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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ix, 110 leaves : ill. ; 28 cm.
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Jung, Tae-Won, 1953-
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Semiconductors   ( lcsh )
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Electrical Engineering thesis Ph. D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 107-109.
Statement of Responsibility:
by Tae-Won Jung.
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Typescript.
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Vita.

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Full Text









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























CSCR(V)


CSCRO

D

Dij


Dp
Ax

e

i(x,O-)

ID

IDC(O)

i first-mode(t)


IFMO

IFO

Ii(s)


LIST OF SYMBOLS
area of a device

XQNBKi/L
characteristic matrix elements of a quasi-neutral region
for large-signal transient normalized surface
recombination velocity

characteristic matrix elements of a subregion for large-
signal transient

space-charge region capacitance of a p-n junction
forward-biased with voltage V

space-charge region capacitance of a p-n junction at
V=O

diffusion coefficient of minority carriers

characteristic matrix elements of a quasi-neutral region
for small-signal low-frequency analysis

diffusion coefficient of minority holes

thickness of a subregion

magnitude of the electron charge

minority carrier current at t = 0-

dark current of an ideal diode

dc steady-state current at x = 0

the first-natural-frequency (first transient mode)
current at x = 0

pre-exponential factor of the first natural-frequency
current at t = 0"

pre-exponential factor of a steady-state current of a
diode with negligible space-charge current

Laplace transform of ii(t)









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:










-p(x,0-) + sP(x,s) = Dp d2P(x,s)/dx2 P(x,s)/Tp ,(2.2)


where

t=co
P(x,s) = f e-tp(x,t)dt s = + jw, j = (-1)1/2 (2.3)
t=O

Solving Eq. (2.2) yields


P(x,s) = p(x,0-)/s + M1 exp(-x/L ) + M2 exp(x/L ) (2.4)


where L = [(D T )/(1 + ST )]12 and where M1 and M2, given below, are

to be determined by the boundary values at the two edges of the quasi-

neutral base region: P(O,s) at x=0, and P(XQNB,s) at x=XQNB.

Substitution of (2.4) into (2.2) yields the steady-state continuity

equation for p(x,O-), 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:


I(0,s) i(0,0")/s 3 L eD e i (2.5)
:_ 1 -1 1 (2.5)
I(XQNBs) i(XQNB,0 )/s p -e QNB/p e QNBp M2

where I(0,0) and I(XQNB,0~) are the initial values of the minority hole

diffusion current at x=0 and x=XQNB. In (2.5), hole current entering

the quasi-neutral base is positive, by definition.










Regarding the minority carrier densities at the two edges as the

excitation terms for a system analogous to a linear 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):

I(0,s) i(0,0")/s A 11 A12 P(0,s) p(0,0)/s
I(XQNBs) i(XQNB,0-)/s A21 A22J P(XQNBS) -p(XQNBO-)/s

(2.6)
where p(O,O0) and p(XQNB,0-) are the initial values of the excess hole
densities. Equation (2.6) extends a similar earlier development [9] by

including initial conditions so that transients may be directly
studied. We call Eq. (2.6) the master equation for the quasi-neutral
base. In Eq. (2.6), A12=A21= -e(D /L *) cosech(XQNB/L *) and A11= A22
e(Dp/L*)coth(XQNB/L*). Figure 2.1 displays the master equation, where

the initial values are included in I(O,s), I(XQNB,s),P(O,S), and

P(XQNB,S).
Transient solutions can be derived from (2.6) by inserting proper
boundary conditions, initial values and constraints imposed by the
external circuit. For example, I(O,s)=O in OCVD open-circuit-voltage-

decay [1], I(O,s)=constant for reverse step recovery [2,3], and P(O,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)=IDC/s + dI(O,s) and

P(O,s)=PDC/s + (edV/kT)PDC. Here the suffix DC denotes a dc steady-
state variable. In later sections, we will show briefly how to get

solutions from the master equation for various of these methods.
























+ z
C

Ix


_


+ o
n_


S
zb
a
x

















S
0


C,









r- 0
0 *r-
co





u
















0





4-
c.I





















=-







4.) J
0 0
.: C1
0)














S-.


i-



. r-
V).



