D) (AeD/Ax)(l + sAx2/6D)-1 (4.9)
for )s| 6D/Ax2.
From (4.7), B^(s) + B^(s) in Fig. 4.2 is realized in terms of
admittance-like elements [9,21]:
Bll(s) + B12(s) = G + sC (4.10)
where
G = AeAx/2x (4.11)
and
C = AeAx/2 (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


BIOGRAPHICAL SKETCH
Tae-Won Jung was born in Seoul, Korea, on May 21, 1953. He
received the 8.S. degree in electronic engineering from the Seoul
National University in 1976 and the M.S. degree in electrical
engineering from the Korea Advanced Institute of Science, Seoul, Korea,
in 1978. Since August 1981, he has been working for the Ph.D. degree at
the University of Florida, Gainesville.
In July 1985, he joined the Linear Product Design group at Harris
Semiconductor, Melbourne, Florida, where he has been designing analog
devices such as operational amplifiers and analog switches. From 1978
to 1981, he was with Kyung-Pook National University where he was engaged
in teaching semiconductor device physics.
His area of interest includes bipolar and MOSFET transistor
modeling and integrated circuit design.
He is a member of the honor society of Phi Kappa Phi.
110


This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy,
May 1986
Dean, Graduate School


REFERENCES
1. S. R. Lederhandler and J. J. Giacoletto, "Measurements of minority
carrier lifetime and surface effects in junction devices," Proc.
IRE, vol. 43, pp. 447-483, 1955.
2. E. M. Pell, "Recombination rate in germanium by observation of
pulsed reverse characteristic," Phys. Rev., vol. 90, pp. 278-279,
1953.
3. R. H. Kingston, "Switching time in junction diodes and
transistors," Proc. IRE, vol. 42, pp. 829-834, 1954.
4. J. E. Mahan, T. W. Ekstedt, R. I. Frank, and R. Kaplow,
"Measurement of minority carrier lifetime in solar cells from
photo-induced open-circuit voltage decay," IEEE Trans. Electron
Devices, vol. ED-26, pp. 733-739, 1979.
5. S. R. Dhariwal and M. K. Vasu, "A generalized approach to lifetime
measurement in pn junction solar cells," Solid-State Electron.,
vol. 24, pp. 915-927, 1981.
6. R. H. Dean and C. J. Nuese, "A refined step-recovery technique for
measuring minority carrier lifetimes and related parameters in
asymmetric p-n junction diodes," IEEE Trans. Electron Devices,
vol. ED-18, pp. 151-158, 1971.
7. H. Y. Tada, "Theoretical analysis of transient solar-cell response
and minority carrier lifetime," J. Appl. Phys., vol. 37, pp. 4595-
4596, 1966.
8. B. H. Rose, "Minority-carrier lifetime measurements on silicon
solar cells using I. and VQC transient decay," IEEE Trans.
Electron Devices, vol. ED-31, pp. 559-565, 1984.
9. F. A. Lindholm and D. J. Hamilton, "A systematic modeling thoery
for solid-state devices," Solid-State Electron., vol. 7, pp. 771-
783, 1964.
10. C. T. Sah, "Tne equivalent circuit model in solid-state
electronics, III: Conduction and displacement currents," Solid-
State Electron., vol. 13, pp. 1547-1575, 1970.
107


Fig.4.3
RLC equivalent-circuit representation for the i
.th
subregion.


61
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 quasi-neutral
region. The thickness of the quasi-neutral region is 96 ym, the
lifetime of the minority carriers is 100 us, the diffusivity is
35 cm^/s, and the surface recombination velocity of the back low-high
junction is 200 cm/s. The results of the 2- and 3- and 15-subregion
equivalent circuits are shown in the linear-linear graph of Fig. 4.4.
The short-circuit current of the 3-subregion equivalent circuit nearly
coincides with that of the 15-subregion equivalent circuit for the time
range where the first-mode current dominates the short-circuit current.
Recall that the first-mode component contains the information about
the parameters t and S. Our earlier work in Chapters Two and Three,
focussed on determining these parameters.
Figure 4.5 displays the short-circuit 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 ys. Thus from Figs. 4.4 and
4.5, a 3-subregion equivalent-circuit suffices for the determination
of t and S for this solar cell. The decay time constant and the
current ratio R are determined from Fig. 4.4 and Fig. 4.5:
xd = 0.98 ys and R = 31
These values coincide with the exact solutions obtained by solving (3.7)
and (A.3) of Appendix A.


+ o
v(0,
o
Fig.4.8
trW'
)
c
x = O
Equivalent circuit of Fig.4.7 augmented by a circuit element that accounts for an arbitrary
value of minority-carrier recombination velocity at the back surface.


24
which accounts for vanishing of minority holes both by volume
recombination within the quasi-neutral base and effectively by surface
recombination, greatly exceeded the time required for the excess hole
density at the base edge of the space-charge 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 electrical-short-circuit-current decay (ESCCD)
method. In general, the current response derives from vanishing of
minority carriers not only in the quasi-neutral base but also in the
quasi-neutral 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 low-injection
conditions for which the response was measured. But for other solar
cells or for higher levels of excitation, the recombination current of
the quasi-neutral 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 in-process control in solar-cell
manufacturing.
This chapter began with a mathematical formulation of the relevant
boundary-value problem that led to a description similar to that of two-


3.3 Experiments 36
3.3.1 Improvements in the Circuit for Short-Circuit-
Current Decay 36
3.3.2 Quality of the Short Circuit of the Switching
Ci rcuit 37
3.3.3 Measurement of the Dark I(V) Characteristics 42
3.4 Experimental Results and Discussions 43
FOUR EQUIVALENT-CIRCUIT REPRESENTATION OF THE QUASI-NEUTRAL BASE,
WITH APPLICATION TO DIODES AND BIPOLAR TRANSISTORS 53
4.1 Introduction 53
4.2 Equivalent-Circuit for Large-Signal Transient 54
4.2.1 Derivation by Two-Port Approach 54
4.2.2 SPICE2 Simulation of the Equivalent Circuit for
ESCCD 61
4.3 Equivalent Circuits for Low-Frequency Small-Signal
Analysis 64
4.3.1 Derivation in Frequency Domain 64
4.3.2 Derivation in the Time Domain for Short-Base
Case 70
4.3.3 Calculation of the Delay Time 77
4.3.4 Modification of the Conventional Hybrid-it Model
by Including the Minority-Carrier Current
Propagation Delay 81
4.3.5 Minority-Carrier Delay Time with Built-In
Electric Field 87
4.3.6 Measurement of Minority-Carrier Delay Time Across
the Quasi-Neutral 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 SHORT-CIRCUIT-CURRENT 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