• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Analysis of minority carrier transport...
 Determination of grain-boundary...
 Potential improvement of polysilicon...
 Aluminum gettering in (cast)...
 Summary, discussion, and recom...
 Appendix A: Numerical solution...
 Appendix B: Mathematical justification...
 Appendix C: Numerical algorithm...
 Appendix D: Polysilicon cell...
 References
 Biographical sketch














Title: Studies of the influence, and its control, of grain boundaries on minority-carrier transport in polysilicon devices /
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 Material Information
Title: Studies of the influence, and its control, of grain boundaries on minority-carrier transport in polysilicon devices /
Physical Description: vii, 212 leaves : ill. ; 28 cm.
Language: English
Creator: Sundaresan, Ravishankar, 1955-
Publication Date: 1983
Copyright Date: 1983
 Subjects
Subject: Grain boundaries   ( lcsh )
Silicon crystals -- Electric properties   ( lcsh )
Semiconductors   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1983.
Bibliography: Bibliography: leaves 205-211.
Additional Physical Form: Also available on World Wide Web
Statement of Responsibility: by Ravishankar Sundaresan.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097426
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000493690
oclc - 11988504
notis - ACR2550

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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    Abstract
        Page vi
        Page vii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
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    Analysis of minority carrier transport in polysilicon devices
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    Determination of grain-boundary recombination velocity from electron-beam-induced-current measurements
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    Potential improvement of polysilicon solar cells by grain-boundary and intragrain diffusion of aluminum
        Page 87
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    Aluminum gettering in (cast) polysilicon
        Page 114
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    Summary, discussion, and recommendations
        Page 135
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    Appendix A: Numerical solution of poisson’s equation in the grain-boundary space-charge region
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    Appendix B: Mathematical justification for using the folding technique
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    Appendix C: Numerical algorithm for solving the two-dimensional continuity equation underlying the ebic response
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    Appendix D: Polysilicon cell fabrication
        Page 201
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    References
        Page 205
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    Biographical sketch
        Page 212
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        Page 214
        Page 215
Full Text





STUDIES OF THE INFLUENCE, AND ITS CONTROL, OF GRAIN BOUNDARIES
ON MINORITY-CARRIER 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 co-chairman 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, Bor-Yuan 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 GRAIN-BOUNDARY RECOMBINATION VELOCITY
FROM ELECTRON-BEAM-INDUCED-CURRENT 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
GRAIN-BOUNDARY 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
GRAIN-BOUNDARY SPACE-CHARGE REGION ............................141

B MATHEMATICAL JUSTIFICATION FOR USING THE FOLDING TECHNIQUE.......164

C NUMERICAL ALGORITHM FOR SOLVING THE TWO-DIMENSIONAL CONTINUITY
EQUATION UNDERLYING THE EBIC RESPONSE .........................168

C.I Numerical Solution of the Steady-State 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 (Small-Area Cells) ............................202
D.5 Thin-Film Process (Large-Area 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 MINORITY-CARRIER TRANSPORT IN POLYSILICON DEVICES

By

Ravishankar Sundaresan

December 1983

Chairman: Dr. Jerry G. Fossum
Co-Chairman: 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

three-dimensional, nonlinear boundary-value problem that defines

minority-carrier transport, including recombination, in polysilicon

devices. These assumptions enable the separation of the grain-boundary

recombination analysis, which is based on quasi-equilibrium, from the

intragrain transport analysis, which is done by partitioning the grain

into subregions in which the minority-carrier 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 minority-carrier density in the adjacent quasi-neutral

grain.

The dependence of the recombination velocity on the carrier density

(excitation) at the grain boundary is experimentally demonstrated using

electron-beam-induced-current (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 computer-aided numerical

analysis. By comparing the theoretical and experimental EBIC responses,

we predict values for the surface-state density at typical grain

boundaries and the minority-carrier diffusion length in the grains.

Experimental results are presented that imply potential

improvements afforded by aluminum diffusion in both bulk and thin-film

polysilicon solar cells. For bulk cells, a high-temperature aluminum

diffusion alloyingg) is shown to increase the minority-carrier diffusion

length by gettering intragrain impurities. For thin-film 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

majority-carrier transport through the grain boundaries [2-6], which

defines the resistivity of the polysilicon, as well as studies of

minority-carrier recombination at the grain boundaries [7-11], which

defines the bipolar characteristics of the polysilicon.

