IMPROVED CHARACTERIZATION OF THE P/N JUNCTION
SPACECHARGE REGION, WITH APPLICATIONS
BY
JUINJEI LIOU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
ACKNOWLEDGMENTS
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. TaeWon Jung, Dr.
Adelmo Ortiz Conde, Mr. MengKai Chen, and Mr. Robert J. McDonald for
their helpful discussions.
I am greatly indebted to my wife, PeiLi, my parents, and my
parentsinlaw for their love and encouragement.
The financial support of the Jet Propulsion Laboratory and the
Semiconductor Research Corporation is acknowledged.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS.....................................................ii
LIST OF SYMBOLS.....................................................vi
ABSTRACT.............................................................x
CHAPTER
ONE INTRODUCTION..................................................1
TWO REVIEW OF EARLIER WORK ON THE CAPACITANCE OF THE
SPACECHARGE REGION.......................................... 5
2.1 Introduction..............................................5
2.2 Properties of the SpaceCharge Region....................7
2.3 Theoretical Development of the QuasiStatic Capacitance
of the SpaceCharge Region................................8
2.3.1 FreeCarrier and SpaceCharge Approaches...........8
2.3.2 Energy Approach ...................................20
2.3.3 Current Approach..................................21
2.3.4 CutOff Frequency Approach.......................22
2.3.5 SmallSignal Approach.............................22
2.4 Measurement Methods of Finding the Capacitance of the
SpaceCharge Region.....................................24
THREE DETERMINATION OF LIFETIME AND SURFACE RECOMBINATION VELOCITY
OF P/N JUNCTION SOLAR CELLS AND DIODES BY OBSERVING
TRANSIENTS...................................................27
3.1 Introduction.............................................27
3.2 OpenCircuit Voltage Decay Method .......................29
3.3 Green's Compensated OpenCircuit Voltage Decay Method...33
3.4 Extension of Compensated OpenCircuit Voltage Decay
Method.................................................. 35
3.5 OpenCircuit Voltage Decay Method with the Effects of
the Junction SpaceCharge Region........................40
3.6 Effect of Heavy Doping Emitter Region on Open
Circuit Voltage Decay Method.............................44
3.7 FrequencyResponse Methods................................46
3.8 Conclusions and Summary ................................. 48
FOUR LIFETIME DETERMINATION IN P/N JUNCTION DIODES AND SOLAR
CELLS FROM OPENCIRCUIT VOLTAGE DECAY INCLUDING JUNCTION
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
SpaceCharge Region............................... 58
4.2.3 Determination of Lifetime........................64
4.3 Illustration............................................ 66
FIVE IMPROVED JUNCTION SPACECHARGE REGION CAPACITANCE BASED
ON TIMEDOMAIN REASONING......................................73
5.1 Introduction............................................. 73
5.2 Derivation by Detailed TimeDomain 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 HighInjection Effects....................86
5.5 Physical Interpretation of the Condition dC/dX = 0......88
5.6 Conclusions.............................................92
SIX FORWARDVOLTAGE CAPACITANCE AND THICKNESS OF P/N JUNCTION
SPACECHARGE REGIONS..........................................93
6.1 Introduction.............................................93
6.2 Importance of the Capacitance of the SpaceCharge
Region...................................................95
6.3 ZerothOrder Model for the Capacitance and Thickness
of the SpaceCharge 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 ZerothOrder 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
APPENDIX
A LEIBNITZ'S RULE FOR THE QUASISTATIC CAPACITANCE
OF P/N JUNCTIONS............................................. 125
B RELATIONS AMONG VARIOUS POTENTIALS........................... 134
REFERENCES.......................... ..............................138
BIOGRAPHICAL SKETCH..................................................144
LIST OF SYMBOLS
A
a
C
C'
CBC
CD
CF
CO
CQNB
CT
Cil C2
C1, C2
D
An
Ap
6V
E
e
e
fTf
G'
G1' G2 Gs
Gd
H
c
area of a device
junction gradient
junction spacecharge region capacitance of diodes or
emitterbase transition region capacitance of bipolar
transistors
electriccircuit counterpart of S
base collector transition capacitance of bipolar
transistors
capacitance resulting from QD
capacitance resulting from QF
C for zero applied voltage
quasineutral base capacitance
total capacitance of a p/n junction
electriccircuit 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 cutoff frequency
electriccircuit counterpart of H
c
electriccircuit counterpart of Hcl Hc2, and H
electriccircuit counterpart of Hd
combinance of Linvill's model for longbase diodes
Hcl Hc2' Hs
Hd
IC
IE
i(t)
i
c
j
k
L
LD
NA
NB
ND
N
n
n
n.
