• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Abstract
 Introduction
 Auger recombination in heavily...
 Recombination through defects in...
 On the deionization of impurities...
 Heavily doped transparent-emitter...
 A first-order engineering-design...
 Computer-aided study of Voc in...
 Summary, conclusions, and...
 Appendices
 References
 Biographical sketch






Title: Device physics for engineering design of heavily doped regions in pn-junction silicon solar cells /
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Title: Device physics for engineering design of heavily doped regions in pn-junction silicon solar cells /
Physical Description: viii, 193 leaves : ill. ; 28 cm.
Language: English
Creator: Shibib, Muhammed Ayman, 1953-
Publication Date: 1979
Copyright Date: 1979
 Subjects
Subject: Solar batteries   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 185-192.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Muhammed Ayman Shibib.
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Bibliographic ID: UF00099523
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 - 000097514
oclc - 06578011
notis - AAL2954

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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
        Page iv
        Page v
        Page vi
    Abstract
        Page vii
        Page viii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
    Auger recombination in heavily doped shallow-emitter silicon PN-junction solar cells, diodes, and transistors
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
    Recombination through defects in heavily doped shallow-emitter silicon PN-junction devices
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
    On the deionization of impurities as an explanation for excess intrinsic carrier density in heavily doped silicon
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Heavily doped transparent-emitter regions in silicon junction solar cells, diodes, and transistors
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
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        Page 54
        Page 55
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        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
    A first-order engineering-design model for heavily doped silicon devices
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
    Computer-aided study of Voc in N=p and p+N silicon solar cells
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
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        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
    Summary, conclusions, and recommendations
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
    Appendices
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
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        Page 179
        Page 180
        Page 181
        Page 182
        Page 183
        Page 184
    References
        Page 185
        Page 186
        Page 187
        Page 188
        Page 189
        Page 190
        Page 191
        Page 192
    Biographical sketch
        Page 193
        Page 194
        Page 195
        Page 196
Full Text











DEVICE PHYSICS FOR ENGINEERING DESIGN
OF HEAVILY DOPED REGIONS IN PN-JUNCTION
SILICON SOLAR CELLS








By

MUHAMMED AYMAN SHIBIB


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA
























To My Parents:

Subhi and Loutfieh Shibib













ACKNOWLEDGMENTS


I am indebted to my family for their encouragement and

parental support that helped me pursue my graduate studies.

I want to express my deep appreciation to the chairman of

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

guidance, encouragement, and support. His research insight and excel-

lent tutoring in physical electronics have been invaluable for develop-

ing and guiding my research interests.

I extend my gratitude to Professor Jerry G. Fossum, the

co-chairman of my supervisory committee, for guiding the development

of this research and contributing significantly to its completion.

Thanks are due to Professor Arnost Neugroschel and Sheng S. Li

for helpful discussions and for their participation on my committee.

I am also grateful to Professor Charles F. Hooper, Jr., for serving on

my committee.

Helpful discussions in solid-state physics with the late

Thomas A. Scott, Professor of Physics, are most appreciated.

I thank professors Allan H. Mlarshak and H. P. D. Lanyon for

their helpful comments.

The support of NASA and DOE is gratefully acknowledged.

I also thank my typist Sofia Kohli for her excellent work.















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . . . . . . . . . iii

ABSTRACT . . . . . . . ... . . . ... .vii

CHAPTER

I. INTRODUCTION . . . . . . . . ... . . 1

II. AUGER RECOMBINATION IN HEAVILY DOPED
SHALLOW-EMITTER SILICON PN-JUNCTION
SOLAR CELLS, DIODES, AND TRANSISTORS . . . . .. 10

2.1 Introduction . . . . . . . . 10
2.2 Derivation . . . . . . . . . 11
2.3 Discussion . . . . . . . ... .15
2.4 Conclusion . . . . . . . ... .20

III. RECOMBINATION THROUGH DEFECTS IN HEAVILY
DOPED SHALLOW-EMITTER SILICON PN-JUNCTION
DEVICES . . . . . . . . ... .. . .23

3.1 Introduction . . . . . . . . 23
3.2 Importance of Surface Recombination . . .. 24
3.3 Shockley-Read-Hall Recombination
via Defects . . . . . . . . . 25
3.4 The Excess Minority-Carrier Charge
Storage in Heavily Doped Regions ...... 30

IV. ON THE DEIONIZATION OF IMPURITIES AS AN
EXPLANATION FOR EXCESS INTRINSIC CARRIER
DENSITY IN HEAVILY DOPED SILICON . . . . .... 34

V. HEAVILY DOPED TRANSPARENT-EMITTER REGIONS
IN SILICON JUNCTION SOLAR CELLS, DIODES,
AND TRANSISTORS . . . . . . . . . . 44

5.1 Introduction . . . . . . . ... .44
5.2 Derivation . . . . . . . . 46
5.3 Heavy Doping Effects . . . . . . 49
5.4 Discussion . . . . . . . . 54











TABLE OF CONTENTS--Continued


5.5 Application to pn-Junction
Silicon Solar Cells . . . . . .
5.6 Perspective . . . . . . . .

VI. A FIRST-ORDER ENGINEERING-DESIGN MODEL
FOR HEAVILY DOPED SILICON DEVICES . . . . .

6.1 Introduction . . . . . . .
6.2 Modeling Approach . . . . . . .
6.3 Development of the Model . . .
6.4 Relating Device Design Parameters
to the First-Order Model's
Parameters . . . . . . .
6.5 Verification of First-Order Model
by Computer Analysis . . . . . .
6.6 Experimental Support . . . . . .
6.7 Summary and Conclusions . . . . .

VII. COMPUTER-AIDED STUDY OF VOC IN
N+P AND P+N SILICON SOLAR CELLS . . . . .

7.1 Introduction . . . .
7.2 Dependence of VOC on the Emitter
Design Parameters N+P and
P+N Silicon Solar Cells . . . . .
7.2A Dependence of VOC on WE . .
7.2B Dependence of VOC on NS . .
7.2C Dependence of VOC on S . .
7.3 Discussion . . . . . . . .
7.4 Experimental Support . . . .
7.5 Conclusions . . . . . . . .


VIII. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . . .


91

91


Summary . . . . . . .
Accomplishments . . . . .
Scope and Limitations . . . .
Recommendations for Future Research


APPENDICES


A. EXTENSION OF SHOCKLEY'S AUXILIARY RELATIONS
FOR HEAVILY DOPED PN-JUNCTION DEVICES . ..


CHAPTER


Page


.


. .









TABLE OF CONTENTS--Continued


APPENDICES Page

B. QUASI-ELECTRIC FIELDS IN HEAVILY DOPED
SEMICONDUCTORS . . . . . . . . .. . 135

C. INCLUSION OF DEGENERACY IN THE ANALYSIS OF
HEAVILY DOPED SEMICONDUCTOR DEVICES . . . . .. 138

D. LISTING OF THE COMPUTER PROGRAM . . . . .... 146

E. CHARACTERIZATION OF THE EMITTER CURRENT
IN HEAVILY DOPED SILICON DEVICES BY
COMPUTER-AIDED NUMERICAL ANALYSIS . . . . .... 165

F. ON THE PARABOLIC DENSITY OF STATES IN
HEAVILY DOPED SILICON . . . . . . . . . 183

REFERENCES . . . . . . . . ... .... . .. .185

BIOGRAPHICAL SKETCH . . . . . . . . ... .. .... .193











Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



DEVICE PHYSICS FOR ENGINEERING DESIGN
OF HEAVILY DOPED REGIONS IN PN-JUNCTION
SILICON SOLAR CELLS

By

Muhammed Ayman Shibib

December 1979
Chairman: Fredrik A. Lindholm
Co-Chairman: Jerry G. Fossum
Major Department: Electrical Engineering

This dissertation presents a quantitative study of the physical

mechanisms underlying the anomolously large recombination current

experimentally observed in heavily doped regions of silicon pn-junction

solar cells and bipolar transistors. The study includes a comparison

of theoretical predictions with a variety of experimental observations

in heavily doped silicon and silicon devices.

A major conclusion is that the simplest physical model that

adequately describes the heavily doped regions must include Fermi-

Dirac statistics, a phenomenological excess intrinsic carrier density

(or deficit impurity concentration), Auger recombination in the bulk,

and recombination at the surface. These mechanisms are incorporated in

a first-order model useful in the design of silicon pn-junction solar

cells. The accuracy of the first-order model is supported by comparing

its results with the results of more detailed models and of a numerical










analysis of the problem. Experimental data are presented that are

consistent with the predictions of the first-order model and of the

numerical solution.


viii














CHAPTER I

INTRODUCTION

Heavily doped regions of silicon pn-junction devices show

experimentally a larger recombination current than is predicted by classi-

cal pn-junction theory [1,2]. This large recombination current in heavily

doped regions is responsible for the low values of the open-circuit

voltage VOC of silicon solar cells [3] and the commion-emitter current

gain hFE in silicon bipolar transistors [4,5].

The main objective of this dissertation is to study quantitatively

large recombination currents in heavily doped regions and to describe

the fundamental limitations imposed by heavy doping on the performance

of silicon pn-junction solar cells and bipolar transistors. The

achievement of this objective involves:

A. Identifying physical mechanisms associated with

heavy doping.

B. Determining the dominant heavy-doping mechanisms

that affect the device performance.

C. Including the dominant heavy-doping mechanisms in

analytic and computer-aided analysis of heavily

doped regions in silicon devices.

D. Establishing accurate and simple first-order

engineering design models based on device physics.










From charge-control theory [6-10], the recombination current

in any heavily doped region, for example, the minority-carrier emitter

current JE, can be expressed by


QE
JE E (1.1)

where QE is the minority-carrier charge storage and TE is the effective

lifetime in the emitter. From classical pn-junction theory [11], QE

increases with decreasing bandgap and decreases with increasing majority

carrier density. The effective lifetime TE may be the average bulk

lifetime of the minority carriers or their transit time, defined as the

average time needed by a minority carrier to cross the semiconductor

region without recombining in the bulk, or a combination of the two.

From (i.l) we can see that an excessive emitter current is due to (a) a

large minority-carrier charge, (b) a short effective lifetime, and

(c) a combination of (a) and (b).

Various physical mechanisms associated with heavy doping in

semiconductors can give rise to (a), (b), or (c). These heavy-doping

mechanisms are outlined, in Table I, ard contrasted to corresponding

mechanisms in lightly doped semiconductors. We will describe briefly

some of the fundamental heavy-doping mechanisms outlined in Table I.