Compared to silicon (majority- and minority-carrier) 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 turn-on characteristic that is beyond the strong-inversion

threshold [51, while in thin-film 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 cost-competitive with silicon cells, we must understand how the

grain boundaries influence the minority-carrier transport and then

devise fabrication techniques to minimize this influence. Although

several approximate models [7-10, 13, 14] for the minority-carrier

transport in polysilicon have been developed, they are inadequate in the







general case. The reason for this is that the models [7-10, 13] are

based on restricting assumptions that limit their ranges of validity;

furthermore, the solution of the minority-carrier transport equation is

obtained by truncating an infinite series [8, 14], and hence the

accuracy of the models is questionable. Many processing (passivating)

techniques [15-19] 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 [15-19].

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 minority-carrier 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 minority-carrier transport

in polysilicon;

(2) to provide experimental corroboration for the transport model

and to determine values for the pertinent grain-boundary

parameters;

(3) to develop experimental techniques to improve the performance

of bulk and thin-film 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 majority-carrier trapping, gives rise to a

potential barrier which influences the conduction properties of the

polysilicon [6]. In minority-carrier devices, such as solar cells, the

grain-boundary states serve as recombination centers for the minority

carriers and hence constitute a source of recombination current. Thus

the development of the minority-carrier transport model for polysilicon

is complicated by the presence of randomly oriented surfaces (grain

boundaries) at which significant recombination can occur.

Impedance measurements [20], current-voltage measurements [15, 16],

and electron-beam-induced-current (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 grain-boundary

recombination. We use the EBIC measurements to infer the grain-boundary

parameters since it facilitates isolation of the grain boundary from the

bulk grain. Such an isolation is beneficial in order to study the

grain-boundary characteristics, e.g., the influence of the excitation on

grain-boundary recombination. Furthermore the EBIC measurements do not

require tedious sample preparation as do current-voltage measurements

[15, 16] to obtain the grain-boundary 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 [16-19] 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 minority-carrier lifetimes. Hence it is

necessary to consider processes that can getter the intragrain

impurities and hence increase the minority-carrier lifetime.

In Chapter 2, we develop an analytic model for the minority-carrier

transport in polysilicon. The general transport problem, which is

three-dimensional and nonlinear [23], is simplified by making key

assumptions with physical justification. The main feature of the model

is the separation of the grain-boundary 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 (minority-carrier 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 single-crystal 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 electron-hole pairs, predominantly in the base region of the

cell. The generation (short-circuit) 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 computer-aided 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

grain-boundary recombination velocity [24-26]. 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 minority-carrier diffusion length in the grains.

Having demonstrated the detrimental influences of the grain-

boundary recombination on the minority-carrier 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 minority-carrier 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 high-temperature

(above eutectic, 577C) aluminum-diffusion process from the back

surface. For the development of thin-film cells, in which the grain-

boundary recombination is dominant, we propose a low-temperature (below

eutectic) aluminum-diffusion process from the front surface of the

polysilicon. We use forward current-voltage and reverse-bias

capacitance measurements to demonstrate the intragrain gettering, and

EBIC measurements to show the grain-boundary passivation. Our results

indicate that the high-temperature aluminum-diffusion process

effectively getters intragrain impurities, whereas the low-temperature








aluminum-diffusion process produces significant grain-boundary

passivation. We also find that the grain boundaries emit impurities

during the high-temperature process and that these emitted impurities

are effectively gettered by the aluminum.

Although we attribute the benefits produced by the high-temperature

aluminum-diffusion process in Chapter 4 to intragrain gettering, the

observed improvement in polysilicon device performance is also

commensurate with the formation of a back-surface field (BSF) [27]. The

BSF, which is nonuniform and nonplanar, is produced as a result of the

preferential dissolution of silicon during the high-temperature

aluminum-diffusion 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 current-voltage

measurements of aluminum-diffused 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 high-temperature aluminum-diffusion process, and that

the nonuniform BSF is ineffective. No improvement in the performance of

semiconductor-grade and solar-grade [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 solar-grade 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 MINORITY-CARRIER TRANSPORT IN
POLYSILICON DEVICES



2.1 Introduction

The performance of bipolar polycrystalline silicon devices can be

limited because of minority-carrier recombination losses at the grain

boundaries [7-11]. For example, the efficiency of thin-film polysilicon

solar cells [12], which can potentially be used to produce cost-

competitive terrestrial photovoltaic energy-conversion systems, is lower

than that of silicon cells because of grain-boundary recombination.

Hence a complete theoretical understanding of the influence of grain

boundaries on minority-carrier 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,

three-dimensional boundary-value problem having nonlinear boundary

conditions [23]. For example, in the forward-biased polysilicon n+p

junction with columnar grains illustrated in Fig. 2.1, the electron

current injected into the quasi-neutral p-type base is governed, for

low-injection conditions, by the three-dimensional electron continuity

equation subject to nonlinear boundary conditions at the grain

boundaries [7-11]. The three-dimensionality results basically because

the carriers are injected at a surface (i.e., the pn junction) that is

not parallel to the grain-boundary surface. The boundary conditions,




10

















ZZ E
<
zz


ro
C.


