Title Page
 Table of Contents
 List of Figures
 Review of earlier work on the capacitance...
 Determination of lifetime and surface...
 Lifetime determination in p/n junction...
 Imorived junction space-charge...
 Forward-voltage capacitance and...
 Summary and recommendations
 Biographical sketch

Group Title: Improved characterization of the p/n junction space-charge region, with applications
Title: Improved characterization of the pn junction space-charge region, with applications
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00082416/00001
 Material Information
Title: Improved characterization of the pn junction space-charge region, with applications
Physical Description: xi, 144 leaves : ill. ; 28 cm.
Language: English
Creator: Liou, Juin-Jei, 1954-
Publication Date: 1987
Subject: Capacitance meters   ( lcsh )
Voltage regulators   ( lcsh )
Junction transistors   ( lcsh )
Bipolar transistors   ( lcsh )
Electric capacity   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1987.
Bibliography: Bibliography: leaves 138-143.
Statement of Responsibility: by Juin-Jei Liou.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082416
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000943145
oclc - 16818176
notis - AEQ4833

Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
        Page 1
        Page 2
        Page 3
        Page 4
    Review of earlier work on the capacitance of the space-charge region
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
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        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
    Determination of lifetime and surface recombination velocity of p/n junction solar cells and diodes by observing transients
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
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        Page 36
        Page 37
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        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
    Lifetime determination in p/n junction diodes and solar cells from open-circuit voltage decay including junction capacitance effect
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
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        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Imorived junction space-charge region capacitance based on time-domain reasoning
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
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        Page 83
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        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
    Forward-voltage capacitance and thickness of p/n junction space-charge regions
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
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        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
    Summary and recommendations
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
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        Page 131
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        Page 137
        Page 138
        Page 139
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        Page 142
        Page 143
    Biographical sketch
        Page 144
        Page 145
        Page 146
Full Text








I wish to express my sincere appreciation to the chairman of my

supervisory committee, Professor Fredrik A. Lindholm, for his guidance

and encouragement during the course of this study. I also thank

Professor Arnost Neugroschel for his guidance in experiments, and

Professor Jerry G. Fossum, Dorothea E. Burk, and Tim Anderson for their

participation on my supervisory committee.

I am grateful to my colleagues and friends, Dr. Tae-Won Jung, Dr.

Adelmo Ortiz Conde, Mr. Meng-Kai Chen, and Mr. Robert J. McDonald for

their helpful discussions.

I am greatly indebted to my wife, Pei-Li, my parents, and my

parents-in-law for their love and encouragement.

The financial support of the Jet Propulsion Laboratory and the

Semiconductor Research Corporation is acknowledged.




LIST OF SYMBOLS.....................................................vi



ONE INTRODUCTION..................................................1

SPACE-CHARGE REGION.......................................... 5

2.1 Introduction..............................................5
2.2 Properties of the Space-Charge Region....................7
2.3 Theoretical Development of the Quasi-Static Capacitance
of the Space-Charge Region................................8
2.3.1 Free-Carrier and Space-Charge Approaches...........8
2.3.2 Energy Approach ...................................20
2.3.3 Current Approach..................................21
2.3.4 Cut-Off Frequency Approach.......................22
2.3.5 Small-Signal Approach.............................22
2.4 Measurement Methods of Finding the Capacitance of the
Space-Charge Region.....................................24


3.1 Introduction.............................................27
3.2 Open-Circuit Voltage Decay Method .......................29
3.3 Green's Compensated Open-Circuit Voltage Decay Method...33
3.4 Extension of Compensated Open-Circuit Voltage Decay
Method.................................................. 35
3.5 Open-Circuit Voltage Decay Method with the Effects of
the Junction Space-Charge Region........................40
3.6 Effect of Heavy Doping Emitter Region on Open-
Circuit Voltage Decay Method.............................44

3.7 Frequency-Response Methods................................46
3.8 Conclusions and Summary ................................. 48

CAPACITANCE EFFECT........................................... 51

4.1 Introduction.............................................51
4.2 Theory...................................................52
4.2.1 Linvill Model and Electrical Counterpart..........52
4.2.2 Generalizations and Inclusion of the
Space-Charge Region............................... 58
4.2.3 Determination of Lifetime........................64
4.3 Illustration............................................ 66

ON TIME-DOMAIN REASONING......................................73

5.1 Introduction............................................. 73
5.2 Derivation by Detailed Time-Domain Reasoning............74
5.2.1 Review and Elaboration...........................74
5.2.2 Interpretation....................................78
5.3 Applicable Voltage Range for Equation (5.7).............83
5.4 Discussion of High-Injection Effects....................86
5.5 Physical Interpretation of the Condition dC/dX = 0......88
5.6 Conclusions.............................................92

SPACE-CHARGE REGIONS..........................................93

6.1 Introduction.............................................93
6.2 Importance of the Capacitance of the Space-Charge
6.3 Zeroth-Order Model for the Capacitance and Thickness
of the Space-Charge Region for High Forward Voltages....99
6.4 Analytical Solution for X(V) Under Very High Forward
Voltages. .............................................. 102
6.5 Iterative Scheme to Extend Applicability and
to Determine the Applicable Range of
the Zeroth-Order Model..................................105
6.5.1 Development .....................................105
6.5.2 Discussion.......................................107
6.6 Development of Analytical Relations for C(V)............113
6.6.1 Reverse, Zero, and Low Forward Voltages.........113
6.6.2 High Forward Voltages............................ 115
6.6.3 Very High Forward Voltages ......................115
6.6.4 Moderate Forward Voltages........................116
6.6.5 Results.......................................... 118
6.7 Conclusions............................................121

SEVEN SUMMARY AND RECOMMENDATIONS ................................. 122


OF P/N JUNCTIONS............................................. 125

B RELATIONS AMONG VARIOUS POTENTIALS........................... 134

REFERENCES.......................... ..............................138

BIOGRAPHICAL SKETCH..................................................144












Cil C2
C1, C2








G1' G2 Gs



area of a device

junction gradient

junction space-charge region capacitance of diodes or
emitter-base transition region capacitance of bipolar

electric-circuit counterpart of S

base collector transition capacitance of bipolar

capacitance resulting from QD

capacitance resulting from QF

C for zero applied voltage

quasi-neutral base capacitance

total capacitance of a p/n junction

electric-circuit counterpart of S1 and S2

diffusion coefficient of minority carriers

excess electron density

excess hole density

incremental V

electrical field

magnitude of electron charge

dielectric constant

forward active cut-off frequency

electric-circuit counterpart of H
electric-circuit counterpart of Hcl Hc2, and H

electric-circuit counterpart of Hd

combinance of Linvill's model for long-base diodes

Hcl Hc2' Hs





















combinances of Linvill's model for short- and intermedium-
base diodes

diffusance of Linvill's model

collector current

emitter current

minority current

incremental current

current density

Boltzmann constant

minority-carrier diffusion length

Debye length

acceptor impurity concentration

base impurity concentration

donor impurity concentration

net impurity concentration

electron density

intrinsic electron density

electron density at thermal equilibrium

hole density

incremental power

hole density at thermal equilibrium


net free-carrier charge at the edge of the space-charge

free-carrier (hole or electron) charge at the edge of the
space-charge region

QF free-carrier (hole or electron) charge in the volume of
the space-charge region

QQNB free-carrier charge in the quasi-neutral base
Rbi intrinsic base resistance

R emitter resistance
S storance of Linvill's model for long-base diodes

s d/dt

SB effective back-surface recombination velocity

S1, S2 storances of Linvill's model for short- and intermedium-
base diodes

a electron conductivity

ON electron quasi-Fermi potential

Op hole quasi-Fermi potential

T absolute temperature

t time

ttr transit time across the space-charge region

tr characteristic time, (e/a)(l + X/XB)-1

r minority-carrier lifetime in the quasi-neutral base

TB base transit time

rd decay time constant of the fundamental mode for short-
circuit current decay method

V separation of the quasi-Fermi potentials in the space-
charge region

V effective junction built-in potential

v(t) transient voltage across a p/n junction

VA applied voltage

VB barrier height of the p/n junction


Vbi junction built-in potential

V critical voltage, Vbi-5VT

V V for linear-graded junction
V. electrostatic potential

Vj voltage drop across the space-charge region

VLB lower-bound voltage, Vbi-7VT

VQNB voltage drop across the quasi-neutral base

VQNE voltage drop across the quasi-neutral emitter

VT thermal voltage

VUB upper-bound voltage, Vbi-0.3

W electric energy

X thickness of the space-charge region

x position

xL front contact of p/n junctions

x n-type side space-charge region edge

XnO x at thermal equilibrium

x p-type side space-charge region edge

xp0 x at thermal equilibrium

xR back contact of p/n junctions

XB thickness of the quasi-neutral region

p space-charge density

p' resistivity

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




May 1987

Chairman: Fredrik A. Lindholm
Major Department: Electrical Engineering

This dissertation focuses on excess electron and hole storage in a

p/n junction space-charge region and its effects on measurement methods

and quasi-static capacitance. After unifying reviews of the many past

treatments of this capacitance and of transient methods for measuring

recombination lifetime, a new method for measuring recombination

lifetime is developed and illustrated. This method utilizes one

existing quasi-static capacitance model to include electron and hole

storage effects in the open-circuit-voltage decay technique for

recombination lifetime measurement in silicon solar cells and diodes.

Illustrative measurements demonstrate that the accuracy of the new

method exceeds that of previous enhanced versions of open-circuit-

voltage decay.

The study then turns to the improvement of existing models for the

quasi-static capacitance, seeking a comprehensive model for the entire

pertinent range of applied voltages, from reverse to small forward to

large forward voltages. Quasi-static junction region capacitance for

large forward voltages, including voltages so large that the junction

barrier height becomes of the order of a thermal voltage, has not

previously received detailed physical or analytical analyses. As a

first effort toward developing a comprehensive model, a previous

theoretical model based on time-domain phenomenological reasoning is

improved by defining its applicable voltage range and by including high-

injection effects in the bordering quasi-neutral regions. The

improvement leads to an analytical model for large forward voltages,

which when combined with existing models for lower and reverse voltages,

yields the comprehensive model sought. Comparisons of the comprehensive

model predictions with results of measurements and device simulation

show excellent agreement. Poor definition of the edges of the junction

space-charge region complicates this comparison. That problem is evaded

in various ways, although an iterative technique yields a rough estimate

of the junction-region thickness dependence on forward voltage, a

dependence that is most accurate at high forward voltage. This part of

the study seeks to provide a comprehensive junction capacitance model

for bipolar circuit and integrated-circuit simulation based on detailed

physical understanding. In principle, the physical understanding will

aid both device and circuit design.


During the past three decades, the use of silicon, rather than

germanium, and the emergence of gallium arsenide as primary materials

for semiconductor p/n junction devices have enhanced the importance of

the space-charge region and its capacitive effects. Such effects are

becoming even more important recently as the technology of large-scale

integrated circuits advances and geometries shrink causing doping levels

to necessarily increase. Accurate device characterization, through

device and circuit simulation, thus requires better modeling of the

junction capacitance C of the space-charge region, which motivates this


The general purposes of this study are: first, to explore the

importance of the capacitance C of the space-charge region in

recombination lifetime measurements by including C in the analysis of

the open-circuit-voltage-decay-method, and, second, to develop a

comprehensive quasi-static space-charge-region capacitance model. The

model developed describes the capacitance for all voltages and the model

is applicable for practical junction profiles. The capacitance model

shows good agreement when compared with measured dependencies, with an

empirical model for circuit simulation, and with device simulation. The

results of this study are directly applicable to any p/n junction

device, including diodes, solar cells, and bipolar transistors.