One of the fundamental heavy-doping mechanisms that yields a

large minority-carrier charge, and consequently a large current by

(1.1), is energy-bandgap narrowing. From a theoretical standpoint,

various mechanism exist that lead to bandgap narrowing [12-19]. More-

over, electrical measurements in heavily doped single-crystal silicon









TABLE I

COMPARISON BETWEEN A LOW IMPURITY CONCENTRATION
SEMICONDUCTOR (LESS THAN = 1016/CM3) AND A HIGH
IMPURITY CONCENTRATION SEMICONDUCTOR.(MORE THAN = 1018/CM3)


Low Impurity Concentration High Impurity Concentration

Effective bandgap= Intrinsic band- Effective bandgap gap gap

(AEG = O) (AEG O)

Maxwell-Boltzmann statistics Maxwell-Boltzmann statistics for
for the majority and minority minority carriers; Fermi-Dirac
carriers statistics for majority carriers

Simple Law of I1ass Action valid Generalized Law of Mass Action

PNO =n2(T) PONO = n? (T,x)


All impurity atoms ionized (at Some impurity atoms deionized (at
room temperature) room temperature)


UN, Pp, Dp and DN constant (in- uN, ip Dp and DO dependent on
dependent of impurity concentra- impurity concentration
tion)

Einstein's relation valid Einstein's relation not valid;
modified relation:
D kT
S-q D kT F1/2 (i)
v q F_1/2 (n)

Built-in electric field in QNR Effective drift field in QNR is not
is same for majority and minor- same for majority and minority
ity carriers carriers

Emaj = Emin maj r Emin






TABLE I--Continued


TABLE I--Continued


Low Impurity Concentration High Impurity Concentration


Recombination of excess minor- Recombination of excess minority
ity carriers via defects, carriers via (a) fundamental band-
(Shockley-Read-Hall) to-band Auger process, (b) position-
dependent recombination centers or
defects (Shockley-Read-Hall)










materials and devices indicate that the forbidden energy gap may be

smaller than the intrinsic energy gap of pure silicon. These experi-

mental measurements are:

A. Measurements of the temperature dependence of the

current gain in silicon bipolar transistors [4,5]

B. Measurements of the collector current and its

temperature dependence in silicon bipolar tran-

sistors [20]

C. Measurements of the temperature dependence of the

photoresponse of silicon pn-junctions and tran-

sistors [21]

D. Measurements of the temperature dependence of the

emitter current in silicon solar cells (in the dark

condition) and bipolar transistors [22]

E. Measurements of the emitter current in different

pn-junction structures [23]

Rather than providing a direct measurement of the bandgap, the

above measurements yield an excess effective intrinsic carrier density

n2e, defined as the thermal equilibrium pn-product in heavily doped

silicon [24]. The measured values of nLe are considerably larger than

n for pure silicon.
1
Another mechanism that can provide an excess intrinsic carrier

density, and a large minority-carrier charge in the heavily doped

region, is deionization of impurities at high doping concentrations

[25]. For low concentrations of shallow impurities in silicon at room

temperature, all impurity atoms are essentially ionized. This is so










because the Fermi level is well below the impurity levels. As the

impurity concentration increases (above 1016 cm-3) some deionization

of impurities occur. The degree of ionization of impurity atoms depends

on the density and position of localized energy levels in the bandgap

and on the position of the Fermi level relative to these levels.

Another fundamental heavy-doping mechanism that must be con-

sidered is the degeneracy of the majority carriers in heavily doped

semiconductors. Majority-carrier concentrations in heavily doped

semiconductors are high enough that the carriers can interact with

each other via the Pauli Exclusion Principle. Thus Maxwell-Boltzmann

statistics are no longer applicable and Fermi-Dirac statistics must be

used instead.

We consider now two other fundamental heavy-doping mechanisms

that can increase JE in (1.1) by decreasing TE. The first mechanism

is Auger band-to-band recombination [26]. It is fundamental in the

sense that it sets an upper bound on the minority-carrier lifetimes

in these regions. The second mechanism is Schockley-Read-Hall

recombination [27-28] via position-dependent defect densities.

We have discussed the fundamental heavy-doping mechanisms

that can limit the performance of heavily doped silicon pn-junction

devices such as solar cells and bipolar transistors. We outline now

our approach in assessing the relative importance of these heavy-doping

mechanisms to tne performance of silicon devices.

Our approach in assessing the importance of heavy-doping

effects is to include each of these effects separately in a simple yet










accurate model for the heavily doped region. We first include Auger

reccmbination alone, neglecting other heavy-doping effects, in a

rigorous analytic model for the emitter region of silicon solar cells

and bipolar transistors. The model includes also a finite surface

recombination velocity at the emitter surface. We find, in Chapter II,

that values of VOC in silicon solar cells, predicted by this model,

are considerably higher than those experimentally observed. So

Auger recombination is not the dominant mechanism that yields the large

emitter current in bipolar devices.

Next, we consider recombination via defects, in Chapter III.

Using a previous detailed study of defect recombination [9], we

argue that if recombination via defects is the main mechanism underlying

the large emitter current, then surface recombination should not

influence the device performance. This implication is inconsistent with

recent experimental data [29] showing the sensitivity of VOC in typical

silicon solar cells to surface treatment. These experimental data

emphasize the importance of surface recombination and an excess in-

trinsic carrier density in heavily doped silicon (i.e., n2 > n?).
le I
The importance of deionization in heavily doped silicon is

assessed in Chapter IV. We investigate whether deionization can be the

main mechanism underlying the experimentally observed effective

intrinsic carrier density in heavily doped silicon. By using a

variety of experimental data, we infer that deionization in heavily

doped silicon at room temperature is not an important mechanism, at least

not in accordance with the recent models of Pooovic and Heasell.










Having established the important fundamental heavy-doping

mechanisms, we proceed to develop engineering design models for

heavily doped regions.

Energy-bandgap narrowing, Fermi-Dirac statistics, and a finite

surface recombination velocity are included in a rigorous analytic

model of the heavily doped emitter of silicon devices in Chapter V.

The emitter is assumed to be transparent to the injected minority

carriers, that is, most of the injected minority carriers can cross

the quasi-neutral emitter region and recombine mainly at the surface

rather than in the bulk. The transparency assumption can be tested

for self-consistency for any given device. We show, in Chapter V,

that the transparent emitter model can explain the experimentally

observed values of VOC in silicon solar cells if the surface recombina-

tion velocity is high and energy-bandgap narrowing and Fermi-Dirac

statistics are included. In agreement with other experimental data,

we also show that reducing the front surface recombination velocity

increases VOC in silicon solar cells.

In Chapter VI, we develop a simple first-order model for the

emitter that includes energy-bandgap narrowing, Fermi-Dirac

statistics, Auger recombination, and a finite surface recombination

velocity. The first-order model is useful as a design tool particularly

for silicon solar cells. The accuracy of the model is checked by a

rigorous computer-aided numerical analysis of the problem. The model

is found to be accurate if the surface impurity concentration is less

than 4 x 1020 cm-3.










To compare the limitations imposed by heavy doping on N+P and
P+N cells, we present, in Chapter VII, the results of a quantitative

parametric study of the dependence of VOC in N+P and P+N silicon solar

cells. These design parameters considered are the surface impurity

concentration, the width of the quasi-neutral emitter region, and

the emitter surface recombination velocity. We show, based on our

computer-aided analysis, that it is more beneficial to have a P emitter

than an N+ emitter to achieve large VOC in silicon solar cells.

Finally, we summarize our studies and review the main conclusions

and accomplishments of this dissertation in Chapter VIII. We also dis-

cuss the scope and limitations of this work and provide suggestions

for future research.














CHAPTER II

AUGER RECOMBINATION IN HEAVILY DOPED SHALLOW-EMITTER
SILICON PN-JUNCTION SOLAR CELLS, DIODES, AND TRANSISTORS


2.1 Introduction

Because the recombination current occurring in the quasi-

neutral emitter can limit the current gain of silicon junction tran-

sistors and the open-circuit voltage of pn-junction silicon solar

cells, the physical origin of this current is of interest. In an at-

tempt to develop a simple physical model that is consistent with

experimental data, a recent study [30] asserted that the inclusion of

Auger recombination in the heavily doped regions of the emitter is

alone sufficient to explain the data, and that it is unnecessary to

include the heavy-doping effect of bandgap narrowing [19]. This con-

clusion is questionable since Auger recombination was accounted for

qualitatively in the study [30], rather than by a rigorous quantitative

treatment.

To remove this uncertainty, we present in this chapter a

rigorous analytic evaluation of a model for the emitter region that

includes Auger recombination but excludes bandgap narrowing and

degeneracy of the charge carriers. We show that for silicon pn-

junction solar cells this model cannot explain the experimentally

observed values of VOC [3,31] and the observed dependence of VOC










on surface treatment [29]. It follows that for bipolar transistors,

the model cannot explain the measured values of the common-emitter

current gain. Inclusion of Fermi-Dirac statistics would worsen the
discrepancy between theory and experiment. Our conclusions are con-

trary to the assertions of [30] but are consistent with previous

numerical treatments of the problem [32-35]. Our analysis, therefore,

reinforces the importance of bandgap narrowing in silicon devices.


2.2 Derivation
We consider an N-type, heavily doped, inhomogeneous emitter
region having a Gaussian net impurity profile:



NDD() = N exp i1E (2.1)
L L J


where NS is the surface impurity concentration and WE is the width of

the quasi-neutral emitter region in thermal equilibrium. The Auger re-
combination lifetime (assuming full ionization of impurity atoms) is


TA(x) = 1 (2.2)
CA N~ (x)

where CA is the Auger coefficient [36]. The relationships (2.1) and

(2.2) are used in the hole continuity equation (for the dark condition),

SdJp (x) = AP(x (2.3)
q dx TA(x)


where AP is the excess hole density. If we let








y(x) = NDD(x) AP(x)


(2.4)


and use the general expression for the hole current density, including
both drift and diffusion components, then, for the usual low-injection
conditions, (2.3) becomes

-x
2 x___ exp[2f E2j -L
2 -E d 2L y =0 (2.5)
dx2 2L2 dx LA


where


2 Dp
CAN


(2.6)


is the square of the hole diffusion length
(2.6), Dp is an average value for the hole
emitter. The solution of the differential
to that given in [37]:


at the emitter surface. In
diffusion coefficient in the
equation (2.5) is similar


i5 14E x L F. E { x1 (2.7)
y = A sinh L erf-f + B cosh erf 2L (.7)


The coefficients A and B are determined by the boundary conditions [II],


(2.8)


=p) N NDDJ rexp(__.)


Jp(WE) = q Sp AP(WE)


(2.9)









In (2.9), Sp is the hole recombination velocity at the emitter sur-
face.
The minority-carrier current can be written (for low-level
injection) as

DD(x) x d2 (2.10)

Substitution of the expressions obtained for A and B into (2.10)
yields the following expression for the emitter saturation current
density:


q D coshA erf ] + (D /S LA) sinh erf
I L rLA L rPLA
JPO
P NSLA snh rf + (D/SLA) cosh iA erf



(2.11)

Equation (2.11) reduces to:

2
q Dp nl 2 T L Wr
JPO =NsL cothLA erf (2.12)

as Sp -> , and to the familiar expression:

2
q Dp ni (213
JPO NSLA coth (2.13)


as Sp -- and L --* = (L --- corresponds to a flat impurity profile
with NDD(x) = NS).










With the charge-control relation,


Jp(O) p (2.14)


we can evaluate Tp, the effective hole lifetime in the emitter. The
charge storage in the emitter is

WE
Qp = q AP(x) dx (2.15)

0

The contribution to Jp(O) due to surface recombination, JpS, and due
to Auger recombination, JPA, can be determined by integrating (2.3) over
the quasi-neutral emitter region:

E
Jp(0) = Jp(WE) + q aP() dx (2.16)

0

SPS + JPA

Using the mean-value theorem, we can write

WE
q AP() dx -= (2.17)
q A(x) TA
0

Thus, from (2.14), (2.16), and (2.17), we obtain


= + (2.18)
SP TA Tt










where


Qt P (2.19)
t JPS

defines the hole transit time, the average time required for a hole to

cross the quasi-neutral emitter region and recombine at the surface.

In the limiting case of negligible bulk recombination (TA -- )' p

is equal to Tt.