n0
P
PC
p0
Q
Q'
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
minoritycarrier 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
QD + QF
net freecarrier charge at the edge of the spacecharge
region
freecarrier (hole or electron) charge at the edge of the
spacecharge region
QF freecarrier (hole or electron) charge in the volume of
the spacecharge region
QQNB freecarrier charge in the quasineutral base
Rbi intrinsic base resistance
R emitter resistance
e
S storance of Linvill's model for longbase diodes
s d/dt
SB effective backsurface recombination velocity
S1, S2 storances of Linvill's model for short and intermedium
base diodes
a electron conductivity
ON electron quasiFermi potential
Op hole quasiFermi potential
T absolute temperature
t time
ttr transit time across the spacecharge region
tr characteristic time, (e/a)(l + X/XB)1
r minoritycarrier lifetime in the quasineutral base
TB base transit time
rd decay time constant of the fundamental mode for short
circuit current decay method
V separation of the quasiFermi potentials in the space
charge region
V effective junction builtin potential
v(t) transient voltage across a p/n junction
VA applied voltage
VB barrier height of the p/n junction
viii
Vbi junction builtin potential
V critical voltage, Vbi5VT
V V for lineargraded junction
g
V. electrostatic potential
Vj voltage drop across the spacecharge region
VLB lowerbound voltage, Vbi7VT
VQNB voltage drop across the quasineutral base
VQNE voltage drop across the quasineutral emitter
VT thermal voltage
VUB upperbound voltage, Vbi0.3
W electric energy
X thickness of the spacecharge region
x position
xL front contact of p/n junctions
x ntype side spacecharge region edge
XnO x at thermal equilibrium
x ptype side spacecharge region edge
xp0 x at thermal equilibrium
xR back contact of p/n junctions
XB thickness of the quasineutral region
p spacecharge 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
IMPROVED CHARACTERIZATION OF THE P/N JUNCTION
SPACECHARGE REGION, WITH APPLICATIONS
BY
JUINJEI LIOU
May 1987
Chairman: Fredrik A. Lindholm
Major Department: Electrical Engineering
This dissertation focuses on excess electron and hole storage in a
p/n junction spacecharge region and its effects on measurement methods
and quasistatic 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 quasistatic capacitance model to include electron and hole
storage effects in the opencircuitvoltage 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 opencircuit
voltage decay.
The study then turns to the improvement of existing models for the
quasistatic capacitance, seeking a comprehensive model for the entire
pertinent range of applied voltages, from reverse to small forward to
large forward voltages. Quasistatic 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 timedomain phenomenological reasoning is
improved by defining its applicable voltage range and by including high
injection effects in the bordering quasineutral 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
spacecharge region complicates this comparison. That problem is evaded
in various ways, although an iterative technique yields a rough estimate
of the junctionregion 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 integratedcircuit simulation based on detailed
physical understanding. In principle, the physical understanding will
aid both device and circuit design.
CHAPTER ONE
INTRODUCTION
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 spacecharge region and its capacitive effects. Such effects are
becoming even more important recently as the technology of largescale
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 spacecharge region, which motivates this
study.
The general purposes of this study are: first, to explore the
importance of the capacitance C of the spacecharge region in
recombination lifetime measurements by including C in the analysis of
the opencircuitvoltagedecaymethod, and, second, to develop a
comprehensive quasistatic spacechargeregion 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 spacecharge
region capacitance, providing insight for a better understanding of the
capacitance.
In Chapter Three, we present a unified view of transient methods,
such as the opencircuit voltage decay and the electrical shortcircuit
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 spacecharge region on the opencircuit 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 spacecharge region during transient.
Recognizing the need for including C in the opencircuit voltage decay,
we propose a new method, which is treated in Chapter Four.
A new more accurate method for determining the baseregion
minoritycarrier 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 spacecharge region
for the forwardvoltage 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 ChawlaGummel model for the space
charge region with the Linvill model for the quasineutral base,
together with a computer program, yields theoretically the opencircuit
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
admittancebridge method, the shortcircuit current decay method, and an
improved shortcircuit 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 spacecharge region
capacitance model, which is the basis of the comprehensive capacitance
model that will be developed in Chapter Six. The timedomain reasoning
employed in deriving the previous capacitance model is elaborated. The
previous model is improved by defining the applicable voltage range and
by including highinjection effects. The treatment in this chapter also
supplements the derivation of the previous model, thus adding physical
insight.
Chapter Six presents a comprehensive analytical model for the
quasistatic capacitance of the p/n junction spacecharge region. The
model describes the capacitance for all voltages and applies for
exponentialconstant doping profiles. In addition to the analytical
model, an iterative technique is developed to yield numerically the
capacitance and the thickness of the spacecharge region. A rigorous
solution for the thickness of the spacecharge region under very high
forward voltage is also presented. This solution serves to verify
4
aspects of the capacitance model described in Chapter Five, which is
part of the comprehensive model. The capacitancevoltage 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.
