STUDIES OF THE INFLUENCE, AND ITS CONTROL, OF GRAIN BOUNDARIES
ON MINORITYCARRIER TRANSPORT IN POLYSILICON DEVICES
By
RAVISHANKAR SUNDARESAN
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
1983
To my parents
ACKNOWLEDGMENTS
I express my sincere gratitude to the chairman of my supervisory
committee, Dr. Jerry G. Fossum, for his expert guidance and
encouragement. His research insight into semiconductor device physics
combined with superb tutoring has been of immense value in developing my
interests. I also express my gratitude to the cochairman of my
supervisory committee, Dr. Dorothea E. Burk, for her guidance and
encouragement. I gratefully acknowledge her assistance and cooperation
in acquiring my experimental data.
I am thankful to the members of my supervisory committee, Drs. Paul
H. Holloway, Fredrik A. Lindholm, Arnost Neugroschel, and Arun K. Varma,
for stimulating discussions and for their participation on my committee.
I express my appreciation to Bill Axson, James Chamblee, Bruce
Chovnick, Victor de la Torre, Jerry Goeke, E. J. Jenkins, Donna Ray, and
Bruce Rushing for their technical assistance and cooperation. I am
thankful to my colleagues, Franklin Gonzalez, BorYuan Hwang, Dersun
Lee, and Adelmo Ortiz, for useful discussions.
I am grateful to my friends, Ashok, Bhaskar, Chaiti, Gautam, Kumar,
Liley, Shiv, Tiru, and Vishu, for their moral support. I am indebted to
my brothers for their constant encouragement and continued support
throughout my graduate study.
The financial support of the Solar Energy Research Institute is
gratefully acknowledged.
I commend my typist, Carole Boone, on her excellent work.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS.................................................... iii
ABSTRACT ............................................................ vi
CHAPTER
1 INTRODUCTION.......................................................1
2 ANALYSIS OF MINORITY CARRIER TRANSPORT IN POLYSILICON DEVICES......9
2.1 Introduction................... .............................9
2.2 Analysis ...................................................14
2.3 Discussion .................................................48
3 DETERMINATION OF GRAINBOUNDARY RECOMBINATION VELOCITY
FROM ELECTRONBEAMINDUCEDCURRENT MEASUREMENTS.................53
3.1 Introduction ...............................................53
3.2 Formulation of the Problem ..................................58
3.3 The Volume Distribution of the Generation Rate............... 63
3.4 Results.................................. .................71
3.5 Discussion .................................................82
4 POTENTIAL IMPROVEMENT OF POLYSILICON SOLAR CELLS BY
GRAINBOUNDARY AND INTRAGRAIN DIFFUSION OF ALUMINUM.............87
4.1 Introduction................................................. 87
4.2 Cell Fabrication...................................... ... 91
4.3 Results ...................................... ................94
4.4 Discussion ................................................109
5 ALUMINUM GETTERING IN (CAST) POLYSILICON ........................114
5.1 Introduction .. ............................................114
5.2 Description of the Mechanisms ..............................117
5.3 Results..................................................122
5.4 Di scussion ................... ...............................131
6 SUMMARY, DISCUSSION, AND RECOMMENDATIONS ........................135
APPENDIX
A NUMERICAL SOLUTION OF POISSON'S EQUATION IN THE
GRAINBOUNDARY SPACECHARGE REGION ............................141
B MATHEMATICAL JUSTIFICATION FOR USING THE FOLDING TECHNIQUE.......164
C NUMERICAL ALGORITHM FOR SOLVING THE TWODIMENSIONAL CONTINUITY
EQUATION UNDERLYING THE EBIC RESPONSE .........................168
C.I Numerical Solution of the SteadyState Continuity Equation..168
C.2 Numerical Evaluation of the Currents........................172
0 POLYSILICON CELL FABRICATION...................................201
D.1 Standard Cleaning Process..................................201
0.2 Bulk Process (Large Area Cells) ............................200
D.3 Photol ithography...........................................202
D.4 Bulk Process (SmallArea Cells) ............................202
D.5 ThinFilm Process (LargeArea Cells) .......................204
REFERENCES........................................................... 205
BIOGRAPHICAL SKETCH................................................ 212
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
STUDIES OF THE INFLUENCE, AND ITS CONTROL, OF GRAIN BOUNDARIES
ON MINORITYCARRIER TRANSPORT IN POLYSILICON DEVICES
By
Ravishankar Sundaresan
December 1983
Chairman: Dr. Jerry G. Fossum
CoChairman: Dr. Dorothea E. Burk
Major Department: Electrical Engineering
In this dissertation we develop an analytic model for minority
carrier transport in polysilicon devices, and provide experimental
corroboration for the model. The model is used to facilitate the
development of experimental techniques, compatible with conventional
device processing, to control the effects of grain boundaries.
Techniques are investigated to reduce bulk recombination current by
gettering intragrain impurities in polysilicon.
Key assumptions are made, with justification, to simplify the
threedimensional, nonlinear boundaryvalue problem that defines
minoritycarrier transport, including recombination, in polysilicon
devices. These assumptions enable the separation of the grainboundary
recombination analysis, which is based on quasiequilibrium, from the
intragrain transport analysis, which is done by partitioning the grain
into subregions in which the minoritycarrier flow is predominantly one
dimensional. The analyses are coupled through the effective minority
carrier recombination velocity at the grain boundary, which generally is
dependent on the minoritycarrier density in the adjacent quasineutral
grain.
The dependence of the recombination velocity on the carrier density
(excitation) at the grain boundary is experimentally demonstrated using
electronbeaminducedcurrent (EBIC) measurements. To facilitate
quantitative interpretation of the EBIC measurements, we solve the
underlying carrier transport problem subject to the nonlinear boundary
condition at the grain boundary using computeraided numerical
analysis. By comparing the theoretical and experimental EBIC responses,
we predict values for the surfacestate density at typical grain
boundaries and the minoritycarrier diffusion length in the grains.
Experimental results are presented that imply potential
improvements afforded by aluminum diffusion in both bulk and thinfilm
polysilicon solar cells. For bulk cells, a hightemperature aluminum
diffusion alloyingg) is shown to increase the minoritycarrier diffusion
length by gettering intragrain impurities. For thinfilm cells, a low
temperature aluminum diffusion is shown to substantially passivate grain
boundaries and hence decrease the recombination velocity. The decrease
is evaluated using EBIC measurements, the interpretation of which is
aided by the numerical carrier transport model developed.
CHAPTER 1
INTRODUCTION
Polycrystalline silicon (polysilicon) is being widely used in the
semiconductor industry to fabricate unipolar as well as bipolar devices
[1]. The emergence of polysilicon devices has motivated studies of
majoritycarrier transport through the grain boundaries [26], which
defines the resistivity of the polysilicon, as well as studies of
minoritycarrier recombination at the grain boundaries [711], which
defines the bipolar characteristics of the polysilicon.
Compared to silicon (majority and minoritycarrier) devices,
polysilicon devices perform poorly. The grain boundaries, in general,
are responsible for the poor performance of the polysilicon devices.
For example, in polysilicon MOSFETs the presence of grain boundaries
causes a turnon characteristic that is beyond the stronginversion
threshold [51, while in thinfilm polysilicon solar cells [12] the
conversion efficiency is lowered due to recombination at the grain
boundaries. The understanding of how grain boundaries influence the
transport of carriers in polysilicon is hence of engineering
signi ficance.
To enable the design and development of polysilicon solar cells
that are costcompetitive with silicon cells, we must understand how the
grain boundaries influence the minoritycarrier transport and then
devise fabrication techniques to minimize this influence. Although
several approximate models [710, 13, 14] for the minoritycarrier
transport in polysilicon have been developed, they are inadequate in the
general case. The reason for this is that the models [710, 13] are
based on restricting assumptions that limit their ranges of validity;
furthermore, the solution of the minoritycarrier transport equation is
obtained by truncating an infinite series [8, 14], and hence the
accuracy of the models is questionable. Many processing (passivating)
techniques [1519] have also been developed to control the grain
boundary recombination. However none of these techniques is easily
adapted to solar cell processing and/or is totally effective [1519].
The purpose of this research is to develop a general minority
carrier transport model for polysilicon that is based on a good physical
understanding of grain boundaries, and to use it to develop experimental
techniques, compatible with conventional device processing, to control
the effects of grain boundaries; for example, to passivate grain
boundaries and thereby improve the efficiency of polysilicon solar
cells. Another limitation on the efficiency of polysilicon solar cells
is due to the base minoritycarrier lifetime, which is typically shorter
than that in silicon cells due to high intragrain defect densities.
Hence another purpose of this research is to develop experimental
techniques to getter intragrain impurities. The primary tasks of the
research are
(1) to develop an analytic model for the minoritycarrier transport
in polysilicon;
(2) to provide experimental corroboration for the transport model
and to determine values for the pertinent grainboundary
parameters;
(3) to develop experimental techniques to improve the performance
of bulk and thinfilm polysilicon solar cells and other bipolar
devices.
Polycrystalline silicon, as the nomenclature implies, has more than
one crystalline orientation (grain), each orientation is separated by a
plane of dislocations (grain boundary). At the grain boundaries are
defects [1], e.g., dangling bonds, which produce localized electron
states (traps) throughout the energy gap. The net charge on these
states, which results from majoritycarrier trapping, gives rise to a
potential barrier which influences the conduction properties of the
polysilicon [6]. In minoritycarrier devices, such as solar cells, the
grainboundary states serve as recombination centers for the minority
carriers and hence constitute a source of recombination current. Thus
the development of the minoritycarrier transport model for polysilicon
is complicated by the presence of randomly oriented surfaces (grain
boundaries) at which significant recombination can occur.
Impedance measurements [20], currentvoltage measurements [15, 16],
and electronbeaminducedcurrent (EBIC) measurements [21, 22] have been
used to characterize the barrier, or the defect density at the grain
boundary, either of which is a measure of the grainboundary
recombination. We use the EBIC measurements to infer the grainboundary
parameters since it facilitates isolation of the grain boundary from the
bulk grain. Such an isolation is beneficial in order to study the
grainboundary characteristics, e.g., the influence of the excitation on
grainboundary recombination. Furthermore the EBIC measurements do not
require tedious sample preparation as do currentvoltage measurements
[15, 16] to obtain the grainboundary parameters. An additional
advantage is that we can infer the lifetime in the grain from EBIC
measurements.
The presence of defect states results in current loss at the grain
boundary and hence deteriorates the performance of polysilicon
devices. To control the detrimental influence of the grain boundaries,
it is necessary to reduce the defect states at the grain boundary. This
reduction can be achieved by selective (preferential) diffusion of
certain impurities [1619] down the grain boundaries or by fabrication
of special structures to eliminate them from the active portions of the
device [15]. Furthermore, polysilicon contains high intragrain defect
densities which cause short minoritycarrier lifetimes. Hence it is
necessary to consider processes that can getter the intragrain
impurities and hence increase the minoritycarrier lifetime.
In Chapter 2, we develop an analytic model for the minoritycarrier
transport in polysilicon. The general transport problem, which is
threedimensional and nonlinear [23], is simplified by making key
assumptions with physical justification. The main feature of the model
is the separation of the grainboundary analysis, which yields an
expression for the effective recombination velocity, from the grain
analysis, which yields an expression for the base recombination
current. The analyses are coupled through the effective recombination
velocity at the grain boundary.
Our results indicate that the effective recombination velocity is,
in general, dependent on the excitation (minoritycarrier density) at
the grain boundary, and in fact decreases with increasing excitation.
This result renders most transport analyses [13, 14], which are based on
the common assumption that the effective recombination velocity is a
constant, not valid in general. Our results also indicate that at low
and intermediate forward voltages, the base recombination current in a
polysilicon diode comprises a component independent of the grain
boundary parameters and a component strongly dependent on the grain
boundary parameters. Contrarily, at high forward voltages, the grain
boundaries have negligible effect on the base recombination current, and
the polysilicon virtually behaves like singlecrystal silicon.
In Chapter 3, we characterize the nonlinear effective recombination
velocity at the grain boundary using EBIC measurements [21, 22]. The
electron beam is incident from the top surface of a polysilicon cell and
generates electronhole pairs, predominantly in the base region of the
cell. The generation (shortcircuit) current is measured directly using
an ammeter. The EBIC response of a grain in the vicinity of a grain
boundary is obtained by measuring the generation current as a function
of the distance of the center of the beam from the grain boundary.
To facilitate quantitative interpretation of the EBIC response, we
develop a computeraided numerical analysis of the underlying minority
carrier transport problem. Several numerical [24] and analytical
[25, 26] solutions have been derived for the transport problem. However
their ranges of validity are limited owing to simplifying assumptions of
a point [26] or a spherical [24, 25] generation source, and a constant
grainboundary recombination velocity [2426]. Our analysis removes
these restricting assumptions, and hence is more general. The computer
aided numerical analysis, in this respect, is advantageous. The
experimentally obtained EBIC response shows excellent agreement with our
numerical analysis and demonstrates clearly the dependence of the
effective recombination velocity on the excitation at the grain
boundary, which provides corroboration for the transport model in
Chapter 2. By comparing the theoretical and experimental EBIC
responses, we predict values for the defect density at typical grain
boundaries and the minoritycarrier diffusion length in the grains.