To summarize the analysis, we have obtained a closed-form

solution of the continuity equation, and have derived an expression

for the hole current density when Auger recombination is the dominant

bulk recombination mechanism in a heavily doped N-type emitter. Bandgap

narrowing and Fermi-Dirac statistics have not been included. The

surface and Auger components of the emitter recombination current and

the associated hole transport parameters in the emitter can be easily

obtained form (2.14), (2.15), (2.16), (2.17), and (2.19).


2.3 Discussion

The preceding development is then an analytic treat-

ment of the minority-carrier transport in the emitter of pn-junction

devices for the case in which Auger recombination is included but

bandgap narrowing is neglected. In this treatment a Gaussian impurity

profile is assumed, and a finite surface recombination velocity Sp

is included. Inclusion of a finite Sp is particularly important for

solar cells [29].

To illustrate quantitative results of this model, we apply

it to an N+P silicon solar cell with a base doping density










NAA = 5 x 1017cm-3 We let NS = 1020cm-3 and WE = 0.25 ur. The

electron diffusion length Ln in the base is assumed to be = 80 pm,
consistent with experimental data [1]. In Figure 2-1, we plot JPO

and its two components, JpSO and JPAO, as functions of Sp. For low

values of Sp (about 103 cm/sec), JPSO << JPAO For the high values

of Sp (above 106 cm/sec) JPSO = 2JPA0. As Figure 2-1 shows, the two

components of JPO are equal when Sp = 10 cm/sec. When the emitter
current is due mainly to Auger recombination (Sp = 103 cm/sec or lower),

for an assumed short-circuit-current density of 23 mA/cm2 [1] we

get VOC = 680 mV. This voltage disagrees markedly with the cor-
responding maximum observed VOC of 610 mV. The disagreement

corresponds to a discrepancy in JPO of about a factor of 15.

We consider now the variations of JpO as a function of WE,
with NS fixed at 1020 c3. In Figure 2-2, we plot JPO versus WE

for Sp = 102 cm/sec andSp = 108cm/sec. The figure shows the dominance
of bulk Auger recombination for large WE, independent of the value

of Sp, and the dominance of surface recombination for small WE and

large Sp. For low values of Sp(Sp = 102 cm/sec in Figure 2-2) the

emitter current is due mainly to Auger recombination (JPO JPAO) and
it decreases slightly for small WE. The largest value of the Auger

recombination current is about 3 x 104A/cm2, corresponding to

VOC = 680 mV. Again, this voltage disagrees appreciably with the
610 mV value that is observed experimentally.

The dominance of the surface recombination is emphasized
in Figure 2-3, where we let Sp = 108 cm/sec and plot the resulting









10-13
Jo


o-14 -PAO
C\J
PSO


Fn- -15
10





0-16
103 J 4 15 o1 17 Jo8

Sp (cm/sec)
Figure 2.1 The emitter saturation current JPO and its surface-recombination component JpSO and Auger-
recombination component JPAO versus the surface recombination velocity Sp






1-13

Sp = 10 cm/sec


3Po



C\J PAO



Qo 7 so








-15 ,1 11.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

WE (Pm)
Figure 2.2 JPO versus WE for S = 102 cm/sec and Sp = 108 cm/sec; JPS is the surface-recombination
component of Jpo and JPAQ is the Auger-recolmbination component











10-8-
I0




T,

Sp =10 cm/sec



7A





10
10
H









I 0 I I

0.0 0.2 0.4 0.6 0.8 1.0

WE (Pm)

Figure 2.3 The effectivehole lifetime Tp and its two components, the
transit tim -rt and the average Auger lifetime TA, versus WE
for Sp = 10 cm/sec










effective hole lifetime and its components as functions of WE. The

surface concentration NS is 1020cm 3 Note that the transit time

essentially determines the emitter recombination current for WE <0.3 im.

This corresponds to a transparent emitter [24] in which the minority

carriers recombine mainly at the emitter surface. The transparent-

emitter model is consistent with experimental observations in con-

ventional pn-junction silicon solar cells [29] and, because of

high recombination velocity at an ohmic contact, the model can be

expected to apply also to shallow-emitter silicon transistors. For

larger WE, the average Auger lifetime is smaller than the transit time

and Auger recombination becomes important.

Finally, Figure 2-4 shows the dependence of VOC on the surface

concentration NS for different values of Sp with WE = 0.25 Um. When NS

is relatively small (about 1019 cm-3), surface recombination is the

dominant recombination mechanism in the emitter, and VOC is very

sensitive to Sp. As NS increases toward 1021 cm-3, VOC is eventually

limited by Auger recombination. The value of VOC that results when

the emitter current is dominated by Auger recombination (NS larger

than 2 x 1020 cm-3) is about 680 mV. Once again, this result cor-

responds to a discrepancy with experiment of about a factor of 15 in

emitter recombination current.

2.4 Conclusion

The analytic results we have presented demonstrate that a

model for the emitter of silicon pn-junction devices based on Auger













700

Sp= 102 cm/sec

4
105
680 ---





660 -



10
0 640





620 -





600
109 1020 !021

Ns (cm-3)
Figure 2.4 VOC versus the surface concentration NS for different
values of Sp with WE = 0.25 pm





22




recombination without bandgap narrowing cannot explain experimental

values of VOC in pn-junction solar cells. It follows that this model

also cannot explain the low values of the common-emitter current gain

measured in bipolar transistors. In this demonstration, we have used

Maxwell-Boltzmann statistics. Use of Fermi-Dirac statistics would

worsen the discrepancies between the model and experimental measure-

ments [1]. Contrary to the conclusions of [30], therefore, we must

conclude that physical mechanisms, e.g., bandgap narrowing, in addition

to Auger recombination are responsible for the large emitter recombi-

nation currents observed in pn-junction devices.














CHAPTER III

RECOMBINATION THROUGH DEFECTS IN HEAVILY DOPED
SHALLOW-EMITTER SILICON PN-JUNCTION DEVICES


3.1 Introduction

Having established in Chapter II that Auger recombination alone

cannot explain the observed experimental values of VOC in silicon solar

cells, we must consider other heavy-doping mechanisms that will bring

theory and experiment into agreement.

In this chapter, we consider recombination mechanisms, other

than the Auger process, that can possibly occur in heavily doped

shallow-emitter regions. These mechanisms are recombination via defects

and surface recombination. We compare the effects of these two

mechanisms with recent experimental observations showing the sensitivity

of the open-circuit voltage in silicon pn-junction solar cells to

surface treatment. These experiments are discussed briefly in Section

3.2.

In Section 3.3, we discuss Shockley-Read-Hall recombination via

defects. We argue that recombination via defects alone yields results

that are inconsistent with experimental observations of VOC in silicon

solar cells. If the Shockley-Read-Hall lifetime is very short (in

the order of picoseconds), it will yield a low value of VOC, as ob-

served experimentally. However, in that case, VOC will be insensitive










to surface treatment (i.e., surface recombination velocity) contrary

to the experimental observations discussed in Section 3.2. For larger

Shockley-Read-Hall lifetimes, which will yield a VOC that is sensitive

to variations in the surface recombination velocity, the calculated VOC

is larger than the experimentally observed values.

From Section 3.3 and Chapter II, we conclude in Section 3.4

that a short minority-carrier lifetime cannot yield an excessive

recombination current that agrees with experiment, and at the same time

be consistent with the recent experimental observations discussed in

Section 3.2. Therefore, other heavy-doping mechanisms that can provide

a large minority-carrier charge storage must be considered. This

conclusion is corroborated by recent experimental measurements of the

short-wavelength quantum efficiency of silicon solar cells [38].

Excessive charge storage in heavily doped semiconductor regions is

attributed to either an effective excess intrinsic carrier density or

an effective deficit majority-carrier density.


3.2 Importance of Surface Recombination

Fossum et al. [29] recently reported a considerable improvement

in the open-circuit voltage in a variety of silicon solar cells by

reducing the recombination velocity at the emitter surface. These

solar cells include diffused, implanted, back-surface-field (BSF),

and high-low-emitter (HLE) structures, of both N-on-P and P-on-N

types [29].

It was observed that the growth of a thin thermal Si02 layer

on the front (emitter) surface increased VOC by about 20 mV in some










cells [29]. Table II, taken from [29], shows the improvement attained

in VOC in the solar cells described above by growing thermal Si02

on the emitter surface.

Other experiments, reported in [29], indicate that the thermal

growth of Si02 reduced the surface recombination velocity rather than

improved the bulk minority-carrier lifetime in the emitter. The reduc-

tion of the surface recombination velocity by thermal Si02 was observed

for both N and P+-emitters. These experiments indicating the sensi-

tivity of VOC to the emitter surface condition imply that the emitter

region is at least partially transparent to minority carriers. That is,

a significant portion of the emitter recombination current occurs at

the surface (prior to the surface passivation).


3.3 Shockley-Read-Hall Recombination via Defects

Defects in doped silicon can result from the diffusion of

impurities and the subsequent heat treatment of pn-junction devices.

These defects are due primarily to vacancies and vacancy complexes

near the surface region of the device [39].

Lindmayer and Allison [40] suggested a dead-layer model for

the emitter of thin diffused silicon solar cells in which the thin

surface layer is strongly damaged by the diffusion of impurities. In

this surface layer, there are large defect densities that result in

very short minority-carrier lifetimes (in the picosecond range). The

short lifetimes yield a large (dark) emitter recombination current

that limits the open-circuit voltage of solar cells to values lower

than those predicted by classical pn-junction theory.










In a treatment of recombination via defects and other heavy-

doping effects, Lindholm et al. [9] studied the effect on the dark

emitter current of process-induced defects in the emitter of silicon

junction solar cells. They derived an impurity-concentration-dependent

defect density NTT [9]:


NTT(X) = K[NDD(x) + NAA]m (3.1)

for an N+-region. The integer m can vary from 1 to 4 depending on the

type of vacancy in the semiconductor. In (3.1), NDD(x) is the

impurity concentration in an Nf-emitter, NAA is the P-type base impurity

concentration, and K is a constant. For an assumed linear impurity

profile near the junction, that is, an impurity concentration that

depends linearly on distance, they found, for an emitter junction depth

of 0.25 im, that the excess minority carriers are packed in a very thin

layer (about 200 A in width) next to the space-charge region. This

situation, illustrated in Figure 3.1, means that most of the minority

carriers could not reach the surface.

The above conclusion is similar to earlier bipolar-transistor

studies [41,42] of the effect of a built-in electric field in a dif-

fused quasi-neutral emitter region, having a constant minority-

carrier lifetime. In [9], the position-dependent minority-carrier

lifetime, due to the position dependence of the defect density,

enhanced the confinement of the excess injected minority carriers

to a region near the junction.

We consider now the position-dependent defect density model

of Lindholm et al. [9] in which the defect density is proportional


















BUILT-IN
ELECTRIC FIELD


P(WE)


S-- IP (WE)
I


I


E X

X O 0.1 WE
(SURFACE) (SCR EDGE)

Figure 3.1 Sketch of minority-carrier concentration in a heavily doped
emitter having a graded impurity profile and a large
defect density, after [9]










to an integer power of the doping density as in (3.1). We assess

the importance of defect recombination by determining if recombination

via defects is alone sufficient to explain the experimentally low

values of VOC and its observed sensitivity to surface treatment. We

consider three cases for the integer m in (3.1): m = 1, m = 2, and

m > 2.