CHAPTER TWO
REVIEW OF EARLIER WORK ON THE CAPACITANCE OF THE SPACECHARGE REGION
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 spacecharge 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 quasineutralregion
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 spacecharge region,
particularly for forward voltages.
The purpose of this chapter is to review models and measurement
methods for the quasistatic capacitance of the spacecharge 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
transistors.
2.2 Properties of the SpaceCharge Region
For the quasineutral regions of a bipolar device, one can obtain
useful results without dealing with the Poisson equation. In contrast,
analysis of the junction spacecharge region always involves the Poisson
equation,
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)
and
n(x) niexp[(ON Vi)/(kT/e)]. (2.3)
Based on (2.1)(2.3), detailed analytical physical theories [1][3]
concerning the spacecharge density can be developed. A rigorous
mathematical analysis, however, requires numerical methods [4][5]. The
theory is further complicated by the observation of nonflat quasiFermi
levels in the spacecharge region [6] for very large forward voltages.
The analysis of the spacecharge region becomes transparent when
the junction is under reverse bias [7][10]. For such bias, in the
depletionapproximation model, the free electrons and holes are depleted
in the spacecharge 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 QuasiStatic Capacitance
of the SpaceCharge Region
2.3.1 FreeCarrier and SpaceCharge Approaches
The quasistatic approximation, which is used in almost all
theoretical capacitance modeling, assumes that the mobilecarrier
densities (also the freecarrier charges) retain, during transients,
their steadystate dependence; this approximation also assumes that the
carriers travel with infinite velocity. In modeling the quasistatic
capacitance C of the junction spacecharge region, the approach used
most often focuses on the change of the excess freecarrier charges dQ
in the spacecharge region resulting from the change of the separation
of the quasiFermi potentials dV in the spacecharge region [7]. This
approach leads to the definition of Shockley [7]:
x
d rp
C = dV e(Ap or An) dx, (2.4)
x
n
where Ap is the excess hole density, An is the excess electron density,
and x and x are the spacecharge region edges of quasineutral emitter
n p
and quasineutral 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 quasineutral regions. Thus
V = VA = V, where VA is the applied voltage and VJ is the voltage drop
across the spacecharge 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 spacecharge region
(p = n = 0), this influx decreases the net ionized impurity charge,
e[ND(Xp) NA(Xp)]dXp. Thus,
1/2
C = [eN /2(Vbi V)] /2, for oneside step junction (2.5)
and
C = [ee2a/12(Vbi VA)]1/3, for lineargraded junction (2.6)
where e is the silicon dielectric permittivity, Vbi is the builtin
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 thermalequilibrium condition.
XL
XR
XP
x
0
Fig. 2.1 Schematic n/p junction, illustrating the edges of the spacecharge 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 (contactto
contact) junction capacitance C for a lineargraded junction (Fig.
2.1):
xR
CT = (d/dV) ep dx. (2.7)
xL
Using symmetry, one can decompose (2.7) into a neutral capacitance C
n
and a spacecharge capacitance C :
x
C = (d/dV) J 2ep dx (2.8)
xR
R
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 spacecharge 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
THEORETICAL DEVELOPMENT FOR C
FREECARRIER
and
SPACECHARGE
APPROACHES
DEPLETION C
Schottky [7]
CURRENT APPROACH CUTOFF FREQ.
APPROACH
Djuric et al. [29]
Lee and
Prendergast [30]
SMALLSIGNAL
APPROACH
Maes and Sah [31]
Green and
Shewchum [41]
4
Fig. 2.2 Flow diagram showing the evolution of the models and methods
for the capacitance of the spacecharge region.
SPACECHARGE REGION PHYSICAL PROPERTIES
Garrett and Brattain [1]
DeWiele and Memoulin [2]
Kittle and Kroemer [3]
DeMari [4]
ENERGY APPROACH
Heard et al. [25]
Parrott and
Leonidou [26]
Parrott [27]
Parrott and Tieng [28]
NONDEPLETION C
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]
MEASURING TECHNIQUES FOR C
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
spacecharge 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 regionalanalysis 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 spacecharge 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 spacecharge region for a given thickness. They then use these
expressions, iterating numerically to obtain a selfconsistent solution
for Q from which they calculate, numerically, the quasistatic
capacitance C. We note that their assumption that the potential varies
as Alx + A2x2 disagrees with the requirement of a cubic function for a
lineargraded 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 SirsiBoothroyd model and the depletion
capacitance is illustrated in Figs. 57 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 lineargraded junction diode.
Sah [12] also derived a model for C for the lineargraded
junction, which considers the effects of the free carriers in the space
charge region. Basically, Sah requires at the edges of the spacecharge
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
chargeregion 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. 1418 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
spacecharge region, by integrating the Poisson equation. Instead of
focusing on the freecarrier charge, he defines C by integrating p(x) in
the spacecharge 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 freecarrier
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 ntype side contact for a p /n asymmetrical
junction. Consequently, they cannot be counted when integrating p(x)
for the ntype side of the spacecharge 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 ntype spacecharge 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 ntype (or donor) contact
and the holes entering or exiting the ptype (or acceptor) contact.
