DEVICE PHYSICS FOR ENGINEERING DESIGN
OF HEAVILY DOPED REGIONS IN PNJUNCTION
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
cochairman 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 solidstate 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
SHALLOWEMITTER SILICON PNJUNCTION
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 SHALLOWEMITTER SILICON PNJUNCTION
DEVICES . . . . . . . . ... .. . .23
3.1 Introduction . . . . . . . . 23
3.2 Importance of Surface Recombination . . .. 24
3.3 ShockleyReadHall Recombination
via Defects . . . . . . . . . 25
3.4 The Excess MinorityCarrier 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 TRANSPARENTEMITTER 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 CONTENTSContinued
5.5 Application to pnJunction
Silicon Solar Cells . . . . . .
5.6 Perspective . . . . . . . .
VI. A FIRSTORDER ENGINEERINGDESIGN 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 FirstOrder Model's
Parameters . . . . . . .
6.5 Verification of FirstOrder Model
by Computer Analysis . . . . . .
6.6 Experimental Support . . . . . .
6.7 Summary and Conclusions . . . . .
VII. COMPUTERAIDED 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 PNJUNCTION DEVICES . ..
CHAPTER
Page
.
. .
TABLE OF CONTENTSContinued
APPENDICES Page
B. QUASIELECTRIC 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
COMPUTERAIDED 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 PNJUNCTION
SILICON SOLAR CELLS
By
Muhammed Ayman Shibib
December 1979
Chairman: Fredrik A. Lindholm
CoChairman: 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 pnjunction
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 firstorder model useful in the design of silicon pnjunction solar
cells. The accuracy of the firstorder 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 firstorder model and of the
numerical solution.
viii
CHAPTER I
INTRODUCTION
Heavily doped regions of silicon pnjunction devices show
experimentally a larger recombination current than is predicted by classi
cal pnjunction theory [1,2]. This large recombination current in heavily
doped regions is responsible for the low values of the opencircuit
voltage VOC of silicon solar cells [3] and the commionemitter 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 pnjunction solar cells and bipolar transistors. The
achievement of this objective involves:
A. Identifying physical mechanisms associated with
heavy doping.
B. Determining the dominant heavydoping mechanisms
that affect the device performance.
C. Including the dominant heavydoping mechanisms in
analytic and computeraided analysis of heavily
doped regions in silicon devices.
D. Establishing accurate and simple firstorder
engineering design models based on device physics.
From chargecontrol theory [610], the recombination current
in any heavily doped region, for example, the minoritycarrier emitter
current JE, can be expressed by
QE
JE E (1.1)
where QE is the minoritycarrier charge storage and TE is the effective
lifetime in the emitter. From classical pnjunction 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 minoritycarrier 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 heavydoping
mechanisms are outlined, in Table I, ard contrasted to corresponding
mechanisms in lightly doped semiconductors. We will describe briefly
some of the fundamental heavydoping mechanisms outlined in Table I.
One of the fundamental heavydoping mechanisms that yields a
large minoritycarrier charge, and consequently a large current by
(1.1), is energybandgap narrowing. From a theoretical standpoint,
various mechanism exist that lead to bandgap narrowing [1219]. More
over, electrical measurements in heavily doped singlecrystal 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)
MaxwellBoltzmann statistics MaxwellBoltzmann statistics for
for the majority and minority minority carriers; FermiDirac
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
Sq D kT F1/2 (i)
v q F_1/2 (n)
Builtin 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 IContinued
TABLE IContinued
Low Impurity Concentration High Impurity Concentration
Recombination of excess minor Recombination of excess minority
ity carriers via defects, carriers via (a) fundamental band
(ShockleyReadHall) toband Auger process, (b) position
dependent recombination centers or
defects (ShockleyReadHall)
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 pnjunctions 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
pnjunction 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 pnproduct 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 minoritycarrier 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 cm3) 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 heavydoping mechanism that must be con
sidered is the degeneracy of the majority carriers in heavily doped
semiconductors. Majoritycarrier concentrations in heavily doped
semiconductors are high enough that the carriers can interact with
each other via the Pauli Exclusion Principle. Thus MaxwellBoltzmann
statistics are no longer applicable and FermiDirac statistics must be
used instead.