In Chapter Two, we review earlier treatments of the space-charge

region capacitance, providing insight for a better understanding of the


In Chapter Three, we present a unified view of transient methods,

such as the open-circuit voltage decay and the electrical short-circuit

decay methods, for the determination of base recombination lifetime r

and the back surface recombination velocity SB for silicon diodes and

solar cells. The discussion in this chapter reveals the importance of

the capacitance of the space-charge region on the open-circuit voltage

decay method. One alternate approach is to use the electrical short-

circuit current decay method, which nullifies the presence of the free-

carrier charges in the space-charge region during transient.

Recognizing the need for including C in the open-circuit voltage decay,

we propose a new method, which is treated in Chapter Four.

A new more accurate method for determining the base-region

minority-carrier lifetime for p/n junction diodes or solar cells is

illustrated in Chapter Four. The method uses the traditional open-

circuit voltage decay measurement but includes the capacitive effects of

the free electrons and holes stored in the junction space-charge region

for the forward-voltage transient. For the capacitance model in this

chapter, we utilize Chawla and Gummel's model, which is valid for

applied voltage less than 600 mV for most junction diodes. Thus the

method developed in this chapter is restricted to applied voltages less

than 600 mV. The combination of the Chawla-Gummel model for the space-

charge region with the Linvill model for the quasi-neutral base,

together with a computer program, yields theoretically the open-circuit

voltage decay response. By adjusting r to obtain agreement between the

theoretical response and measured response, one determines r. The value

of r obtained from this new method agrees with those obtained by the

admittance-bridge method, the short-circuit current decay method, and an

improved short-circuit current decay method. A comprehensive

capacitance model will be developed in Chapter Six. The use of such

capacitance model relaxes the limitation of the voltage range applying

to the method described above.

In Chapter Five, we improve a previous space-charge region

capacitance model, which is the basis of the comprehensive capacitance

model that will be developed in Chapter Six. The time-domain reasoning

employed in deriving the previous capacitance model is elaborated. The

previous model is improved by defining the applicable voltage range and

by including high-injection effects. The treatment in this chapter also

supplements the derivation of the previous model, thus adding physical


Chapter Six presents a comprehensive analytical model for the

quasi-static capacitance of the p/n junction space-charge region. The

model describes the capacitance for all voltages and applies for

exponential-constant doping profiles. In addition to the analytical

model, an iterative technique is developed to yield numerically the

capacitance and the thickness of the space-charge region. A rigorous

solution for the thickness of the space-charge region under very high

forward voltage is also presented. This solution serves to verify


aspects of the capacitance model described in Chapter Five, which is

part of the comprehensive model. The capacitance-voltage dependencies

of the comprehensive model agree well with that obtained from a

measurement method, from an empirical model, and from a method based on

device simulation.

Chapter Seven summarizes the contributions of this dissertation

and presents recommendations for extension of the present study.


2.1 Introduction

Very large scale integration (VLSI) fabrication techniques for

semiconductor devices has increased demand for more accurate device

modeling that can be used in computer simulation. The capacitance of

the space-charge region of the semiconductor junction plays a

significant role in switching speed and frequency response, particularly

for VLSI bipolar devices because they have large doping concentrations

that reduce the relative importance of the quasi-neutral-region

capacitances. Although there have been considerable efforts to model

junction capacitance in the past three decades, there have also been

many controversies in the literature regarding the modeling and the

calculation of the capacitance of the junction space-charge region,

particularly for forward voltages.

The purpose of this chapter is to review models and measurement

methods for the quasi-static capacitance of the space-charge region that

have been developed in the past. The review, we hope, will unify

different approaches and will yield a better understanding of the

subject. We will focus on semiconductor junction diodes and junction


2.2 Properties of the Space-Charge Region

For the quasi-neutral regions of a bipolar device, one can obtain

useful results without dealing with the Poisson equation. In contrast,

analysis of the junction space-charge region always involves the Poisson


d2Vi/dx2 = e[ND NA + p n]/e, (2.1)

where e is the dielectric constant, ND is the donor impurity doping

concentration, NA is the acceptor impurity doping concentration, p is

the hole concentration, and n is the electron concentration. In

general, nonlinear relations describe p and n in terms of the

electrostatic potential V. and the electrochemical potential

p(x) = n.exp[(Vi O )/(kT/e)] (2.2)


n(x) niexp[(ON Vi)/(kT/e)]. (2.3)

Based on (2.1)-(2.3), detailed analytical physical theories [1]-[3]

concerning the space-charge density can be developed. A rigorous

mathematical analysis, however, requires numerical methods [4]-[5]. The

theory is further complicated by the observation of non-flat quasi-Fermi

levels in the space-charge region [6] for very large forward voltages.

The analysis of the space-charge region becomes transparent when

the junction is under reverse bias [7]-[10]. For such bias, in the

depletion-approximation model, the free electrons and holes are depleted

in the space-charge region because of the high electric field.

Therefore, the analysis of the region can be carried out by integrating

the linearized Poisson's equation.

2.3 Theoretical Development of the Quasi-Static Capacitance
of the Space-Charge Region

2.3.1 Free-Carrier and Space-Charge Approaches

The quasi-static approximation, which is used in almost all

theoretical capacitance modeling, assumes that the mobile-carrier

densities (also the free-carrier charges) retain, during transients,

their steady-state dependence; this approximation also assumes that the

carriers travel with infinite velocity. In modeling the quasi-static

capacitance C of the junction space-charge region, the approach used

most often focuses on the change of the excess free-carrier charges dQ

in the space-charge region resulting from the change of the separation

of the quasi-Fermi potentials dV in the space-charge region [7]. This

approach leads to the definition of Shockley [7]:

d rp
C = dV- e(Ap or An) dx, (2.4)

where Ap is the excess hole density, An is the excess electron density,

and x and x are the space-charge region edges of quasi-neutral emitter
n p
and quasi-neutral base, respectively (Fig. 2.1). Note that (2.4), in

general, can be solved rigorously through use of (2.1)-(2.3) provided

that the boundaries (x and x ) can be characterized as functions of V.
p n
To simplify the discussion, we will assume for now that low injection

prevails and that,the current density of the device is not high enough

to cause any significant ohmic drop in the quasi-neutral regions. Thus

V = VA = V, where VA is the applied voltage and VJ is the voltage drop

across the space-charge region.

In developing (2.4), Shockley [7] reasoned that dV provokes an

influx of hole charge, ep(x )dx from the right. For applied reverse

voltage, because free carriers are depleted in the space-charge region

(p = n = 0), this influx decreases the net ionized impurity charge,

e[ND(Xp) NA(Xp)]dXp. Thus,

C = [eN /2(Vbi V)] /2, for one-side step junction (2.5)


C = [ee2a/12(Vbi VA)]1/3, for linear-graded junction (2.6)

where e is the silicon dielectric permittivity, Vbi is the built-in

potential of the junction, and a is the junction gradient. The model is

simple and compact, but the depletion approximation is invalid for

forward voltages, and even for the thermal-equilibrium condition.






Fig. 2.1 Schematic n/p junction, illustrating the edges of the space-charge region, xn
and xp, front contact, xL, and back contact, xR.

Therefore, replacing the depletion approximation and improving

Shockley's capacitance model has been a research issue for the past

thirty years [8], [11]-[39]. A flow diagram (Fig. 2.2) shows the

evolution of the analysis of C since 1949.

To improve Shockley's model, and accommodate forward voltages,

Morgan and Smits [8] developed an expression for the total (contact-to-

contact) junction capacitance C for a linear-graded junction (Fig.


CT = (d/dV) ep dx. (2.7)

Using symmetry, one can decompose (2.7) into a neutral capacitance C
and a space-charge capacitance C :

C = (d/dV) J 2ep dx (2.8)

C = (d/dV) f p dx, (2.9)

where p is the charge density. For reverse bias, C = 0. This yields

CT = C and CT reduces to (2.6).

The physical reasoning underlying the separation is as follows.

The free carriers will be depleted in the space-charge region (p 0)

for reverse bias, which yields CT = C ; on the other hand, the space-

charge density p will approach zero if the junction is under high




Schottky [7]

Djuric et al. [29]
Lee and
Prendergast [30]


Maes and Sah [31]
Green and
Shewchum [41]


Fig. 2.2 Flow diagram showing the evolution of the models and methods
for the capacitance of the space-charge region.

Garrett and Brattain [1]
DeWiele and Memoulin [2]
Kittle and Kroemer [3]
DeMari [4]


Heard et al. [25]
Parrott and
Leonidou [26]
Parrott [27]
Parrott and Tieng [28]


Morgan and Smits [8]
Sirsi and Boothroyd [11]
Sah [12]
Kennedy [13]
Chang [14], [17]
Kleinknecht [15]
Gummel and Scharfetter [16]
Nuyts and Overstraeten [18]
Chawla and Gummel [20]
Lindholm [21]
Ven der Biesen [22]-[23]


Lindmayer and Wrigley [32]
O'Clock [33]
Bouma and Roelef [34]
Rein [35]

forward voltage because of the sizable density of free carriers in the

space-charge region. This yields CT = Cn. Thus the separation will

enable one to visualize physically the meaning of CT at reverse and at

high forward voltages. For intermediate voltages, however, the Morgan-

Smitz model requires numerical procedures and introduces several

auxiliary parameters. Morgan and Smits also derived an approximate

solution for reverse bias and for high forward voltage ((26)-(27) of

Ref. 8), which can be solved analytically. The comparison of the exact

and the approximate models is shown in Fig. 6 of Ref. 8.

Sirsi and Boothroyd [11] introduced a regional-analysis method to

simplify the numerical scheme while keeping the same basic concept as

that used by Morgan and Smits. First, Sirsi and Boothroyd assume an

initial potential distribution of the form A x + A2x2 for the space-

charge region, where Al and A2 are unknown constants. This is followed

by an initial guess of the thickness of the space-charge region, based

on the depletion approximation, from which p(x) is estimated by using

Gauss's law. Combining p(x) with (2.1)-(2.3), they determine a

transcendental expression relating Al and A2, and obtain expressions for

the charge Qj = fp(x)dx and the peak normalized electric field (= A,) in

the space-charge region for a given thickness. They then use these

expressions, iterating numerically to obtain a self-consistent solution

for Q from which they calculate, numerically, the quasi-static

capacitance C. We note that their assumption that the potential varies

as Alx + A2x2 disagrees with the requirement of a cubic function for a

linear-graded junction in the depletion approximation. Thus this

assumption introduces error in their calculations, the magnitude of

which they do not estimate. For very large voltages, however, we expect

that the potential will vary approximately as A1x, for reasons to be

discussed in Chapter Six.

The comparison of the Sirsi-Boothroyd model and the depletion

capacitance is illustrated in Figs. 5-7 of Ref. 11 for several different

junction diodes. The comparison indicates that the capacitance of this

model agrees well with the depletion capacitance for forward voltages

19 -4
less than 0.3 volts and a junction gradient that equals 10 cm and

the capacitance of the model increases rapidly and monotonically for

high forward voltages. Sirsi and Boothroyd do not comment on the

apparent divergence of their model and its resulting infinite

capacitance at large forward voltage. Their model is meant to apply

only for a linear-graded junction diode.