Focusing on the freecarrier 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 falloff
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 quasiFermi potentials in the spacecharge
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 quasiFermi 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 exponentialconstant junction and its two extremes, the
step and lineargraded junctions. The model retains the same form as
that of the Shockley's depletioncapacitance model ((2.5)(2.6)) but
replaces Vbi with an effective builtin voltage so that the effect of
the freecarrier charges in the spacecharge 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 contacttocontact 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 builtin voltage Vbi. They
thus replace Vbi by an effective builtin voltage shown in (43) of Ref.
20 for a lineargraded junction and in (45)(49) of Ref. 20 for an
exponentialconstant 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 spacecharge 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 spacecharge region, whereas CF is the
F
capacitance corresponding to the change of the free carriers within the
volume of the spacecharge 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 spacecharge 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
p
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 lineargraded junction becomes independent of V
defined as the separation of the electron and hole quasiFermi
potentials in the junction spacecharge 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 spacechargeregion
and quasineutralregion capacitances are separated in the computer
simulation.
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 spacecharge 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 spacecharge
region, which appears to be a severe approximation except at very high
forward voltages, a point not made in Ref. 21. That the spacecharge
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 nondegenerate carrier statistics, and
calculated the total quasistatic capacitance of a junction diode by
solving the basic transport equations and Poisson's equation. This is
followed by a smallsignal solution in the frequency domain. By varying
the frequency, Kwok separates the capacitance of the spacecharge region
and the capacitance of the quasineutral 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 spacecharge 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 spacecharge region by using the
socalled "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 spacecharge region
capacitance and the quasineutral 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 spacecharge 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 righthand 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 CutOff Frequency Approach
The junction spacecharge capacitance can also be obtained by
using the forward active cutoff frequency fTf. Based on admittance
parameters developed from the GummelPoon 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
cutoff 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 spacecharge 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 SmallSignal Approach
Maes and Sah [31] use the transmissionline circuit model of Sah
[40] to obtain the forwardvoltage capacitance for p/n junction devices.
The authors use the GreenShewchun [41] simulation method, which they
indicate greatly simplifies the numerical calculation. In general,
Sah's smallsignal transmissionline circuit model consists of a large
number of sections in each section of which the steadystate potentials
can be treated as constant. If the steadystate 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 transmissionline 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
frequency.
As discussed above, the use of the Sah's smallsignal circuit
model requires the steadystate solution of the transport equations for
a given bias. The steadystate 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 NewtonRaphson
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 steadystate values of the node potentials dC of Sah's transmission
line circuit for an incremental applied voltage dVA. By solving the
entire smallsignal transmissionline, 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 smallsignal 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
3
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 smallsignal frequency is much higher than the
reciprocal of the minoritycarrier 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 SpaceCharge Region
In additional to the above methods, which are all solely based on
theoretical analysis, measurement techniques have been used to determine
the junction spacecharge capacitance [32][34].
Lindmayer and Wrigley [32] first developed the following relation:
(2.17)
1/27f = rB + R (C + CBC)
where TB is the base transit time, Re is the emitter resistance, and CBC
is the collector spacecharge 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 forwardactive mode. Thus the measurement of fTf versus IE
yields C.
A method of obtaining the spacecharge region capacitance derived
from the commonemitter 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 spacecharge 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 spacecharge 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 quasineutral capacitance, shows
that the spacecharge region capacitance possesses falloff
characteristics at applied voltages near the builtin voltage of the
junction (Figs. 56 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 commonbase
current gain and P is the commonemitter current gain, in order to avoid
large errors in determining fTf [35].
CHAPTER THREE
DETERMINATION OF LIFETIME AND SURFACE RECOMBINATION VELOCITY OF P/F
JUNCTION SOLAR CELLS AND DIODES BY OBSERVING TRANSIENTS
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 smallsignal admittance vs.
frequency and forward voltage. The treatment will be limited to low
injection conditions in quasineutral 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, opencircuit voltage decay (OCVD)
[42] and reverse recovery (RR) [43] appear unified as a single method in
Fig. 3.1.
The opencircuit 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 opencircuit voltagedecay analysis and reverse
recovery analysis.
v (t)
observes the decay of voltage. The reverserecovery 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
initialboundaryvalue 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 OpenCircuit 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 quasineutral base. Because of the
exponential relation between the minoritycarrier density at the edge of
this region and the nonequilibrium 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 longbase diode, for which
the thickness of the quasineutral 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 quasistatic 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 admittancebridge techniques and by the
shortcircuit current deriving from incident xray 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 cm3, T = 300K
.LOPE = (kT/e)/r
TIME (t)
Fig. 3.2 Transient response of opencircuit voltage decay for Ge.