Having demonstrated the detrimental influences of the grain
boundary recombination on the minoritycarrier transport, we next focus
our attention on experimental methods to improve the performance of
polysilicon solar cells. In Chapter 4, we describe an experimental
technique, using aluminum diffusion, to passivate the grain boundaries,
i.e., to reduce the defect density at the grain boundaries. Aluminum
was chosen owing to its compatibility with solar cell processing, and to
its large diffusion coefficient down the grain boundaries. It can also
concomitantly getter the intragrain impurities, i.e., reduce the defect
density in the grains, which results in an increase in the minority
carrier lifetime. The minoritycarrier lifetime in polysilicon is
typically lower than that in silicon due to high intragrain defect
densities, and hence it is useful to consider processes that improve
1i fetime.
For the development of bulk polysilicon solar cells, in which the
intragrain recombination is dominant, we propose a hightemperature
(above eutectic, 577C) aluminumdiffusion process from the back
surface. For the development of thinfilm cells, in which the grain
boundary recombination is dominant, we propose a lowtemperature (below
eutectic) aluminumdiffusion process from the front surface of the
polysilicon. We use forward currentvoltage and reversebias
capacitance measurements to demonstrate the intragrain gettering, and
EBIC measurements to show the grainboundary passivation. Our results
indicate that the hightemperature aluminumdiffusion process
effectively getters intragrain impurities, whereas the lowtemperature
aluminumdiffusion process produces significant grainboundary
passivation. We also find that the grain boundaries emit impurities
during the hightemperature process and that these emitted impurities
are effectively gettered by the aluminum.
Although we attribute the benefits produced by the hightemperature
aluminumdiffusion process in Chapter 4 to intragrain gettering, the
observed improvement in polysilicon device performance is also
commensurate with the formation of a backsurface field (BSF) [27]. The
BSF, which is nonuniform and nonplanar, is produced as a result of the
preferential dissolution of silicon during the hightemperature
aluminumdiffusion process [283. In Chapter 5, we examine the
mechanisms of gettering and BSF formation in an attempt to isolate the
actual mechanism that is instrumental in improving the polysilicon
device performance. The studies are based on forward currentvoltage
measurements of aluminumdiffused Wacker (cast) polysilicon cells, made
before and after lapping off the back p p junction. Our results
indicate that the observed improvement in the polysilicon device
performance is due to effective gettering of intragrain impurities that
occurs during the hightemperature aluminumdiffusion process, and that
the nonuniform BSF is ineffective. No improvement in the performance of
semiconductorgrade and solargrade [29] silicon cells is observed, and
hence we conclude that the benefits resulting from the aluminum
diffusion process are peculiar to the (cast) polysilicon. The benefits
result possibly because of unique impurities prevalent in the
polysilicon that are readily gettered by the aluminum. We surmise that
aluminum getters oxygen and/or oxygen related defects which are present
in higher densities in Wacker (cast) polysilicon than in semiconductor
grade and solargrade silicon.
8
We summarize our studies and review the main conclusions and
accomplishments of this dissertation in Chapter 6. We also discuss the
scope and limitations of this work and provide suggestions for future
research.
CHAPTER 2
ANALYSIS OF MINORITYCARRIER TRANSPORT IN
POLYSILICON DEVICES
2.1 Introduction
The performance of bipolar polycrystalline silicon devices can be
limited because of minoritycarrier recombination losses at the grain
boundaries [711]. For example, the efficiency of thinfilm polysilicon
solar cells [12], which can potentially be used to produce cost
competitive terrestrial photovoltaic energyconversion systems, is lower
than that of silicon cells because of grainboundary recombination.
Hence a complete theoretical understanding of the influence of grain
boundaries on minoritycarrier transport is essential in order to
optimize the design and performance of polysilicon solar cells and other
bipolar polysilicon devices.
In the general case this influence is described by a complicated,
threedimensional boundaryvalue problem having nonlinear boundary
conditions [23]. For example, in the forwardbiased polysilicon n+p
junction with columnar grains illustrated in Fig. 2.1, the electron
current injected into the quasineutral ptype base is governed, for
lowinjection conditions, by the threedimensional electron continuity
equation subject to nonlinear boundary conditions at the grain
boundaries [711]. The threedimensionality results basically because
the carriers are injected at a surface (i.e., the pn junction) that is
not parallel to the grainboundary surface. The boundary conditions,
10
ZZ E
<
zz
ro
C.
+ .,
0
P0
~o *
Jz
which characterize carrier recombination through energygap states at
the grain boundaries, can be expressed by an effective recombination
velocity that, in general, is dependent on the intragrain electron
density and on position. Approximate solutions [711, 13, 14, 303 for
this and related problems have been derived, but their ranges of
validity have not been checked.
To help advance our understanding of minoritycarrier transport in
polysilicon, and to aid the engineering design of optimal polysilicon
bipolar devices, including the solar cell, we describe in this chapter
the development of an analytic model for the minority electron current
injected into the base of the n+p junction in Fig. 2.1. The model,
which is subject to the uncertainties and inhomogeneities in the
morphology and in the electrical properties of polysilicon, is not
exact, but is, we believe, sufficiently representative of the physics to
ensure its utility. Although we emphasize polysilicon, the analytical
methods used in the model development are applicable to other
polycrystalline semiconductors.
The electron current 1 injected into the base of each grain in
Fig. 2.1 derives from recombination at the grain boundaries (I) and
G)and
within the quasineutral pregion (1 ). A key feature of the model is
the separation of the grainboundary recombination analysis, which
GB
defines the effective recombination velocity S ,f from the quasi
neutral grain transport analysis, which, with S defines The
n(eff)' G'
model differs from previous ones [711, 13, 14, 30] in the following
ways: (a) it is based on a computeraided numerical solution of
Poisson's equation within the grainboundary spacecharge region, which
provides the physical insight necessary to analytically characterize
S 8eff); (b) It unifies previous analyses based on the restricting
assumptions of carrier depletion [7, 9, 10] and of quasiequilibrium
[7, 8] in the grainboundary spacecharge region, as well as those
analyses based on the common assumption that S [13], by
n(eff) [13, by
incorporating into the quasiequilibrium analysis the physical upper
limit on S eff), i.e., the kineticlimit velocity SKL(= 5 x 106 cm/sec
at 3000 K) imposed by the random thermal motion of the conductionband
electrons [31]; (c) it avoids possible errors involved in truncating
infiniteseries solutions for the threedimensional electron continuity
equation in the quasineutral grain [8, 14, 23, 30] by reducing the
problem to coupled onedimensional ordinary differential equations that
are formulated based on "gradualcase" approximations [32]; (d) it is
the first analytic model for minoritycarrier transport in polysilicon
derived from the physical insight provided by computeraided numerical
solutions.
The model development begins with the computer solution of
Poisson's equation in the grainboundary spacecharge region adjoining
quasineutral ptype grains, assumed to be in low injection. The
numerical analysis is based on the quasiequilibriun assumption [7, 8,]
but the inherent limitations are effectively removed by accounting for
the kineticlimit electron velocity. The calculation of the grain
boundary potential barrier height is based on the assumption of an
"effective" monoenergetic density of donortype traps located near the
middle of the energy gap. This assumption is commensurate with the
experimental observation [20] that the "neutral Fermi level" of
polysilicon grain boundaries is near midgap. Once the barrier height is
determined, the composite recombination rate of holeelectron pairs
through both donortype and acceptortype traps is calculated accounting
for the commonly large difference between the carrier capture parameters
for neutral and ionized traps [9]. This recombination rate is then used
to derive S f) which is a function of the electron density.
Empirical dependencies, implied by the numerical analysis, of S, on
n(e ff) on
the grainboundary surfacestate densities, on the grain doping density,
and on the "excitation" of the grain boundary, i.e., the separation of
the electron and hole quasiFermi levels, are used to formulate an
expression for S ef) The expression is supported by qualitative
discussion of the underlying physics, including mention of the effects
of assuming different trap energy levels.
To complete the model, the electron continuity equation in the
quasineutral ptype base of a representative grain, assumed to be a
rightcircular cylinder, is analytically solved subject to the nonlinear
rGB
boundary condition [S ]n(e at the grain boundaries, which varies with
excitation along the boundaries, and to the excess electron density at
the edge of the junction spacecharge region resulting from the forward
bias V. This complicated threedimensional problem is solved
analytically for relatively large grains by partitioning the grain into
subregions in which the electrons are assumed to flow predominantly in
one dimension. The onedimensional solutions are then coupled to give
the composite solution, which is approximate but nevertheless
illustrates well the effects of the nonlinearity. The result is an
analytic expression for I in terms ofV and the parameters defining the
morphology and electrical properties of the grain and the encompassing
grain boundaries.
2.2 Analysis
The assumed representative grain, a right circular cylinder, of the
polysilicon n+p junction in Fig. 2.1 is illustrated in Fig. 2.2(a). The
grain and grain boundary are assumed to be isotropic. Hence rotational
symmetry exists about the central axis, and the carrier transport
problem can be reduced to two dimensions as indicated in Fig. 2.2(b).
Our objective is to analytically characterize the injected electron
current I = I + IN that results from the junction forward bias V.
(Note that we are accounting for recombination at the portion of the
grain boundary adjacent to a quasineutral region. Recombination at the
grain boundary within the junction spacecharge region, which can be
significant [33], is discussed in Section 2.3.) Rigorously this would
require the solution of the complicated twodimensional electron
continuity equation in the quasineutral pregion of the grain subject
to appropriate boundary conditions at the grainboundary and junction
spacecharge regions [23]. The derived electron density N(r,z) must be
B
integrated over the volume of the pregion to determine IQ and its
gradient integrated over the encompassing grainboundary surfaces to
determine IG. Such a rigorous solution would, even with simplifying
assumptions, require a computeraided numerical analysis [13].
To facilitate the engineering design of polysilicon bipolar
devices, and to advance our understanding of the pertinent physics
underlying their performance, an approximate analytic solution is
useful. To derive such a solution, we separate the analysis of the
grainboundary recombination from that of the intragrain transport, even
though the mechanisms are interrelated. This interrelationship
Ir _rG B
I =rG
pI

GRAIN BOUNDARY
?B 1 I \ SPACECHARGE
"SC BR REGION
QUASINEUTRAL
+V pTYPE BASE OF GRAIN
(b)
Fig. 2.2 Representative grain (a) of the forwardbiased n p junction
and its crosssection (b). We assume rGGW SR; hence the
zdependence of W GB
zdependence o W deriving from the zdependence of the
grainboundary excitation is inconsequential and has not been
shown.
materializes when we combine the results of the analyses to determine
N(r,z) and to characterize I (V).
2.2.1 Grain Boundary
We first consider the grainboundary spacecharge region, which is
identified in Fig. 2.2(b). By describing the holeelectron
recombination rate UGB along the grain boundary, assumed to involve only
bandbound transitions, we can define the effective surface
combination velocity SGB
recombination velocity S for minority electrons at the edge of the
grainboundary spacecharge region.
At an arbitrary depth z in the base region, we consider the energy
band diagram in the grainboundary spacecharge region. This diagram,
which follows from the solution of Poisson's equation to be discussed,
is illustrated for nonequilibrium conditions (V > 0) in Fig. 2.3. The
existence of the potential barrier height B is due to majoritycarrier
(hole) charging of the grainboundary surface states and has been
discussed extensively in previous papers [2, 710, 20].
To describe tB of the polysilicon grain boundary adjacent to a
ptype grain, we assume an "effective" monoenergetic density NT(eff) of
donortype grainboundary surface states, or traps, which are either
neutral or positively ionized, is representative of the grainboundary
potential barriers for ptype polysilicon because the "neutral Fermi
level" of polysilicon grain boundaries, which is related to the unknown
nature of the surface states, is experimentally observed to be near
midgap [20]. This means that the electron energy bands will bend down
at the grain boundaries in ptype polysilicon (see Fig. 2.3); this will
be predicted by an analysis based on donortype traps. Indeed N T(eff)
ST(e ff)
U
04C 4
50cw
U, C~
!(0)
o 0) 0
0.m U,
fr~O
C.' 04'
SC
04'
U,
C C
0 04 C)
3 +
04i
040
04 '
0= U,
0C (
C, C. aj
m w4
(0 S
aj 3
0* 0
w
wr1
is the grainboundary trap density that would be inferred from
conductance or capacitance measurements [2, 20] of ptype polysilicon.
Although the assumption of a monoenergetic density of traps at ET=
Ei(O) may not characterize exactly actual grain boundaries in ptype
polysilicon [34], it is sufficiently representative to describe their
basic influence on the minority electron transport. The utility of a
monoenergetictrapdensity analysis in characterizing the electrical
behavior of polysilicon has been previously demonstrated [8, 10, 20,
23].
We recognize however that UGB, which depends on tB, must be
calculated based on the possible existence of both donortype (N) and
) and
acceptortype (NA) trap densities. We later describe this calculation
and the resulting description ofS which depend critically on the
n(eff)' which depend critically on the
large difference between the carrier capture parameters of neutral (Cn)
and ionized (Ci) traps [9]. We also later discuss qualitatively the
effects of moving ET away from midgap.