L-
0











In a solar cell, the back contact is generally characterized in

terms of effective recombination velocity, Seff. The boundary condition

at the back contact is I(XQNB,s) = -eSeffP(XQNBs). From a circuit

viewpoint, this relation is equivalent to terminating Fig. 2.1 by a

resistor of appropriate value dependent partly on Seff. Because Seff in

part determines the transient in the various methods named above, we can

determine Seff from the transient response, as will be shown.

In Secs. 2.4 to 2.6 we consider the utility of the master equation

in characterizing selected measurement methods. The main emphasis will

be placed on the 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.











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 (OCVD)

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(O,s)=O:


i(0,0-)L* 1 + D [coth(X NB/L *)]/L *S
P(O,s) = p(O,0 )/s e P p eff
eD s coth(XQB/L *) + Dp/L *Seff


(2.7)

Here we have assumed that the 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):










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










13 -3
developed, and for which ni 10 cm Here ni 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.











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.











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

10-12s. 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

regions, thus causing the nearly sudden rise of the barrier height to











its equilibrium value. (The physics governing this phenomenon comes

from Maxwell's Curl H = i + aD/at; taking the divergence of both sides

yields 0 = div i + d(div D)/dt, which, when combined with i = aD/e and

divD = p, yields a response of the order of c/a, the dielectric

relaxation time.)

Following this readjustment of the barrier height, the excess holes

and electrons exit the junction 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.










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

eD p(0,0) coth(X /L *) + D /L *S
I(O,s) = i(0,O-)/s QN P p eff
sL 1 + (D/L *Seff)cotn(XQNB/L *)
(2.10)


Cauchy's residue theorem yields the inverse transform of (2.10):

() eD p(0,0-)K cot(KiXQNB/L ) DpK /LSeff i.t
ii(t) Spp p poef2 e)
i= p ( s ) + (XQNB/2Sff)cosec (KiXNB/Lp)


(2.11)
where si is the ith singularity which satisfies the Eigenvalue equation,

D XN
1 + LSeP V1+si T coth( L- 1+sJ ) = 0 (2.12)
p eff L p

and where Ki = (-1-siTp)1/2 > 0, with si < 0.

Truncating (2.11) and (2.12) to include only the first mode (sl),
we obtain


1/2
1 + (Dp/LpSff) V1 + 1Tp coth[(XQNB/Lp)(1 + S1 p) ] = 0 (2. 13)












and



first mode(0)

eDp(0,0-)K1 cot(KlXQNB/Lp) (DpK1/LpSeff)
lLp (T /2K ) + (XQNB/2S ) cosec2(KX /L)
(2.14)


Equations (2.13) and (2.14) contain four unknowns: first mode(0)' Sl*
Tp, and Seff. The parameters, s, and first mode (0) are determined
from the straight-line portion of the observed decay (in Fig. 2.3(c) to
be discussed below) p(0,0O) = (n 2/ND)[exp(eV(0O)/kT)-1]. Here v(O-) is

known and the doping concentration NDD of the base is measured by usual

methods; Dp(NDD) is known, and XQNB is measured. Combining (2.14) and
(2.13) then yields the desired parameters: Tp and Seff*


2.6.2 Experiments and Results

To explore the utility of the ESCCD method, we connect the solar
cell under study to node B of the electronic switching circuit

illustrated in Fig. 2.2.
The circuit works as follows. When V1(t) is high, switching
transistor T1 turns on, which charges the large capacitor in parallel
with it and divides the high voltage Vhigh about equally between the
solar cell and the emitter-collector terminals of the transistor. Thus










19







rC
a)3





EO





o o "
0 C .0

S1C C)


a E 0







co
CC +3>

-- C








cu
0




e l"0 .
u C











a -0






-uo
4-) --
-CE e a
o *o
Su OJa)







Q1) cc C








UCC

C
OL
Sgg 00



E 4-'O
0 (T V)
2 *~ fL *X
u s- e:
r- *rC~- 4>r
"SUA 1











the voltage across the solar cell becomes about 0.6 V, which one may

control by altering (Vhigh), and the variable resistor connected to the

transistor base, or both. In this mode, the 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.











From the above development, these results give DEVICE 1, rp =

119 us, Seff = 25 cm/sec; DEVICE 2, = Tp = 119 us, Seff = 60 cm/s;

DEVICE 3, Tp = 213 us, Seff = 100 cm/s. These results agree favorably

with those obtained for the same devices by using the more time-

consuming methods detailed in [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 ni discussed in Sec. 2.4.