Sah [12] also derived a model for C for the linear-graded

junction, which considers the effects of the free carriers in the space-

charge region. Basically, Sah requires at the edges of the space-charge

region that the first and second derivatives of the potential vanish and

solves Poisson's equation using (2.2) and (2.3). We note that assuming

that the second derivative vanishes implicitly neglects that the space-

charge-region edge itself has a thickness of about an extrinsic Debye

length. Thus this approximation works well for junction thicknesses

much greater than a Debye length but introduces error for very large

voltages where the thickness of the junction region becomes of the order

of a Debye length. For large forward voltages, Sah's approximations

yield a junction capacitance that is proportional to exp(5eV/8kT). In

Chapters Six, our capacitance.varies as exp(eV/4kT) for V less than a

critical voltage; for larger V, the capacitance decreases rapidly. Sah

was the first author to include the effects of ohmic drops in the quasi-

neutral regions, a subject we treat in Chapter Six and Appendix B. His

model is shown in (40)-(47) of Ref. 12. The model again requires

numerical methods, although asymptotic solutions can be obtained for

some extreme cases. Figures. 14-18 of Ref. 12 illustrate that the model

agrees well with the experimental results.

Since the validity of the Shockley's model in the depletion

approximation is questionable at thermal equilibrium, Kennedy [13]

studied C for a step junction for reverse voltages and for thermal

equilibrium. Kennedy derives an equation for the electric field in the

space-charge region, by integrating the Poisson equation. Instead of

focusing on the free-carrier charge, he defines C by integrating p(x) in

the space-charge region, C = (d/dV)Jp(x)dx, and employs electric field

as the boundary conditions for the integration. The method, in general,

requires numerical method. But the model presents an analytical

solution based on assuming a negligible accumulation layer of mobile

carriers for a symmetrical step junction and by assuming that one type

of mobile carriers is much greater than the other type for an

asymmetrical step junction. He indicates (Fig. 6 of Ref. 13) that for

large reverse voltages Shockley's model is accurate, and that at thermal

equilibrium Shockley's model can have maximum error of 30% due to the

depletion approximation.

For junction capacitance subject to reverse bias, Chang [14]

indicated that focusing on p(x) as in Ref. 13, instead of free-carrier

charge, does not yield (2.5), nor do its results agree with experimental

results for reverse bias. Chang's argument is based on exact

calculations of C from the calculation of total space charge by solving

Poisson's equation. Chang concludes that (2.5) agrees well with exact

calculation for symmetrical step junctions yet they disagree by as much

as an order of magnitude for asymmetrical step junctions.

Kleinknecht [15] suggested that Chang's approach can be corrected

by reasoning that, on variation of the bias voltage, the holes do not

flow in and out through the n-type side contact for a p /n asymmetrical

junction. Consequently, they cannot be counted when integrating p(x)

for the n-type side of the space-charge region. The model, which

follows Chang's approach but incorporates this aspect of the physics,

agrees with experimental data. Gummel and Scharfetter [16] also

suggested that if the contribution of holes in the p+/n asymmetrical

junction to the n-type space-charge region is considered when

integrating p(x), the resulting capacitance is incorrect. We note that

Shockley explicitly pointed out that the charge dQ calculated in (2.4)

comes from the charge flowing in the external circuit; that charge

results from electrons entering or exiting the n-type (or donor) contact

and the holes entering or exiting the p-type (or acceptor) contact.

Focusing on the free-carrier charge, Chang later improved his

model for all biasing conditions and for symmetrical and asymmetrical

step junctions [17]. The analysis in Chang's later model [17] is done

by expanding the integrand in a Taylor series and numerically

integrating (2.4). The results from Chang's model show excellent

agreement with that of Sah's model [12] for reverse, small, and medium

forward voltages (Fig. 5 of Ref. 17). But the two models disagree at

high forward voltages at which Chang's model indicates a fall-off

characteristic whereas Sah's model shows an increasing characteristic.

We have previously suggested that the origin of this discrepancy may be

Sah's boundary condition on the second derivative of the potential.

Nuyts and Overstraeten [18] did computer calculations that justify

the assumption that the quasi-Fermi potentials in the space-charge

region are almost constant, for symmetrical and asymmetrical step

junctions. Their method is similar to that of Gummel [19]. Making use

of the property of constant quasi-Fermi potentials, an analytical method

for calculating C is carried out. The results of the analytical method,

the authors indicate, completely agree with the exact computer

calculation, but contradict those calculated by Chang [14], [17].

Chawla and Gummel [20] introduced a simple analytical model which

is valid for reverse, zero, and small forward voltages. The model is

valid for the exponential-constant junction and its two extremes, the

step and linear-graded junctions. The model retains the same form as

that of the Shockley's depletion-capacitance model ((2.5)-(2.6)) but

replaces Vbi with an effective built-in voltage so that the effect of

the free-carrier charges in the space-charge region is taken into

account. Chawla and Gummel develop the capacitance model by first

converting the Morgan and Smits model into an asymptotic form for large

reverse bias (large K). Then they compare the asymptotic form with the

result of their contact-to-contact numerical simulation. The numerical

results agree with the asymptotic form provided the junction gradient a
14 -4
> 10 cm and provided K > 20. Thus they conclude that the asymptotic

form of the Morgan and Smits model is valid for junction gradients
14 -4
greater than 10 cm and for applied voltage up to a small forward

voltage (K > 20). Next they compare the asymptotic form with the

expression for the depletion capacitance; the only difference between

these two is the term corresponding to the built-in voltage Vbi. They

thus replace Vbi by an effective built-in voltage shown in (43) of Ref.

20 for a linear-graded junction and in (45)-(49) of Ref. 20 for an

exponential-constant junction.

Lindholm [21] derived an analytical model for C for high forward

voltages for a general junction based on phenomenological reasoning. He

decomposes the capacitance into the dielectric capacitance CD and the

free capacitance CF:

CD e/X (2.10)

CF = e f ap/aV dx, (2.11)

where X is the thickness of the space-charge region and the limits of

integration are x and x The separation stemmed from the physical
n p
reasoning that CD is the capacitance corresponding to the change of the

free carriers at the edges of the space-charge region, whereas CF is the

capacitance corresponding to the change of the free carriers within the

volume of the space-charge region. For reverse bias, because p = 0, C

reduces to CD, which is the depletion capacitance in the classical

treatment. By reasoning that the derivative of C with respect to the

thickness of the space-charge region equals zero, Lindholm obtained C as

C = (eV T1eni)1/2exp(VA/4VT), (2.12)

where VT is the thermal voltage.

Recently, Van Den Biesen [22]-[23] decomposed the total junction

capacitance CT into C and C the same approach as that used by Morgan
and Smits. Instead of solving for C numerically, Van Den Biesen solves

C asymptotically at two extremes: reverse bias and very high forward

bias. For very high forward voltages, his model predicts that the

capacitance for a linear-graded junction becomes independent of V

defined as the separation of the electron and hole quasi-Fermi

potentials in the junction space-charge region. Except for a

symmetrical step junction, for which an analytical model for all

voltages is presented, the model does not give details between these two

extremes. The model, which is intended to describe symmetrical step,

asymmetrical step, and linear junctions, is claimed to agree with the

computer simulation (TRAP) at both reverse and high forward voltages.

Van Den Biesen gives no details concerning how the space-charge-region

and quasineutral-region capacitances are separated in the computer


To summarize the treatments of the the junction capacitance C

discussed to this point, we first consider C for reverse voltage. The

depletion approximation introduced by Shockley [7] makes the Poisson

equation linear and leads to a simple analytical model, which Kittel and

Kroemer [9] have improved for the very small reverse voltages and

thermal equilibrium. All of the analyses for C under forward voltage

discussed to this point must confront the problem of specifying the

edges of the space-charge region, which appear as integration limits.

This problem is often treated by changing the variables of integration

and introducing approximations, the validity of which is not completely

apparent. To some degree, Ref. 21 avoids this problem, but at the

expense of the approximation that n(x,V) = p(x,V) in the space-charge

region, which appears to be a severe approximation except at very high

forward voltages, a point not made in Ref. 21. That the space-charge-

region edges are not abrupt, but rather have a thickness of the order of

an extrinsic Debye length, further complicates the problem.

Kwok [24] presented a numerical method, using the iterative scheme

of Gummel [19]. Kwok assumed non-degenerate carrier statistics, and

calculated the total quasi-static capacitance of a junction diode by

solving the basic transport equations and Poisson's equation. This is

followed by a small-signal solution in the frequency domain. By varying

the frequency, Kwok separates the capacitance of the space-charge region

and the capacitance of the quasi-neutral regions from the total

capacitance. Results are presented for both step and linear junction

profiles in the forward biased region. Figure 3 of Ref. 24 shows that

the space-charge region capacitance falls at high forward voltages.

2.3.2 Energy Approach

Heard et al. [25] proposed a method of calculating the quasi-

static capacitance C of the junction space-charge region by using the

so-called "energetic" definition, which involves the total energy stored

in the electric field of the junction. This approach defines C as

C = (1/V)(dW/dV), (2.13)

where W is the energy of the electric field. The capacitance obtained

from this method is claimed to show excellent agreement for symmetrical

step junctions with that obtained from selected papers of Sec. 2.3.1

[7], [13], [17] (Fig. 5 of Ref. 25).

Parrot and Leonidou [26] proposed a similar treatment to find the

total capacitance CT of junction diodes. They derive C from the

following equation:

V(t)J(t) = GV2(t) + (C /2)*(d/dt)[V2(t)], (2.14)

where J is the current density and G is the conductance of the diode.

The separation of this total capacitance into the space-charge region

capacitance and the quasi-neutral capacitance can be carried out based

on the distinction between the time dependence of the dielectric

relaxation and carrier recombination.

Parrott later used the theory of Ref. 26 to derive a formula for

the space-charge region capacitance of an abrupt asymmetrical junction

[27]-[28]. The author claims that the depletion capacitance yielded by

this approach is in better agreement with experiment than that from

Shockley formula.

2.3.3 Current Approach

Djuric et al. [29] suggest that the capacitance C can be defined

using the current response to voltage excitation:

C = iC/(dV/dt) = (id + icc)/(dV/dt), (2.15)

where iC is the capacitive current, ide is the displacement current, and

ice is the convection current. They define x xM as the position where

the displacement current reaches its maximum and the convection current

has its minimum. Thus at x = x (2.15) reduces to

C = id /(dV/dt) aeE/8V I (2.16)
dc xxM4

in which E is the electric field. The term on the right-hand side of

(2.16) is (-p/E) which is relevant to the Poisson equation. The authors

suggest that the use of (2.16) and Poisson equation greatly simplifies

numerical calculations for C, especially for abrupt junctions. The

exact (numerical) solution and the approximate (analytical) solution of

the capacitance obtained from this method [29] are compared with the

depletion capacitance (Fig. 4 of Ref. 29) for step junctions, which

indicates large deviations at medium and high forward voltages and shows

agreement otherwise.

2.3.4 Cut-Off Frequency Approach

The junction space-charge capacitance can also be obtained by

using the forward active cut-off frequency fTf. Based on admittance

parameters developed from the Gummel-Poon transistor model, Lee and

Prendergast [30] developed an analytical model C ((8) of Ref. 30) in

terms of the collector saturation current IC0 and WT(-), which is the

cut-off frequency for infinite collector saturation current and is the

reciprocal of delay time of the quasineutral regions and the collector

depletion. They then obtained 1/2rfTf versus l/ICO from the MEDUSA

device simulator, and, thus, by extrapolation, they obtained wT(() and

the desired capacitance C. In their Fig. 6, they plot C as a function

of the voltage V which is the difference between the applied voltage

and the ohmic drops in the base and emitter resistances (Fig. 1 of Ref.

30). The capacitance of the space-charge region rises and then falls as

the applied voltage increases. The details given here will be important

in our treatment in Chapter Six.

2.3.5 Small-Signal Approach

Maes and Sah [31] use the transmission-line circuit model of Sah

[40] to obtain the forward-voltage capacitance for p/n junction devices.