OCVD
Voc (t)
4.
O
c
U
S SLOPE = (kT/e)/T
0
TIME (t)
Fig. 3.3 Transient response of opencircuit 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 spacecharge region are
proportional to n.; these mobile carrier densities contribute to the
quasistatic 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 quasineutral
base, whose recombination is the central feature of conventional OCVD,
2
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 admittancebridge techniques
[45].
3.3 Green's Compensated OpenCircuit 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 quasistatic analysis [47][48].
0.4 299K 
0  "^T=4.4psec
a
> 6"K
0.3
376'K
0.2 = 7gsec ~
0.2
0.1 I i I I I I I
0 10 20 30 40 50 60
TIME (g sec)
Fig. 3.4 Opencircuit 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 backsurfacefield (BSF) solar cells.
Our experience is that the method requires subjective judgement.
For some (longbase) 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 OpenCircuit 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 shortcircuit 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 opencircuit voltage decay to
explore r and SB.
C
II
t > 0:
'SC (t)
RO0
Fig. 3.5 Schematic illustration for the shortcircuit current decay analysis.
In this method, the very small external resistance used to monitor
the current i(t) nearly provides a shortcircuit across the junction
spacecharge 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 quasineutral 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 quasineutral base. The response is
described by an infinite series of exponential decays
"ot/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 ith mode, rd is the decay
time constant of the lth 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 quasineutral emitter
contributes only to the initial part of the transient.
(a)
Cell Oscilloscope
v 
R
t=o
(b)
loge i(t)
If
t
0
Slope a'Td
9
1(0) '
Fig. 3.6(a) The electronic switching circuit used in shortcircuit
current decay method.
(b) Transient response of the shortcircuit current decay on
a semilogarithmic scale.
From a mathematical standpoint, the poles corresponding to
relaxation in the quasineutral emitter mix the poles corresponding to
the fast relaxation of the quasineutral base. Following the near
vanishing of the resulting fast part of the response, only the dominant
pole response remains. This contributes the straightline portion of
the response shown in Fig. 3.6. from which SB and r are determined.
The circuit used for electrical shortcircuit current decay (Fig. 3.6)
utilizes a MOStransistor 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 zerobias 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 straightline 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 fourpoint probe will yield the most accurate value of v(t <
0).
3.5 OpenCircuit Voltage Decay with the Effects of the
Junction SpaceCharge Region
As discussed in foregoing sections, opencircuit 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 spacecharge 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 junctionregion barrier height
referenced to the equilibrium height, for the assumed lowinjection
conditions. We may make the quasistatic assumption that q[v(t)] has
the same dependence on voltage as in the timeinvariant 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 spacecharge region. The result for C is
called the quasistatic 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 spacecharge region. This capacitance model
will be discussed in details in Chapter Five and Chapter Six of this
study.
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 dominantpole or quasistatic
part of the response v(t) is not falling straight line. Rather, it will
show curveture. Then, if r in the model of the quasineutral 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 builtin
potential, C follows the powerlaw functional dependence of the
depletion capacitance with the exception that an effective builtin
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 spacecharge region. The
resulting expression is
r
C = C0[l + v/V ] (3.2)
where CO is the spacecharge region capacitance for zero applied voltage
and 1/3 s r s 1/2. For a step junction, r = 1/2; for a lineargraded
junction, r = 1/3. For a lineargraded junction, V can be expressed as
[20]
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 spacecharge 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 spacecharge region below the values predicted by the
depletion approximation. Thus, at equilibrium (v = 0), V < Vbi. This
inequality continues as v increases. For an exponentialconstant
profile, which approximates a profile resulting, for example, from
phosphorous diffusion into a constant background ptype 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 lineargraded junction with junction gradient a = 10 cm
TABLE 3.1
Capacitance C of Junction SpaceCharge Region for a LinearGraded
Junction with Junction Gradient a = 1023 cm4 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
(108f/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 quasineutral 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 upperbound voltage of
slightly less than 600 mV for many silicon solar cells and diodes [20].
Thus the ChawlaGummel 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 dominantpole
or quasistatic model of the quasineutral base. We use either a
Linvill model [52], which emphasizes minoritycarrier density, or a Sah
equivalent circuit [40], which emphasizes the minoritycarrier
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 OpenCircuit
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
quasineutral emitter and the quasineutral base edges of the junction
spacecharge region. Thus the poles in the complex frequency plane
corresponding to the quasineutral emitter simply add to those
corresponding to the quasineutral base. Because the poles associated
with the emitter produce short relaxation times relative to the
dominantpole 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 socalled heavy doping effects, which may yield
a large steadystate recombination current of the quasineutral 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
quasineutral base edge of the junction spacecharge region and the
densities at the quasineutral 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 longbase devices in which the thickness of
the quasineutral base greatly exceeds the minoritycarrier 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 quasineutral 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 spaceaveraged recombination
lifetime and the minoritycarrier transit time in the heavily doped
emitter.