If m = 1, the Shockley-Read-Hall lifetime for low-level

injection is


TSRH CSRH N(x) (3.2)


where CSRH is a constant. Fossum [43] considered this case in a de-

tailed numerical study of silicon solar cells. He showed that in order

to predict a value of VOC comparable to experimentally observed values

in silicon solar cells, very short minority-carrier lifetimes in the

emitter region (~ 1 nsec) are needed. Such short lifetimes yielded

a VOC that was insensitive to variations in the surface recombination

velocity at the emitter surface [43], contrary to experimental obser-

vations [29]. Therefore, although short Shockley-Read-Hall lifetimes

that vary inversely with N(x) can yield VOC in the range of experimental

values, they cannot account for the sensitivity of VOC to the surface

treatment.

For m = 2, the Shockley-Read-Hall lifetime has the same

dependence on the doping density N(x) as for Auger recombination,

except that the coefficient CSRH is different than CA:










1
TSRH: C (3.3)
SRH =CSRH [N(x)]2 (3.3)

In this case, for Shockley-Read-Hall recombination to dominate over

Auger recombination, CSRH must be larger than the Auger coefficient CA.

This yields TSRH < A

We used the analytic model of Chapter II to calculate VOC,

the transit time, and the average bulk lifetime for this case (with

m = 2). We found that for CA < CSRH 5 4CA, the transit time is smaller

than the average bulk lifetime for high values of surface recombination

velocity S (about 108 cm/sec), and consequently V0C is sensitive to

variations in S. However, the magnitude of V0C we obtained (> 670 mV)

was considerably higher than the experimentally observed value of

about 600 mV. For CSRH > 4CA, our calculations yielded a bulk lifetime
in the order of a fraction of a nanosecond, which is much smaller than

the transit time. Hence, although the short lifetimes yield lower

values of V0C (< 670 mV), they result in a V0C that is insensitive to

variations in S.

For basically the same reason discussed above, we infer that

for m > 2 in (3.1), the emitter is expected to be insensitive to surface

recombination. As the lifetime varies faster than [N(x)]-2, the injected
minority-carrier concentration falls sharply with increasing distance
away from the junction. Thus, most of the injected minority carriers

cannot reach the emitter surface before they recombine in the bulk


via defect centers.










3.4 The Excess Minority-Carrier
Charge Storage in Heavily Doped Regions

We discussed Shockley-Read-Hall recombination via defects

in the last section and showed that for Shockley-Read-Hall lifetimes

that have an inverse-power functional dependence on the doping density,

recombination via defects cannot alone be the dominant mechanism

limiting VOC in silicon solar cells. This is because such a recom-

bination mechanism cannot yield results that are simultaneously in

agreement with the low values of VOC observed in silicon solar cells

and with the sensitivity of VOC to surface treatment. Furthermore,

recent experimental investigation of the short wavelength spectral

response of silicon solar cells indicates that heavy-doping effects,

other than short lifetimes, are required to obtain agreement between

experimental and theoretical results [38].

Combining the above conclusion with the conclusion of Chapter

II, we deduce that short minority-carrier lifetimes cannot be the only

cause of the large emitter current in silicon solar cells. A similar

conclusion is anticipated for bipolar transistors. From Equation (1.1),

we infer that the minority-carrier charge storage in heavily doped

regions must then be large to account for the large current.

We now show qualitatively that a large minority-carrier charge

storage is due either to an excess intrinsic carrier density or to a

deficit majority-carrier concentration.

The minority-carrier charge storage Qp, in an N-type region

for example, is









WE

Qp = q AP(x) dx (3.4)

0


where AP(x) is the excess minority-carrier (hole) concentration. For

non-equilibrium conditions in a non-degenerate semiconductor,


2 FN EF
PN = n expFN EF
L kT

where EFp and EFN are the hole and electron quasi-Fermi levels.

From (3.5), the condition of low-injection,


(3.5)


(3.6)


N = NO

and the fact that


P= AP


(3.7)


we get


ni
AP
"0


If we assume full ionization of impurities,


N = NDD

and (3.8) becomes

2
n.
AP N--
NDD


(3.8)


(3.9)


(3.10)










If, because of some physical mechanism, e.g., energy-bandgap narrowing,

n? is larger than its intrinsic value (in pure silicon), or N0 is

smaller than the doping concentration because of, e.g., incomplete

ionization of the impurity atoms, then AP, and consequently Qp in (3.4),

will be larger than anticipated, and thus, by (1.1), the emitter current

will be excessively large.
2
The two effects mentioned above, ni larger than its intrinsic

value and N0 < NDD, can be phenomenologically incorporated into an

effective excess intrinsic carrier density n? such that (3.10) can be

written as


2
IP nie (3.11)
NDD

for a non-degenerate N-type semiconductor. For a degenerate semiconductor,

we show later that n2 can include the effects of Fermi-Dirac statistics,
ie
and hence (3.11) is still applicable. Note that, in this way, heavy-

doping effects can be incorporated in a single parameter, n .

In the next chapter, we will consider deionization of impurities,

as discussed by the recent papers of Heasell [25,44] and Popovic [45],
2 2
as a possible mechanism that results in ni > n-.
ie V~









TABLE II

MEASURED VOC OF A VARIETY OF SILICON SOLAR
CELLS WITH AND WITHOUT A FRONT-SURFACE
THERMAL Si02 LAYER (T = 250C)


Cell Cell
Designation Description (VOC) without SiO2 (VOC) with SiO2

A N+-P (diffused) 596 mV 605 mV

B N+-N-P HLE 599 mV 619 mV

C N+-P (implanted) 625 mVa 6441lVa
D P+-N-N BSF 594 mV 626 mV

aMeasurements for cell C were made at AMO whereas those for the other cells were made at AMI.

Source: [29]














CHAPTER IV

ON THE DEIONIZATION OF IMPURITIES AS AN EXPLANATION FOR
EXCESS INTRINSIC CARRIER DENSITY IN HEAVILY DOPED SILICON


Experimental measurements of the collector currents in silicon

bipolar transistors [20] have indicated that n the square of the

effective intrinsic carrier concentration, increases to values well
2
above n., the classical intrinsic carrier concentration, as the doping

level in the quasi-neutral base region increases. This was interpreted

by Slotboom and DeGraaff to be due to energy-bandgap narrowing in the

quasi-neutral base region, which was heavily doped in the devices used

for their study [20].

Recently, Popovic [45] presented an alternate interpretation

for the increase of n with the doping level. He argued that, in

the range of impurity concentrations of 1017 cm-3 to 109 cm-3 in
2
silicon, the increase of nie can be attributed to a Fermi energy

difference AE corresponding to different density-of-states models,

whose physical significance will be discussed later. In recent papers [25,

44], Heasell gave another interpretation based on the assumption of

a constant impurity activation energy at high doping levels. The

models of Popovic and Heasell resemble each other in that they both

emphasize the importance of deionization of impurity levels, which

results in the majority carrier density being substantially lower than

the net doping density.










We discuss here in detail the models of Popovic and Heasell,

and show that their underlying assumptions are inconsistent with a

variety of experimental observations in heavily doped silicon. Hence

we conclude that these interpretations do not dislodge energy-bandgap

narrowing as a possible mechanism underlying the increase in ne in

heavily doped silicon.

As the basis for his model, Heasell assumed (a) that the

bandgap of silicon is not affected by heavy doping, and (b) that the

impurity energy-levels in the bandgap due to the presence of shallow-

level impurity atoms are independent of the impurity concentration.

For heavily doped silicon, these assumptions lead to a small ionization

ratio, which we define as the ratio of the concentration of ionized

impurity atoms to the total concentration of impurity atoms. To show

this, consider a donor energy level of ED. Then the majority-carrier

concentration N (electrons for an N+ region), which equals the ionized

impurity concentration, is related to the total donor concentration,

NDD, for uncompensated N+ silicon by f46]

N = [N0DNC/gD] /exp[-(EC ED)/2kT] (4.1)

for NDD > 1018 cm-3. Here gD is the impurity energy-level degeneracy

factor, NC is the conduction-band effective density of states, and

EC is the conduction-band edge. Thus, in the Heasell model, where ED

does not depend on the impurity concentration, the majority-carrier

concentration varies in proportion as the square root of the impurity

concentration. If NDD increases by two orders of magnitude, N increases










by only one order of magnitude. Hence, the NP (electron-hole) product

(given, at equilibrium for low doping concentrations, by the law of

mass action) implies a larger minority-carrier density than is

predicted classically. This has the same effect in device analysis
2 2
as would nie being larger than ni-

The above assumption of a fixed discrete impurity energy level

disagrees with the experimental observations of Pearson and Bardeen

[47] and Penin et al. [48]. They found that the ionization energy

of the impurity atoms vanishes as the impurity concentration approaches

about 3 x 1018 cm3 fcr both n-type and p-type silicon. Therefore,

Heasell's assumption of a fixed impurity energy level in the bandgap

of heavily doped silicon conflicts with long established experimental

findings.

To modify Heasell's approach to bring it into harmony with

the results of these experiments, we have calculated the ionization

ratio for heavily doped silicon using an impurity-concentration

dependence of the ionization energy based on the experimental results

of Pearson and Bardeen [47]. We found, in agreement with [49], that

the lowest ionization ratio at room temperature is about 90% for

phosphorus-doped silicon, and about 70% for boron-doped silicon.

This difference in the degree of ionization arises from the difference

in the effective density of states in the conduction and valance bands

and in the degeneracy factors applying to the donor and acceptor

impurity energy levels. This deionization of impurities, based on the

experimental results of Pearson and Bardeen [47], is not large enough










2
to account for the experimentally observed values of n. quoted by
ie
Heasell [25]. Thus Heasell's assertion that deionization is the

first-order mechanism underlying the large values of ne observed in

heavily doped silicon is not justified.

We discuss now the treatment of Popovic [45]. Popovic

related the increase of ne observed in the base region of a bipolar

transistor as the impurity concentration increases above 1017 cm-3

[20] to a difference in Fermi energy AE. The energy difference AE

originates from calculations based on two different models for the

density of states in heavily doped silicon.

The first (classical) model assumes parabolic conduction and

valance bands; i.e., the density of states is assumed proportional to

the square root of energy, and full impurity ionization is assumed.

This model yields a Fermi energy EF, calculated from the quasi-

neutrality condition, that increases monotonically as the doping level

increases. Above an impurity concentration of about 1019 cm3, EF

enters the conduction band (for n-type material) or the valance band

(for p-type material).

The second model assumes that (a) the conduction-band and

valance-band density of states have exponential tails that extend

into the forbidden bandgap, (b) the impurity density of states has a

Gaussian form, and (c) the total density of states is the envelope

of the density of states in (a) and (b) [16]. When used with the

quasi-neutrality condition in the base region, this model predicts

a different Fermi energy EF than that (EF) calculated from the classical










model. The Fermi energy EF saturates for high impurity concentrations

at the low-impurity-concentration value of the impurity energy level

in the bandgap [16].

Popovic claims that the energy difference,


AE = EF EF (4.2)


is an alternate interpretation to energy-bandgap narrowing in silicon

in the doping range of 1017 to 1019 cm-3, an interpretation that can

account for the increase in n2 in the same way as bandgap narrowing,

i.e.,


n2~ = n? exp(AE/kT) (4.3)
le = 1


for Maxwell-Boltzmann statistics. The energy difference AE increases

as the impurity concentration increases because, in the second model

[16] for the density of states, EF saturates at the impurity energy

level, while, according to the classical (first) model, EF increases

monotonically as the impurity concentration increases.
2
To estimate quantitatively the effect of AE on nie, we con-

sider a concentration of 1019 cm-3 phosphorus atoms in silicon at room

temperature. Then, AE = 0.045 eV since EF is very close to the con-

duction band edge, while EF is nearly at the impurity energy level

of phosphorus in silicon--0.045 eV below the conduction band edge.