For steadystate conditions, i.e., gB timeindependent, the energy
band diagram in Fig. 2.3 and UGB, which defines the rate of flow of
electrons and holes to the grain boundary, are determined by the
condition that the grainboundary charge that produces overall
neutrality in the grainboundary spacecharge region be commensurate
with equal net capture rates for electrons and holes at the grain
boundary surface. Mathematically, the quasiequilibrium assumption
enables us to express this condition by combining Poisson's equation for
the grainboundary spacecharge region with ShockleyReadHall capture
emission statistics [35] for the grainboundary surface states.
Referring to Fig. 2.3, in which the grain boundary is located at
x = 0, we write Poisson's equation in the spacecharge region
(0 < x < WGJR) as
d2E.) a2
_l = [P(x) N(x) NAA] (2.1)
dx
where P and N are the hole and electron densities, NAA is the grain
(acceptor) doping density, q is the electron charge, e is the dielectric
constant of silicon, and Ei is the intrinsic Fermi level, whose
derivative with respect to x defines the electric field. The electric
field at x = 0 is determined by the grainboundary surfacestate charge
through Gauss' law and provides one of the two boundary conditions
necessary to solve (2.1):
2D
x=0 q2NT(eff)(1 f)(2.2)
_T x1 =0 = 2 (2 .2)
where f is the steadystate probability of electron occupancy for the
surface states. The factor of 1/2 in (2.2) reflects our implicit
assumption of symmetrical energy bands on either side of the grain
boundary, i.e., identical adjacent grains.
Assuming a unity degeneracy factor for the grainboundary surface
states, we get from the ShockleyReadHall theory [35]
Ei (O)ET
C N(O) + Cpnexp (2.3
pnW e i T
SFETEiO1 rEi(O).)E (2.
CN(0) + Cn nexp + C P(0) + C exp
which simplifies under our assumption that ET = Ei(O). In (2.3), ni is
the intrinsic carrier density in silicon; C, is the capture coefficient
for electrons, which we assume is Ci = 107cm3/sec (at 300"K) because
the trap is (positively) ionized prior to the electron capture [36]; and
Cp is the capture coefficient for holes, which we assume is
Cn 10"9cm3/sec (at 300K) because the trap is neutral prior to the
hole capture [36].
To write the second boundary condition, we assume low injection in
the uniformly doped quasineutral pregion:
S GB 0 (2.4)
SWSCR
This assumption also means that
P(WR) NAA ( 2.5)
Within the grainboundary spacecharge region, P and N are related to Ei
through the respective quasiFermi levels EFp and EFN:
FE (x)E 1
P(x) = niexp  (2.6)
SEFNEi(x)
M(x) = n exp !E (2.7)
The difference between EFN and EFp reflects the "excitation" of the
grain boundary that results from the forward bias on the n+p junction.
To facilitate the separation of the grainboundary recombination
analysis from the intragrain transport analysis, we assume that
conditions of quasiequilibrium exist in the grainboundary spacecharge
region. This assumption implies that EFN and EFp are nearly flat (i.e.,
that their variations are less than kT) in this region, and will be
valid if N and P are sufficiently large and the electron and hole
currents sufficiently small, as we discuss later. This will be the case
for sufficiently high excitations (see Fig. 2.3),
qVB EFN EFP (2.8
and sufficiently low potential barriers,
qB Ei (WsR) E (0) (2.9)
Note that VB defined in (2.8) decreases with z in accordance with
N(rG,z), and that necessarily VB(z) is everywhere less than V because of
the electron density gradient in the grain.
As we will see, the quasiequilibrium assumption enables us to
characterize UGB without having to describe the mechanism, e.g.,
thermionic emission [10], by which majority carriers (holes) are
transported from the quasineutral region to the grainboundary
surface. This simplification is made without loss of generality because
when the quasiequilibrium assumption is invalid, for example, when the
grainboundary spacecharge region is virtually depleted of free
carriers [7, 9, 10], the minority electrons are approaching the grain
boundary surface with velocities about equal to their kinetic limit
SKL (= 5 x 106 cm/sec at 3000K) [31]. Hence, as we show later, when
(2.8) becomes inapplicable, the grain boundary can be adequately modeled
as a surface having a recombination velocity equal to SKL. A detailed
analysis of grainboundary recombination based on the depletion
approximation thus is unnecessary and, in fact, is invalid.
The solution of (2.1)(2.7), with arbitrary excitation Vg in (2.8),
is impossible to derive analytically. Thus to aid the development of an
analytic model for SGB
analytic model for eff) in terms of VB and the polysilicon
properties, we use computeraided numerical techniques. A Harwell
subroutine [37], which represents (2.1) using finitedifference
approximations for the derivatives, has been slightly modified to solve
(2.1)(2.9). Details of the numerical algorithm and the listing of the
computer program are given in Appendix A. We now discuss results of
this numerical analysis of the grainboundary region.
Consider first the thermalequilibrium case (V = 0) for which
Vg = 0 and EFN and EFp are coincident with the Fermi level EF.
Equilibrium barrier heights 0BO at polysilicon grain boundaries have
been previously calculated [7] and measured [20, 34, 38], and comparison
of our numerical results with these earlier ones provides an assessment
of the assumptions underlying the analyses. For example, it implies the
validity of the depletion approximation for equilibrium conditions. In
Fig. 2.4 we have plotted OBO versus NAA for representative values of the
effective grainboundary surfacestate density NDT(ef. When N Def
is very low (< 1010 cm2), BO is extremely small, and electrically the
polysilicon resembles singlecrystal silicon. For intermediate values
of NT(eff) (~ 101 cm'2)' BO decreases monotonically with increasing
NAA. This is explained by the fact that BO is small enough that EF is
below ET (see Fig. 2.3), and hence virtually all of the (donor) surface
states are empty (positively ionized) for all NAA. Thus as NAA
increases, the spacecharge region width, which contains charge
(including ionized acceptor impurities) that neutralizes the grain
boundary charge, diminishes and hence iBO decreases. When NT(eff) is
high (> 1012 cm2), BO is large enough that EF is essentially fixed
at, or slightly above, ET (see Fig. 2.3). Thus as NAA increases and EF
moves closer to the valenceband edge in the quasineutral grain [i.e.,
farther away from Ei as described by (2.5) and (2.6)], ET follows EF at
x = 0, and hence 0BO increases slightly.
We note that the calculated values of 3BO in Fig. 2.4 are generally
higher than those derived from measurements [34] of the activation
energy for the zerobias conductance of grain boundaries in ptype
polysilicon. This discrepancy results because of our assumption
ET = Ei(0), which is not, as we mentioned earlier, strictly valid.
Assuming ET < Ei(O) brings the BO predictions and measurements into
better agreement, but does not significantly augment the physical
insight provided by our analysis. It is interesting to note that for
ntype polysilicon, analogous calculations based on an assumed
monoenergetic density of acceptortype traps at midgap yield values of
gBO that are roughly consistent with those derived experimentally [38].
Consider now nonequilibrium cases (V > 0) for which VB > 0 in
accordance with the quasiequilibrium assumption in the grainboundary
spacecharge region. Nonequilibrium conditions for which this
assumption fails, alluded to earlier, will be treated explicitly
later. The relationship between VB(z) and V is complex and follows from
the electron continuity equation in the quasineutral pregion of the
grain, which we solve in the next subsection.
1011
0
'o 101
10 o'l
NAA (cm3)
Fig. 2.4 Calculated thermalequilibrium barrier height versus grain doping
density (ptype) for representative values of effective grain
boundary surfacestate density (T = 300K).
Figs. 2.5 and 2.6 show plots of calculated barrier heights i.
versus VB for representative values of N0T(ef and for NAA 1015 cm3
and 1017 cm3 respectively. These plots are restricted to conditions of
low injection in the quasineutral pregion. They imply that
N5B 1
v (2.10)
for VB greater than a few thermal voltages (kT/q = 0.026 V), a condition
that validates the quasiequilibrium assumption. The monotonic
reduction of nB with increasing VB is consistent with the results in [8]
and is qualitatively explained as follows. The electron capture rate at
the grain boundary, which is proportional to N(O), increases with
increasing excitation. Consequently the net charge at the grain
boundary and hence the grainboundary barrier height, which is directly
proportional to the net charge, decrease with increasing excitation..
The slope of 1/2 in (2.10) has a physical basis as we now describe.
For steadystate conditions with sufficient excitation, the
electron and hole capture rates at the grainboundary surface states,
including both donor and acceptor types, are nearly equal. Thus, for
either type of state, we can write
t [C"(1 f)N(O)] = [CpfP(O)] (2.11)
From (2.3), for the assumed conditions, we see that f = CnN(0)/[CnN(0) +
CpP(O)], which is always between zero and one. Because the change in f
induced by a small variation in Vg will be much smaller than the
corresponding changes in N(0) and P(0), (2.11) implies
ST(off)
NAA= 1015cm3
0.30
V,(V)
Fig. 2.5 Calculated nonequilibrium barrier height versus grain
boundary excitation for representative values of effective
grainboundary surfacestate density and for a grain doping
density of 10 s cm3 (T = 3000K).
ik
La
o C
LW
0
: O"
50
o a
.5n3
LWa
o r r
o as
C 0
LW
CO
0 0 L
o .I~C
o 0 Ci
04s 44
( I
o 41
(U yi
>C'
(A (
SS3
O 'i Ol
E (U
0 * II
c "+
^*'"
o D +
1 aN(0) 1 aP(0) (2.12)
CT37g PiTy *(2.12)
NT7 TD73 TM m a
The relation in (2.12) is a general result of invoking the steadystate
condition (dOg/dt = 0) at a sufficiently excited grainboundary
surface. Combining it with
P(0)N(0) = nexpI (2.13)
which follows from (2.6)(2.8), reveals that
1 aP(0) q (2.14)
Then substituting
P(0) = NAA exp (2.15)
which results from (2.5), (2.6), and (2.9) with the assumption that EFp
is nearly flat, into (2.14) yields (2.10). (Note that this analytic
description of the grainboundary physics, as well as others to follow,
are facilitated by the numerical solutions.)
The important conclusion to be drawn from (2.10) and the associated
relationships mentioned above is that
P(0), N(0) exp (2.16)
and consequently, as we detail later, UGB exp(qVg/2kT). We now
discuss the dependencies of Bg, P(O), and N(O) on NT(eff) and on NAA as
derived from the numerical solutions of (2.1).
These solutions, those plotted in Figs. 2.5 and 2.6 and others,
imply further that
aB kT (2.17)
SoNST(eff)]
for sufficient excitation (VB) as defined earlier. The implications of
(2.17) regarding P(O) and N(O) follow from (2.13) and (2.15):
P) N ) log(e) ) 1/2 (2.18)
P(o) [sT(effl [IST(eff)] (2.18)
and
l D 1og(e) 1/2
N(0) NT(efflog(e) T(eff 12 (2.19)
which are also seen directly from the numerical results.
A physical explanation for (2.18) and (2.19) can be given, again
based on the condition of nearly equal electron and hole capture rates
through the (effective) grainboundary traps:
C(1 f)NTeff)N(O) C fN TeffP(0 (2.20)
Cs ( T( ff) p ST(eff)
For sufficient VB, partly because of the large discrepancy between
C, ( Ci) and Cp (= Cn), f = 1; i.e., most of the traps are filled with
electrons. Therefore the factor (1 f)N in (2.20) must be a
)ST~e ff) in (2.20) must be a
sublinear function of NT(eff); i.e., f must increase slightly with
increasing NsT(eff). Thus (2.20) suggests that
N(0) ND
PTOT ST(eff) (2.21)
Then (2.13) and (2.21) give approximately the observed dependencies
(2.18) and (2.19).
Finally, the numerical solutions of (2.1) for sufficient excitation
reveal that
"B 3kT
30_B[T 3kT (2.22)
7[1ogNA]A 2q
and that
[1 log(e)] /
P(0) Z NAA N 3A (2.23)
and
[1 1log(e)]
N() AA N (2.24)
These approximate dependencies can be supported by physical argument as
follows. From (2.15), we get
aP(0)
B kT aNAA
NAA q I 1 j. (2.25)
"AA
For B > 0, the second term within the brackets of (2.25) is less than
one, and, typically, considerably so as evidenced by (2.15), Fig. 2.3,
and the exponential relationship between P and EFp, which defines the
modification in the energyband diagram resulting from a change in
NAA. Thus (2.25) implies that
T_ kT' (2.26)
"'AA 
which is approximately equivalent to the observation (2.22).
Our computeraided determination of the dependencies of pB and of
P(0) and N(O) on Vg, NST(eff), and NAA enables the characterization of
the steadystate holeelectron recombination rate UGB through the grain
boundary surface states, provided the active surfacestate densities are
specified. This calculation is based on the ShockleyReadHall capture
emission model [35] for recombinationgeneration through localized
states in the energy gap.
Because the exact nature of the grainboundary surface states in
polysilicon is not known, we allow for the possible existence of both
donortype (NT ) and acceptortype (NST) states in our calculation of
UGB. We assume that all states are located at ET = Ei(O). Note
that FT and NST are different from N T(ef which was used to
calculate g, and P(O) and N(O). Physically, based on this
model, N TT and NST produce grainboundary charge that is effectively
characterized by N D(f.