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 minoritycarrier 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 shortbase 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
admittancebridge 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 minoritycarrier diffusion length L is given as [54]
10 SB= 90 cm/s
o 10
S1/2
S Gaf
0 1 III>I
103 3 1 104 105
FREQUENCY (Hz)
Fig. 3.7 The frequency response, from the admittancebridge method, of the conductance of
a n+/p/p diode with diffusion length L and surface recombination velocity SB
specified.
2XBD /2 D 1 + (S L2/DXB)
WL) ( [ (D/XB) + Sg
where D is the diffusivity of the minoritycarrier 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. Opencircuit voltage and reverse recovery serve adequately to
determine the base recombination parameters such as minoritycarrier
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 quasistatic capacitance C of the junction spacecharge
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. Energygap 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 longbase 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 shortcircuit
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 admittancebridge 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 steadystate 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
dominantpole or quasistatic model for the base region enables a
determination of r for longbase devices. The method may find use
for inprocess control in manufacturing, where it is desirable to
employ contact or pressure probes rather than ohmic contacts.
CHAPTER FOUR
LIFETIME DETERMINATION IN P/N JUNCTION DIODES
AND SOLAR CELLS FROM THE OPENCIRCUIT VOLTAGE DECAY
INCLUDING JUNCTION CAPACITANCE EFFECT
4.1 Introduction
The opencircuit voltage decay (OCVD) technique is a simple method
by which, in principle, the minoritycarrier 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 quasineutral 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 spacecharge region.
Thus the nonlinear capacitance C of a junction spacecharge 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 opencircuit and shortcircuit
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 spacecharge region capacitance as well as the
quasineutral 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 networklike 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 spacecharge region and the carrier densities at
the two edges of this regions. For low injection, which we assume
prevails, linear relations exist between the minoritycarrier density
and the minoritycarrier current in the quasineutral 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 minoritycarrier 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 longbase solar cell or diode having a position
independent base doping density and negligible contributions from the
quasineutral 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
spacecharge region:
An(x=0,t) = n0exp[v(t)(kT/e)], n0 = ni /NB (4.1)
where NB is the base doping concentration. For the longbase diode
under study, the element values of the storance S and the combinance H
c
are
S eL, H S/r. (4.2)
c
These dependencies on diffusion length L and recombination lifetime r
follow from the usual quasistatic 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).
(4.3)
N P
v_(t)
S H c
I I
r 1]
An (x=O, t)
Fig. 4.1 The simplest Linvill model for a longbase 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).
c
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)
and
v(t)/(kT/e) = ln[An(0,0)/n0] t/r, t > 0. (4.6)
This is the standard quasistatic result for OCVD.
Using the Linvill model, we may include the effect of the
minoritycarrier transient in the quasineutral emitter by adding a
combinance and storance to the left side of the box representing the
junction spacecharge region in Fig. 4.1. The currents through these
networklike elements are driven by the minoritycarrier density Ap at
the emitter edge of the spacecharge region, which is related to An(O,t)
by the ratio of the minoritycarrier equilibrium densities in the two
regions. Thus the effect of the minoritycarrier 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 minoritycarrier density at the emitter
edge of the junction spacecharge 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
reasoning.
To allow direct use of circuitanalysis computer codes, we note
that for every Linvill model there exists a nonlinear electriccircuit
model. As an example, the electriccircuit 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 (quasiFermi) potential 0p at the
base edge of the junction spacecharge region as the reference potential
(ground). Then the minority electron electrochemical potential qN at
v(t)
C1 I Lo
C' G
Fig. 4.2 The electriccircuit 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 zerocurrent 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 SpaceCharge Region
The OCVD circuit diagram is shown in Fig. 4.3. We will utilize
Linvill's model [52], [56] for the quasineutral base. For the junction
capacitance C we utilize Chawla and Gummel's model, which is valid for
all voltage up to an upperbound voltage slightly less than 600 mV for
most silicon junction diodes [20]. Fig. 4.4 illustrates the resulting
model which we call the OCVDCAP (opencircuit voltage decay with
capacitance) model. The effects of the quasineutral 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
I
a U
I
I
a
I
*I I
I a
I I
HI
Fig. 4.3 Schematic illustration of the opencircuit voltage decay analysis.
I
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 quasistatic 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 backsurface 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(xXB,t)/(kT/e), (4.17)
G1 = Cl(kT/e)/[roN(x=O,t)],
(4.18)
I
.j1C gC1 G1 IC2 G2 Gs
Gd
ON (x=O, t) = v(t) ,N (x=X B, t)
Fig. 4.5 The counterpart equivalent electriccircuit 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 longbase diode or if a minoritycarrier 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 spacecharge 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 quasineutral
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 longbase diode, and that for a shortbase diode
ON(x=XB,t) (kT/e).n(l + (exp[ON(x=O,t)/(kT/e)] 1).