If the electronic switch providing the short circuit closes fast

enough, the mobile holes and electrons stored for negative time in the

junction 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,











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 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.





















z

x














0.


04












'.

(


Au

*1
SI


C
co
04-)

*C- L


C 0
0 U
u

S.r
*0 S






E
0
0




LU
'O
*r
E






















a ."
x





0"
Si












4-







4- C
3 I-
0).





Lr-

0

4L-) r
C0.
*r_





0





34-J
0S -
oil










C: Z
cyrr











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 = XQNB, I(XQNB,s) = -AeSP(XQNB,s), removes 12(s) and

P2(s) in Fig. 3.1 from consideration. Figure 3.2 displays the resulting

two-port network representation of the quasi-neutral base region of a

solar cell. Here








30










0
.- VI
U




0 I

z S-
o 0 r-

0 1-




II ') C. r


s- 0
Sr
0 0 I
.U II

C- O
-1a
1 4"" X
*r- 0
= (-f


U4
C tv
r- S- S
*r-







I+- u r.-
4 0) Iw
UC
0 0







-, mut



0 ii S
r-- 0

O
-.C U S-
0- 0
*r- LS U












C *r-




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)












ADni sinh(X NB/L) + acosh(XQNB/L)
whe NBO LNDD cosh(XNB/L) + asinh(XQNB/L) 36)


LS
and where a = -. Here a is the ratio of the normalized surface

recombination velocity to the diffusion velocity [16]. Thus the ratio


IFMO
R -
QNBO


2K1 cotA1 + tanA1
21 12
Sl (XQNB/A) /D + (XQNB/S) csc2A1




cosh(XQNB/L) + tsinh(XQNB/L)
sinh(XQNB/L) + acosh(XQNB/L) (3.7)


where Al = K1XQNB/L and where K1 and AI are obtained by solving (A.3) of

Appendix A. The ratio R will be utilized for the determination of the

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






























































*b~
S


-o


r_
0



0
c-


*r- O

- *r-

,-) 0










C
I -
gO
0
c0







ON




c..
E(
.w- U
*0 .-






E) 0
U

-o+-





*1 -i-

*r ,c











I =D + Ish (3.8)




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


Here IFO is the 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











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 Td larger

than 100 ns. This switching circuit provides a sudden shorted path

across the two terminals of a solar cell in a manner similar to that of

the bipolar switching circuit described in Chapter Two.



3.3.2 Quality of the Short Circuit of the Switching Circuit

We now consider the quality of short circuit provided by the

switching circuit of Fig. 3.4. The discharging path has a series

resistance of a few ohms instead of a perfect 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 Td of the first-term

natural-frequency current. In Fig. 3.5, rs and rsh are the series and

shunt resistance of a solar cell, rd detects the discharging current,

and il(t) is the first-term natural-frequency current.
































cuJ



b~I r



'V)i

Th


bI 0





o4


I0
0


& I


(A
I-


-il


_ _


"cM


,_ l
PCZi

















i(t)


0+




C (t)
rMOS SCR i (t) rsh 12



0-
2



< rd






Fig.3.5 Equivalent-circuit representation of the measurement
circuit of Fig.3.4 when the first mode dominates
electrical-short-circuit-current decay.










The current i(t) flowing through rd in Fig. 3.5 is


i(t) = CSCR(dvl2(t)/dt) + vl2(t)/rsh + il(t) (3.15)


where

dQsc
CSCR
CCR dv1 f12) (3.16)


in which QSCR/e is the integrated 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.











3.3.3 Measurement of the Dark I(V) Characteristics

The measurement of the dark I(V) characteristics of a solar cell is

straightforward. One first measures the terminal I(Vout)

characteristics in the dark condition and then corrects for the effects

coming from the existence of the series resistance.

This method is based on the assumption that the main deviation of

the diode current from the ideal exp(qV/kT) behavior at high currents

can be attributed solely and relatively simply to series resistances

[19].

From combining the measured I(Vout) characteristic with idealized

diode theory, we obtain



(Vout)i = Iirs + (kT/q)loge(Ii/IFo) (3.24)


where rs is the series resistance, IFO is the idealized 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 = r1AI + (kT/e)loge (12/1) (3.25)


upon subtraction. This determines rs, which we may thus ignore in the

subsequent discussion. To determine IFO, we use the procedure of

Ref. [19].