The authors use the Green-Shewchun [41] simulation method, which they

indicate greatly simplifies the numerical calculation. In general,

Sah's small-signal transmission-line circuit model consists of a large

number of sections in each section of which the steady-state potentials

can be treated as constant. If the steady-state potentials Vi, ON, and

Op are known for a given bias point, then the circuit elements in each

section can be computed and the entire transmission-line model can be

solved subject to the boundary conditions at ohmic contacts. Thus the

input admittance of a device can be solved, which yields the small-

signal capacitance and conductance of the device as a function of


As discussed above, the use of the Sah's small-signal circuit

model requires the steady-state solution of the transport equations for

a given bias. The steady-state solution however, is difficult. For

example, the solution of the Poisson equation for nonabrupt junction

devices and devices with nonuniform recombination center requires

lengthy numerical calculation. Green and Shewchun [41] propose a method

of reducing the numerical procedures. They employ the Newton-Raphson

iteration technique to solve nonlinear equations. The iteration method

requires only a trial solution of the dc potential versus distance at a

particular bias. A few iterations can then find the accurate changes of

the steady-state values of the node potentials dC of Sah's transmission-

line circuit for an incremental applied voltage dVA. By solving the

entire small-signal transmission-line, one can find d( for each of the

node potentials. Adding these d( to the original node potentials (

yields a solution close to the exact solution. At the same time, the

values of the circuit elements in each section are obtained. Thus the

circuit is ready for small-signal admittance computation, which yields

the capacitance. For a p /n diode with constant background

14 -3 19
concentration of 5x10 cm and a emitter surface concentration of 109
cm the capacitance, for 100 kHz, has a value of 20 pf at thermal

equilibrium, has a peak value of 350 pf at VA = 0.55 volts, and

decreases rapidly toward zero for larger voltages. Maes and Sah also

compare theoretical calculations with experimental results for

frequencies of 10 to 10 Hz and for temperatures of 77 to 300 K.

Excellent agreements are obtained.

Maes and Sah's method is carried out for the entire junction.

Therefore, the capacitance obtained is the total junction capacitance.

However, if the small-signal frequency is much higher than the

reciprocal of the minority-carrier lifetime in the base and in the

emitter region, the free carriers in these regions can not keep up with

the voltage variation and thus the capacitances of the base and the

emitter region can be neglected. This enables one to find the space-

charge region capacitance.

2.4 Measurement Methods for Finding the Capacitance
of the Space-Charge Region

In additional to the above methods, which are all solely based on

theoretical analysis, measurement techniques have been used to determine

the junction space-charge capacitance [32]-[34].

Lindmayer and Wrigley [32] first developed the following relation:


1/27f = rB + R (C + CBC)

where TB is the base transit time, Re is the emitter resistance, and CBC

is the collector space-charge region capacitance. Differentiating

(2.17) with respect to 1/IE, where I is the emitter current, yields

d(l/2nfTf)/d(l/IE) = kT/e-(C + CBC) = (kT/e)C, (2.18)

for the forward-active mode. Thus the measurement of fTf versus IE

yields C.

A method of obtaining the space-charge region capacitance derived

from the common-emitter input impedance has been reported by O'Clock

[33]. This impedance measurement, however, is frequently inaccurate

owing to the presence of the base resistance, the emitter resistance,

and the lead inductance in the measuring circuit.

Bouma and Roelofs [34] developed a measuring circuit that can

determine fTf and C in a manner based on the same ideas as those used by

Lee and Prendergast. They suggest that C can be expressed as

C = (elI/kT)[(l/2fTf) reff], (2.19)

where IC is the collector current and reff is the delay time associated

with all regions of the device except the space-charge region. Using

the same concept as that in Ref. 30 and the fact that the variation of

1/27fTf and reff with biasing current IC is different, they demonstrate

that measurement together with (2.19) determines C. Because the fTf

measurement requires large conductance, Bouma and Roelofs indicate that

at reverse and forward voltages lower than about 600 mV the values of

the space-charge region capacitance are obtained by bridge measurement

of the common emitter input impedance at a frequency of 1.5 MHz. For

higher applied voltages (V > 600 mV), they measure f T This method,

which includes the correction for the quasi-neutral capacitance, shows

that the space-charge region capacitance possesses fall-off

characteristics at applied voltages near the built-in voltage of the

junction (Figs. 5-6 of Ref. 34).

The fTf measurement, nonetheless, requires the proper choice of

the measuring frequency, which is characterized by the condition that

fTf = af = constant or fT = ff = constant, where a is the common-base

current gain and P is the common-emitter current gain, in order to avoid

large errors in determining fTf [35].


3.1 Introduction

We shall assess various existing methods and introduce new ones

that utilize electrical transients for determining base recombination

lifetime r and back surface recombination velocity S.B Transient

measurements have the advantages of being simple and rapid and of

possibly being applicable to determining recombination lifetime

following key processing steps in manufacturing. The assessment will

involve comparing results against those obtained by other methods such

as those deriving from the dependence of small-signal admittance vs.

frequency and forward voltage. The treatment will be limited to low-

injection conditions in quasi-neutral base regions.

Historically, the first transient methods were developed in 1950's

when germanium devices were prominent. This is not simply a fact of

history but rather a consideration that will prove important in the

unfolding of ideas. The two methods, open-circuit voltage decay (OCVD)

[42] and reverse recovery (RR) [43] appear unified as a single method in

Fig. 3.1.

The open-circuit voltage decay method sets the initial conditions

by exciting the device with forward voltage or with incident light in

negative time. Then, at time t = 0, one switches off the excitation and

f (t) = I REVERSE

S1 = 0, OCVD
} 0, RR

Fig. 3.1 Schematic illustration for both open-circuit voltage-decay analysis and reverse
recovery analysis.

v (t)

observes the decay of voltage. The reverse-recovery method differs only

in that an external current source, I(reverse), consisting of an ideal

voltage source in series with a relatively large resistance, speeds the

removal of the excess holes and electrons stored within the device. As

the main observable, this method employs the duration of time that the

current stays constant, which is approximately the duration that the

voltage across the junction region stays in the forward direction. In

OCVD, I(reverse) = 0, and the voltage v(t) is observed. Because the

incremental current or time variation in current is zero, that is,

AI(reverse) 0, the two methods, OCVD and RR, are alike: the device is

connected to an incremental open circuit. Thus mathematically, the

initial-boundary-value problems for the two methods have the same

natural frequencies (or relaxation times or poles in the complex

frequency plane). In what follows, we mainly shall concentrate on open-

circuit voltage decay, understanding that parallel discussions will

holds for reverse recovery.

These two methods still are widely used alone or with other

measurement methods to determine r and SB in the base region not only

for silicon solar cell but also for silicon power diodes [44].

3.2 Open-Circuit Voltage Decay Method

In the theoretical (ideal) OCVD described in the Appendix of Ref.

42, v(t) shows an initial phase of rapid decay corresponding to an

infinite number of large natural frequencies or short relaxation times

characteristic of a distributed system. Then v(t) follows the response

from the dominant natural frequency associated with the pole on the real

axis nearest the origin. This response, in terms of the minority-

carrier density, is proportional to exp(-t/r), where r is the

recombination lifetime of the quasi-neutral base. Because of the

exponential relation between the minority-carrier density at the edge of

this region and the non-equilibrium component of the junction voltage,

the voltage at the terminals is a falling straight line having slope = -

(kT/e)/r. This simple relation applies for a long-base diode, for which

the thickness of the quasi-neutral base greatly exceeds the minority-

carrier diffusion length. As we shall see, this is the only case for

which OCVD has practical utility. Figure 3.2 illustrates the response,

from which one infers the value of r. Figure 3.3 shows the dominant-

pole or quasi-static response, v(t) vFR(t), where the subscript FR

means the parts of the response deriving from the fast relaxation times.

As indicated in Ref. 42, this response is observed in Ge diodes,

which enables determination of T. It is not observed in Si diodes and

solar cells, despite the widespread assumption to the contrary both for

Si photovoltaics and Si power devices. Apparently, the first work to

recognize the invalidity of the direct use of OCVD for Si was that of

Neugroschel et al. [45] in 1978 in which results for r from OCVD were

compared with those obtained by admittance-bridge techniques and by the

short-circuit current deriving from incident x-ray irradiation. The

invalidity of the direct use of reverse recovery was also noted in the

paper. Later authors, including Mahan and Barnes in 1981 [46] and Green

in 1983 [47], also recognized the invalidity of the direct use of OCVD

Ge :EG 0.7eV
n 1013 cm-3, T = 300K

.LOPE = -(kT/e)/-r

TIME (t)

Fig. 3.2 Transient response of open-circuit voltage decay for Ge.


Voc (t)



S SLOPE = (kT/e)/T

TIME (t)

Fig. 3.3 Transient response of open-circuit voltage decay for Ge, excluding the part of
the fast transient which results from the fast relaxation times.

for Si devices and presented modified versions of the method designed to

extract the value of 7.

That OCVD works for Ge devices but not for Si devices stems from

the differences in the energy gap and hence in the intrinsic carrier

density n.. The intrinsic density of Ge exceeds that of Si by about

three orders of magnitude. From the theory of p/n junctions, the excess

hole and electron densities in the junction space-charge region are

proportional to n.; these mobile carrier densities contribute to the

quasi-static capacitance C of this region. This capacitance is

neglected in the ideal or conventional OCVD interpretation. In

contrast, the excess hole and electron densities in the quasi-neutral

base, whose recombination is the central feature of conventional OCVD,
are proportional to n. The pertinent ratio is n.. For Ge, the ratio

is so large that C can be neglected. For Si, and even more so for GaAs,

C markedly slows the relaxation of the excess holes and electrons,

producing a curving deviation from the ideal falling straight line. The

experimental.data for Si p/n junction diodes in Fig. 3.4 illustrate this

[45]. There the ideal OCVD decay, neglecting C, appears for contrast,

the value of r being determined by use of admittance-bridge techniques


3.3 Green's Compensated Open-Circuit Voltage Decay Method

The essence of Green's method is conveyed in Fig. 6 of Ref. 47.

The value of an external shunt resistance is adjusted until the

curvature in v(t) vanishes. Then Green uses this value in the

calculation of r, using a quasi-static analysis [47]-[48].

0.4 299K -
0 - "^T=4.4psec

> 6"K

0.2 = 7gsec ~

0.1 I i I I I I I
0 10 20 30 40 50 60
TIME (g sec)
Fig. 3.4 Open-circuit voltage decay response for a diode with base doping concentration
of 1.25 x 10 cm at 299 K and 376 K. Dash lines correspond to the lifetime T
determined by the capacitance method [45].

This approach significantly improves the accuracy of OCVD for the

determination of r. This is Green's experience, as well as ours. Our

experience consists in use of this method on six different types of

solar cells; we compare the value of r thus obtained with that derived

by others' methods and we assessed the method by another approach to be

described in the next section of this chapter. We had no success,

however, in extending compensated OCVD to enable determination of 7 and

SB, as would be desirable for back-surface-field (BSF) solar cells.

Our experience is that the method requires subjective judgement.

For some (long-base) devices, we could determine only that the optimal

value of the external resistor lay in a range large enough to produce

significantly larger error bars on the determined value of r than those

obtainable by other methods, such as those described recently by

Neugroschel [49] and by Jung et al. [50].

3.4 Extension of Compensated Open-Circuit Voltage Decay Method

Consistently with Green's observations, we noted that the near

absence of curvature in the voltage decay v(t) prevailed only for a

short interval of time. This finding, as indicated by Green, owes its

origin to the voltage dependence of capacitance C. Note that this

difficulty vanishes in the limit of small external resistance. This

limit leads to electrical short-circuit current decay (ESCCD) (Fig.