((D/XB)/[SB + (D/XB)]}),
(4.23)
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
procedures.
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
longbase 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
circuitanalysis computer programs.
We use finitedifference 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 opencircuit 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(x0,t)/dt + S2dAn(x=XBt)/dt +
HclAn(x=0,t) + Hc2An(x=XB,t) + HsAn(x=XB,t).
(4.26)
In any time interval,
jck = (Vk+l
 k)/(tk/Ck) (tk+/Ck+l)].
Note that subscript k and k+l represents ttk 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],
(4.28)
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,
(4.29)
which reduces to Hcl for longbase diodes. Note that j is determined
cl ck
(4.27)
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 makeup, 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 minoritycarrier 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 admittancebridge method [45], the shortcircuit
currentdecay method [50], and an improved shortcircuitcurrentdecay
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
c
COMPUTER PROGRAM FLOW CHART
DEFINE TIME INTERVAL At
tK < t < tK+1
START COMPUTATION OF VOLTAGE AT t=O+ (K=O)
ASSUME A VALUE FOR V(tK+l) AND FOR T
SOLVE CURRENT jCK+1 THROUGH C(SCR) AT tK+1
SOLVE An(X=0, tK+1) BY USING LINVILL'S MODEL
SOLVE V' (t K+I) FROM An(X=0, t K+1)
V' (tK+1) = V(tK+1)
YES
GO TO NEXT TIME INTERVAL (K * K+1)
Fig. 4.6 Flow chart of the OCVDCAP numerical procedures.
H
H
H
68
TABLE 4.1
Device Makeup (NB, XB, A), the Back Surface Recombination Velocity SB
(Determined by the ShortCircuitCurrent Decay Method), and the Minority
Carrier Lifetime T Determined by Four Methods:
* TOCVD by the Present Method
* TSCCD by the ShortCircuitCurrent Decay Method
* TISCCD by an Improved ShortCircuitCurrent Decay Method
* TAB by the AdmittanceBridge Method
A
(cm2)
XB
(Um)
NB
(cm3)
1015
5X1016
5X1016
1.25X1015
SB
(cm/s)
TOCVD
(us)
TSCCD
(us)
TISCCD
(us)
(u) (us) (u~s)
TAB
(lS')
380 50
1540 25
104 12
4
Solar
Cell
v(t)
(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.
O
0.4 o
0.2
III I I I I t(lS)
0.0 20 40 60 80
Fig. 4.7 Opencircuit 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 opencircuit
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 admittancebridge method of Ref. 45.
The OCVDCAP method determines either minoritycarrier lifetime r
or the back surface recombination velocity S The method has direct
utility in two cases: (a) for longbase diode, for which S is
irrelevant, and (b) for a backohmic contact shortbase 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.
I I I I I I I I
 Model with R = 4kK
Model without RC
T = 4.0 s 15
NB = 1.25x101 cm
Boron Diff.
2F
I N
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
dv
dt

The version of OCVDCAP presented has assumed implicitly that the
recombination current of the junction spacecharge region and through
shunt conductances are small, which are good assumptions for many
silicon devices for voltages in the 500600 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
positionindependent r in driftfield 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 opencircuit voltage involves negligible
current flow through the contacts.
CHAPTER FIVE
IMPROVED FORWARDVOLTAGE P/N JUNCTION SPACECHARGE REGION CAPACITANCE
BASED ON TIMEDOMAIN REASONING
5.1 Introduction
The quasistatic capacitance C of a p/n junction spacecharge
region plays an important role in Si and GaAsbased 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 spacecharge 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 quasistatic
capacitance because it departs from the usual steadystate 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
73
becomes important at high forward voltages where highinjection
conditions and resistive effects can exist.
In this chapter, we correct these defects, discuss the insight
provided by timedomain 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 TimeDomain Reasoning
5.2.1 Review and Elaboration
A timedomain analysis enables separation of the junction
capacitance into dielectric and freecharge components and leads to a
simple picture of the junction capacitance not obtained by previous
steadystate analyses. To illustrate this, we review briefly the
reasoning used in Ref. 21, including some details not present in Ref.
21.
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 spacecharge 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
dVA
40t
edp
t~tr
dn,dp
1I\ 1
t tr+ ttr
Fig. 5.1(a)
(b)
(c)
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
distribution.
edn
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 onedimensional model, a fact we will use
later. Second, a characteristic time enters from the resistivity and
geometry of the quasineutral 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 spacecharge 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 lowinjection and negligible series resistance, dVA = dV
 dV dVB, where VB is the junction barrier height, V is the voltage
drop across the spacecharge region, and V is the separation of the
quasiFermi potentials in the spacecharge region. In Ref. 21, the
meaning of V is clear only for lowinjection. 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 spacecharge region (Fig.