If we neglect boundbound carrier transitions between the donor and
acceptor traps, the composite recombination rate through NT and NT is
simply
UGB = UDNT + A (N(T
UGB NGB(NST) + UGB(NST)
(2.27)
where UD and UB are ShockleyReadHall representation [35] of the
recombination rates through the donor and acceptor traps respectively:
P(0)N(O)n2
UD (2.28)
GB P(O) + n () + n. (228
ST T
P(O)N(0)n2
A 1
UGB = P(O) + n. N(0) + n (2.29)
+
ST MST
In (2.28) and (2.29) we have used the previously assumed capture
parameter values Ci = 107 cm3/sec and Cn = 109 cm3/sec for the ionized
(positively or negatively) and neutral states of the traps.
The dependencies of UGB on the excitation Vg and on the grain
boundary surfacestate densities, NDT(eff)' N T, and NT', and on the
grain doping density NAA are implicitly given by (2.27)(2.29) and the
explicit dependencies of P(0) and N(O) derived from the solutions of
(2.1). For example, (2.16) and (2.27)(2.29) show
qV8
UGB =exp (2.30)
for the sufficient excitations needed to formulate the analytic
relationships discussed earlier. The result (2.30) agrees with [7] and
[8], but is not equivocal like [7] and [8] because of questionable
assumptions, which, in fact, are shown to be invalid by our numerical
results. Furthermore (2.30) has been derived here for the general case
in which both donor and acceptortype traps are present at the grain
boundary surface, and the constant of proportionality for (2.30) is
related correctly to the assumed model for the surface states, in
contrast to [7] and [8].
If we neglect recombination in the grainboundary spacecharge
region (see Fig. 2.3), UGB defines the effective recombination velocity
for minority electrons at the edge of the spacecharge region [7, 8]:
GB UGB2.31)
n(eff) (2.31)N(
where, from (2.5)(2.8),
2
GB 2 qVB
N(W ) = exp ; (2.32)
SCR AA
VB and N(WSR), both dependent on z, define the grainboundary
excitation and are determined from the coupled solution to the electron
continuity equation to be discussed in the following subsection. Note
the nonlinearity associated with S as defined
n(eff) s defined by (2.30)(2.32):
2 1/2
SGB s[N D NDINA ] p oL 0 C(2.33)
Sneff) o ST(eff) l ST. AA exp = AAN G (233
where the preexponential coefficient S0 reflects the functional
dependencies derived from the numerical solutions. An expression for
SO, derived empirically from the numerical solutions, is given by
C ni NA NDST
So 2n Cn AA log2 5x1010 1/2 +i115 2 N 1/2
A l \ ST +C NAA 5x101
NA
i[ NAA\10g2 5x100 1/2 n 1015 log2 (2.34)
C NT + C ST
i0r' NA + (AA
T ST
/j  T AA S5x10^
Values of fS have been numerically calculated for two cases
D annef
involving assumed trap densities NST and NAT. In the first case, we let
NST NST(eff and ST = 0, which corresponds to the simple common
assumption that only donortype traps are effective on a polysilicon
grain boundary between ptype grains [23]. Plots of S versus VB
nCeff) Bersus VB
for this case are shown in Figs. 2.7 and 2.8 for NAA = 1015 cm3 and
1017 cm3 respectively. In the second case, we let NST = 2NT(eff)
and NAT = N T(eff to approximately represent a possible situation
involving both donor and acceptortype traps at the grain boundary.
Figs. 2.9 and 2.10 show Sneff for this case, and emphasize the
possibility of certain traps influencing UGB but not significantly
affecting *g or the conductance or capacitance associated with the grain
boundary.
Superimposed on the plots in Figs. 2.72.10 is an indication of the
kineticlimit velocity SKL (~ 5 x 106 cm/sec at 300K), which is defined
by the random thermal motion of electrons in the conduction band [31].
The velocity SKL is the average directional thermal velocity defined by
MaxwellBoltzmann statistics for conductionband (free) electrons, and
is the physical upper limit for SGeff). We therefore recognize that
our plots ofS eff) in Figs. 2.72.10 must be truncated at SKL. This
recognition effectively removes any uncertainty or restriction of our
model due to the possible invalidity of the quasiequilibrium
assumption. We now demonstrate this.
The quasiequilibrium assumption is valid if the variations in the
quasiFermi levels across the grainboundary spacecharge region (see
Fig. 2.3) are less than kT [8, 9]. A quantitative selfconsistency
check for the validity of this assumption is described in [8]. Using
this check and our results shown in Figs. 2.72.10, we find that
generally the quasiequilibrium assumption is justified provided
sGB GB
n(eff) is less than SKL. Since S eff) cannot physically exceed SKL,
our model is then generally valid if, as indicated in Figs. 2.72.10, we
stipulate that
MAX[S eff] = SKL (2.35)
Hence for all conditions under which the grainboundary spacecharge
region is depleted of free carriers, which obviously preclude quasi
equilibrium, SGB
equili n(eff) SKL and a depletionapproximation analysis
[7, 9, 10] is unnecessary. In fact, a recent such analysis [10] yielded
results that comply with these conclusions.
Note in Figs. 2.72.10 that for low values of NT depending
o GB
on NAA n(eff) as calculated from the numerical solutions of (2.1) is
insensitive to low excitations Vg. Our analytic model (2.33) is
inapplicable for these conditions because of the underlying assumption
of "sufficient excitation." This insensitivity reflects the
insensitivity of B on VB for low N (eff) illustrated in Figs. 2.5 and
2.6. Fig. 2.4 reveals that OB = BO for these cases and explains
10o4Cm2 N e
. 2 = "1 T(f,,t)
NAA=1015cm3
SST N T=NST(e f)
1 N 0=o0
ST 103
1012
1010
00 oo10 o..o 0.30 o0.o .o50 0.6o
B (v)
Fig. 2.7 Calculated effective recombination velocity for minority electrons
at the edge of the grainboundary spacecharge region versus grain
boundary excitation and surfacestate density. The kineticlimit
velocity, which is the physical upper limit for the recombination
velocity, is indicated here and in Figs. 2.8 2.10.
Fig. 2.8 Calculated effective recombination velocity for minority electrons
at the edge of the grainboundary spacecharge region versus grain
boundary excitation and surfacestate density.
E
SAN A1015cm3
AA
NT2 NS T(Tf) 1013
SNT= NST (elf)
10"
1010
0.00 0.10 O. 0.30 oo0 0.50 0.6s
B (v)
Fig. 2.9 Calculated effective recombination velocity for minority electrons
at the edge of the grainboundary spacecharge region versus grain
boundary excitation and surfacestate density.
U.u 0. 0.~0 0.30 O.O 0.50 0:60 0.70 0.80
VB(V)
Fig. 2,10 Calculated effective recombination velocity for minority electrons
at the edge of the grain boundary spacecharge region versus grain
boundary excitation and surfacestate density.
why SG is independent of the grainboundary excitation. The
n(e ff1
excitation is insufficient to create a high enough electron density to
alter the barrier height from its low equilibrium value [N(O) <<
P(0)]. Thus, when the electron emission rate is negligible,
Seff) ~ CNT for these cases.
2.2.2 Grain
We now derive an approximate solution for the twodimensional
electron continuity equation in the quasineutral pbase of the grain
illustrated in Fig. 2.2(b). This solution, which is governed
by SGB
by S(ff) given in (2.33) and (2.35), yields the electron
nB B B i
current I B + injected into the base of the forwardbiased n p
junction.
Because of the nonlinear boundary condition at the grain
boundaries, a rigorous derivation of this solution would require a
computer. To simplify the problem such that an analytic solution,
useful for engineering design and for demonstration of the pertinent
physics, can be obtained, we partition the pregion as shown in
Fig. 2.11. The implicit assumption made in doing this is that in
certain subregions of the base, the electron flow is predominantly one
dimensional; that is, the divergence of the current density has one
predominant term. Thus the resulting onedimensional forms of the
continuity equation can be solved, and the solutions then coupled to
give N(r,z), which defines I. This simplification is similar to
Shockley's "gradualcase" approximation in his analysis of the unipolar
transistor [32] and was qualitatively mentioned in [23]. Its utility
was established in a numerical study [39] of the twodimensional
minoritycarrier flow in a forwardbiased planar diffused pn junction.
As indicated in Fig. 2.11, we assume that in the central portion of
the grain (r < r1), the electron flow is not influenced by the grain
boundary recombination, and is hence in the downward zdirection.
Contrarily, we assume that in the vicinity of the grain boundary (rI <
r < rG), the electron flow is strongly influenced by the grainboundary
recombination, and is thus primarily in the lateral rdirection. These
assumptions are commensurate with letting
rI = rG L (2.36)
where Ln is the electron diffusion length. Note that (2.36) restricts
our quasitwodimensional analysis to cases for which rg > Ln. Although
B B
this inequality implies that IB is not dominated by IB, the solutions
for these cases nevertheless serve to illustrate well the effects of the
nonlinearity introduced by S GB If r < L,, which implies a
n(eff) G n,
B
predominance of IGB, the electron flow is truly threedimensional
everywhere, and this case requires a computer solution. However, from a
practical viewpoint, this case may be undesirable because the electron
current is dominated by grainboundary recombination (IGB > I it
could be avoided perhaps by increasing NAA to reduce L., while still
B
decreasing I.
In the central region of the grain, the electron density N(z) for
lowinjection conditions is defined by
d2N(z) N(z)
d, (2.37)
with the boundary conditions
z
+V
=rG
T
Fig. 2.11 Partitioned pregion of the representative grain showing predom
inantly onedimensional electron flow in the subregions (rG>Ln).
2
M(0) = exp( (2.38)
and
N(Wg) = 0 (2.39)
the latter being due to the ohmic contact on the back of the junction.
The solution of (2.37)(2.39) which is obvious, then provides one
boundary condition, i.e., N(rl,z), for the coupled problem in the
vicinity of the grain boundary:
[r r 2 0 (2.40)
The general solution of (2.40) is an infinite series. To obtain an
approximate solution that is consistent with our partitionedbase model,
we initially neglect the second term, but later account for the
recombination current it defines by integrating qN(r,z)/Tn (n is the
electron lifetime) over the quasineutral base of the grain. This
results in a In(r) dependence for N.
The second boundary condition for (2.40) is that defining the
grainboundary recombination discussed in the previous subsection:
1 (rGz) =Dn NN(r z) (2.41)
SN r G rzr n=eff) G f '
GB
where Dn is the electron diffusion coefficient and where Sn(eff), which
depends on z, is given either by (2.33), in which N(W s) = N(rGz), or
by (2.35). The determination of which expression to use involves an
iterative process. We initially use (2.33), solve (2.37)(2.41), compare
(2.33) and (2.35) everywhere on the grain boundaries, and then use
(2.35) where necessary. This complicated process is typically
simplified because unrealistic values ofS f) >S imply, through
the solution of the electron continuity equation in the grain, that
Vg = 0 at the point being considered; this is also implied by the
realistic value of Seff) = SKL provided SKL > Dn/Ln, which is common.
The electron density N(r,z) derived from (2.37)(2.41) can now be
used to calculate the injected electron current I The component due
to recombination at the grainboundary surfaces is
W
IB = 2rqD rG J N(r dz (2.42)
0 a r=rr
and that due to recombination in the quasineutral pbase is
I NB 2q B OG N(r,z) r dr dz (2.43)
in which the integration must be done in accordance with the
partitioning of the pregion.
The general form of the resulting expression for the electron
current, (IB + QN), is
I(V) = 101exp. + IOnexp I l (2.44)
In (2.44) the exp(qV/kT) component is the electron current that would be
injected in the absence of grain boundaries. For example, if Ln < WB,
then 01 A qn D /NAALn where A = G is the area of the grain. In
the exp(qV/nkT) component, which in general comprises both grain
boundary and intragrain recombination, IOn depends on SO, given in
(2.34) in addition to the pregion parameters that define I01 The
reciprocal slope factor n(V), which also depends on SO and the pregion
parameters, increases from one to two as V increases, albeit in the
range corresponding to lowinjection conditions in the pregion.
The general shape of the I,(V) characteristic in (2.44) for typical
values of the grain and grainboundary parameters is illustrated in
Fig. 2.12. The shape resembles that of the dark currentvoltage
characteristic of a BSF solar cell [40], e.g., a p nn+ cell, in which
the inflection in the curve results because of the relative significance
of different components of current having different reciprocal slope
factors.
In Fig. 2.12, for low V, I G exp(qV/kT) because n(V) = 1. This
occurs because the electron injection level is very low, and hence
S which is proportional to [N(WG )]1/2 ( [N(rgz)]1/2 as
n(eff) 'S
shown by (2.33), is high and fixed at SKL everywhere (see Figs. 2.7
2.10). The nonlinearity is thereby removed and both IB and I vary as
exp(qV/kT). As V increases, N(W R) tends to increase also, and
exp~qV/T). SsR
thus S B f begins to fall below SKL at portions of the grain
n( eff)
boundaries. The nonlinearity is hence apparent, and n(V) exceeds unity;
the slope of I~(V) decreases as shown in Fig. 2.12. At higher V, the
exp(qV/nkT) component becomes insignificant, and I (V) I01exp(qV/kT),
resulting in the inflection in the curve, above which the grain
boundaries are inconsequential because S is low where the electron
n(e ff) is low where the electron
density is appreciable.