3.4 Experimental Results and Discussions

In the most general case, the ratio XQNB/L is arbitrary. For this

case, we generated the S(T) locus corresponding to the measured value of

the decay time constant Td. This locus is generated by solving the

transcendental equation of (A.3) of Appendix A.

We consider the following ratios, for reasons that will become

apparent:


R = IFMO/QNBO = f(r,S) (3.26)



R = IQNEO /QNBO = FO IQNBO)/IQNBO (3.27)



RM = IFMO/IFO (3.14)


The relation among these parameters is


R = (1 + RI)RM. (3.28)



The ratio R is determined by theory for any assumed values of S and

T lying on the S(T) locus corresponding to the measured value of the

decay time constant Td. The ratio Ry is determined by the measured

value of IFO and by the value of IQNBO which is obtained from (3.6) for

any assumed values of S and T. The ratio RM is determined by

measurement. Thus (3.28) enables a determination of S and T by an

iterative procedure.











To determine RM, we use the ESCCD method as discussed in connection

with (3.14) to determine the ratio (not the individual components of the

ratio).

Having formed the three ratios above by a combination of experiment

and theory, we search for the values of S and T that satisfy (3.29).

Completion of this search yields the actual values of S and T for the

solar cell under study. It also yields the ratio of the emitter to the

base components of the total current, and hence these components

separately if the base doping concentration is determined in the usual

manner.

As an illustrative example of the above, we consider a particular

cell fabricated on a 0.3 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





48









104



S R=3.5 R=3.56 R=3.58

locus 2: +
S10 ohm-cm n+/p/p, Sff, max
O d = 0.74 us, XQNB = 101 Um.




0
1103


O-
C
Z R=3 5 R=3 6
R=32 -
locus 1: S '
0 2 10 ohm-cm n+/p/p+, Seff, max









0 100 200 300 400
MINORITY CARRIER LIFETIME (92s)








Fig.3.7 S(T) loci for two different solar cells with thin base.
0 -


0 100 200 300 400

MINORITY CARRIER LIFETIME (ps)


Fig.3.7 S(T) loci for two different solar cells with thin base.











the extreme case of negligible surface recombination at the back contact

during the ESCCD transient. For the 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.







Table 3.1 ESCCD MEASUREMENTS

NAME RESISTIVITY THICKNESS
(ohm-cm) (pm)

SPIRE .31 374

ASEC1 .15 301

ASEC2 .15 267

BSF#1 10 240

BSF#2 10 260

BSF#3 10 284

BSF#4 10 96

BSF#5 10 91

BSF#6 10 107

BSF#7 10 102

LEU#1++ 8 328

LEU#2++ 1.5 325

POLY1 2 203

POLY2++ 2 208

++ denotes p+/n/n+ BSF solar ce
* denotes Smax. POLY1-2 have po


FOR VARIOUS SOLAR

Td(vs) T(Ps)


15

7

13

35+

45+

75+

20+

25+


6.5

3.6

4.0

5.3

6.3

7.85

.98

.9

.7

.73

28.0

25.7

2.7

8.8


11. + denotes Tmin
ly-Si layers at the


CELLS.

S(cm/s)


L(pm)


1


5

3

6

0+

0+
2+

5+

5+


I


1300 18!

ohmic 9

ohmic 12(

100-400* 35(

100-350 40(

100-225 51

100-380 26!

100-290 29!

4500*

3200

40-80* 41

40-150 34

2000*

100-400* 161

or Lmin.
back surfaces.


~


8+


7+

7+


145+

105+











Finally, we present one more method to determine the recombination

parameters of solar cells. In this method, one fabricates two different

solar cells out of the same wafer, one BSF solar cell and one ohmic-

contact solar cell. Then one first estimates the lifetime of the cells

by measuring Td of the ohmic-contact solar cell and by using (3.31):



Td-' = [(XQNB/r)2/D]-1 + T-1 (3.33)


Second, one measures rd of the BSF solar cell and generates the S(T)

locus on the same plot. Since the lifetimes of the two cells are the

same, S of the BSF cell can be obtained from the corresponding S(r)

locus.