Using this value of SE, we obtain


ne 6n?
le I


(4.4)










Since 0.045 eV is the maximum value of AE in the range of doping con-
17 -: 19 -3 ?
sidered (from 101 cm-3 to 10 cm3), tne maximum value of n2 due to
ie
the difference in Fermi energy is less than an order of magnitude

larger than n?. If we use this value of n2 for the range of doping
1 ie'
specified above, add the effect of the bandgap narrowing model of

Vol'fson and Subashiev [50] (for NDD ? 2 x 1019 cm-3), as suggested by

Popovic, and include Fermi-Dirac statistics, we get values of the

open-circuit voltage in silicon solar cells and the common-emitter

gain in silicon bipolar transistors that are much higher than experi-

mentally observed values. Thus the increase of n? due to the satura-
le
tion of EF, assuming, in this part of the discussion, that it saturates,

is too small to account for these experimental findings in silicon

devices.

We now turn to the basic assumptions of Popovic's treatment:

the saturation of the Fermi level EF and the resulting tacit assump-

tion of impurity deionization in heavily doped silicon.

From a theoretical viewpoint, saturation of the Fermi level

in the bandgap is due primarily to impurity band widening, that is, to

the spread of impurity energy levels from discrete to quasi-continuous

levels in the bandgap at high impurity concentrations (in the range

of 1017 to 1019 m-3). Impurity band widening introduces a large

number of energy levels in the bandgap that can be populated by the

majority carriers. This lessens the density of carriers in the

majority allowed band, and makes the Fermi level tend to saturate.

Deionization of impurities is tacitly assumed in Popovic's

treatment because the charge carriers populating the impurity band










are regarded to be localized spatially. This is so because, if the

carriers occupying the impurity band were not localized, i.e., if their

wavefunctions were to extend all over the crystal, there would be,

in effect, energy-bandgap narrowing since allowed band-like states

would exist at lower energies (for N-type material) than those present

in pure silicon. However, since Popovic asserts in his model that

the bandgap does not change because of the presence of impurities

in the doping range of 101 to 101 cm3, Popovic's model must assume

that the impurity band states are localized spatially. Such localiza-

tion results in a majority carrier concentration N that is considerably

less than the impurity concentration NDD. Thus, popovic's model leads

to the deionization of impurity atoms, which was a central physical

mechanism also in the Heasell model. In the Popovic model, impurity

deionization and Fermi-level saturation go hand-in-hand.

But we now note that Fermi level saturation is not the trend

observed experimentally in heavily doped silicon. A variety of ex-

perimental observations in heavily doped, single-crystal silicon

indicates that the localization of charge carriers is greatly reduced

by the increase in impurity concentration and the electrons (or holes)

show, experimentally, physical properties that are in good agreement

with the degenerate free-electron gas model in metals. These experi-

mental observations in heavily doped silicon involve measurements of:

resistivity and Hall effect [47-48, 51-55], electron spin resonance

(ESR) [53], magneto-resistance [54], electronic specific heat [56],

and nuclear magnetic resonance (NMR) [57-58]. A thorough discussion










of these mechanisms in group IV semi-conductors is given in [58]. We

consider briefly here the implication of measurements of the electronic

specific heat [56] and nuclear magnetic resonance [57-58] as they

relate to the position of the Fermi level in heavily doped silicon.

Measurements of the electronic specific heat in phosphorus-

doped silicon (NDD > 6 x 1018 cm ) show that the electrons behave as

a degenerate electron gas [56]. The electronic specific heat of the

heavily doped silicon samples (a) has a linear dependence on tempera-

ture, (b) increases considerably as the impurity concentration

increases, and (c) has an NDD dependence on the impurity concentration

in agreement with the dependence predicted by the degenerate free-

electron gas model for metals [59]. The above results indicate that,

for NDD > 6 x 1018, a parabolic density of states in the conduction

band is probably a good approximation. Furthermore, although no

quantitative estimates of the degree of ionization can be concluded from

measurements of the electronic specific heat, these measurements do in-

dicate that almost all donor electrons occupy delocalized states;

otherwise the specific heat would not exhibit the metallic character

described above.

Experimental determination of the position of the Fermi level

and of the delocalization of donor electrons (that is, of the ioniza-

tion of impurity atoms) is provided by nuclear magnetic resonance (NI1R)

experiments in phosphorus-doped silicon [57-58]. These experiments

indicate, via the spin interactions of electrons and impurity nuclei,

that, at an impurity concentration of about 3 x 1018 cm3, electrons in










heavily doped silicon are delocalized, in agreement with the results

of Pearson and Bardeen [47]. Furthermore, at a phosphorus concentration

of about 2 x 1019 cm-3, the Fermi level in heavily doped silicon is at

the conduction band edge of silicon [58], that is, the Fermi level

enters the conduction band. Thus NMR experiments indicate that the

Fermi level does not saturate in the bandgap of silicon [58]. They

indicate also that at impurity concentrations above 2 x 1019 cm3

the impurity band merges almost completely with the conduction band

[58], resulting in a composite density of states having, to a good

approximation, a parabolic dependence on energy (E /2) [57].

Therefore, we conclude that the basic assumptions of Popovic's

treatment are not justified since experiments predict that the Fermi

level does not saturate, and impurity deionizationis not a first-order

mechanism in heavily doped silicon.

Our approach has been to dispute the conclusions of Heasell

and Popovic because their initial premises conflict with experimental

observations. In the case of Heasell's model, we also showed that a

version of it, modified to be consistent with experiment, still failed
2
to explain the observed variation of nie with doping level. In this

chapter we have not discussed theoretical grounds for the model

advanced earlier to explain the variation of ne that is, the model

of energy-bandgap narrowing. Our position is that bandgap narrowing

is a possible valid mechanism influencing n2 --and thus the common-

emitter current gain of silicon transistors and the open-circuit

voltage of silicon solar cells. But we believe that the stage now










reached in theoretical and experimental work on bandgap narrowing does

not allow a consensus of agreement to be reached that it is the dominant

mechanism underlying the observed variations in ne.

In accordance with the above comment, the expression "bandgap

narrowing" will mean, in the following chapters, aphenomenological

narrowing of the bandgap characterized by an excess intrinsic carrier

concentration ne > n That is, our use of the term "bandgap

narrowing" does not necessarily refer to the true shrinkage in the

bandgap of the semiconductor. As shown in Chapter II and later in

this dissertation, use of a phenomenclogical bandgap narrowing in the

analysis of heavily doped silicon devices is important to bring theory

and experiment into agreement.

In the next chapter, we incorporate bandgap narrowing into

the analysis of heavily doped emitters of silicon pn-junction devices.














CHAPTER V

HEAVILY DOPED TRANSPARENT-EMITTER REGIONS
IN SILICON JUNCTION SOLAR CELLS, DIODES, AND TRANSISTORS


5.1 Introduction

Excess minority carriers injected into the emitter of pn-

junction devices recombine in the bulk and at the surface of the

emitter. If the emitter junction is shallow enough, the minority

carriers can cross the quasi-neutral emitter region without appreciable

bulk ecombination. The minority carriers then recombine at the

emitter surface. For this case the emitter is transparent to the

injected minority carriers, and an important parameter then is the

surface recombination velocity S at the emitter surface.

This parameter is particularly important for pn-junction

silicon solar cells in which most of the illuminated surface is not

covered by metal. In devices in which thermal Si02 covers this

nonmetallized portion of the surface, experiments show that S can be

less than 104 cm/sec for both P-cells (P-type substrate) [29] and

N-cells [29]. This value of S is orders of magnitude less than that

of an ohmic contact and is consistent with values determined earlier

by different experimental methods [60]. Furthermore, recent experi-

ments involving P-on-N and N-on-P cells without thermal SiO2 demon-

strate that the emitter can be completely transparent [29].










The purpose of this chapter is to provide an analytical

treatment of transparent emitter devices, particularly solar cells,

that is more complete than treatments previously available

In this treatment, we include the effects of (a) bandgap narrowing

[19,61], (b) Fermi-Dirac statistics, (c) built-in field due to the

impurity profile, and (d) a finite surface recombination velocity S.

Detailed numerical studies including these various effects have been

done [32,62-63], but they have not treated the case of the transparent

emitter.

A major result of this chapter is the demonstration that the

transparent-emitter model can predict experimental values of VOC

observed on N+P thin diffused junction silicon solar cells made on

low-resistivity (0.1 0-cm) substrates. Thus, the transparent-emitter

model is shown to provide an explanation for the discrepancy between

the prediction of simple classical theory (VoC = 700 mV) and the

measured maximum value (VoC = 600 mV). The transparent-emitter model

gives VOC 0 600 mV for high values of Sp (Sp > 10 cm/sec) provided

the effects of bandgap narrowing (modified by Fermi-Dirac statistics)

are included. This result suggests that VOC can be increased toward

the classical value of 700 mV if Sp is decreased and the effects of

bandgap narrowing are reduced. This is accomplished in the HLE solar

cells, early versions of which have shown increases in VOC to the

640-650 mV range [64-65].

In addition to the development of the theory for the

transparent-emitter device, and its application to solar cells, this










chapter will include a test for the self-consistent validity of the

transparent-emitter model. This test compares the calculated transit

time of minority carriers across the emitter with the Auger-impact

minority-carrier lifetime within the emitter region.


5.2 Derivation

We consider an N-type heavily doped quasi-neutral emitter

region; analogous results apply to P-type emitters. The minority

carrier current density in the N-type region is


Jp(x) = qUpAP(x) E(x) qD dAP(x (5.1)
P dx

in which E(x) is the thermal equilibrium value of the effective field

which, for low-level injection, is given by

Dp 1 dPO(X)
E(x) = dx (5.2)
5P PJ(X) dx

We now define an effective intrinsic density nie such that


n e(x) = Po(X) No(x) (5.3)

in which PO(x) and NO(x) are the hole and electron concentrations in

thermal equilibrium. The parameter ne depends on position for two

reasons:

1. The influence of Fermi-Dirac statistics, and

2. The influence of bandgap narrowing

These influences are discussed in Section 5.3. For Maxwell-
2
Boltzmann statistics and no bandgap narrowing, n. is the square of
ie









the intrinsic carrier concentration in silicon and is a function of
temperature only.
If the expressions is (5.2) and (5.3) are used in (5.1), we
get, after some manipulations,

Nn(x) N 1]"
Jp(x) (x) dx = -qDpd AP() -(54)
nie(x) n ne(x)

If we integrate (5.4) over the quasi-neutral emitter region, we get


,E NO(x) _NO(x)WE
Jp(x) --dNe x = -qDp AP(x) N(-- (5.5)
"ie(x) nie(x)

where Dp is some average value of Dp. If the emitter is transparent
(transit-time limtied), that is, if the minority carrier transit time
Tt is much less than the minority carrier lifetime Tp (for an N-type
emitter), then Jp is constant independent of position in the emitter.
Use of the minority carrier boundary conditions [11]

AP(O) = PO(O)(exp(qV/kT)- 1) (5.5)

at the edge of the emitter space charge region, and [11]

Jp(WE) = q Sp AP (WE) (5.7)

at the emitter surface, enable (5.5) to be expressed as

qDp (exp(qV/kT)- 1)
ip = (5.8)
fE NO(x) p NO(WE)
J dx +
J ne(x) Sp n ie(WE)









Equation (5.8) is the general experssion for the minority-carrier
current in a transparent emitter.
To check the condition, .t << Tp, required for transparancy,
we must determine the steady-state transit time Tt, which is defined
by the charge control relation,


(5.9)


Qp
Lt Jp


Here Qp is the excess minority carrier charge storage in the emitter:


W"E
Qp = q AP(x) dx

0


Using (5.4), (5.8), and (5.10) to express Qp, and
we find the following expression for the minority
time:


It

0


rWE
*


NO(x)
- --- dx +
nie(x)


combining with (5.9),
carrier transit


2
nie(x)
]{ )- dx
N0(x)


SN (x') n2 (x) d
Dp (x) n2 (x')
0 ie
0 0


(5.11)


(5.10)










Some special cases are of interest. For a flat impurity

concentration profile, the above expression reduces to



W + (5.12)
t 2Dp Sp

If, furthermore, Sp is infinite, (5.12) reduces to the familiar ex-

pression,


2
Tt (5.13)



5.3 Heavy-Doping Effects

In thermal equilibrium, heavy-doping concentrations of shallow

level impurities affect the minority-carrier concentration in a quasi-

neutral region by two mechanisms: bandgap narrowing and Fermi-Dirac

statistics. These two mechanisms affect the minority-carrier

concentration in opposite ways. For any given position of the Fermi

level relative to the band edges, bandgap narrowing tends to increase

the minority-carrier concentration, while inclusion of Fermi-Dirac

statistics tends to decrease the minority-carrier concentration below

the value calculated using Maxwell-Boltzmann statistics. The dominance

of either of the two effects, at any specific impurity concentration,

depends on the model of bandgap narrowing adopted.