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0



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~o *


-Jz








which characterize carrier recombination through energy-gap 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 [7-11, 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 minority-carrier 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 quasi-neutral p-region (1 ). A key feature of the model is

the separation of the grain-boundary 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 [7-11, 13, 14, 30] in the following

ways: (a) it is based on a computer-aided numerical solution of

Poisson's equation within the grain-boundary space-charge 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 quasi-equilibrium

[7, 8] in the grain-boundary space-charge region, as well as those

analyses based on the common assumption that S [13], by
n(eff) [13, by
incorporating into the quasi-equilibrium analysis the physical upper

limit on S eff), i.e., the kinetic-limit velocity SKL(= 5 x 106 cm/sec

at 3000 K) imposed by the random thermal motion of the conduction-band

electrons [31]; (c) it avoids possible errors involved in truncating

infinite-series solutions for the three-dimensional electron continuity

equation in the quasi-neutral grain [8, 14, 23, 30] by reducing the

problem to coupled one-dimensional ordinary differential equations that

are formulated based on "gradual-case" approximations [32]; (d) it is

the first analytic model for minority-carrier transport in polysilicon

derived from the physical insight provided by computer-aided numerical

solutions.

The model development begins with the computer solution of

Poisson's equation in the grain-boundary space-charge region adjoining

quasi-neutral p-type grains, assumed to be in low injection. The

numerical analysis is based on the quasi-equilibriun assumption [7, 8,]

but the inherent limitations are effectively removed by accounting for

the kinetic-limit electron velocity. The calculation of the grain-

boundary potential barrier height is based on the assumption of an

"effective" monoenergetic density of donor-type 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 hole-electron pairs








through both donor-type and acceptor-type 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 grain-boundary surface-state densities, on the grain doping density,

and on the "excitation" of the grain boundary, i.e., the separation of

the electron and hole quasi-Fermi 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

quasi-neutral p-type base of a representative grain, assumed to be a

right-circular 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 space-charge region resulting from the forward

bias V. This complicated three-dimensional 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 one-dimensional 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 quasi-neutral region. Recombination at the

grain boundary within the junction space-charge region, which can be

significant [33], is discussed in Section 2.3.) Rigorously this would

require the solution of the complicated two-dimensional electron

continuity equation in the quasi-neutral p-region of the grain subject

to appropriate boundary conditions at the grain-boundary and junction

space-charge regions [23]. The derived electron density N(r,z) must be
B
integrated over the volume of the p-region to determine IQ and its

gradient integrated over the encompassing grain-boundary surfaces to

determine IG. Such a rigorous solution would, even with simplifying

assumptions, require a computer-aided 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

grain-boundary 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 \ SPACE-CHARGE
"SC BR REGION

QUASI-NEUTRAL
+V p-TYPE BASE OF GRAIN
(b)

Fig. 2.2 Representative grain (a) of the forward-biased n p junction
and its cross-section (b). We assume rGGW SR; hence the
z-dependence of W GB
z-dependence o W deriving from the z-dependence of the
grain-boundary 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 grain-boundary space-charge region, which is

identified in Fig. 2.2(b). By describing the hole-electron

recombination rate UGB along the grain boundary, assumed to involve only

band-bound transitions, we can define the effective surface
combination velocity SGB
recombination velocity S for minority electrons at the edge of the

grain-boundary space-charge region.

At an arbitrary depth z in the base region, we consider the energy-

band diagram in the grain-boundary space-charge region. This diagram,

which follows from the solution of Poisson's equation to be discussed,

is illustrated for non-equilibrium conditions (V > 0) in Fig. 2.3. The

existence of the potential barrier height B is due to majority-carrier

(hole) charging of the grain-boundary surface states and has been

discussed extensively in previous papers [2, 7-10, 20].

To describe tB of the polysilicon grain boundary adjacent to a

p-type grain, we assume an "effective" monoenergetic density NT(eff) of

donor-type grain-boundary surface states, or traps, which are either

neutral or positively ionized, is representative of the grain-boundary

potential barriers for p-type 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 p-type polysilicon (see Fig. 2.3); this will

be predicted by an analysis based on donor-type traps. Indeed N T(eff)
ST(e ff)













U














04-C 4-

50cw

U,- C~
!-(-0)
-o 0)- 0-
0.m- U,
fr~O





C.' 04-'


SC-
0-4-'


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 grain-boundary trap density that would be inferred from

conductance or capacitance measurements [2, 20] of p-type polysilicon.