An illustrative example is shown in Fig. 3.8. In this example, we

used a wafer which is 10 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 XQNB/L decreases. For example, for a 10 ohm-cm

n+p/p+ solar cell ohmicc contact) with a thickness of 350 3 mn, the

error in the lifetime is estimated to be about 20%. In doing this

calculation, we assumed that the lifetime is 50 us (XQNB/L = 0.84) and

that 0 = 35 cm2/s. For a 0.3 ohm-cm n+/p/p+ solar cell ohmicc contact)

with a thickness of 350 3 in, this error is estimated to be about

5%. Here for this calculation, we assumed that the lifetime is 20 us

(XQNB/L = 1.64) for diffusivity D = 23 cm2/s.














107


-


10

O
0j locus 1

10
UO -







I 104
0


U locus 2



102 1
10

0 100 200 300 400
MINORITY CARRIER LIFETIME (ps)

Fig.3.8 Illustration of the determination procedure of S and T using
one ohmic contact solar cell and one BSF solar cell from the
same material. Locus 1: ohmic contact solar cell(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.











Along 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-w 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:





Ii(s) ii(O-)/s Bi B12 Pi(s) Pi(O-)/s


(4.1)



Ii+1(s) ii+1(0-)/s B21 B22 Pi+ Pi+1(0-)/s


Y





























m
z
0
x


I
4-

C




*r
















4-
U)








0

J






0


S-
I-
U)














U

C
4-













S.0

0



CO
*r- "U
*0-..1
IC)













U 0

*r- 0.
Sa
t-+-
(0
C *i
0r~
*- -

vi a>
C:
uvl










where B11(s) = B22(S) = (AeD/L*)coth(Ax/L*) (4.2)


and B12(s) = B21(S) = -(AeD/L*)csch(Ax/L*) (4.3)


Figure 4.2 displays the matrix equation (4.1), where pi(0-)/s and

Pi+1(0~)/s are the minority carrier densities and ii(0")/s and
ii+1(O)/s are the minority carrier currents, respectively, at the two
edges of the ith subregion. The designation, t = 0-, means the instant
before we apply the excitation to start the transient. The circuit
elements Y1 and Y2 in Fig. 4.2 are related to B11 and B21 as follows:


Y1 = B11 + B12 (4.4)


Y2 = -B12 (4.5)


To realize an RLC 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











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 Isl, 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.













O .
0 IB0


+
cn






































oc
-I




















0
W


+

II
x


C
0

L
.C

0


-x


A

0.
Y

aT











4.2.2 SPICE Simulation of the Equivalent Circuit for ESSCD

We have carried out the SPICE simulations of RLC equivalent-

circuits with different numbers of subregions for a given quasi-neutral

region. The thickness of the quasi-neutral region is 96 um, the

lifetime of the minority carriers is 100 us, the diffusivity is

35 cm2/s, and the surface recombination velocity of the back 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,

focused 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 us. 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 Td and the

current ratio R are determined from Fig. 4.4 and Fig. 4.5:



Td = 0.98 Ps and R = 31



These values coincide with the exact solutions obtained by solving (3.7)

and (A.3) of Appendix A.


















TIME (p.s)


+-10


--40


Fig.4.4 SPICE2 simulation of 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.


-+-3













1 (b)
(c)






I- rd -0.98 ps
Z0.5
W

(a)





2 -

0
z








0.1 I I
0 0.5 1 1.5

TIME (vs)

Fig.4.5 SPICE2 simulation of 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.











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 TD 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 many-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:













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.











r
All(S) A12(s)





A21(s) A22(s)


ev(O,s)p (
kT PDC(0)




ev(XQNB ,s)
kT PDC(XQNB)


where Iv(O,s) < and V(XQN,S) < kand v is the hole
e e
electrochemical potential and P, as before, is excess hole density.
This result is derived by using the approximation,


exp[(VDC + v)/(kT/e)] = exp[VDC/(kT/e)]exp[1 + v/(kT/e)]


= exp[vDC/(kT/e)][1 + v/(kT/e)]


Rewriting (4.16), we have


i(0,s)






i(XQNB,S)


D11(s) D12(s)






D21(s) 022(s)


v(0,s)






V(XQNB.S)


(4.17)


In (4.17),


i(0,s)






i(XQNB,s)


(4.16)










D11(s) = KIA11(s)


012(s) = K2A12(s)