The general shape of the IB(V) curve predicted by our analysis and
shown in Fig. 2.12 is consistent with measured currentvoltage
characteristics of gated n+p diodes fabricated on Wacker polysilicon
[16]. The characteristics of diodes containing substantial grain
boundarysurface area show an inflection similar to that in Fig. 2.12
whereas those of diodes having little or no grainboundarysurface area,
e.g., diodes fabricated within a grain, shown no inflection.
If SO + 0, then In 0, and the I~(V) characteristic approaches
that of a singlecrystal silicon junction. If S GB ) everywhere,
n(eff) + SKL here,
then n 1, and, as in the SO = 0 case, I(V) = exp(qV/kT), but with a
preexponential coefficient greater than I01 because of the grain
boundary recombination. This discrepancy is commonly described in terms
of an "effective" minority electron lifetime in the grain, which is
smaller than T, [1, 8, 13]. It is important to note however that such a
description pertains only to IG, and not necessarily to other responses
of the junction, e.g., the shortcircuit photocurrent, and is meaningful
only when n = 1.
nexpqV (VnkT n= \0 x
Oflxp nkT)
i01exp V ) 1 exp
LkT/ nkT/
Fig. 2.12 Theoretical representation of the electron current injected into
the base of the forwardbiased polysilicon n+p junction.
2.3 Discussion
In this chapter we have developed, using computeraided numerical
solutions, an approximate model for the electron current I injected
into the base of a representative grain of the forwardbiased
polysilicon n+p junction illustrated in Fig. 2.1. Key assumptions have
been made to simplify the general threedimensional, nonlinear boundary
value problem, i.e., the electron continuity equation, and to make this
development possible. We now discuss these assumptions and, where
necessary, the model limitations they imply.
We based the grainboundary recombination analysis on the
assumption of quasiequilibrium, i.e., nearly flat quasiFermi levels,
in the grainboundary spacecharge region. This assumption facilitates
the separation of the grainboundary recombination analysis from the
intragrain electron transport analysis, the results of which define the
grainboundary excitation. The assumption further enables a complete
analysis of the grainboundary recombination without having to ascertain
the mechanism, e.g., thermionic emission [10], by which majority holes
are transported from the quasineutral grain to the grainboundary
sur face.
The limitations associated with the quasiequilibrium assumption
are effectively removed by recognizing that when conditions obtain that
negate quasiequilibrium, the minority electrons flow to the grain
boundary surface with velocities about equal to the kineticlimit
velocity SKL [31]. Thus we simply truncated in (2.35) our quasi
equilibriumbased prediction (2.33) for the effective electron
recombination velocity Seff at SKL when (2.33) yields S G
n( e ff ) KL n (2.33) yields Seff) = SKL"
Even with the quasiequilibrium assumption, the grainboundary
analysis, i.e., the solution of Poisson's equation (2.1), is
formidable. Thus we resorted to a computeraided numerical solution of
(2.1) to facilitate the derivation of the analytic model for S Gef
n(eff)
Empirical relationships implied by the numerical solutions were
supported by qualitative discussion of the underlying physics.
We also assumed monoenergetic densities of surface states at the
grain boundary. First, in accord with the position of the "neutral
Fermi level" of polysilicon grain boundaries being near the middle of
the energy gap, we postulated the existence of effective donortype
(adjacent to ptype grains) states near midgap to calculate the grain
boundary potential barrier height nB. Then, based on our assumed
existence of both donortype and acceptortype traps at midgap, OB was
used to calculate the grainboundary recombination and ultimately
SGB
n(eff)
The utility of an analysis based on a monoenergetic density of
grainboundary surface states, which is not realistic, has been
discussed before [8, 10, 20, 23]. Such a model provides physical
insight and possibly simulates well actual surfacestate distributions
in the energy gap that significantly influence carrier recombination at
the grain boundary. A recent analysis [11], which assumes the surface
states to be distributed in the energy gap, has yielded results that are
consistent with ours. To generalize our results somewhat, we studied
the effects of moving the trap level ET away from midgap. Provided ET
remains relatively deep in the energy gap, i.e., = 5kT above or below
midgap, the functional dependencies in our model prevail. That
is, S eff3 as described in (2.33) for sufficient excitation of the
n(e ff)
grain boundary depends on the electron density in the adjacent grain and
hence manifests a nonlinear boundary condition for the electron
continuity equation in the quasineutral pregion. When ET is shallow,
near either the conduction or the valence band, the carrier emission
(exponential) terms in (2.3) tend to dominate and render f, and hence 0g
nearly insensitive to the grainboundary excitation. Thus in this case,
GB
the nonlinearity does not occur, and S ,eff is nearly constant equal
either to SKL (e.g., for ET near the conduction band) or to lower values
(e.g., for ET near the valence band).
However the observation [20, 34, 38] that the neutral Fermi level
of polysilicon grain boundaries is near midgap implies that ET is not
shallow. Thus the electron transport within the quasineutral pregion
is generally complicated by the nonlinear boundary condition defined by
S ff) in (2.33). To enable the derivation of an analytic solution
for the electron continuity equation, we partitioned the pregion into
subregions in which we assumed the electron flow is predominantly one
dimensional. In the central portion of the grain, we assumed that the
electron flow was not significantly influenced by the grainboundary
recombination. The onedimensional solution in this subregion then
provided a boundary condition for the coupled transport problem in the
subregion adjacent to the grain boundary where the strong influence of
the grainboundary recombination, simulated by S GB [i.e., (2.33) and
n(e ff)
(2.35)], was assumed to cause the electrons to flow onedimensionally
toward the grain boundary.
The general results of the intragrain analysis, valid for rG > Ln,
was the expression (2.44) for IB(V), which includes an exp(qV/nkT)
component (1 < n < 2) that occurs because of the nonlinearity introduced
by Seff) The occurence of this component complicates the
interpretation of measured currentvoltage characteristics of
polysilicon pn junctions. Because the current component deriving from
recombination in the junction spacecharge region exhibits the same
dependence on the forward bias [41], it is generally impossible to
distinguish between the two components unless special structures are
used.
It is likely however that the junction spacecharge region
recombination current could be predominantly due to recombination
through grainboundary surface states within the junction spacecharge
region [33]. The voltage dependence of this current I G can be derived
by applying to this part of the grain boundary those portions of our
analysis related to (2.10) and (2.30). Generalizing the quasi
equilibrium assumption to mean nearly flat quasiFermilevel planes in
the spacecharge region surrounding the intersection of the grain
boundary with the metallurgical junction [8, 33], we have for this
case VB = V, and hence from (2.30)
SSCR SCR expl 2) (2.45)
GB GB" G
where the reciprocal slope factor is exactly two. Thus, (2.45) possibly
may facilitate the decomposition of the measured currentvoltage
characteristic of a polysilicon pn junction and thereby, with the
minoritycarrier transport model (2.44) developed herein, provide
physical insight into the performance of polysilicon bipolar devices and
their optimal designs.
52
From the grainboundary analysis described in this chapter, it is
clear that the boundary condition at the grain boundary adjacent to a
quasineutral grain is, in general, nonlinear. We will experimentally
demonstrate in the next chapter the nonlinearity in SGB using EBIC
n(eff) using EBIC
measurements [21, 22] interpreted quantitatively via a computeraided
numerical solution of the underlying transport problem. Numerical
simulation of the EBIC is used because of the complexity of the
transport problem, i.e., the threedimensional continuity equation
subject to the nonlinear boundary conditions. With the aid of the
grainboundary model developed in this chapter, we will obtain typical
values for T(ef) and ET at a grain boundary in Wacker (cast)
polysil icon.
CHAPTER 3
DETERMINATION OF GRAINBOUNDARY RECOMBINATION VELOCITY
FROM ELECTRONBEAMINDUCEDCURRENT MEASUREMENTS
3.1 Introduction
The minoritycarrier transport model developed in Chapter 2 reveals
that the grainboundary recombination velocity S is, in general,
n(eff) is, in general,
dependent on the excitation level (carrier density) at the grain
boundary. This nonlinearity can cause unique currentvoltage
characteristics for polysilicon diodes. To provide some experimental
support for the model, we investigate in this chapter the recombination
properties of grain boundaries using the scanning electron microscope
(SEM) in the electronbeaminducedcurrent (EBIC) mode [42]. The EBIC
technique has been widely used to measure bulk diffusion length [21, 22,
4345] and surface properties [21, 22, 24, 4649] in semiconductor
devices.
There are numerous advantages in using EBIC over other forms of
excitation to characterize semiconductor devices. The EBIC probe volume
has been well investigated and is well defined. For example, the
penetration depth of an electron beam is dependent on the atomic number
of the semiconductor and is independent of the energy gap [21]. This is
in contrast to optical excitation [25], in which the absorption
coefficient depends strongly on the energy gap and possibly on the
impurity concentration. The EBIC technique is well suited to probe
finite semiconductor regions since the excitation volume is small. For
example, in polysilicon the EBIC technique facilitates isolation of a
grain boundary from the adjacent quasineutral grain without tedious
sample preparation, such as fabrication of smallarea diodes [16].
Furthermore EBIC is a potentially powerful tool for measuring transport
properties near semiconductor surfaces [46, 49], especially in
integrated circuits.
In this chapter we will determine SGB at a grain boundary from
n(eff)
an analysis of the EBIC response in a grain adjacent to it. The grain
boundary is assumed to be perpendicular to the collecting (n+p)
junction, and the electron beam traverses it as illustrated in
Fig. 3.1. The quantitative interpretation of the EBIC response requires
the solution of the underlying minoritycarrier (electron) transport
problem in the ptype base, which, in general, is threedimensional and
has nonlinear boundary conditions. The threedimensionality arises
because the carriers are generated over only a finite region in the
semiconductor, while the nonlinearity in the boundary condition is due
to the dependence of SGB
to the dependence eff) on the excitation as shown in Chapter 2.
The existence of a finite holeelectronpair excitation volume, which,
in general, is described by a Gaussian function [47, 50, 51], further
complicates the analysis.
The conventional methods to solve the transport problem treat the
grain boundary as a surface, which can be justified using the "folding
technique" [261. The EBIC response in the vicinity of a surface
perpendicular to the collecting junction was first derived by
Van Roosbroeck [52] for the case of a point generation source. Later
Berz and Kuiken [25] developed an analytic model by assuming a spherical
generation source (volume) and using the method of images. These
55
0
CC
U+
uj c
mm
coc
m~ n m
w w
0S 
I I )
analyses have been used in the determination of minoritycarrier
lifetime and grainboundary recombination velocity from EBIC
measurements [21, 22, 24, 47]. However the point [52] or the spherical
[25] generationsource models do not give a realistic description of the
actual electronbeam generation [50]. Furthermore none of the analyses
[21, 22, 2426, 47, 52] account for the nonlinear SGBe at the grain
n(eff) at the grain
boundary, the effects of which depend critically on the generation
source.
The approach we take is to solve numerically the minoritycarrier
transport problem underlying the EBIC response subject to the nonlinear
boundary condition at the grain boundary. The threedimensional,
steadystate electron continuity equation is reduced to two dimensions
by recognizing that the grains are sufficiently wide (in the
ydirection) that the ydependence of the electron concentration is
insignificant (see Fig. 3.2). The solution of the twodimensional
continuity equation yields the electron density, N(x,z). The EBIC
collected at the junction, IEBIC, is subsequently calculated. Both the
spherical and the Gaussian generationsource models have been considered
in the numerical analysis.
Our results, obtained by numerically solving the twodimensional
continuity equation, are in agreement with published analytic solutions
[25, 26]. Experimental data obtained on Wacker (cast) polysilicon cells
demonstrate the dependence of SGB
n(effdemonstrate the dependence on the excitation level, which is
consistent with the results in Chapter 2. With the aid of the numerical
analysis, we predict values for the effective grainboundary trap
density N D
density ST(eff) at passivated (see Chapter 4) and unpassivated grain
boundaries in Wacker polysilicon. We also find that using the
57
r'r
c
0 ~0
Cl.)
I oVI
I 0
r 0
a:
Io 5
(n C
I [I
0 0n
o
CK C
II 'J
U0 C
I m
ICC
o 4
"5. 0i
\^> ") >! oj 1
0^AS "" E^'S
o ~ ~
U) c "
2 S S^ ,
L~ OCI
en,
,,
assumption of a simplified spherical generation source to predict the
EBIC response is valid provided the beam is farther than 2zo from the
grain boundary, where zo is the penetrating range of the primary
electrons [21].
3.2 Formulation of the Problem
The configuration of an assumed representative grain of the
polysilicon n+p junction is shown in Fig. 3.2. The grain is assumed to
extend infinitely in the positive x and zdirections and in the
positive and negative ydirections. The metallurgical junction is
located at a distance xj from the SCR edge top surface, and WSCR is the
width of the (nearly onesided) junction spacecharge region. The
electron beam is incident from the top surface and creates electronhole
pairs over a finite region in the semiconductor. The collection of the
electrons, which are assumed to flow only by diffusion, by the junction
gives rise to IEBIC. Our objective is to characterize IEBIC, which
decreases as the beam moves closer to the grain boundary because of
recombination losses there. From the EBIC response we can determine
GB
n(eff at the grain boundaries as well as the electron diffusion
length Ln in the grains, which we assume to be everywhere constant.