Although the assumption of a monoenergetic density of traps at ET=

Ei(O) may not characterize exactly actual grain boundaries in p-type

polysilicon [34], it is sufficiently representative to describe their

basic influence on the minority electron transport. The utility of a

monoenergetic-trap-density 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 donor-type (N) and
) and

acceptor-type (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 steady-state conditions, i.e., gB time-independent, 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 grain-boundary charge that produces overall

neutrality in the grain-boundary space-charge region be commensurate

with equal net capture rates for electrons and holes at the grain-

boundary surface. Mathematically, the quasi-equilibrium assumption

enables us to express this condition by combining Poisson's equation for

the grain-boundary space-charge region with Shockley-Read-Hall capture-

emission statistics [35] for the grain-boundary surface states.







Referring to Fig. 2.3, in which the grain boundary is located at
x = 0, we write Poisson's equation in the space-charge 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 grain-boundary surface-state 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 steady-state 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 grain-boundary surface
states, we get from the Shockley-Read-Hall theory [35]

Ei (O)-ET
C N(O) + Cpnexp (2.3
p-nW e i T
SFET-EiO1 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 quasi-neutral p-region:



S GB 0 (2.4)
SWSCR

This assumption also means that


P(WR) NAA ( 2.5)


Within the grain-boundary space-charge region, P and N are related to Ei

through the respective quasi-Fermi levels EFp and EFN:


FE (x)-E 1
P(x) = niexp ---- (2.6)


SEFN-Ei(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 grain-boundary recombination

analysis from the intragrain transport analysis, we assume that
conditions of quasi-equilibrium exist in the grain-boundary space-charge







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 quasi-equilibrium 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 quasi-neutral region to the grain-boundary

surface. This simplification is made without loss of generality because

when the quasi-equilibrium assumption is invalid, for example, when the

grain-boundary space-charge 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 grain-boundary 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 computer-aided numerical techniques. A Harwell

subroutine [37], which represents (2.1) using finite-difference
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 grain-boundary region.

Consider first the thermal-equilibrium 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 grain-boundary surface-state density NDT(ef. When N Def
is very low (< 1010 cm-2), BO is extremely small, and electrically the

polysilicon resembles single-crystal 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 space-charge 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 cm-2), 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 valence-band edge in the quasi-neutral 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 zero-bias conductance of grain boundaries in p-type

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

n-type polysilicon, analogous calculations based on an assumed

monoenergetic density of acceptor-type traps at midgap yield values of

gBO that are roughly consistent with those derived experimentally [38].
Consider now non-equilibrium cases (V > 0) for which VB > 0 in

accordance with the quasi-equilibrium assumption in the grain-boundary

space-charge region. Non-equilibrium 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 quasi-neutral p-region of the

grain, which we solve in the next subsection.



































1011
0


'o 101
10 o'l

NAA (cm-3)
Fig. 2.4 Calculated thermal-equilibrium barrier height versus grain doping
density (p-type) for representative values of effective grain-
boundary surface-state 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 cm-3

and 1017 cm-3 respectively. These plots are restricted to conditions of

low injection in the quasi-neutral p-region. 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 quasi-equilibrium 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 grain-boundary 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 steady-state conditions with sufficient excitation, the

electron and hole capture rates at the grain-boundary 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= 1015cm-3


0.30
V,(V)


Fig. 2.5 Calculated non-equilibrium barrier height versus grain-
boundary excitation for representative values of effective
grain-boundary surface-state density and for a grain doping
density of 10 s cm-3 (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 4-1
(U yi

>C'












(A (
SS3






O 'i Ol



E (U
0 *- II







c "+

^*'"
o D +








1 aN(0) 1 aP(0) (2.12)
CT-37g- PiT-y *(2.12)
NT7 -TD73 TM m a


The relation in (2.12) is a general result of invoking the steady-state

condition (dOg/dt = 0) at a sufficiently excited grain-boundary

surface. Combining it with


P(0)N(0) = nexp-I (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 grain-boundary 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) grain-boundary 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 energy-band 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 computer-aided determination of the dependencies of pB and of

P(0) and N(O) on Vg, NST(eff), and NAA enables the characterization of

the steady-state hole-electron recombination rate UGB through the grain-

boundary surface states, provided the active surface-state densities are

specified. This calculation is based on the Shockley-Read-Hall capture-

emission model [35] for recombination-generation through localized

states in the energy gap.

Because the exact nature of the grain-boundary surface states in

polysilicon is not known, we allow for the possible existence of both

donor-type (NT ) and acceptor-type (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 grain-boundary charge that is effectively

characterized by N D(f.