D21(s) = K1A21(s)


D22(s) = K2A22(s)


where


K1 = (e/kT)(ni2/NDD)exp(eV(O)/kT


K2 = (e/kT)(ni2/NDD)exp(eV(XQNB)/kT)


If we assume a thin 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












C1 = eAK1XQNB/2 (4.25)


R= XQNB/eAK2D (4.26)


L= XQNB3/62eAK2 (4.27)


G2= eAKXQNB/2T (4.28)


C2 = eAKlXQNB/2 (4.29)


and


G(s) = -eA(K1 K2)(XQNB/D SXQNB/6) (4.30)


In Fig. 4.7, the nodal variable has the dimensions of voltage.
This contrasts with our earlier models, for 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










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















a
0


0











0









P1














II0


(A
61

Ij


L.
r-
4-)



Co
4.-

0
c,


0



4--

U


Co
CI






J4J


4-
c4-,
E






cr-,
CO







0
O
u






r--)






-3 4--
(0 U


-
X N








U- L4
C-)
0)


5--




or,-
>0)



r0 0


LJJ>
0C,
-I--









U-








L.










i(O,s)


IVVV vv


v(,s) I C G



-n___K ,____,__ _________


x=-


(a)


i(O,s) L R
I


v(Os)


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.











region. Thus we need to comment on the physical mechanisms occurring

within the base region that give rise to this inductance. From the

viewpoint of frequency response, the inductance causes the current

phasor at x = XQNB to lag (in phase angle) the voltage phasor, v(0,jw).

In the time domain, this lagging phase angle corresponds to a delay in

the current at the back contact following the sudden application of

voltage at x = 0. This delay is emphasized in the 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.










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]:










T(delay) = f te(t)dt/ f e(t)dt (4.35)
0 o

where e(t) is the impulse response of the current at x = XQNB to vl(t)

A6(t). In the complex frequency domain, this becomes [26]


T(delay) = lim{-[dF(s)/ds]/F(s)} .(4.36)
s+O

F(s) is the system function



Vbe(s)/Vbe(s)


the inverse transform of which is the impulse response of the system.

The equality of (4.35) and (4.36) is shown in Appendix C.

To determine in the time domain the minority-carrier delay time of

the thin quasi-neutral base region with an ohmic contact at x = XQNB, 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 XQNB 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 = XQNB. We derive the delay time r(delay) from this

simulation by using (4.35) and numerical integration:


T(delay) = XQNB2/60 (4.37)
(4,37













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We substitute (4.37) into (4.34) to get the inductance. This

inductance in (4.40) is larger, by a factor (6 + 1), than the inductance

in Fig. 4.9 as we indicated at the end of Sec. 4.3.2:



L' = (B+ 1)(RXQNB2/6D) = ( + 1)L (4.38)



where L' is the inductor for the 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 vbe based on the conventional quasi-static

approximation. Note the instantaneous readjustment implied by this

model. Contrast this with the transient response of the minority

carriers determined by solving the continuity equation displayed in

Fig. 4.13(b). This figure illustrates the physical origin of the

delay. From the results of the previous section, the input port of the





























-lAAA


.0)
E
-o













0,
I






























t = 0+


emitter SCR base SCR collector


Fig.4.13(a)


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





























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.





































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f0



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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):











2
T(delay) = XNB (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 wm if we consider the influence of the space-

charge region width and of the overlapping of the buried layer and the

base region. The estimation of the base width is in Appendix D. The

parameter K is about 6. We put these parameter values into (4.42). The

delay time is estimated to be about 5 ns.

We measured the phase angle of the common emitter hybrid parameter

h21 of transistors at IC = 10 pA, and 100 iA. The results of these

measurements are shown in Fig. 4.17(a)-(b). The delay times estimated

from these figures are 5.3 ns and 5.6 ns. (The frequency range used to

estimate the delay time should be much less than the reciprocal value of

the delay time; only then can we regard the actual delay across the

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.
































































C,,

U,





U,


0
0o


















20
10




emitter n+


r^ 18
E 10
buried layer n
E
0

LU
I-
LL


C 16
S10
Base p




5 1
z-S















0 10 20 30(pm)
DISTANCE FROM THE SURFACE

Fig.4.16 Impurity profile of the measured transistors. The dashed line
is for the exponential fucntion approximation.

































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