In order to interpret the. EBIC measurements quantitatively, we
solve, numerically, the electron transport problem in the ptype base
subject to the appropriate boundary conditions. The steadystate
continuity equation for electrons when an incident beam of electrons
creates g(x,y,z) electronhole pairs (cm3sec1) is
7 3n qU + qg = 0
In (3.1) Jn is the electron current density, q is the electron charge,
and U is the net thermal recombination rate (cm3sec1). For a
homogeneously doped base under lowlevelinjection conditions, (3.1) can
be written as
2N+ + 2 + g(x,y,z) 0 (3.2)
7x ay az L n
where N(x,y,z) is the excess electron density at any point in the base,
D is the electron diffusion coefficient, and L = (Dn n)1/2 for an
electron lifetime n.
Complex analytical solutions of (3.2) have been published for the
cases of point [26], spherical [25], and Gaussian [53] generation
sources. These solutions are not completely general since they do not
account for the nonlinearity in the boundary condition at the grain
boundary. The nonlinearity occurs due to the dependence of S on
n(eff) on
the excitation, i.e., N at the grain boundary, given by (2.33). The
incorporation of the nonlinear boundary condition complicates the
solution of (3.2), and hence we must resort to a computeraided
numerical solution. The direct numerical solution of (3.2) in three
dimensions, however, would involve excessive computer times.
Fortunately, we can reduce (3.2) to a twodimensional equation that
faithfully represents the actual electron transport if the grains are
sufficiently wide, i.e., the surfaces in the ydirection are far away
(~ ). Ascertaining that the EBIC response is invariant along the
ydirection (due to the symmetry), Donolato [53] proved mathematically
the validity of the simplification of the transport problem from three
to two dimensions. He concluded [53] that the detailed distribution of
the generation function g(x,y,z) along the ydirection does not
influence the resulting value of the EBIC, and hence it is sufficient to
solve, as we will, the twodimensional (in x and z) continuity equation
to calculate IEBIC. We will show qualitatively that IEBIC in the actual
(threedimensional) case differs from that calculated from our numerical
(twodimensional) analysis by a constant factor for all excitation
conditions. This difference does not hinder our quantitative
interpretation of the EBIC measurements, e.g., evaluation of S Gff ,
since we do not use absolute values of current in our interpretation.
The simplification to two dimensions (x and z) is valid if the flux
of electrons in the ydirection is inconsequential. This indeed is
generally true with regard to relative value of IEBIC. When the
electron beam is far away from the grain boundary, the electrons that
diffuse (initially) in the ydirection have virtually the same
probability of being collected by the junction, rather than recombining
at the grain boundary, as those electrons diffusing in the xdirection
toward the grain boundary. For this condition, the ydependence in
(3.2) can be neglected. As the beam is moved close to the grain
boundary, the number of electrons that diffuse in the ydirection is
small compared to the number that diffuse in the x (and z)
directions. This is true because the grainboundary surface (and the
junction) is an effective sink for the minority electrons, and hence
they tend to diffuse directly towards the grain boundary (or the
junction). For this condition also, the ydependence in (3.2) is not
important.
Therefore integrating (3.2) over the ydirection, we get
82N a2 i I 1
S a dy+ d N dy + g(x,y,z) = 0
 dx az L a n 
(3.3)
where, in accord with the above conclusions, we have neglected the
ydependence of the electron concentration, which implies a two
dimensional solution for IEBIC that differs from the actual EBIC by a
constant factor. We then simplify (3.3) to
S2N(x,z) 2 N(x,z) + G'(x,z)
K 2ax + Ky zN K 0 (3.4)
nL
where Ky is an effective ydirection width that serves as a normalizing
constant. In (3.4), N(x,z) is the electron concentration obtained from
the twodimensional numerical analysis, and
G'(x,z) = f g(x,y,z) dy (3.5)
y="
is the twodimensional generationrate function. Dividing (3.4)
throughout by Ky, we obtain
a N(x,z) + aN(x,z) N(x,z) + G(x,z) (3
ax az L n
which is the differential equation that we solve to obtain the EBIC
response. Detailed expressions for G(x,z) for spherical and Gaussian
distributions will be given in the next section.
We have thus demonstrated the validity of the simplification of the
electron transport problem (3.2) from three to twodimensions. Further
support for our twodimensional model is obtained when we compare the
normalized EBIC responses calculated from the three [25] and the two
dimensional analysis. Excellent agreement between the two (normalized)
responses is achieved.
The boundary conditions for this problem are N(z=O) r 0 at the edge
of the (shorted) junction spacecharge region; N(z + ) = 0 since Ln <
Wg where Wg is the thickness of the base; N(x + ) = 0 since the grain
is semiinfinite (>> Ln) in the xdirection; and at the edge of the
grainboundary spacecharge region (x=O),
D N sGB
n x n(ef N (3.7)
where SGB
n(eff) is given by (2.33). In (3.7) we have implicitly accounted
for the flux of electrons that do not recombine at the grain boundary
but flow through it to the adjacent grain. Qualitatively this can be
visualized by "folding over" the region to the left of the grain
boundary onto that to the right [26], and noting that the electron
distributions in the adjacent grains add to yield a solution of (3.6)
provided 1) the transport problem is linear, 2) Ln is the same in both
grains, and 3) quasiequilibrium prevails across the grainboundary
spacecharge region. When these conditions obtain, the grainboundary
recombination is properly characterized by (3.6) and (3.7) provided
N(x,z) is recognized to be the folded sun of the left and rightside
solutions. This is shown to be mathematically valid in Appendix B.
From the numerical solution N(x,z) of (3.6) we calculate the EBIC
collected at the junction:
IEBIC = qKy D 0 z=0 dx (3.8)
3.3 The Volume Distribution of the Generation Rate
Before we solve (3.6), it is important to model the volume
distribution of the electronbeaminduced holeelectronpair generation
rate G(x,z). The simplest representation is a pointsource excitation
[21, 26, 523, i.e., the electronhole pairs are generated at a point in
the base. The distance of the pointsource from the junction is
determined by the energy of the incident electron beam. It is obvious
that the pointsource generation model does not represent the actual
generation.
Improved models for the excitation volume have been proposed. A
spherical generation source [24, 25] has been suggested as a reasonable
representation of the actual generation distribution. More
sophisticated models such as the radial Gaussian [50] and the spherical
Gaussian [513 have been proposed and have been shown to be more
generally reliable than the spherical generationsource model. In our
analyses we consider two cases: 1) a spherical generation source
[24, 25]; and 2) a more complicated radial Gaussian generation source
[50]. The reason for analyzing the spherical generation source is that
it is easier to implement it numerically than the Gaussian source, and,
as we see later, the results obtained by assuming a spherical generation
source are in agreement with those obtained by assuming a Gaussian
source, provided the beam is far away from the grain boundary. Any
uncertainties resulting from assuming a spherical source model,
particularly related to the dependence ofS on the excitation,
will be removed b the nan ce mod ff)
will be removed by the Gaussian source model.
3.3.1 Spherical Generation Source
The electron beam incident from the top surface, is assumed to
generate electronhole pairs in a spherical volume [25]. In our
numerical analysis, in which we have reduced the transport problem to a
twodimensional differential equation (in the xzplane), the source is a
circle as shown in Fig. 3.3. The radius of the circle, rG, is much
smaller than the electron diffusion length in the base, and the
generation rate at any point inside the circle is assumed to be
independent of position [25]:
G = G/wr Ky (3.9)
where Go is the total number of electronhole pairs per unit time
generated by the electron beam, and Ky is the normalizing constant in
(3.4).
The center of the circle is located at a distance zo/3 from the top
surface, where [21]
S= E75 (3.10)
is the primary electron range, i.e., the penetration of the primary
electrons in the zdirection. In (3.10) Eo is the energy of the beam in
kiloelectronvolts (keV), and C is a constant which is dependent on
properties of the semicondcutor [21]:
C = 4 x 106
p
(3.11)
65
IcI
0
 VV
0o
0
No
'K~ CD
'K?
II C
'IL
E
where p is the (mass) density of the semiconductor. For silicon
C = 0.0171 x 104, which yields zO = 8.6 um for Eo = 35 keV. The
physical reason for using a circular generation source centered at a
distance zo/3 from the top surface will be explained in the next
subsection.
3.3.2 Gaussian Generation Source
When highenergy electrons penetrate the semiconductor, the
electronhole pairs are generated over a characteristic pearshaped
volume rather than over a spherical volume [50]. The reason for this is
that the energy dissipation of the incident electrons has been
experimentally shown to be given by the Gruen's function [54] in the
depth coordinate, i.e., in the zdirection. The Gruen's function is a
thirddegree polynomial in z and has a maximum at a depth determined by
the energy of the incident beam. These results have also been verified
by Monte Carlo simulations [55]. Hence, in our analysis, we will assume
that the distribution in the zdirection is adequately described by the
Gruen's function.
In the radial (x) direction, we will assume that the carrier
generation follows a Gaussian distribution [50]. The Gaussian
distribution has been used to reliably model carrier generation due to
Xrays produced by an electron beam [56] and successfully in electron
lithography studies [50]. The amplitude A and the width o of the
Gaussian function will depend on z and zo. The generation rate an any
point can then be expressed as [50]
A(z,zo) (xxo)2
G(x,z) = eexp i0 (3.12)
Ky2[2o(z,z)] 2 (zz,)] 2a (z,zo)
where Ky is the normalizing constant in (3.4). The expression for G in
(3.12) has been proposed by Donolato [50]. Expressions for A and o will
be given shortly. Note in (3.12) that G is independent of the y
direction since the transport problem has been reduced to two
dimensions. The actual generation in the ydirection is accounted for
implicitly by matching the experimentally measured and the theoretically
calculated values of the maximum EBIC (at a point well inside the grain)
to obtain the theoretical EBIC response, IEBIC(X).
A property of (3.12) is
f G(x,z)dx = A(z,z )/K (3.13)
x=
From (3.10) and (3.13) we can infer that A(z,zo) represents the relation
between the penetration depth of the electron beam and its energy
[54]. An expression for A(z,zo) has been experimentally obtained [50,
54],
G
A(z,z ) = [A(z/z )] (3.14)
oy
where A(z/zo) is the Gruen's function [50, 54]:
A(z/z ) = 0.6 + 6.21(z'/z ) 12.4(z'/z) 2
+ 5.69 (z'/z )3 for 0 < z' < 1.1 z
= 0 for z' > 1.1 z ;
0
in (3.15), z' A z + (x + WSCR). Since this form for A yields a unity
integral between 0 and 1.1 zo, integration of (3.15) in the range
0 < z' < correctly yields Go/Ky. In Fig. 3.4 we have plotted A(z,zo)
for E = 35 keV. We see that the maximum value of A(z,zo) occurs at
zo/3 from the top surface (z' = zo/3), where zo = 8.6 um. This
observation qualitatively justifies the earlier assumption that the
center of the approximate circular generation volume is at z' = zo/3.
The width of the Gaussian for a finite value of the beam diameter
is given by [50]
o2(z,z ) = 0.36 d2 + 0.11 (z')3/z (3.16)
where d is the beam diameter. For EBIC measurements, typically
d = 5001000 A. The complete expression for G adopted in our numerical
analysis is (0 < z < 1.1 zo)
G /K A(z/z ) (x x )2
G(x,z) = [2 Z)]~ exp (3.17)
0o [2no(z,z0) L 2o(z,z)j
where A(z,ZO) and a2(z,zo) are described by (3.15) and (3.16). In
Fig. 3.5 we have plotted G as a function x for various values of z.
These plots have been drawn for x. = 30 pm and Eo = 35 keV.
3.4 Results
The computeraided numerical solution of (3.6), subject to the
given boundary conditions, was obtained by using a finitedifference
approximation technique [57, 58]. The grain is divided into rectangular
grids and the differential equation (3.6) is approximated by its finite
0.20
0.18 
0.1 
EO=35 keV
0.14
0
.12 
0.28,
0.06
0.22
0.00
0 1 2 3 4 5 6 7 8 9
(Xj +WsCR) Z/(pm)
Fig. 3.4 The preexponential coefficient A(z,z ) of the Gaussian generation
distribution versus the depth coordinate. The shaded area
indicates the total generation in the emitter and the junction
spacecharge region. For our cells, (xj + WSCR) = 0.8 um, within
which only about 6% of the total generation occurs.
1.00
0.90
0.80
0.70 Xo=30 pm
._ E0=35 keV
0.60
S=0
0.50
= 0.18
0.30 z0 .
0.20 
=0.39
0.10 0 .
=0.55
0
0.00
28 28.5 29 29.5 30 30.5 31 .5 32
x (p m)
Fig. 3.5 The generation rate (assumed to be a radial Gaussian) as a function
of the distance x from the grain boundary at different depths z.
difference form at each grid point. The resultant set of algebraic
equations is combined with that at the boundaries and solved
simultaneously using the GaussSeidel elimination method with successive
overrelaxation [57, 58]. The EBIC current (3.8) is calculated by
numerically differentiating N(x,0) and integrating it over the area of
the junction. Salient features of the numerical algorithm are described
in Appendix C, wherein we have also listed the Fortran computer program.