If we neglect bound-bound 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 Shockley-Read-Hall 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 = 10-7 cm3/sec and Cn = 10-9 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 surface-state 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 acceptor-type 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 grain-boundary space-charge
region (see Fig. 2.3), UGB defines the effective recombination velocity
for minority electrons at the edge of the space-charge 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 grain-boundary
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 pre-exponential 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 donor-type traps are effective on a polysilicon
grain boundary between p-type 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 cm-3 and
1017 cm-3 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 acceptor-type 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. 2.7-2.10 is an indication of the
kinetic-limit 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
Maxwell-Boltzmann statistics for conduction-band (free) electrons, and
is the physical upper limit for SGeff). We therefore recognize that
our plots ofS eff) in Figs. 2.7-2.10 must be truncated at SKL. This







recognition effectively removes any uncertainty or restriction of our

model due to the possible invalidity of the quasi-equilibrium

assumption. We now demonstrate this.

The quasi-equilibrium assumption is valid if the variations in the

quasi-Fermi levels across the grain-boundary space-charge region (see

Fig. 2.3) are less than kT [8, 9]. A quantitative self-consistency

check for the validity of this assumption is described in [8]. Using

this check and our results shown in Figs. 2.7-2.10, we find that

generally the quasi-equilibrium 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. 2.7-2.10, we

stipulate that


MAX[S eff] = SKL (2.35)


Hence for all conditions under which the grain-boundary space-charge

region is depleted of free carriers, which obviously preclude quasi-
equilibrium, SGB
equili n(eff) SKL and a depletion-approximation analysis
[7, 9, 10] is unnecessary. In fact, a recent such analysis [10] yielded

results that comply with these conclusions.

Note in Figs. 2.7-2.10 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























10o4Cm-2 N e
.- 2 = "1 T(f,,t)

NAA=1015cm-3
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 grain-boundary space-charge region versus grain-
boundary excitation and surface-state density. The kinetic-limit
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 grain-boundary space-charge region versus grain-
boundary excitation and surface-state density.































E

SAN A1015cm3
AA
NT-2 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 grain-boundary space-charge region versus grain-
boundary excitation and surface-state 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 space-charge region versus grain
boundary excitation and surface-state density.








why SG is independent of the grain-boundary 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 two-dimensional

electron continuity equation in the quasi-neutral p-base 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 forward-biased 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 p-region 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 one-dimensional 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 "gradual-case" 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 two-dimensional

minority-carrier flow in a forward-biased 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 z-direction.

Contrarily, we assume that in the vicinity of the grain boundary (rI <

r < rG), the electron flow is strongly influenced by the grain-boundary

recombination, and is thus primarily in the lateral r-direction. These

assumptions are commensurate with letting



rI = rG L (2.36)


where Ln is the electron diffusion length. Note that (2.36) restricts

our quasi-two-dimensional 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 three-dimensional

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 grain-boundary 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

low-injection conditions is defined by


d2N(z) N(z)
d, (2.37)


with the boundary conditions


















z








+V
=rG


T


Fig. 2.11 Partitioned p-region of the representative grain showing predom-
inantly one-dimensional 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 partitioned-base 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 quasi-neutral base of the grain. This

results in a In(r) dependence for N.

The second boundary condition for (2.40) is that defining the

grain-boundary 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 grain-boundary surfaces is



W
IB = 2rqD rG J N(r dz (2.42)
0 a r=rr



and that due to recombination in the quasi-neutral p-base 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 p-region.







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 p-region parameters that define I01 The
reciprocal slope factor n(V), which also depends on SO and the p-region

parameters, increases from one to two as V increases, albeit in the

range corresponding to low-injection conditions in the p-region.

The general shape of the I,(V) characteristic in (2.44) for typical
values of the grain and grain-boundary parameters is illustrated in

Fig. 2.12. The shape resembles that of the dark current-voltage

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 current-voltage

characteristics of gated n+p diodes fabricated on Wacker polysilicon

[16]. The characteristics of diodes containing substantial grain-

boundary-surface area show an inflection similar to that in Fig. 2.12

whereas those of diodes having little or no grain-boundary-surface 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 single-crystal 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

pre-exponential 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 short-circuit 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 forward-biased polysilicon n+p junction.








2.3 Discussion
In this chapter we have developed, using computer-aided numerical

solutions, an approximate model for the electron current I injected

into the base of a representative grain of the forward-biased

polysilicon n+p junction illustrated in Fig. 2.1. Key assumptions have

been made to simplify the general three-dimensional, 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 grain-boundary recombination analysis on the

assumption of quasi-equilibrium, i.e., nearly flat quasi-Fermi levels,

in the grain-boundary space-charge region. This assumption facilitates

the separation of the grain-boundary recombination analysis from the

intragrain electron transport analysis, the results of which define the

grain-boundary excitation. The assumption further enables a complete

analysis of the grain-boundary recombination without having to ascertain

the mechanism, e.g., thermionic emission [10], by which majority holes

are transported from the quasi-neutral grain to the grain-boundary

sur face.