5.1b). Recognizing the similarity of this charge distribution to that
of a parallelplate 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 spacecharge region with
holes and electrons, constituting a storage phenomena that also
contributes to the capacitance of the spacecharge region (Fig. 5.1c).
According to Ref. 21, the freecarrier charge QF of either holes or
electrons stored in the spacechargeregion 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
aQF Q
C (5.3)
F aV constant 2T
where CF denotes the freecarrier 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 spacecharge 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)
and
C(V) = 2CD = 2CF [2een VT1 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 steadystate treatments that underlie almost all
previous work on the junction quasistatic capacitance. The reasoning
in the time domain focuses on a short time interval, approximately the
+ CF
d 0WA
dC
dX
Fig. 5.2 Qualitative illustration of the dependence of CD, CF, and CF + CD versus X.
sum of t and t after a stepfunction 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 spacecharge region with free carriers
(charge QF).
To clarify the physics involved, we employ the nonlinear
transmissionline model of Sah [40], which is a finitedifference
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
recombinationtrapping mechanisms and focus on the spacecharge 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 spacecharge region (Fig. 5.3). This
treatment of the quasineutral regions is adequate for the qualitative
discussion of this section. This threeline 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 foursection model corresponding to
NSIDE
Fig. 5.3 Sah's network representation for the spacecharge 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.
PSIDE
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 majoritycarrier storage are
proportional to the majoritycarrier density [40]; thus CN n at the
emitter edge and Cp c p at the base edge of the spacecharge region are
large compared with the horizontally directed capacitances and for t <
t could be replaced by short circuits without introducing error.
r
Therefore, current initially follows the path shown in Fig. 5.3.
Applying a stepfunction voltage dVA to this model, we see that
the resulting initial impulse of current deposits timeindependent
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 spacecharge 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 quasistatic
capacitance C, and the main points of the timedomain 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 nonquasi
static effect, ignored by quasistatic analysis, that gives the
frequencydomain junction admittance a real as well as an imaginary
part. In the steadystate condition eventually reached after step
function voltage is applied, the Sah transimissionline 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 steadystate analysis used by most past studies
fails to yield a simple analytical model for the forwardvoltage
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 spacecharge 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 onedimensional model is independent of position x. Thus, directly
after the stepfunction voltage is applied, the total current has only a
convection component in the quasineutral regions and only a displacement
component in the spacecharge 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
QuasiNeutral SpaceCharge QuasiNeutral
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 lineargraded junction. The hole concentration in the
spacecharge region is
p(x) = niexp([Op Vi(x)]/VT), (5.8)
where 0p is the hole quasiFermi potential and Vi is the electrostatic
potential in the spacecharge 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 onedimensional program for p/n junction devices. Their Fig. 8
indicates that dp(x), resulting from dV, is sizable at the edges of the
spacecharge region but is small in the middle of the spacecharge
region. Physically, this is expected because most practical junctions
reduce to a symmetrical, lineargraded junction for large forward
voltages. For such a junction, the electron and hole concentrations are
about equal near the middle of the spacecharge region and each greatly
exceeds the net ionized doping density there; thus, dp almost vanishes
except near the edges of the spacecharge 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 zerothorder model for X (X(0)) and
calculate C and X iteratively.
The results of the iterations for silicon lineargraded 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, Vbi7VT T V
5 Vbi5VT, where Vbi is the junction builtin potential. Because many
practical junctions reduce to a lineargraded junction when subjected to
large V, we suggest that the above relation holds for all junction
profiles.
5.4 Discussion of HighInjection 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 quasiFermi levels in the spacecharge region. We now explore
the relations among V, V and VJ and discuss (5.7) for low and high
injection.
From Marshak and van Vliet [58] and Warner and Grung [59], for
Boltzmann gases in a onedimensional model,
V V + VTn[p(Xp)/pO(xpO)] + VTn[n(xn)/nO(xnO)],
(5.10)
where subscript 0 represents thermal equilibrium and x and x are
n p
defined in Fig. 5.5.
For a onesided junction, (5.10) reduces to
V = V + VT2n[p(Xp)/p(xp0)]
 Vbi VB + VTn[p(xp)/p(xp0)].
(5.11)
For lowinjection, p(x ) = P0(x ), we have V = V For moderate
injection
(5.12)
p(Xp) = NB/2 + G/2,
where NB is an assumed constant base doping concentration and
B
G (N 2 + 4n exp(V/V ))/2
(5.13)
For highinjection
p(x ) = n(x ) = n.exp(V/2VT).
(5.14)
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
agreement
VA
II
I I
I I
I I
I I
I I
I I
Xn Xp
0
Fig. 5.5 Schematic n/p junction, illustrating the edges of the spacecharge region, x
and xp.
ExponentialConstant 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)
O
20 
O
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 alco stant junction with a = 10
cm and base doping concentration Ng 106 cm3.