Typically EBIC measurements are performed on shallowjunction
devices. In such devices we can assume that the generation in the
quasineutral emitter and the junction spacecharge region is much
smaller than the total generation. Hence the EBIC due to generation of
carriers in the quasineutral emitter and the junction spacecharge
region, which is a constant, can be neglected. For example, in the
Wacker polysilicon cells used in our experiments xj = 0.4 uand
WSCR 0.4 pm. The value of x was inferred from sheet resistivity
measurements, while the value of WSCR was determined from the zerobias
capacitance measurements performed on the completed cell. From Fig. 3.4
we note that the generation in the emitter and the junction spacecharge
region is about 6% of the total generation (in the entire cell), which
is negligible.
The calculated IEBIC as a function of the beam position is plotted
in Figs. 3.6 and 3.7 for two different values of the background EBIC
excitation level, i.e., maximum IEBIC produced when the beam is far
removed from the grain boundary:
I MAX N(x Z)
I C qK f [G(x,z) N z)] dx dz (3.18)
oo n
where G is given by (3.17). These plots have been drawn subject to the
condition that SGB
condition that Sn(eff) is dependent on the excitation as described by
(2.33). Note that the theoretical EBIC responses in Figs. 3.6 and 3.7
as well as the experimental data pertain to the same grain.
Superimposed on the theoretical plots in Figs. 3.6 and 3.7 are the
measured EBIC responses of a typical grain in a standard (unpassivated)
polysilicon n+p cell, the fabrication of which will be described in the
D
next chapter. The two parameters L and N T(eff) are determined
simultaneously by fitting the experimental data at one current level;
the fits at other current levels provide a check on the model, i.e., the
nonlinear Sef and the constant Ln. By comparing the theoretical and
experimental responses, we infer that Ln = 120 um and that N T(eff) =
1 x 1012 cm2 for the grain measured.
The error bars shown in Figs. 3.6 and 3.7 as well as in the other
figures to follow reflect the uncertainty in the measurement of
thedistance, xo, of the generation volume from the center of the grain
boundary. This uncertainty is due to the difficulty in being able to
locate the center of the grain boundary in our measurements.
In Figs. 3.6 and 3.7, we have also plotted theoretical EBIC
GB
responses based on a constant S The assumption of a
n(e ff T s
constant SGB
constant S seems to be a reasonable one when the beam is
sufficiently far away from the grain boundary. However, as is clear
from Figs. 3.6 and 3.7, the EBIC responses for the constant S case
n(e ff)
deviate from the experimental data as the beam approaches the grain
boundary. The physical reason for this deviation is evident from
Fig. 3.8, where we have plotted S n versus x for the two different
excitation levels in Figs. 3.6 and 3.7. As the beam nears the grain
excitation levels in Figs. 3.6 and 3.7. As the beam nears the grain
1.0
0.9
^ Eg= 35 keV
x Nonlinear / G0=1013/sec
SM G / .. .
S 0.8 S~iel
0.7 /Constant SGff
I
/
I
I
0 .6 __ __ __ __ i i l _
0 5 10 15 20 25 30 35 40
x0 (pm)
Fig. 3.6 Calculated and measured (e) EBIC (normalized) from a grain in the
vicinity of an unpassivated grain boundary (G = I0 /sec). The
EBIC has been calculated for constant () aid nonlinear ()
SeGB
n(eff)*
1.0
0.9
xC
a0
0.7
r
K
Nonlinear
/
/
/
/
/
/
/ Cons
1
Eo=35 keV
G=1014/sec
ta S
tant Sneff
nietti
I I I I I I I I
0 5 10 15 20 25 30 35 40
xo (pm)
Fig. 3.7 Calculated and measured (A) EBIC (normalized) fromla grain in the
vicinity of an unpassivated grain boundary (G = 10 /sec). Note
the significance of the nonlinearity when the electron beam is near
the grain boundary.
r
nr
boundary we find that S GB decreases due to the increase in
n(eff)
excitation, i.e., N(O,z), at the grain boundary. Consequently the
actual EBIC collected by the junction at any beam position is higher
GB
than that predicted by the unrealistic constantS ef curves. This
n(eff)
demonstrates the significant nonlinearity in S f in accord with our
n(eff) in accord with our
theoretical results in Chapter 2.
We also note from Figs. 3.6 and 3.7, which pertain to the same
grain, that the value of SGB
n(eff) inferred by comparing the theoretical
(constant SGB
(constant S ff)) and experimental responses, is not unique; different
values are implied at the same grain boundary at different current
GB
levels when S ef f is assumed to be a constant. This further
GB
demonstrates the nonlinearity and shows that a constantS eff) model is
physically unreasonable. We note that when the nonlinearity in S GB
n(eff)
is accounted for, comparison of theoretical and experimental results
yields a unique value for N (ff); the nonlinear model is thus
physically reasonable.
The nonlinearity in SGB
The noni nearity in Seff) is further confirmed when we compare
the experimental data measured at the two different excitation levels
(Figs. 3.6 and 3.7). The increased excitation (GO) was obtained by
increasing the beam current and keeping the beam energy constant. (The
increase in the beam diameter with excitation level [42] is
inconsequential since the second term on the right side of (3.16) is
always dominant.) If SGB
n(always dominant.) I eff) were a constant then the percentage EBIC
collected by the junction at any given xo should be independent of the
excitation level [25, 47]. However the experimental data in Figs. 3.6
and 3.7 show a strong dependence on the excitation level. For the same
value of xo we observe that the percentage EBIC lost at the grain
boundary at the lower excitation level (Fig. 3.6) is higher than that at
the higher excitation level (Fig. 3.7). The reason for this is obvious
from Fig. 3.8 wherein we have plotted SGB as a function of x at the
n(eff) as a function of xo at the
two excitation levels for the grain boundary in the standard cell. From
Fig. 3.8 we note that S ff) at the grain boundary in the standard
cell is lowered by about an order of magnitude at the higher excitation
level. Hence we conclude that the nonlinearity in S GB influences
n[e ff)
the EBIC response when the beam is close to the grain boundary and
should be accounted for to ensure accurate quantitative interpretation
of EBIC measurements.
In Figs. 3.9 and 3.10 we compare the theoretical plots with the
experimental data obtained in a grain of an aluminumpassivated (active)
n+p cell. The passivation technique will be described in the next
chapter. By comparing the experimental data with the theoretical plots,
which were obtained by assuming G to be a Gaussian given by (3.17), we
infer that N T(eff 101 cm2 at a passivated grain boundary. Once
again we observe that S is nonlinear. However the nonlinearity is
less pronounced than at the unpassivated grain boundaries in a standard
cell, particularly at the low excitation level. An explanation for this
can be provided using the grain boundary analysis in Chapter 2. For
low N Teff)' B which implies that SGBff (= CN ) is
lo ST(e ff), 1 @BO' n(eff) i ST(eeff)
virtually constant.
The theoretical EBIC response obtained in a grain in the vicinity
of an unpassivated grain boundary is compared to the experimental data
in Fig. 3.11. We infer that at this grain boundary,
ST(eff) 101 cm which suggests that this particular grain
boundary is a highangle boundary. This value ofN T(eff) is unusually
S GO 10i/sec
S10 / Standard Cell
GO=10 /sec
E
Passivated Cell
GO=1013/sec
3 105
E0=35 keV
104i
0 5 10 15 20 25 30 35
X0 (pm)
Fig. 3.8 The grainboundary effective recombination velocity as a function
of xo at unpassivated and passivated grain boundaries. These
GB
plots show how Sn(eff) changes as the electron beam moves toward
(excites) the grain boundary.
1.0
0.9
Nonlinear
SGB EO=35 keV
nieffl 3
nx" 1 Go=1013/sec
5w / nGB
S0.8 //Constant Snieff
w
0.7
0.6 I I
0 5 10 15 20 25 30 35 40
Xo (i m)
Fig. 3.9 Calculated and measured (0) EBIC (normalized) from grain in
the vicinity of a Dassivated grain boundary (G0 =10 /sec).
1.0
Nonlinear SG
0.9 
/
/
/
/
/GB
x U /Constant SGB
S/ EO=35 keV
,,0.8 / 14/sec
L)G0 10 /sec
0.7
0.6 I I
0 5 10 15 20 25 30 35 40
xo (p m)
Fig. 3.10 Calculated and measured (A) EBIC (normalized) from a grain
in the vicinity of a passivated rain boundary (Go 1014/sec).
high, possibly because of our assumption that ND(eff) is monoenergetic
situated at midgap. The large value of N T(ef causes e to be
high. Consequently the value ofSG does not vary significantly
with the excitation level and is at the kinetic limit as described in
Chapter 2.
In Fig. 3.12 we have plotted the theoretical EBIC response obtained
by assuming a circular generation source. Comparing it to the Gaussian
generationsource case, we observe that the circulargeneration
assumption is good provided we interpret data far away ( 2zo) from the
grain boundary. As the beam approaches the grain boundary the predicted
EBIC responses in the two cases differ from each other. If we interpret
the experimental data in Fig. 3.6 using the circular generation source
assumption we find that ND 1 2 x 10 2, which is a factor of
ST(eff) 2 x 10 cm which is a factor of
two higher than that obtained by assuming a Gaussian generation source.
3.5 Discussion
In this chapter we determined S at typical passivated and
n(eff) at typical passivated and
unpassivated grain boundaries in Wacker (cast) polysilicon from EBIC
GB
measurements. We demonstrated the dependence of 5n( on the
excitation level as predicted by the transport model developed in
Chapter 2.
In order to facilitate quantitative interpretation of EBIC
measurements we solved, using computeraided numerical analysis, the
underlying transport problem. The general transport problem in three
dimensions was reduced to one in twodimensions by neglecting the
variations in the (y) direction normal to the plane defined by the
moving electron beam. This assumption is justified since the surfaces
1.0
0.9
EO=35 keV
xo G=1014/sec
\ 0.8 
0.7
0.6 I I I I
0 5 10 15 20 25 30 35 40
X0 (p m)
Fig. 3.11 Calculated and measured (A) EBIC (normalized) from a grain in
the vicinity of an unpassivated (highangle) grain boundary
(G= 1014/sec); SGeff) is virtually constant (=SKL) due to
high N ,, (=10n 2
hioh ND (=1014cm2).
ST(eff)
1.0
0.9 
/// EO=35 keV
Spherical// G=10 13/sec
I //
0.7
0 6 1 1_ I I I I I
0 5 10 15 20 25 30 35 40
xo (G m)
Fig. 3.12 Comparison of the calculated EBIC (normalized) as a function
of xo for two different models for the holeelectronpair
generation volume.
in the ydirection are far removed from the excitation volume. We found
that the discrepancy between the actual solution and ours is a constant
factor in IEBIC provided the ydependence in (3.2) can be neglected,
which was justified qualitatively. Our quantitative interpretation
GB
of n(eff) is not hindered due to this constant difference since we
compare only relative values of EBIC current. The results of the two
dimensional analysis agree well with published analytic solutions [25,
26, 52]. The simplification of the transport problem from three to
twodimensions has also been shown to be mathematically valid [53].
In our analyses, we have incorporated two different models for the
holeelectronpair excitation volume: 1) the spherical generation
source model [24, 25]; and 2) the Gaussian generationsource model
[50]. The utility of the sphericalsource model results from its ease
of implementation in the numerical analysis. The EBIC response obtained
by using this model far away from the grain boundary agrees well with
that obtained by using the sophisticated and hence more complicated
Gaussiansource model. Thus the sphericalsource model is useful for
interpreting EBIC data far away from the grain boundary when S is
n(eff) is
GB
a constant (e.g., when Sn(eff) SKL or at passivated grain boundaries).
In the sphericalsource model the incident electron beam generates
holeelectron pairs in the base in a sphere (circle in the two
dimensional analysis) whose radius is much smaller than the electron
diffusion length and within which the generation rate is a constant
[25]. To remove any uncertainties associated with the sphericalsource
assumption [47, 50, 51], we also modeled the excitation volume using a
radial Gaussian distribution [50], i.e., at any depth z, G is given by a
Gaussian function. In the zdirection the distribution is given by the
Gruen' s function [50, 54]. This description of the generation source is
more generalized and better represents the actual generation than the
simple i fied spherical source model.
The analytic solution of the twodimensional continuity equation
(3.6) in the general case of nonlinear S5 e and a Gaussian generation
n(e ff] and a Gaussian generation
source is formidable. Hence we resorted to a computeraided numerical
solution. The solution was obtained by approximating (3.6) using finite
differences [57, 58]. The resultant set of algebraic equations was
solved by the GaussSeidel elimination method [57, 58]. The EBIC
current collected at the junction was calculated as a function of xo,
SGBi and D
n(eff)', N T(eff). In the analyses we have treated the grain
boundary as a surface, which can be justified by using the folding
technique [26].
The EBIC response of (al uminum) passivated and unpassivated Wacker
(cast) polysilicon cells were measured experimentally. The cell
fabrication and the passivation technique will be described in the next
chapter. When the beam is situated close to the grain boundary we
observed that IEBIC measured experimentally was higher than predicted by
the theoretical plots for constant S This early demonstrates,
n(eff)" This clearly demonstrates,
subject to our assumption of constant Ln which was indeed supported by
our results, that the dependence of S,) on the excitation level as
predicted in Chapter 2. We also observed that the current loss at the
grain boundary for a given xo decreased with increasing excitation
level. This provides further evidence for the nonlinearity.