The limitations associated with the quasi-equilibrium assumption

are effectively removed by recognizing that when conditions obtain that

negate quasi-equilibrium, the minority electrons flow to the grain-

boundary surface with velocities about equal to the kinetic-limit

velocity SKL [31]. Thus we simply truncated in (2.35) our quasi-

equilibrium-based 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 quasi-equilibrium assumption, the grain-boundary

analysis, i.e., the solution of Poisson's equation (2.1), is

formidable. Thus we resorted to a computer-aided 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 donor-type

(adjacent to p-type grains) states near midgap to calculate the grain-

boundary potential barrier height nB. Then, based on our assumed

existence of both donor-type and acceptor-type traps at midgap, OB was

used to calculate the grain-boundary recombination and ultimately
SGB
n(eff)
The utility of an analysis based on a monoenergetic density of

grain-boundary 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 surface-state 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 quasi-neutral p-region. 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 grain-boundary 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 quasi-neutral p-region

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 p-region 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 grain-boundary

recombination. The one-dimensional 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 grain-boundary recombination, simulated by S GB [i.e., (2.33) and
n(e ff)
(2.35)], was assumed to cause the electrons to flow one-dimensionally

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 current-voltage characteristics of

polysilicon pn junctions. Because the current component deriving from

recombination in the junction space-charge 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 space-charge region

recombination current could be predominantly due to recombination

through grain-boundary surface states within the junction space-charge

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 quasi-Fermi-level planes in

the space-charge 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 current-voltage

characteristic of a polysilicon pn junction and thereby, with the

minority-carrier transport model (2.44) developed herein, provide

physical insight into the performance of polysilicon bipolar devices and

their optimal designs.




52


From the grain-boundary analysis described in this chapter, it is
clear that the boundary condition at the grain boundary adjacent to a

quasi-neutral 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 computer-aided

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 three-dimensional continuity equation

subject to the nonlinear boundary conditions. With the aid of the

grain-boundary 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 GRAIN-BOUNDARY RECOMBINATION VELOCITY
FROM ELECTRON-BEAM-INDUCED-CURRENT MEASUREMENTS



3.1 Introduction
The minority-carrier transport model developed in Chapter 2 reveals

that the grain-boundary 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 current-voltage

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 electron-beam-induced-current (EBIC) mode [42]. The EBIC

technique has been widely used to measure bulk diffusion length [21, 22,

43-45] and surface properties [21, 22, 24, 46-49] 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 quasi-neutral grain without tedious

sample preparation, such as fabrication of small-area 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 minority-carrier (electron) transport

problem in the p-type base, which, in general, is three-dimensional and

has nonlinear boundary conditions. The three-dimensionality 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 hole-electron-pair 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 minority-carrier

lifetime and grain-boundary recombination velocity from EBIC

measurements [21, 22, 24, 47]. However the point [52] or the spherical

[25] generation-source models do not give a realistic description of the
actual electron-beam generation [50]. Furthermore none of the analyses

[21, 22, 24-26, 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 minority-carrier

transport problem underlying the EBIC response subject to the nonlinear

boundary condition at the grain boundary. The three-dimensional,

steady-state electron continuity equation is reduced to two dimensions

by recognizing that the grains are sufficiently wide (in the

y-direction) that the y-dependence of the electron concentration is

insignificant (see Fig. 3.2). The solution of the two-dimensional

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 generation-source models have been considered

in the numerical analysis.

Our results, obtained by numerically solving the two-dimensional

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 grain-boundary 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 z-directions and in the

positive and negative y-directions. The metallurgical junction is

located at a distance xj from the SCR edge top surface, and WSCR is the

width of the (nearly one-sided) junction space-charge region. The

electron beam is incident from the top surface and creates electron-hole

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 p-type base

subject to the appropriate boundary conditions. The steady-state

continuity equation for electrons when an incident beam of electrons

creates g(x,y,z) electron-hole pairs (cm-3-sec-1) 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 (cm-3-sec-1). For a

homogeneously doped base under low-level-injection 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 computer-aided

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 two-dimensional equation that

faithfully represents the actual electron transport if the grains are

sufficiently wide, i.e., the surfaces in the y-direction are far away

(~ -). Ascertaining that the EBIC response is invariant along the

y-direction (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 y-direction does not

influence the resulting value of the EBIC, and hence it is sufficient to

solve, as we will, the two-dimensional (in x and z) continuity equation

to calculate IEBIC. We will show qualitatively that IEBIC in the actual

(three-dimensional) case differs from that calculated from our numerical

(two-dimensional) 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 y-direction 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 y-direction have virtually the same

probability of being collected by the junction, rather than recombining

at the grain boundary, as those electrons diffusing in the x-direction

toward the grain boundary. For this condition, the y-dependence in

(3.2) can be neglected. As the beam is moved close to the grain

boundary, the number of electrons that diffuse in the y-direction is

small compared to the number that diffuse in the x- (and z-)

directions. This is true because the grain-boundary 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 y-dependence in (3.2) is not

important.