In this treatment, we assume that bandgap narrowing occurs

without changing the parabolic dependence on energy of the density of










states in the conduction and valance bands. This is the rigid-band

approximation; it is discussed in Appendix F.

The effects of bandgap narrowing and Fermi-Dirac statistics

can be lumped into a position-dependent effective intrinsic carrier

concentration at thermal equilibrium given by


S, ( 2 ni exp('EG(x)/kT)
n2 n (5.14)
ie = 1+C(n) exp (n) (5


where


n = C = (EF EC)/kT (5.15)

for N-type material, and


n = nV = (EV EF)/kT (5.16)

for P-type material. The derivation of (5.14) appears in Appendix C.

In (5.15) and (5.16), EC and EV are the edges of the conduction and

valence bands, respectively, and EF is the Fermi level. The factor
20 -3
C(n) is a function of n, which, for n 5 4 (e.g., NO 5 2 x 1020 cm-3 in
N-type silicon), is [24]


C(n) W -0.04n + 0.3 (5.17)

The above approximation of C(n) gives values of the Fermi-Dirac

integral of order 1/2 with less than 4% error.

In nonequilibrium conditions, bandgap narrowing increases the
minority-carrier current by:










1. Increasing the minority-carrier concentration

2. Decreasing the retarding built-in electric field

acting on the minority carriers
The increase in the minority-carrier concentration P results
from the increase in n e(x). The decrease of the built-in electric
field is due to the position dependence of n2 (x) (and hence of the
le
effective bandgap) in the inhomogeneously doped emitter.
To develop a simple expression illustrating the reduction of
the electric field, we now include only bandgap narrowing excluding the
effect of Fermi-Dirac statistics for the present. Then the effective
electric field acting on the minority carriers, given in (5.2), can be

expressed by using (5.3) and (5.14) as:

DP 1 ( (5.18)
E(x) = R(NO) p N (x dx(518)



where

NOi(x) dAEG (xx
R(NO) kT dNo(x) (5.19)



The factor R(NO) measures the reduction of the built-in electric field
due to bandgap narrowing. For any model of bandgap narrowing, R(N,)
is always less than one. Figure 5.1 shows R(NO) as a function of the
electron (majority carrier) concentration for three models of bandgap
narrowing: Slotboom and DeGraaff [20], Hauser [12], and Lanyon and
Tuft [13].


































Figure 5.1 The bandgap-narrovwing reduction factor R(N) versus the electron (majority-carrier) concentration
N for: (A) Lanyon-Tuft model, (B) Hauser model, and (C) Slotboom-DeGraaff model






1.0





0.5





0.0





-0.5 -





-1.0
1017 1018 1019

N (cm-3)










In the absence 3f bandgcp oarrowir.;, the holes experience

a retarding electric field in an n -type diffused emitter. Eqations

(5.18) and (5.19) indicate that the position dependence of the bandgap

narrowing, in effect, decreases the retarding electric field. The

more it is decreased the smaller is the transit time for a specific

surface recombination velocity. In (5.11) the transit time is shown

to be a function of n. In Figure 5.2 the transit time is plotted
in"
as a function of WE, the width of the quasi-neutral emitter region,

in two cases: neglecting bandgap narrowing, and including bandgap

narrowing (Slotboom and De Graaff model). Note that inclusion of

bandgap narrowing makes the transit time close tc the value it has

if the impurity profile is flat. In general, bandgap narrowing

decreases the transit time if the impurity profile is not flat.

Conversely, inclusion of Fermi-Dirac statistics increases the

transit time as can be seen in Figure 5.2 Inclusion of Farmi-Oirac

statistics shifts the value of the transit time closer to that calcu-

lated when heavy-doping effects are neglected.

5.4 Discussicn

From (5.8), the minGrity-carrier saturation current for a

transparent emitter is

J- PO (5.20

E ni(x) dx + OPiOWE)
J ie ^ E) Spnie`1,E,






























10
/ /S =5 xO1 cm/sec






10-l 1 -- -I-I-I
0.2 0.4 0.6 0.8

WE (0m)
Figure 5.2 The transit time -t versus the width of the emitter region
WE for S = 5 x 105 cm/sec and a Gaussian profile wi:h: no
heavy doping (NHD), bandgap narrowing (Slotboom-DeGraaff
model) and Fermi Dirac statistics (BGN + FD), bandgap
narrowing (Slotbocm-DeGraaff model) only (BGN), and for a
flat profile (NHD + flat)









If


Sp >> (5.21)
Sn2WE(WE)
ie E NO(x)
2 dx
nie(x)
0

then (5.20) reduces to


J qD (5.22)


n0 (x)


which is the exact expression for an infinite surface recombination
velocity. For


Sp < N(E) (5.23)
p n2i (WE) rWE
J 0 xdx
n (x)


(5.20) reduces to

2
qSpnie(lJE)
PO No(WE) (5.24)



It is desirable to make JPO small for the bipolar transistor.
This results in a large emitter efficiency. For the pn-junction solar
cell, if the emitter recombination current JPO is small compared with










the base recombination current, the value of VOC can approach the

classical theoretical limit.

To illustrate the dependence of JPO on Sp, consider the
desirable case in the transparent-emitter model in which Sp is small

enough to satisfy (5.23). Figures 5.3 and 5.4 show the variation of

the emitter saturation current density, JPO, and the transit time, Tt,

as a function of Sp for three models of bandgap narrowing: Slotboom

and De Graaff [20], Hauser [12], and the recent model of Lanyon and

Tuft [13], which has the form

AEG = 22.5 x 10-3 (N/1018)1/2 eV

for non-degenerately doped silicon, and (5.25)


AEG = 162. x 10-3 (N/1020)1/6 eV

for degenerately doped silicon. A Gaussian impurity profile is assumed

with a surface impurtiy concentration of 1020 cm-3 and a junction depth

of .25 1m. Full ionization of the impurity atoms is also assumed.

For values of Sp below 106 cm/sec, JP0 and Tt vary rapidly with

variations in Sp, while for values of Sp above 106 cm/sec, both JPG

and Tt saturate. JPO saturates to its largest value, and Tt saturates

to its lowest value. The largest value of JPO at any Sp occurs for

the Lanyon-Tuft model of bandgap narrowing.

The validity of the transparent-emitter model is based on the

condition that the minority-carrier transit time is much smaller than

the minority-carrier lifetime: Tt << T To test this condition, T








10-II







10-1



oio




104 105 106 107 108


S, (cm/sec)
Figure 5.3 The emitter saturation current density JpO as a function of the surface recombination velocity
Sp, for E = 0.25pm, Fermi-Dirac statistics and bandgap narrowing included: (A) lanyon-Tuft,
(B) Hauser, and (C) Slotboom-DeGraaff. For low values of Sp (less than about 10' cm/sec) the
self-consistency test yields Tt > Tp so the emitter current is then due to Auger recombination
and may be larger than values reported above.








10-8


SA

4-


10-


10-10 -..... 7 ............ ..-,-... .. .---,
104 105 10 107 108

Sp (cm/sec)
Figure 5.4 Tt as a function of surface recombination velocity Sp for VE = 0.25 m, Fermi-Dirac statistics and
bandgap narrowing are included: (A) Lanyon-Tuft, (B) Hauser, and (C) Slotboom-DeGraaff










is calculated from (5.11). Values of Tt are plotted in Figures 5.4

and 5.5. In Figure 5.4 -t is plotted as a function of Sp for the

three models of bandgap narrowing (assuming WE = .25 Iim), while in

Figure 5.5, values of Tt are plotted as a function of the quasi-neutral

emitter region width, WE, for Sp = 5 x 105 cm/sec. The recombination

lifetime Tp has an upper bound determined by the Auger band-to-band

recombination at high impurity concentration.

To illustrate the self-consistency test for transparency, we

assume the surface concentration of a diffused emitter to be 1020 cm-3.

With the impurity profile assumed Gaussian, this corresponds to an

average Auger lifetime of A = 2.4 x 10 sec. In Figures 5.4 and

5.5, we compare this lifetime with Tt for each of the three bandgap

narrowing models (assuming that WE = 0.25 pm). Note that the emitter

is completely transparent if Sp exceeds 105 cm/sec and is opaque if

Sp is below 104 cm/sec.

5.5 Application to pn-Junction Silicon Solar Cells

In this section, we apply the transparent-emitter model to

calculate the open-circuit voltage of silicon pn-junction solar cells

having low substrate resistivity.

The open-circuit voltage is given by


V k kIn JSC (5.26)
OC q JO


1The average Auger recombination lifetime was calculated from a
model similar to that of W. W. Sheng [37], in which the emitter is
divided into two regions; one region has Shockley-Read-Hall recombina-
tion, and the surface region is dominated by Auger recombination.












I1 I- - 1 -


S =5x O5 cm/sec


0.1 0.2


VIE (-Im)
Figure 5.5 T1 versus WE for S = 5 x 105 cm/sec.
statistics and bandgap narrowing are
(A) Lanyon-Tuft, (B) Hauser, and (C)


Fermi-Dirac
included:
Slotboom-DeGraaff


z
0
C3

e


10-10


10- 1


0.8


I
i t I I


I0-8


**










where JSC is the short-circuit current density and JO is the saturation

current of the solar cell in the dark. The saturation current density

JO of the diode is


JO = J + JNO (5.27)

where JPO is the emitter minority-carrier saturation current density

and JNO is the base minority-carrier saturation current density.

The base saturation current density is

9
qniDN (
NO NALN (5.2

Consider low-resistivity silicon solar cells with base doping concen-

tration of NAA = 5 x 1017 cm-3. Measurements made on such cells

indicated the minority carrier diffusion length, LN, to be 80 im [66],

corresponding to JNO = 6.2x10-14 A/cm2. The general expression for

the transparent-emitter current density, given in (5.8), has yielded

JPO as a function of Sp, as shown in Figure 5.3.
Combining these characterizations for JNO and Jp with JSC

23 mA/cm2 (AMO conditions) [1], we plot, in Figure 5.6, VOC versus Sp.