Physically, N(0,z) increases as the beam approaches the grain boundary
or as the incident excitation level is increased. Consequently,
S(eff)' which is proportional to [N(O,Z)]1/2, decreases and hence
IEBIC is higher than predicted by constant S f curves (see
n(eff) curves (see
Figs. 3.6, 3.7, 3.9, 3.10).
By comparing the experimental data with the theoretical plots we
inferred that 101 cm2 at a typical unpassivated grain
boundary and tht 11 2
boundary and N (ef = 101 cm at a passivated grain boundary. The
comparison of the theoretical and experimental EBIC responses also
yielded values for the diffusion lengths in the grains simultaneously.
For the grain of the unpassivated cell, L = 120 rn, while for the grain
of the passivated cell, Ln 110 um. The determinations of NT(eff
were based on the assumption that the trap level, ET, is at midgap (ET
Ei). If ET were below Ei [34], the value of NT(eff) would be higher
than that predicted by our numerical analysis. The dependence of
GB
Sn(eff) on the excitation level, however, remains unaltered and is
given by (2.33), provided ET is located relatively deep in the energy
gap, i.e., = 5 kT below Ei (see Chapter 2).
Our analyses predict that the spherical generationsource
assumption adequately describes the EBIC response provided x > 2z .
For xo < 2zo, the response for a spherical source differs from that for
a Gaussian source and could cause at least a factor of two difference in
the determination ofNT(eff). The reason for the discrepancy is that
the spherical source cannot accurately describe the nonlinearity
in S ef, which becomes important as the beam approaches the grain
boundary. For xo 2 zo, the influence of the grain boundaries on the
EBIC response is minimal and hence the response is independent of the
choice of the generation source. The use of the sphericalsource model,
which is easy to implement and is widely used in most analyses [24, 25,
47], is recommended to interpret EBIC data far away from the grain
boundary. The model yields reliable results when S G is a constant,
n(e ff)
e.g., at a passivated grain boundary or when Si SGKL
In Chapter 2 we studied grainboundary recombination and its
influence on the base recombination current of a polysilicon diode. In
this chapter we described an experimental technique to determine
S f, which is a measure of the grainboundary recombination. It is
n(e ff)I
obvious that grainboundary recombination deteriorates the performance
of bipolar polysilicon devices. For example in a polysilicon solar
cell, the grainboundary recombination both in the junction spacecharge
region as well as adjacent to the quasineutral base can limit the cell
efficiency [8]. In order to improve polysilicon device performance it
is necessary to suppress grainboundary recombination. Such suppression
will also likely reduce carrier scattering by grain boundaries which
lowers 4g at the grain boundaries. This results in an improvement of
the carrier mobility in polysilicon [5], for example that used in
integrated circuits [3].
In the next chapter we suggest an experimental method to reduce
grainboundary recombination by passivating grain boundaries using
preferential diffusion of aluminum. The passivation is assessed by
comparing the EBIC response in the vicinity of an unpassivated and a
passivated grain boundary. The value of SGB is determined from the
n(eff)
EBIC measurements with the aid of the numerical solution of the
transport problem described in this chapter. Possible additional
benefits of aluminum, such as gettering, are also examined.
CHAPTER 4
POTENTIAL IMPROVEMENT OF POLYSILICON SOLAR CELLS
BY GRAINBOUNDARY AND INTRAGRAIN DIFFUSION OF ALUMINUM
4.1 Introduction
The conversion efficiency of polycrystalline silicon (polysilicon)
solar cells, as pointed out in Chapter 2, is limited by high defect
densities within the grains as well as at the grain boundaries. On the
one hand, the efficiency of bulk polysilicon cells having relatively
large grains is generally limited by high densities of intragrain
defects that cause short minoritycarrier lifetimes; the grainboundary
recombination is hence rendered virtually unimportant. The efficiency
of thinfilm polysilicon cells having relatively small grains may, on
the other hand, be generally limited by high densities of grainboundary
traps that produce high recombination velocities. It is therefore of
interest to consider processes that could possibly passivate the grain
boundaries, i.e., reduce the grainboundary trap densities, and also
getter the grains, i.e., remove defects (impurities) from the active
portions of the grains. Such a process is perhaps the diffusion of a
specific impurity whose grainboundary diffusion coefficient is higher
than its bulk coefficient, which implies preferential grainboundary
diffusion [59], and whose presence at the silicon surface increases the
solubility of common defects, which implies bulktosurface transfer of
defects.
Phosphorous, which is an effective gettering agent in silicon [60],
also has the potential to be an effective grainboundary passivating
agent [16]. However preferential grainboundary diffusion of
phosphorous much deeper than ~ 10 um is not practical because it must be
done at a temperature (> 1000C) considerably higher than the normal
solar cell junctionformation temperature (< 900C). It has been shown
that lithium tends to suppress grainboundary recombination, but its
stability and effectiveness are dubious [17]. Monatomic hydrogen has
been shown to be extremely effective in passivating grain boundaries
[18], but the necessity of inordinate sample preparation and the
unorthodox nature of the hydrogenation process make it impractical in
solar cell fabrication. Passivation of grain boundaries has also been
achieved using deuterium [19] but, as is hydrogenation, the
incorporation of deuterium into the polysilicon is incompatible with
conventional solar cell processing. Hence it would be useful to design
a process that 1) is compatible with conventional solar cell
fabrication; 2) passivates most of the grain boundaries in a relatively
short time; and 3) produces concomitant gettering of intragrain defects.
There are indications that aluminum, an acceptortype dopant in
silicon, could be an ideal agent for improving polysilicon solar cell
efficiency. Evidence of anomalous diffusion of and gettering by
aluminum in silicon dates back to the initial development of the
aluminumalloyed n+pp' backsurfacefield (BSF) silicon solar cell
[61]. More recently unusually high grainboundary diffusion
coefficients for aluminum in polysilicon have been measured [62], and
unexpected improvements in the spectral response of "bicrystalline"
silicon cells having aluminum "BSFs" have been observed [63].
The passivation of a grain boundary by preferential aluminum
diffusion might occur via 1) reduction of the grainboundary (deep
level) trap density resulting from removal of dangling bonds;
2) formation of a highlow junction at the grain boundary, which reduces
the effective recombination velocity S ff (see Chapter 2); and/or
3) formation of a pn junction at the grain boundary, which effectively
increases the collecting junction area. It is possible then that
aluminum diffusion, which is compatible with solar cell fabrication,
could improve polysilicon solar cells by passivating grain boundaries as
well as by gettering intragrain impurities and by creating an effective
BSF.
In this chapter we present results of experiments designed to study
the effects of aluminum diffusion on the efficiency of both bulk and
thinfilm polysilicon solar cells. In the case of bulk polysilicon,
which possibly may soon be a cheap substitute for silicon in solar cell
technology, we focus our attention on the conventional AlBSF process
[61, 64], which should ultimately yield the highest attainable
efficiencies as it has in silicon cells [65]. Possible additional
benefits due to the back aluminum diffusion alloyingg), e.g., gettering
and grainboundary passivation, are investigated. With regard to thin
film polysilicon, we examine gettering and grainboundary passivation
resulting from aluminum diffusion at low temperatures (below the
eutecticformation temperature, 577C) from the front surface. Such
lowtemperature diffusion has been shown to yield preferential aluminum
spikes down dislocations [66].
To assess the grainboundary passivation, we use the electronbeam
inducedcurrent (EBIC) technique. A qualitative analysis is done
initially to detect grainboundary passivatidn. Quantitative estimates
GR
of grainboundary parameters, e.g., SG ,B based on EBIC data are
n(eff)'
derived using the computeraided numerical simulation, details of which
were disclosed in Chapter 3, of the minoritycarrier transport
underlying the EBIC response of a pn junction in the vicinity of a grain
boundary. Currentvoltage characteristics and reversebias capacitance
measurements, which yield the base doping density of the cell, are used
to determine the minoritycarrier diffusion length in the base, which
reflects the benefit produced by the AlBSF process.
Our experimental results show that the conventional AlBSF process
results in longer diffusion lengths because of (bulktosurface)
gettering of intragrain impurities, irrespective of indications that the
grain boundaries emit impurities during the hightemperature process.
Our results also suggest that the minoritycarrier lifetime is longer
when measured using lightbias than when measured in the dark. This
suggestion is consistent with the theory [67] that defect clusters in
polysilicon are effectively negated by high carrierinjection levels.
No grainboundary passivation near the front junction is apparent after
the BSF process. However we find significant passivation near the front
junction resulting from lowtemperature aluminum diffusion from the
front surface, but no observable gettering. Consequently, it is
possible that aluminum diffusion, when optimally done, could
substantially increase the efficiencies of both bulk and thinfilm
polysilicon solar cells.
4.2 Cell Fabrication
The cells used in this study were fabricated on ptype Wacker
(cast) polysilican substrates. After standard cleaning and chemical
polishing of the front surface, the substrate thickness, WB, is about
350 pm. The doping density, NAA, varies from 5x1015cm3 to xl016cm3,
as determined by reversebias capacitance measurements on the completed
cells.
Three different kinds of cells were fabricated for comparison in
both the bulk and thinfilm polysilicon studies. Active cells were
subjected to aluminum diffusion while control cells were not. However,
the control cells were subjected to the same thermal stress as were
their aluminumdiffused counterparts. Several standard cells, which
were neither aluminumdiffused nor subjected to any abnormal thermal
stress, were also fabricated.
4.2.1 Bulk Process
For the studies applicable to the development of bulk polysilicon
solar cells, we used the conventional A1BSF process for silicon [64] to
fabricate the active cells. About 1.5 um of aluminum was evaporated
onto the back surface of the cell after the front phosphorous (n+)
diffusion. The aluminum was then diffused into the active cell during
the alloying process at 800C for 1 hour in forming gas. The control
cell underwent an identical heat treatment but without aluminum. Both
largearea (0.4 0.45 cm2) and smallarea (30mildiameter) cells were
fabricated.
Fabricating smallarea cells enables control of the grainboundary
density in each cell. The surface and edge leakage currents, which are
predominant in largearea cells, can be effectively controlled by means
of an MOS guardring overlapping the edge of the emitter in the small
area cells [16]. In addition, by comparing the diffusion lengths,
determined from the currentvoltage characteristics, of the largearea
(measured under light) and the smallarea (measured in dark) cells, we
can deduce the effect of the lightbias on minoritycarrier lifetime.
The n+p junction in the largearea cells was formed by diffusing in
phosphorous at 875C for 30 minutes using POC13 as the source. The
resulting junction depth is about 0.4 um. Subsequent to the aluminum
diffusion, which followed the phosphorous (n+) diffusion, about 0.6 pm
of aluminum was evaporated onto the back surface to serve as a
contact. The control and the standard cells were then sintered at 4500C
for 15 minutes in fonning gas. Following a mesa etch that isolated the
cells, the front contact was formed by evaporating a 300Athick
aluminum "dot". Details of these processing steps as well as those of
the smallarea and the thinfilm cells are given in Appendix D.
The junction in the smallarea cells was formed by a phosphorous
predeposition at 950C for 40 minutes using POC13 as the source followed
by a drivein diffusion at 1000C for 27 minutes. This resulted in a
junction depth of approximately 0.7 um. Aluminum was then diffused into
the active cells from the back surface. The front and backcontact
aluminum evaporations, 5000A and 6000Athick respectively, were
performed subsequently. As in the case of the largearea cells, the
control and the standard cells were sintered at 450C for 15 minutes
after the backcontact evaporation.
In order to suppress surface and edge leakage currents in the
smallarea cells, a silicondioxide layer (~ 1700Athick) was thermally
grown on the top surface during the drivein step, and a metal gate was
deposited overlapping the edge of the diffused region [16]. A negative
gate bias (20V to ensure surface accumulation in the base) was used
while the currentvoltage characteristic of the cell was being measured.
4.2.2 ThinFilm Process
For the studies applicable to the development of thinfilm
polysilicon solar cells, the n+p junction was formed by diffusing
phosphorus at 875C for 30 minutes using POC13 as the source. The
resulting junction depth is approximately 0.4 un. A mesa etch was used
to isolate the cells. The active and control cells were fabricated on
sequential wafers from the Wacker ingot.
We use bulk (largegrain) polysilicon to imply the effects of
aluminum diffusion in thinfilm (smallgrain) cells. This material
facilitates electrical isolation of a grain boundary (using EBIC) and
enables direct quantitative assessment of the passivation yielded by the
aluminum diffusion. We anticipate that the lowtemperature diffusion
will similarly passivate grain boundaries in thinfilm polysilicon.
However because the grain boundaries may differ significantly from those
in Wacker polysilicon, the extent of the passivation might be different.
We diffused aluminum into the active cells from the front surface
at temperatures below the SiAl eutectic temperature (577C) to avoid
irreparable damage to the junction region. ICgrade aluminum, 1.5pm
thick, was evaporated onto the front surface before the phosphorous (n+)
diffusion. The aluminum diffusion was then done at 4500C for 36 hours
in forming gas. Following the removal of excess aluminum, a thin layer
of silicon was etched off to avoid shunting of the n+p junction that is