Therefore integrating (3.2) over the y-direction, 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

y-dependence 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 y-direction width that serves as a normalizing

constant. In (3.4), N(x,z) is the electron concentration obtained from

the two-dimensional numerical analysis, and



G'(x,z) = f g(x,y,z) dy (3.5)
y=-"

is the two-dimensional generation-rate 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 two-dimensions. Further

support for our two-dimensional 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 space-charge region; N(z + -) = 0 since Ln <

Wg where Wg is the thickness of the base; N(x + -) = 0 since the grain

is semi-infinite (>> Ln) in the x-direction; and at the edge of the

grain-boundary space-charge 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) quasi-equilibrium prevails across the grain-boundary

space-charge region. When these conditions obtain, the grain-boundary

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 right-side

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 electron-beam-induced hole-electron-pair generation

rate G(x,z). The simplest representation is a point-source excitation

[21, 26, 523, i.e., the electron-hole pairs are generated at a point in

the base. The distance of the point-source from the junction is

determined by the energy of the incident electron beam. It is obvious

that the point-source 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 generation-source 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 electron-hole pairs in a spherical volume [25]. In our

numerical analysis, in which we have reduced the transport problem to a

two-dimensional differential equation (in the xz-plane), 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 electron-hole 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 z-direction. In (3.10) Eo is the energy of the beam in

kilo-electron-volts (keV), and C is a constant which is dependent on

properties of the semicondcutor [21]:


C = 4 x 10-6
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 10-4, 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 high-energy electrons penetrate the semiconductor, the

electron-hole pairs are generated over a characteristic pear-shaped

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 z-direction. The Gruen's function is a

third-degree 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 z-direction 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

X-rays 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) (x-xo)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 y-direction 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 = 500-1000 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 computer-aided numerical solution of (3.6), subject to the
given boundary conditions, was obtained by using a finite-difference

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 pre-exponential 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
space-charge 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 z-0 .


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 Gauss-Seidel 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 shallow-junction

devices. In such devices we can assume that the generation in the

quasi-neutral emitter and the junction space-charge region is much

smaller than the total generation. Hence the EBIC due to generation of

carriers in the quasi-neutral emitter and the junction space-charge

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 zero-bias

capacitance measurements performed on the completed cell. From Fig. 3.4

we note that the generation in the emitter and the junction space-charge

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 cm-2 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 constant-S 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 constant-S 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 aluminum-passivated (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 high-angle 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 grain-boundary 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-

generation-source case, we observe that the circular-generation

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 computer-aided numerical analysis, the

underlying transport problem. The general transport problem in three

dimensions was reduced to one in two-dimensions 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 (high-angle) grain boundary
(G= 1014/sec); SGeff) is virtually constant (=SKL) due to
high N ,, (=10n 2
hioh ND (=1014cm-2).
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 hole-electron-pair
generation volume.








in the y-direction 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 y-dependence 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

two-dimensions has also been shown to be mathematically valid [53].

In our analyses, we have incorporated two different models for the

hole-electron-pair excitation volume: 1) the spherical generation-

source model [24, 25]; and 2) the Gaussian generation-source model

[50]. The utility of the spherical-source 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

Gaussian-source model. Thus the spherical-source 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 spherical-source model the incident electron beam generates

hole-electron 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 spherical-source

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 z-direction 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 two-dimensional 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 computer-aided 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 Gauss-Seidel 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 generation-source

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 spherical-source 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 grain-boundary 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 grain-boundary recombination. It is
n(e ff)I
obvious that grain-boundary recombination deteriorates the performance

of bipolar polysilicon devices. For example in a polysilicon solar

cell, the grain-boundary recombination both in the junction space-charge

region as well as adjacent to the quasi-neutral base can limit the cell

efficiency [8]. In order to improve polysilicon device performance it

is necessary to suppress grain-boundary 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

grain-boundary 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 GRAIN-BOUNDARY 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 minority-carrier lifetimes; the grain-boundary

recombination is hence rendered virtually unimportant. The efficiency

of thin-film polysilicon cells having relatively small grains may, on

the other hand, be generally limited by high densities of grain-boundary

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 grain-boundary 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 grain-boundary diffusion coefficient is higher

than its bulk coefficient, which implies preferential grain-boundary

diffusion [59], and whose presence at the silicon surface increases the

solubility of common defects, which implies bulk-to-surface transfer of

defects.