We consider now two other fundamental heavydoping mechanisms
that can increase JE in (1.1) by decreasing TE. The first mechanism
is Auger bandtoband recombination [26]. It is fundamental in the
sense that it sets an upper bound on the minoritycarrier lifetimes
in these regions. The second mechanism is SchockleyReadHall
recombination [2728] via positiondependent defect densities.
We have discussed the fundamental heavydoping mechanisms
that can limit the performance of heavily doped silicon pnjunction
devices such as solar cells and bipolar transistors. We outline now
our approach in assessing the relative importance of these heavydoping
mechanisms to tne performance of silicon devices.
Our approach in assessing the importance of heavydoping
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 heavydoping 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 heavydoping
mechanisms, we proceed to develop engineering design models for
heavily doped regions.
Energybandgap narrowing, FermiDirac 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 quasineutral emitter region and recombine mainly at the surface
rather than in the bulk. The transparency assumption can be tested
for selfconsistency 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 energybandgap narrowing and FermiDirac
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 firstorder model for the
emitter that includes energybandgap narrowing, FermiDirac
statistics, Auger recombination, and a finite surface recombination
velocity. The firstorder model is useful as a design tool particularly
for silicon solar cells. The accuracy of the model is checked by a
rigorous computeraided numerical analysis of the problem. The model
is found to be accurate if the surface impurity concentration is less
than 4 x 1020 cm3.
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 quasineutral emitter region, and
the emitter surface recombination velocity. We show, based on our
computeraided 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 SHALLOWEMITTER
SILICON PNJUNCTION 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 opencircuit voltage of pnjunction 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 heavydoping 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 commonemitter
current gain. Inclusion of FermiDirac 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 [3235]. Our analysis, therefore,
reinforces the importance of bandgap narrowing in silicon devices.
2.2 Derivation
We consider an Ntype, 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 quasineutral 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 lowinjection
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 erff + 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 minoritycarrier current can be written (for lowlevel
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 chargecontrol 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 quasineutral emitter region:
E
Jp(0) = Jp(WE) + q aP() dx (2.16)
0
SPS + JPA
Using the meanvalue 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 quasineutral 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 closedform
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 Ntype emitter. Bandgap
narrowing and FermiDirac 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 minoritycarrier transport in the emitter of pnjunction
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 1017cm3 We let NS = 1020cm3 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 21, 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 21 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 shortcircuitcurrent 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 22, 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 22) 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 23, where we let Sp = 108 cm/sec and plot the resulting
1013
Jo
o14 PAO
C\J
PSO
Fn 15
10
016
103 J 4 15 o1 17 Jo8
Sp (cm/sec)
Figure 2.1 The emitter saturation current JPO and its surfacerecombination component JpSO and Auger
recombination component JPAO versus the surface recombination velocity Sp
113
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 surfacerecombination
component of Jpo and JPAQ is the Augerrecolmbination component
108
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 pnjunction silicon solar cells [29] and, because of
high recombination velocity at an ohmic contact, the model can be
expected to apply also to shallowemitter silicon transistors. For
larger WE, the average Auger lifetime is smaller than the transit time
and Auger recombination becomes important.
Finally, Figure 24 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 cm3), surface recombination is the
dominant recombination mechanism in the emitter, and VOC is very
sensitive to Sp. As NS increases toward 1021 cm3, 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 cm3) 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 pnjunction devices based on Auger
700
Sp= 102 cm/sec
4
105
680 
660 
10
0 640
620 
600
109 1020 !021
Ns (cm3)
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 pnjunction solar cells. It follows that this model
also cannot explain the low values of the commonemitter current gain
measured in bipolar transistors. In this demonstration, we have used
MaxwellBoltzmann statistics. Use of FermiDirac 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 pnjunction devices.