3.5), first demonstrated by Jung et al. [50]. A previous related

method, due to Rose and Weaver [51], employed optically excited short-

circuit current decay combined with open-circuit voltage decay to

explore r and SB.


t > 0:

'SC (t)


Fig. 3.5 Schematic illustration for the short-circuit current decay analysis.

In this method, the very small external resistance used to monitor

the current i(t) nearly provides a short-circuit across the junction

space-charge region. Thus the junction region attains a nearly

equilibrium condition in which the barrier voltage closely approaches

its equilibrium value and in which the excess hole and electron

concentrations practically vanish. Hence, in principle, only the

relaxation of excess charge in the quasi-neutral regions determine the

observed i(t). From this response, illustrated in Fig. 3.6, one can

determine both the recombination lifetime r and the back surface

recombination velocity SB of the quasi-neutral base. The response is

described by an infinite series of exponential decays

"o-t/rdi -t/Tr
i(t) = i4 ii(0)e = i(0)e +

2 ii(0)e (3.1)

where rdi is the decay time constant of the i-th mode, rd is the decay

time constant of the l-th mode and i.(0) is corresponding initial value

at t = 0. As shown in Ref. 50, the first mode dominates and the higher

decay modes can be neglected. One measures the slope of the straight-

line part of log[i(t)], the intercept of its extrapolation, and the

forward voltage, v(t < 0), which sets the initial condition. These

observables suffice to determine SB and r. The component of i(t)

deriving from conduction and recombination in the quasi-neutral emitter

contributes only to the initial part of the transient.

Cell Oscilloscope

v ---


loge i(t)



Slope a'Td

1(0) '

Fig. 3.6(a) The electronic switching circuit used in short-circuit
current decay method.
(b) Transient response of the short-circuit current decay on
a semilogarithmic scale.

From a mathematical standpoint, the poles corresponding to

relaxation in the quasi-neutral emitter mix the poles corresponding to

the fast relaxation of the quasi-neutral base. Following the near

vanishing of the resulting fast part of the response, only the dominant-

pole response remains. This contributes the straight-line portion of

the response shown in Fig. 3.6. from which SB and r are determined.

The circuit used for electrical short-circuit current decay (Fig. 3.6)

utilizes a MOS-transistor switch, which provides faster switching (- 1

ns) and a simpler configuration than the bipolar circuit of Ref. 50.

The switch must have a characteristic switching time one order of

magnitude smaller than the decay time that is observed. This decay time

may be as much as one order of magnitude smaller than the recombination

lifetime because the excess holes and electrons vanish from the quasi-

neutral base not only by volume recombination but by conduction through

the junction region on one side of the base and through the back surface

on the other. Further, the external resistance R(ext) interacts with

the equilibrium or zero-bias value of the junction capacitance C, and

the resulting time constant, R(ext)C, must be an order of magnitude or

more smaller than the decay time of the straight-line portion of

log[i(t)]. In this regard, note that R(ext) is the sum of the

resistance used in the external circuit for monitoring the current

decay, the resistance of the contacts to the device being measured, and

the series resistance of the device. Thus attaining the inequality,

R(ext)C << d, demands the exercise of some precautions. If the series

resistance of the solar cell or diode is large, other methods exist to

evade the difficulty.

One more precaution deserves comment. If the experimental

configuration uses contact probes rather than true ohmic contacts, a

standard four-point probe will yield the most accurate value of v(t <


3.5 Open-Circuit Voltage Decay with the Effects of the
Junction Space-Charge Region

As discussed in foregoing sections, open-circuit voltage decay

(OCVD) fails to accurately determine r of the base region because of the

accumulation of excess holes and electrons in the p/n junction space-

charge region under the forward voltage v(t) slows the decay of this

voltage. In circuit terms, we may think of this charge as constituting

a capacitance, dq(v)/dv C, in which q = jAn(x) dx fAp(x) dx. The

integration is over the thickness of the space-charge region, in a one-

dimension model, and A signifies the excess carrier concentration over

the equilibrium value. Note that the current of this capacitance

follows from dq/dt = (dq/dv)(dv/dt), where v is the both the external

voltage at the terminals and the junction-region barrier height

referenced to the equilibrium height, for the assumed low-injection

conditions. We may make the quasi-static assumption that q[v(t)] has

the same dependence on voltage as in the time-invariant steady state

provided the change in normalized voltage Av/(kT/e) is small in any time

interval At that is large compared with the transit time of holes or

electrons across the junction space-charge region. The result for C is

called the quasi-static approximation. Note that we have put only the

charge of mobile carriers, not the charge of immobile ions, into the

definition of C, in the space-charge region. This capacitance model

will be discussed in details in Chapter Five and Chapter Six of this


Green's compensated OCVD method tries to evade the influence of C

by the introduction of an external shunt resistor R(ext). If R(ext) -

0, one obtains Jung's ESCCD method, in which a short circuit wholly

removes the influence.

As an approach alternate to Green's, we now consider inserting C =

f[v(t)]. If C is accurately modeled, the dominant-pole or quasi-static

part of the response v(t) is not falling straight line. Rather, it will

show curveture. Then, if r in the model of the quasi-neutral base

region is adjusted to agree with the actual lifetime of the base region,

one will obtain by theory the response v(t) that is measured. A simple

computer program can accomplish the adjustment of r. This is the

essence of a new method (OCVDCAP) for determining r, which we now

briefly discussed. The detailed development of the OCVDCAP method will

be given in next chapter.

The adequacy of the method depends on the accuracy with which C =

f[v(t)] is modeled. Although widely used, the depletion approximation,

which yields the Schottky or depletion capacitance holds only for

reverse voltages of magnitude much greater than kT/e. For the

approximate range, -3kT/e < v < Vbi 0.3, where Vbi is the built-in

potential, C follows the power-law functional dependence of the

depletion capacitance with the exception that an effective built-in

potential V replaces Vbi. Chawla and Gummel [20] have demonstrated

this and provided expressions for V which accounts for the presence of

mobile carriers in the volume of the junction space-charge region. The

resulting expression is

C = C0[l + v/V ] (3.2)

where CO is the space-charge region capacitance for zero applied voltage

and 1/3 s r s 1/2. For a step junction, r = 1/2; for a linear-graded

junction, r = 1/3. For a linear-graded junction, V can be expressed as


V = (2kT/3e)(n(EkTa2/8e 2n.3), (3.3)

where a is the junction gradient and e is the dielectric permittivity.

As their main effect, the electrons and holes in the space-charge region

reduce the net charge fp(x) dx, including the immobile ionized charge,

to a level below the charge calculated by the depletion approximation.

This decreases the equilibrium barrier height and thus the equilibrium

thickness of the space-charge region below the values predicted by the

depletion approximation. Thus, at equilibrium (v = 0), V < Vbi. This

inequality continues as v increases. For an exponential-constant

profile, which approximates a profile resulting, for example, from

phosphorous diffusion into a constant background p-type dopant

concentration, V is a mild function of v [20]. Table 3.1 demonstrates

C for GaAs, Si, and Ge over the range of validity of Chawla and Gummel

23 -4
model for a linear-graded junction with junction gradient a = 10 cm


Capacitance C of Junction Space-Charge Region for a Linear-Graded
Junction with Junction Gradient a = 1023 cm-4 for GaAs, Si and Ge

C v = 0.0 v = 0.1 v = 0.2 v = 0.3 v = 0.4 v = 0.5 v = 0.6 v = 0.7
(10-8f/cm2) (volts) (volts) (volts) (volts) (volts) (volts) (volts) (volts)

C for GaAs 11.6 11.9 12.3 12.6 13.3 14.0 14.7 15.7
C for Si 12.2 12.8 13.5 14.4 15.5 17.2 19.6
C for Ge 18.8 20.7 23.7 29.9 --

Note that the ratio of the capacitance of the quasi-neutral base to C is

approximately proportional to n 2: 10 cm for GaAs, 1020 cm for Si,
26 -6
and 10 cm for Ge. This demonstrates the increasing importance of C

to OCVD as energy bandgap increases.

The range of validity of the analytical expressions of Chawla and

Gummel, indicated above, corresponds to an upper-bound voltage of

slightly less than 600 mV for many silicon solar cells and diodes [20].

Thus the Chawla-Gummel expressions will suffice because OCVD measurement

can be done for v < 600 mV.

In the OCVDCAP approach, we combine C vs v(t) with a dominant-pole

or quasi-static model of the quasi-neutral base. We use either a

Linvill model [52], which emphasizes minority-carrier density, or a Sah

equivalent circuit [40], which emphasizes the minority-carrier

electrochemical potential. This combination, together with a computer

program, yields the OCVD response, v(t), including the effect of C.

This theoretical response is a function of r. The computer program

adjusts r to obtain agreement between the theoretical and measured

responses. From this procedure, one determines r. Details appear in

Chapter Four.

3.6 Effect of Heavy Doping Emitter Region on Open-Circuit
Voltage Decay Method

In ESCCD method, the short circuit present across the junction

depresses the excess hole and electron densities to zero at both the

quasi-neutral emitter and the quasi-neutral base edges of the junction

space-charge region. Thus the poles in the complex frequency plane

corresponding to the quasi-neutral emitter simply add to those

corresponding to the quasi-neutral base. Because the poles associated

with the emitter produce short relaxation times relative to the

dominant-pole relaxation time of the base, which is the observable used

in determining r and SB, the transient response from the emitter has no

effect on ESCCD method. In this sense, the response from the heavily

doped emitter is decoupled from that of the base. This statement holds

despite the presence of so-called heavy doping effects, which may yield

a large steady-state recombination current of the quasi-neutral emitter.

This issue is more complicated for OCVD and RR methods. In these

methods, the forward voltage present during the entire transient causes

a correlation to exist between the excess hole and electron densities at

quasi-neutral base edge of the junction space-charge region and the

densities at the quasi-neutral emitter edge. Thus a coupling exists

between the poles (or natural frequencies) in the two regions. We have

emphasized, however, that OCVD, RR, compensated OCVD, and OCVDCAP have

practical utility only for long-base devices in which the thickness of

the quasi-neutral base greatly exceeds the minority-carrier diffusion

length of that region. For such solar cells and diodes, Jain and

Overstraeten [53] have demonstrated mathematically that the relaxation

of the excess carriers in the quasi-neutral emitter negligibly

influences the determination of r by OCVD, RR, and related methods. An

implicit assumption underlies their conclusion: the base recombination

lifetime r greatly exceeds both the space-averaged recombination

lifetime and the minority-carrier transit time in the heavily doped


From a physical or phenomenological viewpoint, one can see that

the key to this conclusion is the existence of a dominant pole (at -

1/r). By dominant pole, we mean a pole that is much nearer to the

origin of the complex frequency plane than all the other poles. For

such a system, following a time interval characterized by fast

relaxation times, the transient response will very closely follow a

response proportional to exp(-t/r), for the minority-carrier density

despite the coupling between the excess densities in the emitter and the

base. The decay rate of the system is limited by the dominant pole.

For a short-base device, the various parameters of the heavily doped

emitter enter to influence the OCVD or RR response [53] because of the

absence of a dominant pole.