Phosphorous, which is an effective gettering agent in silicon [60],

also has the potential to be an effective grain-boundary passivating

agent [16]. However preferential grain-boundary 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 junction-formation temperature (< 900C). It has been shown

that lithium tends to suppress grain-boundary 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 acceptor-type 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

aluminum-alloyed n+pp' back-surface-field (BSF) silicon solar cell

[61]. More recently unusually high grain-boundary 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 grain-boundary (deep-

level) trap density resulting from removal of dangling bonds;

2) formation of a high-low 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

thin-film 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 Al-BSF 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 grain-boundary passivation, are investigated. With regard to thin-

film polysilicon, we examine gettering and grain-boundary passivation

resulting from aluminum diffusion at low temperatures (below the

eutectic-formation temperature, 577C) from the front surface. Such

low-temperature diffusion has been shown to yield preferential aluminum

spikes down dislocations [66].

To assess the grain-boundary passivation, we use the electron-beam-

induced-current (EBIC) technique. A qualitative analysis is done







initially to detect grain-boundary passivatidn. Quantitative estimates
GR
of grain-boundary parameters, e.g., SG ,B based on EBIC data are
n(eff)'
derived using the computer-aided numerical simulation, details of which

were disclosed in Chapter 3, of the minority-carrier transport

underlying the EBIC response of a pn junction in the vicinity of a grain

boundary. Current-voltage characteristics and reverse-bias capacitance

measurements, which yield the base doping density of the cell, are used

to determine the minority-carrier diffusion length in the base, which

reflects the benefit produced by the Al-BSF process.

Our experimental results show that the conventional Al-BSF process

results in longer diffusion lengths because of (bulk-to-surface)

gettering of intragrain impurities, irrespective of indications that the

grain boundaries emit impurities during the high-temperature process.

Our results also suggest that the minority-carrier lifetime is longer

when measured using light-bias than when measured in the dark. This

suggestion is consistent with the theory [67] that defect clusters in

polysilicon are effectively negated by high carrier-injection levels.

No grain-boundary passivation near the front junction is apparent after

the BSF process. However we find significant passivation near the front

junction resulting from low-temperature 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 thin-film

polysilicon solar cells.








4.2 Cell Fabrication

The cells used in this study were fabricated on p-type 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 xl016cm-3,

as determined by reverse-bias capacitance measurements on the completed

cells.

Three different kinds of cells were fabricated for comparison in

both the bulk and thin-film 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 aluminum-diffused counterparts. Several standard cells, which

were neither aluminum-diffused 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 A1-BSF 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

large-area (0.4 0.45 cm2) and small-area (30-mil-diameter) cells were

fabricated.

Fabricating small-area cells enables control of the grain-boundary

density in each cell. The surface and edge leakage currents, which are







predominant in large-area cells, can be effectively controlled by means

of an MOS guard-ring overlapping the edge of the emitter in the small-

area cells [16]. In addition, by comparing the diffusion lengths,

determined from the current-voltage characteristics, of the large-area

(measured under light) and the small-area (measured in dark) cells, we

can deduce the effect of the light-bias on minority-carrier lifetime.

The n+p junction in the large-area 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 300-A-thick

aluminum "dot". Details of these processing steps as well as those of

the small-area and the thin-film cells are given in Appendix D.

The junction in the small-area cells was formed by a phosphorous

predeposition at 950C for 40 minutes using POC13 as the source followed

by a drive-in 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 back-contact

aluminum evaporations, 5000-A and 6000-A-thick respectively, were

performed subsequently. As in the case of the large-area cells, the

control and the standard cells were sintered at 450C for 15 minutes

after the back-contact evaporation.

In order to suppress surface and edge leakage currents in the

small-area cells, a silicon-dioxide layer (~ 1700-A-thick) was thermally







grown on the top surface during the drive-in 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 current-voltage characteristic of the cell was being measured.


4.2.2 Thin-Film Process

For the studies applicable to the development of thin-film

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 (large-grain) polysilicon to imply the effects of

aluminum diffusion in thin-film (small-grain) 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 low-temperature diffusion

will similarly passivate grain boundaries in thin-film 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 Si-Al eutectic temperature (577C) to avoid

irreparable damage to the junction region. IC-grade aluminum, 1.5-pm-

thick, was evaporated onto the front surface before the phosphorous (n+)

diffusion. The aluminum diffusion was then done at 4500C for 3-6 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




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