For low Sp (about 103 cm/sec), VOC is limited by the base current, and

for higher values of Sp, VOC is limited by the emitter current, as

has been observed experimentally [1]. Note that for Sp > 105 cm/sec,

VOC saturates to its lowest value. Note also that the Lanyon and

Tuft model of bandgap narrowing gives lower values of VOC (for any

given value of Sp) than those given by the Slotboom and De Graaff and

Hauser models.



































Figure 5.6 Vr versus Sp for W = 0.25 11m. Fermi-Dirac statistics and bandgap narrowing are included:
(A Lanyon-Tuft, (B Hauser, and (C) Slotboom-DeGraaf. In (D) heavy-doping effects are not
included. For low values of Sp (less than 104 cm/sec) VOC is limited by the Auger-
recombination current in the emitter because Tt > Tp and VOC may be lower than values sug-
gested in this figure.













E 640-

620

B
600

C
580 -

A
560 C
10 10 10 10

Sp (cm/sec)










So far we have assumed, for simplicity, that all of the emitter

surface is characterized by a single value of Sp. We now consider

a more realistic structure of silicon solar cells, Figure 5.7. The

emitter saturation current JPO is the sum of three components from

regions 1, 2, and 3 shown in Figure 5.7(b), [67]. The components of

the current denisty from the metal-covered surface, region 1, and

the nonmetal-covered surface, region 2, are given by (5.8). In region

3, the flow of minority carriers is two-dimensional since the minority

carriers within about a diffusion length from region 1 are much more

influenced by the high value of Sp of region 1 than they are by the

relatively low value of Sp of region 2. To avoid the complexity of

two-dimensional analysis, we make the first-order approximation that

the component of JpO from region 3 is essentially the same as that

from region 1 (JP3 = JPl) because Sp of the unmetallized surface can

be made orders of magnitude smaller than Sp of the ohmic contact.

The emitter saturation is then


IpO = (Al + A3) Jp01 + A2JP02 (5.29)

where Al is the metallized surface area and (A2 + A3) is the unmetal-

lized surface area and JP01 and JP02 are the corresponding currents.

The area A3 is approximately equal to

A3 = 2nLGLp (5.30)

where n is the number of metal grid lines (or fingers), LG is the
length of the grid lines (see Figure 5.7), and A2 = AT (A3 + Al),

AT being the total area of the cell.














(a)
Self Sp(metal)
Sp (non-merfi)
oxide
meta I -- 7 -
,(metal) Jp143 2

7----~


(b)
Figure 5.7 (a) The structure of a pn-junction solar cell
(b) The three components of the emitter current:
JPIi JP2' and JP3










As a numerical example, let A = 4 cm2, L = 2 cm, n = 6,

Lp 1 pm, and assume 10% metal coverage. Then A1 0.4 cm2

A2 3.59 cm2, and A3 = 0.0024 cm2. In this case, A3 is negligible,
and

kT JSC
VOC q n A- (5.31)
q Al A2
AT JP01 + A P02 + JNO
T T

This expression can be used to estimate Sp of the nonmetallized

surface from experimental values of VOC. For diffused, thin-junction

pn-junction solar cells made on low resistivity (= 0.1 n-cm) material,

the maximum observed open-circuit voltage is about 600 mV [1]. As

one example, if we consider the Lanyon-Tuft model of bandgap narrowing,

and let Sp of the ohmic contact be 106 cm/sec, let the doping concen-
tration be Gaussian with a surface concentration of 1020 cm-3. then by

using (5.31), with AI/AT = 0.1 and VOC = 600 mV, we get Sp (nonmetal) =

5 x 104 cm/sec. Thus the value for Sp could result in the low V0C

seen in conventional, diffused, thin-junction solar cells. Note that

Figures 5.4 and 5.5 indicate the self-consistent validity of the
transparency assumption for this device in the Lanyon-Tuft model

which permits use of (5.20).
Although the preceding discussion has focused on the

transparent-emitter model applied to N+P silicon solar cells, the
model can also be applied to P+N cells. It is straightforward to

show that heavy-doping effects (bandgap narrowing and Fermi-Dirac










statistics) degrade N+P cell performance more than that of P+N cells

because the effective mass of electrons in silicon is greater than

the effective mass of holes. The resulting different effective

densities of states in the conduction and valence bands (NC and NV)

cause the onset of degeneracy to occur at lower impurity concentrations

in P-type material than in N-type material [32], if both N-type and

P-type regions have the same bandgap narrowing. Thus the net effect

of bandgap narrowing and Fermi-Dirac statistics is to degrade the

N-type heavily doped region more than the P-type region with the same

impurity concentration. This may, in part, be responsible for the

high efficiency P+NN+ cells that have been observed experimentally
[68-69].


5.6 Perspective

This chapter has dealt with the transparent-emitter model of

a solar cell, which is defined by the condition that the minority

carriers in the dark quasi-neutral emitter recombine mainly at the

surface rather than in the bulk. Surface recombination can predominate

over bulk recombination if the emitter junction depth is shallow enough

and if the surface recombination velocity is high enough. In fact,

this occurs in typical pn-junction silicon solar cells, as demonstrated

by recent experiments showing the sensitivity of VOC to the surface

recombination velocity [29]. From a theoretical standpoint, the self-

consistency test in Section 5.4 can determine the validity, for a given

solar cell, of the transparent-emitter model, provided the emitter




69




recombination center density is low enough for the Auger process to

dominate over the Shockley-Read-Hall process.

Although the transparent-emitter model may describe many

conventional shallow pn-junction silicon solar cells, the high value

of the surface recombination velocity S necessary to validate the

transparent-emitter model is not necessarily desirable from a design

point of view. Growth of a thermal Si02 layer on the emitter surface

can substantially decrease S and increase VOC. For such devices, the

dark emitter recombination current is determined mainly by bulk

recombination.














CHAPTER VI

A FIRST-ORDER ENGINEERING-DESIGN MODEL
FOR HEAVILY DOPED SILICON DEVICES


6.1 Introduction
In the previous chapter, we presented a rigorous analytic model

for the emitter that included bandgap narrowing, Fermi-Dirac statistics,

and a finite surface recombination velocity at the emitter surface.

The main assumption of that treatment is the transparency of the

emitter to the injected minority carriers, that is, the transit time

of the minority carriers is much less than their average bulk lifetime.

This case can be physically realized if the emitter junction depth is

shallow, and the surface recombination velocity is high.

In this chapter, we present a first-order engineering-design

model that includes all the mechanisms of the transparent emitter model

except that we relax, in this treatment, the transparency assumption

and include a position-dependent Auger lifetime. Thus we provide a

general first-order model that includes all the fundamental heavy-

doping effects. This model can be easily used to characterize the

minority-carrier current in the heavily doped emitter.

There is no analytic solution for the minority-carrier con-

tinuity equation when a position-dependent lifetime, impurity

concentration, and ne (x) are included. To provide a numerical solu-

tion, we developed a computer program that solves the continuity










equation. This computer solution serves two purposes. First, it

provides information needed to characterize Lhe first-ord.r moJel.

Second, it verifies the accuracy of the first-order model.


6.2 Modeling Approach

The key approximation of the first-order model is the assump-

tion of a negligible net drift component in the minority-carrier current

expression in the heavily doped emitter (see Equation (B.1) in Apoendix

B). The rationale for this approximation is discussed in [29].

To provide further justification, we note that the approxi-

mation is consistent with calculations, which include bandgap narrowing

and Fermi-Dirac statistics, of the effective doping density, i.e.,

(n'/n e)NDD. Figures 6.1 and 6.2 snow the effective doping densities

for the Lanscn-Tuft model of bandgap narrowing [13] versus the actLai

doing densities NDD and NAA. Note that, in the range of doping from

10 to about 10 cm the effective dopig density is nearly

independent of the actual doping density, which correspords to nearly

zero net electric field for minority carriers.

In the first-order model, therefore, the minority carriers flow

primarily by diffusion. We characterize the position-dependent dif-

tusion length for minority carriers by an average value i.E:


A [D ]/2 (5.1)

here D is an average value for the oiffusion coefficient and TA

an average Auger lifetime for the minority carriers. We will later







1020





1019
E

4-
4+-

!018
C!


IoIo'


n-type Si /,


/
/
/"
[ ~//


I.

2 11111111 I 11111111


10 1


1019


I I 1 I I I


/





I ii


1020


1021


NDD (cm-3)
Figure 6.1 The effective doping density NDn versus the actual doping density NDD fur N-type silicon.
Lanyon-Tuft model of energy-ban ga narrowing and Fermi-Dirac statistics a;'e included.


CII~PI~PIIU~I~UUI-~II~~LIEI L-U -l--~-~--r~yRn~g~_IIULL-S~


1017







20




ro 19 / /
F-"
10 -- f/ -
p- iype Si -


-a. / /
,- /
o 10






17

1017 10' 1019 1020 102'

NAA (cm-3)
Figure b.2 The Offective doping density NAAeff versus the actual doping density NAA for P-type silicon.
Lanyon-Tuft model of encrj.y-bandgap narrowirng and Fermi-Dirac statistics are included.










verify that the use of Ln is consistent with the results of the coc.puter

solution.


6.3 Development of the Model

Using the approximations of Section 6.2, we solve the minority-

carrier continuity equation in the heavily doped emitter and obtain

the following expression for the minority-carrier saturation current

(JPo in an NF-emitter, for example):
9--

q n p SpLA +D tanh(WEC A) (6.2)
PO i'ODeff LA Dp + SpLA tan h(WE )

where Sp is the emitter surface recomtination velocity for holes, We

is the emitter quasi-neutral region width, and DDeff is an average

effective doping aersity.

",We can simolify (6.2) by considering the following special

cases:

A. if '; >> LA, then (6.2) reduces to


q nc D
J q n P (6.3)
"DDeff A


which is the familiar expression of the minority-carrier saturation

current for a completely opaque emitter.

B. If A >> '!E, then (5.2) reduces to


2
P _IPD- f- Ll/ ,- (6.4)
P ODDE 1f-p + W:/Dp]










Wef further simplify the expression iQ (6.4) by considering different

values of Sp. If Sp = 0, then (6.4) reduces to

9
q ni A
PO -(6.5)
JO NODefr 'A

In this case JpO is directly proportional to WE. If Sp is very high

(Sp -t ), then (6.4) reduces to


q ni Dp(5
JPO = -E (5.5)
'DDeff

and JPO is inversely proportional to 'IJ. For the case wnen


C >> a and p <<
LA SpP W


then (6.4) reduces to


2
q ni
JPO f P (6.7)
DDeff

and JPO does not depend on WE.


6.4 Relating Device Design Parameters
to the First-Order Model's Parameters

In this section, we provide means for characterizing the

parameters of the first-order model for any set of design parameters

for the emitter. At least three design parameters need to be known

to evaluate the recombination current of the emitter: the surface










impurity concentration NS, the emitter quasi-neutral region width 'c,

and the surface recombination velocity S. The parameters of the

first-order model are Neff, ', rA' WE, and S.

We now describe how Neff, D, and TA are determined. W?

assume that the impurity profile is Gaussian. The parameter Neff is

obtained from a spatial average using Figures 6.1 and 6.2. in

Figures 5.3 and 6.4, we plot average effective and actual doping

densities, NiDneff and NDD for N-type and NAAeff and NAA for P-type,

as functions of the surface impurity concentration NS.

To find 3, we first evaluate the average coping density in

the emitter (by averaging over the Gaussian profile), and find from

taoles the mobility of the minority carriers corresponding to N.

By using the Einstein relation in it: non-degenerate or degenerate

[70] form (depending cn N) we car calculate 0.