3.7 Frequency Response Methods

This chapter has emphasized transient responses for the

determination of r and SB. But, as we have indicated, we have used

admittance-bridge or frequency response methods, where applicable, to

assess the adequacy of results determined by methods involving

transients. Figure 3.7 illustrates one of the useful frequency response

methods. The intersection at I shown there depends on r and SB for BSF

cells or other solar cells employing a partially blocking contact at the

back surface. The minority-carrier diffusion length L is given as [54]

10 SB= 90 cm/s
o 10-

S Gaf

0 1 III>I

103 3 1 104 105


Fig. 3.7 The frequency response, from the admittance-bridge method, of the conductance of
a n+/p/p diode with diffusion length L and surface recombination velocity SB

2XBD /2 D 1 + (S L2/DXB)
WL) ( [ (D/XB) + Sg

where D is the diffusivity of the minority-carrier and XB is the base

thickness. Details of this method appears in the work of Gonzalez and

Neugroschel [54].

3.8 Conclusions and Summary

We list the main points of this chapter as follows:

1. Open-circuit voltage and reverse recovery serve adequately to

determine the base recombination parameters such as minority-carrier

lifetime r for Ge but not for Si (or GaAs) devices. This statement

holds for any related methods of measurement that focus on a

transient during which the junction remains under forward voltage.

2. The inadequacy of OCVD and RR for Si (and GaAs) arises from the

relaxation of the charge of mobile holes and electrons associated

with the quasi-static capacitance C of the junction space-charge

region, which is a nonlinear function of forward voltage v(t). The

decay of this charge has a strong influence for Si (or GaAs) because

the intrinsic density of these materials is orders of magnitude less

than that of Ge. Energy-gap contraction and impact (Auger)

mechanisms in the heavily doped emitter will not influence OCVD or

RR provided the dominant pole or natural frequency of the base

greatly exceeds the dominant pole or natural frequency of the

emitter. This condition will hold for most silicon solar cells

having moderate or high power conversion efficiency.

3. Because of the complications involved in compensating the influence

of C or in including C = f[v(t)] in the analysis, OCVD, RR and

related methods have practical utility only for determining r (not

for determining S also). This means that these methods apply only

to long-base devices. From a practical viewpoint, for moderate or

high efficient Si solar cells, this implies that these methods apply

only for base doping concentrations of the order of magnitude of
17 -3
10 cm Because of the usual dependence of r or diffusion length

on doping concentration, the back surface will affect the observed

decay for lower doping concentrations and the lifetime will be so

short for higher concentrations that the conversion efficiency will

be low.

4. Green's use of a compensating resistor to offset the influence of C

in the OCVD response is an approach in a direction toward more

accurate determination of r.

5. An extension of Green's method, in which one lets the compensating

resistor become a short circuit, yields the electrical short-circuit

current decay method. ESCCD can provide an accurate determination

of both S and r. One can get SB very accurately if XB/L < 1. The

converse of this inequality must hold to get very accurate

determination of r. If this ratio is of order of unity, both r and

SB can be determined with good accuracy by ESCCD.

6. ESCCD yields good agreement with the results of admittance-bridge or

frequency response measurement. The frequency response methods

require an alteration of the structure to yield both S and 7,

whereas ESCCD allows this determination on an unaltered solar cell

or diode.

7. Once one has obtained S and r for the base from ESCCD, a straight

forward decomposition of the dark or illuminated I(V) characteristic

will yield the steady-state recombination current for the quasi-

neutral emitter. This then can be used to explore the effect of the

heavily doped emitter on designing the conversion efficiency.

8. A new method, OCVDCAP, presented here for the first time, uses

an analytical characterization of C = f[v(t)]. This combined with a

dominant-pole or quasi-static model for the base region enables a

determination of r for long-base devices. The method may find use

for in-process control in manufacturing, where it is desirable to

employ contact or pressure probes rather than ohmic contacts.


4.1 Introduction

The open-circuit voltage decay (OCVD) technique is a simple method

by which, in principle, the minority-carrier lifetime r in the base

region can be obtained [42], [55]. The early papers [42], [55] dealt

with Ge diodes for which the OCVD transient results almost entirely from

the quasi-neutral base (QNB) region. But a difficulty arises for Si

diodes and solar cells resulting from a large influence from the storage

of the free holes and electrons in the p/n junction space-charge region.

Thus the nonlinear capacitance C of a junction space-charge region

subjected to forward voltage enters into the analysis of the OCVD

response, thus making the analysis more difficult.

Several methods have been developed that try to overcome this

difficulty [46]-[47], [51]. Manhan and Barnes [46] include a constant

depletion capacitance C(DEP) in the conventional OCVD analysis. This

enables an analytic solution for the transient, but C(DEP) inaccurately

accounts for the nonlinearity of C. Green [47] develops a circuit which

he indicates can compensate for the effect of C. The idea is good in

principle but it suffers, in part, in our experience, from the absence

of a clearly defined local minimum portion of the transient from which

Green infers the lifetime. Rose and Weaver [51] propose a method

involving laser photoexcitation of holes and electrons. They seek the

value of r by examining the resultant open-circuit and short-circuit

decay characteristics. The method does not include the effect of C.

The purpose of this chapter is to describe a method that improves

the accuracy with which OCVD analysis can determine r. The method

utilizes the traditional OCVD measurement. The interpretation of the

measurement differs, however, in the sense that the device model

includes the junction space-charge region capacitance as well as the

quasi-neutral base region. The model for the capacitance contains its

nonlinear dependence on the forward voltage [20]. The presence of this

nonlinearity requires the use of a simple computer program to derive the

value of T from the observed transient.

4.2 Theory

4.2.1 Linvill Model and Electrical Counterpart

In developing the theory for the method, we use the Linvill lumped

model [52], [56]. The Linvill model is a network-like representation of

the continuity equations and the hole and electron current equations,

and of the relations between the nonequilibrium junction voltage v(t)

across the p/n junction space-charge region and the carrier densities at

the two edges of this regions. For low injection, which we assume

prevails, linear relations exist between the minority-carrier density

and the minority-carrier current in the quasi-neutral base and emitter

regions. The Linvill model exploits this linearity by use of elements

like electrical circuit elements but different from such elements

because they relate minority-carrier current and density, thus avoiding

the nonlinearities between current and voltage.

In Fig. 4.1, we show a simple version of the Linvill model for the

base region of a n/p long-base solar cell or diode having a position-

independent base doping density and negligible contributions from the

quasi-neutral emitter region to either the static or transient response.

The box shown in this figure represents the relation between the excess

minority electron density An(0,t) at the base edge of the junction

space-charge region:

An(x=0,t) = n0exp[v(t)(kT/e)], n0 = ni /NB (4.1)

where NB is the base doping concentration. For the long-base diode

under study, the element values of the storance S and the combinance H

S eL, H S/r. (4.2)

These dependencies on diffusion length L and recombination lifetime r

follow from the usual quasi-static approximation that the minority

electron density An(x,t) retains during transients the same x dependence

that it has in the steady state:

An(x,t) = An(0,t)-exp(-x/L).




S H c
r 1]

An (x=O, t)

Fig. 4.1 The simplest Linvill model for a long-base diode. The excess electron density
obeys the usual exponential relation to the voltage v(t). The storance S is
analogous to a capacitor C' in the sense that is = S(d/dt)An(0,t); and the
combinance Hc is analogous to a conductance G': iH = HcAn(0,t).

To demonstrate that the Linvill model enables simple electrical

circuit theory to be applied to a problem of semiconductor device

physics, we now derive the standard result for the OCVD experiment. We

assume a current source i(t) in the forward direction to be applied to

the terminals for t < 0; for t > 0, the current is set zero. Thus

0 = i(t) (Hc + sS)An(O,t), s = d/dt (4.4)

from which s -H /S = -1/r. Thus

n0exp[v(t)/(kT/e)] = An(0,t) = An(0,0)exp(-t/r) (4.5)


v(t)/(kT/e) = ln[An(0,0)/n0] t/r, t > 0. (4.6)

This is the standard quasi-static result for OCVD.

Using the Linvill model, we may include the effect of the

minority-carrier transient in the quasi-neutral emitter by adding a

combinance and storance to the left side of the box representing the

junction space-charge region in Fig. 4.1. The currents through these

network-like elements are driven by the minority-carrier density Ap at

the emitter edge of the space-charge region, which is related to An(O,t)

by the ratio of the minority-carrier equilibrium densities in the two

regions. Thus the effect of the minority-carrier transient in the

emitter follows from a Linvill model exactly like that shown in Fig.

4.1, except that the element values become

S S + [POE/nO]SE, Hc Hc + [POE/nO]HcE (4.7)

where pOE is the equilibrium minority-carrier density at the emitter

edge of the junction space-charge region, SE is the storance of the

emitter, and HcE is the combinance of the emitter. SE and HcE are

calculated using the physics appropriate to a heavily doped emitter. We

see that the emitter will have no substantive effect on the observed

OCVD transient provided

Hc/HcE and S/SE P0E/n0 (4.8)

This condition holds for Si devices unless the base region is very thin,

a fact determined by Jain and Van Overatraeten [53] by different


To allow direct use of circuit-analysis computer codes, we note

that for every Linvill model there exists a nonlinear electric-circuit

model. As an example, the electric-circuit counterpart of Fig. 4.1,

shown in Fig. 4.2, follows by setting currents in the storance and the

combinance equal to currents in the capacitance and conductance and by

regarding the hole electrochemical (quasi-Fermi) potential 0p at the

base edge of the junction space-charge region as the reference potential

(ground). Then the minority electron electrochemical potential qN at


C1 I Lo
C'- G

Fig. 4.2 The electric-circuit counterpart of Fig. 4.1 in which, in contrast to S and Hc,
C' and G' depend nonlinearly on v(t).

this edge obeys the relation: (N = v(t). This procedure yields the

dependence of the circuit elements in Fig. 4.2:

C' = eLn0exp[v(t)/(kT/e)]/(kT/e), (4.9)

G' = C'(kT/e)/rv(t). (4.10)

Applying the zero-current condition of the OCVD experiment to the

electrical circuit of Fig. 4.2, we find from Kirchhoff's nodal law that

(l/v(t))(dv(t)/dt) = -G'/C' -(kT/e)/rv(t), t > 0 (4.11)

from which follows the slope in (4.6) which one uses determine T from

the observed transient.

4.2.2 Generalizations and Inclusion of'the Space-Charge Region

The OCVD circuit diagram is shown in Fig. 4.3. We will utilize

Linvill's model [52], [56] for the quasi-neutral base. For the junction

capacitance C we utilize Chawla and Gummel's model, which is valid for

all voltage up to an upper-bound voltage slightly less than 600 mV for

most silicon junction diodes [20]. Fig. 4.4 illustrates the resulting

model which we call the OCVDCAP (open-circuit voltage decay with

capacitance) model. The effects of the quasi-neutral emitter region are

neglected for the reason stated in Sec. 4.2.1. The applied voltage at t

< 0 for OCVDCAP analysis must be less than 600 mV for Chawla and

V t=O+ sec

a U

*I I

I a



Fig. 4.3 Schematic illustration of the open-circuit voltage decay analysis.


N p

C v(t)
c HO
jS1 'HC1 NS2 NHC2 |HS

S1 I I I I I

An (x=0, t) Hd An (x=XB, t)

Fig. 4.4 A quasi-static model for a diode having an arbitrary effective recombination velocity at
the back surface. The current (density) j(Hd) through the diffusion obeys the relation:
j(Hd) Hd[An(0,t) An(XB,t)]. All circuit elements, except C, are linear.

Gummel's model to suffice. The elements of the circuit in Fig. 4.4 are

[52], [56]

S1 = S2 = eL[l cosh(-XB/L)]/sinh(-XB/L), (4.12)

Hcl Hc2 S1/r, (4.13)

Hs = eSg, (4.14)

Hd = eD/XB, (4.15)

where L is the minority diffusion length, r is the minority lifetime, XB

is the base thickness, D is the minority diffusivity, and SB is the

effective back-surface recombination velocity. These expressions hold

for any ratio of XB/L; the relations of Sec. 4.2.1 hold only for long-

base diode.

Following the concepts explained in Sec. 4.2.1, we obtain for the

electric circuit counterpart (Fig. 4.5)

C1 AS1An(x=O,t)/(kT/e), (4.16)

C2 ASlAn(x-XB,t)/(kT/e), (4.17)

G1 = Cl(kT/e)/[roN(x=O,t)],



.j1C gC1 G1 IC2 G2 Gs


ON (x=O, t) = v(t) ,N (x=X B, t)
Fig. 4.5 The counterpart equivalent electric-circuit model of the Linvill's model in Fig.
4.4. The terminal corresponding to *N(x=XB,t) is not an actual device
terminal. It represents the excess electron density at x=XB and becomes zero
for a long-base diode or if a minority-carrier ohmic contact exists at x=XB.
For t=0, C << C1 for the diode explored in Fig. 4.7; for t > 5 T for the diode,
C > C1, as implied in Fig. 4.7.

G2 C2(kT/e)/[roN(x=XB,t)], (4.19)

Gs AeSBAn(x=XB,t)/ON(x=XB,t), (4.20)

where A is the area of the diode space-charge region and where, as noted

earlier, N (x=0,t) = v(t). The transformation of Hd into conductor Gd

follows from equating currents in Hd and Gd:

AHd[An(x=0,t) An(x=XB,t)] An[qN(x=0,t) ON(X=XB,t)]/XB

Gd1[N(x=0,t) N(X=XB,t)]. (4.21)

This yields

Gd = (AeD/XB)([An(0,t) An(Xg,t)]/[4N(0,t) N (XB,t)]}, (4.22)

where D is the diffusivity of the minority carrier in the quasi-neutral

base region.

In general, the circuit of Fig. 4.5, and the definitions in (4.16)

to (4.20) and (4.22), can be solved numerically. For example, we note

that 4N(XB,t) = 0 for a long-base diode, and that for a short-base diode

ON(x=XB,t) (kT/e).-n(l + (exp[ON(x=O,t)/(kT/e)] 1).

((D/XB)/[SB + (D/XB)]}),


which is derived by combining,

-Ddn(x,t)/dx = D[An(0,t) An(XB,t)]/XB = SBAn(XB,t), (4.24)

where the derivative takes place at XB, and

An(x=XB,t) = {(D/XB)/[SB + (D/XB)]}.An(0,t). (4.25)

For these special cases, the results above may simplify the numerical


4.2.3 Determination of Lifetime

We now outline the procedure for determining r by using the

OCVDCAP model of Sec. 4.2.2. We assume that SB is either known at the

outset, as it would be for an ohmic contact, or that the diode obeys the

long-base condition: XB/L > 1. The procedures are based on the model

of Linvill's form (Fig. 4.4); similar procedures hold for the electric-

circuit counterpart, which admits direct use of existing nonlinear

circuit-analysis computer programs.

We use finite-difference approximations of the nonlinear equations

corresponding to the model of Fig. 4.4, choosing time intervals At small

enough that C is constant, to a good approximation, in each time

interval. This removes the nonlinearity of the equations. We start in

the first time interval defined by to < t < tl, and use the same

procedure for any subsequent time interval tk < t < tk+l, where k is a

positive integer.

For the open-circuit condition under study, the circuit of Fig.

4.4 implies that the current density j flowing through C equals the sum

of the current density jS1 through Sl, jHcl through Hcl jS2 through S2'

jHc2 through Hc2, and jHs through Hs. Thus [52], [56]

je = S dAn(x-0,t)/dt + S2dAn(x=XBt)/dt +

HclAn(x=0,t) + Hc2An(x=XB,t) + HsAn(x=XB,t).


In any time interval,

jck = (Vk+l

- k)/(tk/Ck) (tk+/Ck+l)].

Note that subscript k and k+l represents t-tk and t=tk+l, respectively.

Solving (4.26) for An(x=0,t) at t=tk+l (Ank+l), we get

Ank+l = (ck/Hc') + (Ank ck/H')exp[-(tk+ tk)Hc'/Sl],


where we have exploited the linearity between An and j In (4.28)

H Hcl + ([Hs + Hc2 + (S2SB/XB)]((D/X )/[SB + (D/XB))l,


which reduces to Hcl for long-base diodes. Note that j is determined
cl ck


by an assumed value of vk+l. Therefore, Ank+l is determined for an

assigned trial value for T. We obtain a corresponding voltage (v') from

this value of An by using (4.1):

v'k+l n[(Ank+NBg/n 2) + l].(kT/e). (4.30)

Thus, by comparing the assumed vk+l and calculated v'k+l and iterating,

we obtain the correct value for v at each time interval. Figure 4.6

illustrates the numerical procedure.

Thus, given the observed OCVD transient v(t) and knowledge of the

thickness XB, doping concentration N and other parameters of the

device make-up, we vary r in the numerical procedure until the computed

v(t) agrees with the observed v(t). The value of r yielding this

agreement determines the minority-carrier lifetime in the base region.

4.3 Illustration

We now illustrate the OCVDCAP method by its application to four

different types of solar cells. The value of r obtained from the

OCVDCAP method agree with those obtained by other experimental

techniques: the admittance-bridge method [45], the short-circuit-

current-decay method [50], and an improved short-circuit-current-decay

method [57] (see Table 4.1). Fig. 4.7 illustrates the OCVDCAP method

for one of the solar cells.

To compare the OCVDCAP method with Green's compensate OCVD method

[47], we connect a compensating resistor R across the junction (Fig. 6


tK < t < tK+1





SOLVE V' (t K+I) FROM An(X=0, t K+1)

V' (tK+1) = V(tK+1)



Fig. 4.6 Flow chart of the OCVDCAP numerical procedures.






Device Makeup (NB, XB, A), the Back Surface Recombination Velocity SB
(Determined by the Short-Circuit-Current Decay Method), and the Minority-
Carrier Lifetime T Determined by Four Methods:

* TOCVD by the Present Method
* TSCCD by the Short-Circuit-Current Decay Method
* TISCCD by an Improved Short-Circuit-Current Decay Method
* TAB by the Admittance-Bridge Method









(u) (us) (u~s)


380 50
1540 25
104 12


(V) 0 O Measurement
0.6 -- OCVD Without C
-- OCVD With C
0 NB = 1.25x1015 cm'3
O *T = 4.0 ps, Boron Diff.
0.4- o


III I I I I t(lS)
0.0 20 40 60 80
Fig. 4.7 Open-circuit voltage decay transient from the OCVDCAP method with T = 4 psec
(solid line) and from OCVD measurement (circles). The device has a base doping
concentration of 1.25 x 101 cm -. The junction profile is assumed linear.

of Ref. 47), which is the same as connecting Rc parallel with C in Fig.

4.4. Fig. 4.8 shows a transient response of the compensated circuit for

the same device as that used in Fig. 4.7.

For a trial value, r 4.0 ps, the measured response nearly

coincides with the response calculated (Fig. 4.8) for the composite

circuit formed by adding the compensating circuit of Fig. 6 of Ref. 47

to the model of Fig. 4.4. Thus we demonstrate that the combination of

Fig. 4.4 and the computational procedure of Fig 4.6 is applicable to

various problems apart from the OCVDCAP problem emphasized here; in

particular this combination replicates experimental data derived from

using the method of Green's compensated differential open-circuit

voltage decay [47].

To explore the sensitivity of r on the nonlinear transition

capacitance, we use a depletion capacitance in the present method to

determine r for the same device as that for Fig. 4.7. We obtain r 6.5

ps which compares with the correct r of 4.0 ps obtained by our method in

agreement with the admittance-bridge method of Ref. 45.

The OCVDCAP method determines either minority-carrier lifetime r

or the back surface recombination velocity S The method has direct

utility in two cases: (a) for long-base diode, for which S is

irrelevant, and (b) for a back-ohmic contact short-base diode, for which

SB approaches infinity. In either case, OCVDCAP can determine r. It

may also find use in a more unusual case in which the back surface

recombination current greatly exceeds the bulk or volume recombination

current. Then OCVDCAP determines SB.


-- Model with R = 4kK
----Model without RC
T = 4.0 s 15
NB = 1.25x101 cm
Boron Diff.



t (/s)
Fig. 4.8 The simulated transient response of Green's compensate OCVD circuit
compensating resistor R = 4 kn and with R o . The deviee is the
used in Fig. 4.7.

with a
same as that



The version of OCVDCAP presented has assumed implicitly that the

recombination current of the junction space-charge region and through

shunt conductances are small, which are good assumptions for many

silicon devices for voltages in the 500-600 mV range relevant for

OCVDCAP. To include these effects, elaborate versions of OCVDCAP may be

obtained by the addition appropriate circuit elements to the model of

Fig. 4.4. Other modifications of the model enable determination of a

position-independent r in drift-field solar cells.

In comparison with other accurate methods for determining r,

OCVDCAP has an advantage in monitoring the influence of processing steps

in fabrication. OCVDCAP requires only pressure or probe contacts on the

sample because measurement of open-circuit voltage involves negligible

current flow through the contacts.


5.1 Introduction

The quasi-static capacitance C of a p/n junction space-charge

region plays an important role in Si and GaAs-based bipolar device

switching speed and frequency response, particularly for GaAs because of

its relatively large energy gap. Because injected electron and hole

densities, n and p, in the volume of the space-charge region contribute

to C and because n and p depend nonlinearly on the electrostatic

potential V., most previous treatments of C for forward voltages rely on

numerical analysis [4]-[5], [8], [11], [14], [17]-[18], [20], [24]. The

analytical solution for C can be carried out if the junction is under

reverse bias so that the mobile holes and electrons can be neglected

[7]-[8]. Reference 21 gives the simplest analytical model for C under

forward voltage, and is one of only a few analytical models [12], [21]-

[23]. Reference 21 differs from other work on the quasi-static

capacitance because it departs from the usual steady-state analysis in

its use of reasoning in the time domain.

The capacitance model of Ref. 21 suffers the following defects:

(a) It assumes n(x) = p(x) in the junction region, without

indicating the range of applicable applied voltages.

(b) It fails to distinguish between the applied voltage and the

voltage that appears across the junction region, an issue that


becomes important at high forward voltages where high-injection

conditions and resistive effects can exist.

In this chapter, we correct these defects, discuss the insight

provided by time-domain reasoning, fill in some details not present in

Ref. 21, and compare the capacitance model against experimental results

and those deriving from device simulation.

5.2 Derivation by Detailed Time-Domain Reasoning

5.2.1 Review and Elaboration

A time-domain analysis enables separation of the junction

capacitance into dielectric and free-charge components and leads to a

simple picture of the junction capacitance not obtained by previous

steady-state analyses. To illustrate this, we review briefly the

reasoning used in Ref. 21, including some details not present in Ref.


Consider the application of a step.-function voltage dVA at the

terminals of a diode (Fig. 5.la). For t = 0 holes and electrons

redistribute themselves rapidly, congregating at the base edge and the

emitter edge of the junction space-charge region, respectively (Fig.