;e characterize TA as a function of NS by Figures 6.5 and

6.6 for N-type and P-type silicon, respectively. These characteriza-

tions are derived from the rigorous computer-aided numerical solution.


6.5 Verification of First-Order Model
by Computer Analysis

A computer program has been developed that solves the

minority-carrier continuity equation witn heavy-doping mechanisms

such as energy-bandgap nar-owing, Fermi-Dirac statistics, and a

posicion-dependent lifetime included. This program (see Appendix D

for a listing) is used to verify the accuracy of our first-order model

and to characterize the average bulk Auger lifetime in the emitter














20
10







!9 DD





/a DDeff



0 -

12L





10z -

1019 1020 1021

NS ( cm-3)
Figure 6.3 The average actual doping density -NDD and the average ef-
fective doping density NDDeff versus the surface impurity
concentration NS for N-type silicon. Lanyon-Tuft model of
energy bandgap harrowing and Fermi-Dirac statistics are
included.













20 ___






SN,,
So19 A A .


11

g 18







7 p- type Si

10 1
19 20 21
10 ;0 10

N, (cm 3)
Figure 6.4 The average actual dcping density NAA and the average ef-
fective doping density NDDeff versus the surface impurity
concentration NS for P-type silicon. Lar.yon-Taft model
of energy-bar,dgap narrowing and Fermi-Dirac statistics
are included.













10-7


0
108












io-0 l . ,, I , L ,r

IO19 1020 1021

N (Cm-3)

Figure 6.5 The average Auger lifetime T (calculated by computer-
aided analysis) versus N,- for N-type silicon




80












W, = 0.25 j.m
(p -region)

IO-7

o F


<< \


10-







I0O9
1019 i020 1021

N, (cm3)

Figure 6.6 The average Auger lifetime TA (calculated by computer-
aided analysis) versus NI for P-tvoe silicon










as a function of the surface impurity concentrsticn. Sample results

of the computer-aided analysis aie presented and discussed in Append;i

E.

In Figures 5.7 and 6.8, we plot the minority-carrier saturation

currents (and their surface and bulk components), JrO in a P'-emitter

and JPO in an N -emitter, as functions of SN and Sp, respectively,

for NS = 1020 cm-3 and Wr = 0.25 un. The curves are derived from our

computer-aided analysis and the points are derived from calculations

using the first-order model as described earlier in this chapter.

Note the good agreement between the results of the first-order model

and those of the computer-aided analysis.

In Tables III, IV. and V, we compare values of the minority-

carrier saturation current obtained from the first-order model and the

numerical analysis for N+- and P+-emitters having NS -= 1019 m-3

(Table III), 1020 cm-3 (Table IV), and 4 x !020 cm-3 (Table V). Note

that good agreement is obtained for NS = 101 crm- and 1022 cm-3 in

both N+- and P+-emitters. For Ns = 4 x 1020 cm-3, fair agreement is

obtained for the N+-emitter, but poor agreement is obtained for the

P -emitter. The inadequacy of the first-order model for this latter

case can be explained by noting the differences between Figures 6.1

and 6.2 In Figure 6.1 (N'-emitter), the range of flatness of iDDeff

extends up to about NDD = 4 x 1020 cr-3, whereas in Figure 6.2

(P+-emitter) the flatness of NAAeff extends up to only about NA =

2 x 1020 en-3. Hence, at about NS = 4 x 1020 c-3, the first-order

model does not give accurate results i'or PL-regions, although it gives


































Figure 6.7 The electron saturation current JNO dnd its surface and Auger components (JNSO and JNA) versus
the electron surface recombination velocity SN for a P+-emiLter. Solid curves are f r the
numerical solution and the points are for first-order model,






























SN (cm/sec)


10-I


10-13



































Figure 6.8 The hole saturation current JpO and its surface and Auger components (JpSO and JPAO) versus
the hole surface recombination velocity Sp for an N+-emitter. Solid curves are for the
ruumnrical solution and the points are for first-order model.










10 "


10-12






IO-13
i0


1014 L
102


Sp (cm/sec)











TABLE III

COMPARISON OF EMITTER SATURATION CURRENT
CALCULATED FROM FIRST-ORDER MODEL AND COMPUTER-AIDED
ANALYSIS FOR NS = 1019 cm-3--EXCELLENT AGREEMENT
IS OBTAINED FOR BOTH N+P AND P+N DEVICES.


WE= 0.25 m NS = 1019 cm-3
N+P: JpoA/cm2 P+N: JNOA/cm2

S Numerical First-Order Numerical First-Order
(cm/sec) Solution Model Solution Model

102 1.9 x 10-14 1.9 x 10-14 8.2 x 10-15 8.6 x 10-15

103 6.3 x 10-14 6.7 x 10-14 4.4 x 10-14 5.0 x 10-14

104 4.5 x 10-13 4.7 x 10-13 3.8 x 10-13 4.1 x 10-13

105 2.3 x 10-12 2.3 x 10-12 2.6 x 10-12 2.9 x 10-12

106 3.8 x 10-12 3.4 x 10-12 6.1 x 10-12 6.3 x 10-12

107 4.1 x 10-12 4.0 x 10-12 7.0 x 10-12 7.2 x 10-12










TABLE IV

COMPARISON OF EMITTER SATURATION CURRENT
CALCULATED FROM FIRST-ORDER MODEL AND COMPUTER-AIDED
ANALYSIS FOR NS = 1020 cm-3--GOOD AGREEMENT
IS OBTAINED FOR BOTH N+P AND F+N DEVICES.


WE = 0.25 pm NS = 1020 cm-3
N+P: JpOA/cm2 P+N: JNOA/cm2

S Numerical First Order Numerical First-Order
(cm/sec) Solution Model Solution Model

102 1.2 x 10-12 9.9 x 10-13 1.2 x 10-13 8.2 x 10-14

103 1.2 x 10-12 1.0 x 10-12 1.3 x 10-13 1.0 x 10-13

104 1.6 x 10-12 1.4 x 10-12 2.0 x 10-13 3.1 x 10-13

105 3.2 x 10-12 3.0 x 10-12 8.0 x 10-13 1.6 x 10-12

106 5.1 x 10-12 4.5 x 10-12 2.7 x 10-12 3.4 x 10-12

107 5.4 x 10-12 4.4 x 10-12 3.8 x 10-12 3.8 x 10-12











TABLE V
COMPARISON OF EMITTER SATURATION CURRENT
FROM FIRST-ORDER MODEL AND COMPUTER-AIDED
NUMERICAL ANALYSIS FOR NS = 4 x 1020--FOR THIS CASE
FAIR ACCURACY IS OBTAINED FOR N+P BUT
POOR ACCURACY IS OBSERVED FOR P+N.


WE = 0.25 4m NS = 4 x 1020 cm-3
N+P: JPoA/cm2 P+N: JNOA/cm2

S Numerical First-Order Numerical First-Order
(cm/sec) Solution Model Solution Model

102 1.9 x 10-12 6.5 x 10-13 6.9 x 10-14 1.9 x 10-16
103 1.9 x 10-12 6.5 x 10-13 6.9 x 10-14 2.4 x 10-16

104 2.0 x 10-12 2.3 x 10-13 6.9 x 10-14 7.2 x 10-16

105 2.0 x 10-12 1.0 x 10-12 6.8 x 10-14 3.9 x 10-15

106 2.2 x 10-12 1.3 x 10-12 6.8 x 10-14 8.4 x 10-15

107 2.3 x 10-12 1.3 x 10-12 6.8 x 10-14 9.6 x 10-15











reasonably accurate results for N+-regions. The physical reason

for the differences in Figures 6.1 and 6.2 is that the onset of

degeneracy in P+-emitters occurs at a lower doping density than for

N -emitters; NAAeff rises sharply due to degeneracy of the charge-

carriers at a lower doping density than that at which NDDeff rises.


6.6 Experimental Support

There are limited experimental data that support the first-

order model. These data [29], given in Table II, indicate that a

reduction in the emitter surface recombination velocity for a variety

of silicon solar cells increases the open-circuit voltage. The

emitter currents predicted by the first-order model, and verified

by the computer-aided analysis, yield values of VOC in rough

agreement with these experimental data. We will consider these data

further in the next chapter.


6.7 Summary and Conclusions

In this chapter, we established a first-order engineering-

design model for heavily doped silicon devices and solar cells

that accounts for fundamental heavy-doping mechanisms. After

developing the model and discussing its assumptions, we related the

emitter design parameters to the parameters of the first-order model.

We then verified the accuracy of the first-order model for both N+-

and P -emitters by using results of a rigorous computer-aided analy-

sis. We found that the first-order mcdel is valid for surface con-

centrations up to about 2 x 1020 cm-3 for P+-emitters and up to about




90




4 x 1020 cm-3 for N+-emitters. The difference in the range of

validity of the first-order model betweenN and P+- emitters is

attributed to the onset of degeneracy in P+- regions occurring at a

lower impurity concentration than in I+-regions.

The first-order model is useful in the design of heavily

doped silicon bipolar devices and solar cells. The model provides

a simple, but sufficiently accurate, evaluation of the emitter re-

combination current that can be effectively used in such designs.














CHAPTER VII

COMPUTER-AIDED STUDY OF VOC
IN +P AND P+N SILICON SOLAR CELLS

7.1 Introduction

In this chapter, we present the results of a rigorous computer-

aided numerical analysis of the heavily doped emitter in silicon solar

cells. The purpose of this treatment is to study the dependence of

VOC on the design parameters of the emitter in N P and P N silicon
solar cells.

We include the following fundamental heavy-doping mechanisms:

bandgap narrowing, Fermi-Dirac statistics, and Auger recombination.

We also include a finite surface recombination velocity at the emitter

surface. We assume low-level injection in the heavily doped emitter;

this assumption does not significantly limit the applicability of the

analysis.

7.2 Dependence of VOC
on the Emitter Design Parameters
in NiP and PTN Silicon Solar Cells

The open-circuit voltage VOC is given by

kT n JSC (7.1)
VOC q JBO JEO

where JSC is the short-circuit current density and JBO and JEO are the

base and emitter saturation current densities. In this treatment, both










JSC and JBO are determined by experimental measurements [1,66],
and JEO is characterized from our computer-aided analysis (see Appendix

E).

In studying the dependence of VOC on JEO, we assume that

JSC and JBO do not change. This is a reasonable assumption since
variations in the emitter design do not affect JBO and do not change

JSC enough to cause considerable changes in VOC. We let JSC =
23 mA/cm2 [1] and JB0 = 6.2 x 10-14 A/cm2 for an N+P cell and

3.1 x 10-14 A/cm2 for a P+N cell, which correspond to a diffusion

length in the base of about 80 im [66] and a doping density of about

5 x 1017 cm-3 for N-type and P-type substrates. Note that our assumed

value of JSC is not essential for this study and any experimentally

measured value can be incorporated into our results below by adjusting

the values of VOC in this chapter accordingly.

7.2A Dependence of V0C on WE

We have studied the dependence of VOC on WE for several values

of the surface recombination velocity S, in both N+- and P+-emitters.

Two values of the surface impurity concentration, NS = 1019 cm-3 and

NS = 1020 cm-3, have been used. The impurity profile in all cases

considered is Gaussian.

In Figures 7.1 and 7.2, we plot VOC versus WE for values of

S ranging from 102 to 107 cm/sec for NS = 1019 cm-3. These curves,

for N+P and P+N cells, imply the following:
A. The surface recombination velocity significantly

affects VOC in both NrP and P+N cells. Thus,




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