5.1b). The time required for this first phase of the transient is

determined by two considerations. First, a characteristic time enters

from Maxwell's equation, V X H = j + aD/8t; taking the divergence of

both sides of this equation yields, if we neglect the diffusion

component of current, that the holes and electrons redistribute within

the order of a dielectric relaxation time, e/a. This procedure also






1I\ 1

t tr+ ttr

Fig. 5.1(a)

Application of a step voltage dVA to a diode.
Resulting change in charge density dp for t ~ tr.
Resulting changes in electron and hole density for t ~
tr + ttr. The dash line represents electron
distribution and the solid line represents hole


demonstrates that the total current, which is the sum of the

displacement and the convection currents, has no divergence and is

position independent in a one-dimensional model, a fact we will use

later. Second, a characteristic time enters from the resistivity and

geometry of the quasi-neutral regions which introduces an RC time

constant: R c 1/a and C c e. The constant of proportionality depends on

geometrical factors: the thickness of the space-charge region X and the

base thickness XB. Thus the composite characteristic time t for the

first transient phase following the sudden change in the applied voltage

dVA is (e/a)(l + X/Xg) 1, instead of e/a as stated in Ref. 21. If for

now we assume low-injection and negligible series resistance, dVA = dV

- dV -dVB, where VB is the junction barrier height, V is the voltage

drop across the space-charge region, and V is the separation of the

quasi-Fermi potentials in the space-charge region. In Ref. 21, the

meaning of V is clear only for low-injection. We will discuss high-

injection in Sec. 5.4.

Therefore, for t tr, charge dQD resulting from excess majority

carriers piles up at the edges of the junction space-charge region (Fig.

5.1b). Recognizing the similarity of this charge distribution to that

of a parallel-plate capacitor, we obtain

CD = dQD/dV = E/X. (5.1)

This capacitance CD is called dielectric capacitance because the only

material parameter appearing in the expression is E.

Following the first time phase, which for typical junctions

happens very fast relative to subsequent phenomena, the majority

carriers on both sides of the junction start to diffuse over the lowered

barrier height. Within a time of the order of a transit time ttr across

the junction region, this mechanism fills the space-charge region with

holes and electrons, constituting a storage phenomena that also

contributes to the capacitance of the space-charge region (Fig. 5.1c).

According to Ref. 21, the free-carrier charge QF of either holes or

electrons stored in the space-charge-region volume is

QF en .Xexp(V/2VT), (5.2)

where VT is the thermal voltage and V is the separation of the quasi-

Fermi levels in the region. Thus

C (5.3)
F aV constant 2T

where CF denotes the free-carrier capacitance which results from dQF.

Equations (5.2)-(5.3) rely on the assumption [21]

n(x) = p(x) = niexp(V/2VT). (5.4)

Therefore, (5.3) is meant to describe the capacitance only for large

forward voltages, a point not made explicit in Ref. 21.

According to Ref. 21, the total charge variation dQ is dQD + dQF.

Thus the junction space-charge region capacitance C is [21]:

C dQ/dV = CD + CF = e/X + yX, (5.5)

where 7 is a function of V determined by (5.2) and (5.3). For reverse

voltage, CF = 0 and C CD = e/X, which is the conventional depletion

capacitance. Equation (5.5), together with the condition dC/dX = 0

(Fig. 5.2), leads to simple expressions for C(V) and X(V) [21]:

X(V) e 1/2exp(-V/4VT) (5.6)


C(V) = 2CD = 2CF [2een VT-1 1/2exp(V/4VT). (5.7)

The physics which underlies the condition dC/dX 0 will be discussed in

Sec. 5.5.

5.2.2 Interpretation

Reasoning in the time domain makes possible the separation of QD

and QF' charges that enter the physics that determines C; the charge QD

does not appear in the steady-state treatments that underlie almost all

previous work on the junction quasi-static capacitance. The reasoning

in the time domain focuses on a short time interval, approximately the

+ CF

d 0WA

Fig. 5.2 Qualitative illustration of the dependence of CD, CF, and CF + CD versus X.

sum of t and t after a step-function voltage is applied (Fig. 5.1).
r tr
During this time, current from the external circuit lowers the barrier

height (charge QD) and fills the space-charge region with free carriers

(charge QF).

To clarify the physics involved, we employ the nonlinear

transmission-line model of Sah [40], which is a finite-difference

network representation of the two continuity and current equations, the

Poisson equation, and equations describing the generation,

recombination, and trapping of holes and electrons. In the simplified

version of this model shown in Fig. 5.3, we neglect generation-

recombination-trapping mechanisms and focus on the space-charge region

by treating the quasineutral regions of the junctions as if they were

perfect conductors for 0 < t < (tr + t ), an approximation that places

short circuits at the edges of the space-charge region (Fig. 5.3). This

treatment of the quasineutral regions is adequate for the qualitative

discussion of this section. This three-line transmission line is

driven, from top to bottom, by the hole electrochemical potential, the

electrostatic potential, and the electron electrochemical potential.

The longitudinally directed capacitances correspond to net charge

storage (Poisson's equation). The vertically directed capacitances

correspond to the electron and hole storage (an/8t and 8p/8t in the

continuity equations). The conductances correspond to dependence of the

hole and electron currents on the product of the particle conductivities

and the gradient of the particle electrochemical potentials. For

simplicity, Fig. 5.3 shows a four-section model corresponding to


Fig. 5.3 Sah's network representation for the space-charge region of a n/p diode,
where CN = n(x)Ax and Cp p(x)Ax. The direction of the current for the
first time phase is indicated by arrows.


dividing the thickness X into four equal slices, each of thickness Ax =

X/4, although good accuracy would require many sections. The vertically

directed capacitors that correspond to majority-carrier storage are

proportional to the majority-carrier density [40]; thus CN n at the

emitter edge and Cp c p at the base edge of the space-charge region are

large compared with the horizontally directed capacitances and for t <

t could be replaced by short circuits without introducing error.
Therefore, current initially follows the path shown in Fig. 5.3.

Applying a step-function voltage dVA to this model, we see that

the resulting initial impulse of current deposits time-independent

charge dQD on the longitudinally directed capacitors, the series

connection of which yields a capacitance CD = e/X for unit cross-

sectional area. Subsequently, the electron and hole currents in the

resistors charge the vertically directed capacitors, which corresponds

to a more gradual filling of the space-charge region with charge dQF =

ef(dn or dp)dx on the vertically directed capacitors. The sum of these

two charges thus corresponds to the charge on the quasi-static

capacitance C, and the main points of the time-domain argument agree

with the qualitative conclusions reached using the Sah network

representation. The Sah transmission line also emphasizes that space-

charge region fills gradually with electrons and holes, a non-quasi-

static effect, ignored by quasi-static analysis, that gives the

frequency-domain junction admittance a real as well as an imaginary

part. In the steady-state condition eventually reached after step-

function voltage is applied, the Sah transimission-line model for the

junction region reduces to two parallel resistors driven by the quasi-

Fermi potentials N and 0p. No capacitance appears in this model. That

fact suggests why the steady-state analysis used by most past studies

fails to yield a simple analytical model for the forward-voltage

junction capacitance. An alternate method is to incorporate Leibnitz

rule from calculus and the definition of C, together with the physics

that the minority carriers at the edges of the space-charge region does

not contribute to C. The details of this alternate method appear in

Appendix A.

Figure 5.4 displays this junction physics in another manner. As

observed earlier, the sum of the displacement and convection currents in

a one-dimensional model is independent of position x. Thus, directly

after the step-function voltage is applied, the total current has only a

convection component in the quasineutral regions and only a displacement

component in the space-charge region. But 8D/at caE/8t, implying that

at first the current from the external circuit serves only to change the

voltage across the junction region, thus lowering the barrier height.

Subsequently the convection components an/at and ap/at in the space-

charge region increase, leading ultimately to a charge dQF for t = t +

ttr. Figure 5.4 plots the total current density normalized so that it

is independent of time, enabling the lesser current at later times to be

conveniently displayed.

5.3 Applicable Voltage Range for Equation (5.7)

In this section, we briefly outline an iterative scheme for

determining the applicable voltage range for (5.7). The iteration,

displacement current

_total (normalized)

t Increase \
\ '-- Convection
I Current

Quasi-Neutral Space-Charge Quasi-Neutral
Emitter Region Base

Fig. 5.4 Qualitative illustration of the normalized total current, the convection
current, and the displacement current densities versus position.

which will be discussed in details in Chapter Six, relaxes the

assumption, p(x) = n(x), and employs (5.6)-(5.7) as the zeroth order

(0) (0)
model, X and C

Consider a linear-graded junction. The hole concentration in the

space-charge region is

p(x) = niexp([Op Vi(x)]/VT), (5.8)

where 0p is the hole quasi-Fermi potential and Vi is the electrostatic

potential in the space-charge region. For large applied voltage and

thus a small barrier height, we make the approximation that V. depends

linearly on x:

V.(x) = Vi(0) VBx/X, (5.9)

where VB is the junction barrier height. Hachtel et al. [60] developed

a one-dimensional program for p/n junction devices. Their Fig. 8

indicates that dp(x), resulting from dV, is sizable at the edges of the

space-charge region but is small in the middle of the space-charge

region. Physically, this is expected because most practical junctions

reduce to a symmetrical, linear-graded junction for large forward

voltages. For such a junction, the electron and hole concentrations are

about equal near the middle of the space-charge region and each greatly

exceeds the net ionized doping density there; thus, dp almost vanishes

except near the edges of the space-charge region. This implies that the

electrostatic potential depends nearly linearly on position, a fact we

just used for large forward voltage.

Incorporating (5.8)-(5.9) and the treatment of Sec. 5.2.1, we

obtain an equation for C in terms of V V, and X. The equation can be

solved by using an iterative technique, if an initial value for X is

provided. We use (5.6) as the zeroth-order model for X (X(0)) and

calculate C and X iteratively.

The results of the iterations for silicon linear-graded junctions
22 -4 23 -4 24 -4
of junction gradient a = 10 cm a = 10 cm and a = 10 cm

indicate that (5.7) is applicable in the approximate range, Vbi-7VT T V

5 Vbi-5VT, where Vbi is the junction built-in potential. Because many

practical junctions reduce to a linear-graded junction when subjected to

large V, we suggest that the above relation holds for all junction


5.4 Discussion of High-Injection Effects

Equations (5.6)-(5.7) implicitly include the effects of moderate

and high injection because the independent variable V is the separation

of the quasi-Fermi levels in the space-charge region. We now explore

the relations among V, V and VJ and discuss (5.7) for low and high


From Marshak and van Vliet [58] and Warner and Grung [59], for

Boltzmann gases in a one-dimensional model,

V V + VTn[p(Xp)/pO(xpO)] + VTn[n(xn)/nO(xnO)],


where subscript 0 represents thermal equilibrium and x and x are
n p
defined in Fig. 5.5.

For a one-sided junction, (5.10) reduces to

V = V + VT2n[p(Xp)/p(xp0)]

- Vbi VB + VTn[p(xp)/p(xp0)].


For low-injection, p(x ) = P0(x ), we have V = V For moderate-



p(Xp) = NB/2 + G/2,

where NB is an assumed constant base doping concentration and

G (N 2 + 4n exp(V/V ))/2


For high-injection

p(x ) = n(x ) = n.exp(V/2VT).


Figure. 5.6 illustrates a comprehensive model, part of

based on the model we discussed here. The comprehensive

details of which will be presented in Chapter Six, shows good

which is

model, the




Xn Xp


Fig. 5.5 Schematic n/p junction, illustrating the edges of the space-charge region, x
and xp.

Exponential-Constant Junction
a=10 23cm'4

Comprehensive Model

30 NB=1016 c.3 / (Low Injection)

E Vbl =0.82 V [Ref. 20] Comprehensive Model
I(High Injection)
20 -
S Bouma &
Z Roelofs [34]
DeGraff [39]
0 10 -

Lee &
Prendergest [30]
0 I I I I
0.0 0.2 0.4 0.6 0.8 1.0
VJ (Volts)
Fig. 5.6 Comparison of the comprehensive model for high injection and low injection wth
results from other methods for an exponent al-co stant junction with a = 10
cm and base doping concentration Ng 106 cm3.

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