CHAPTER III
RECOMBINATION THROUGH DEFECTS IN HEAVILY DOPED
SHALLOWEMITTER SILICON PNJUNCTION 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 heavydoping 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
shallowemitter 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 opencircuit voltage in silicon pnjunction solar cells to
surface treatment. These experiments are discussed briefly in Section
3.2.
In Section 3.3, we discuss ShockleyReadHall 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 ShockleyReadHall 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
ShockleyReadHall 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 minoritycarrier 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 heavydoping mechanisms that can provide
a large minoritycarrier charge storage must be considered. This
conclusion is corroborated by recent experimental measurements of the
shortwavelength 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 majoritycarrier density.
3.2 Importance of Surface Recombination
Fossum et al. [29] recently reported a considerable improvement
in the opencircuit voltage in a variety of silicon solar cells by
reducing the recombination velocity at the emitter surface. These
solar cells include diffused, implanted, backsurfacefield (BSF),
and highlowemitter (HLE) structures, of both NonP and PonN
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 minoritycarrier 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 ShockleyReadHall Recombination via Defects
Defects in doped silicon can result from the diffusion of
impurities and the subsequent heat treatment of pnjunction 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 deadlayer 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 minoritycarrier lifetimes (in the picosecond range). The
short lifetimes yield a large (dark) emitter recombination current
that limits the opencircuit voltage of solar cells to values lower
than those predicted by classical pnjunction 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 processinduced defects in the emitter of silicon
junction solar cells. They derived an impurityconcentrationdependent
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 Nfemitter, NAA is the Ptype 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 spacecharge 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 bipolartransistor
studies [41,42] of the effect of a builtin electric field in a dif
fused quasineutral emitter region, having a constant minority
carrier lifetime. In [9], the positiondependent minoritycarrier
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 positiondependent defect density model
of Lindholm et al. [9] in which the defect density is proportional
BUILTIN
ELECTRIC FIELD
P(WE)
S IP (WE)
I
I
E X
X O 0.1 WE
(SURFACE) (SCR EDGE)
Figure 3.1 Sketch of minoritycarrier 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 ShockleyReadHall lifetime for lowlevel
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 minoritycarrier 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 ShockleyReadHall 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 ShockleyReadHall 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 ShockleyReadHall 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
minoritycarrier 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 MinorityCarrier
Charge Storage in Heavily Doped Regions
We discussed ShockleyReadHall recombination via defects
in the last section and showed that for ShockleyReadHall lifetimes
that have an inversepower 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 heavydoping 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 minoritycarrier 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 minoritycarrier charge storage in heavily doped
regions must then be large to account for the large current.
We now show qualitatively that a large minoritycarrier charge
storage is due either to an excess intrinsic carrier density or to a
deficit majoritycarrier concentration.
The minoritycarrier charge storage Qp, in an Ntype region
for example, is
WE
Qp = q AP(x) dx (3.4)
0
where AP(x) is the excess minoritycarrier (hole) concentration. For
nonequilibrium conditions in a nondegenerate semiconductor,
2 FN EF
PN = n expFN EF
L kT
where EFp and EFN are the hole and electron quasiFermi levels.
From (3.5), the condition of lowinjection,
(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., energybandgap 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 nondegenerate Ntype semiconductor. For a degenerate semiconductor,
we show later that n2 can include the effects of FermiDirac 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 FRONTSURFACE
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+NP HLE 599 mV 619 mV
C N+P (implanted) 625 mVa 6441lVa
D P+NN 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 quasineutral base region increases. This was interpreted
by Slotboom and DeGraaff to be due to energybandgap narrowing in the
quasineutral 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 cm3 to 109 cm3 in
2
silicon, the increase of nie can be attributed to a Fermi energy
difference AE corresponding to different densityofstates 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 energybandgap
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 energylevels 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 majoritycarrier
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 cm3. Here gD is the impurity energylevel degeneracy
factor, NC is the conductionband effective density of states, and
EC is the conductionband edge. Thus, in the Heasell model, where ED
does not depend on the impurity concentration, the majoritycarrier
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 (electronhole) product
(given, at equilibrium for low doping concentrations, by the law of
mass action) implies a larger minoritycarrier 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 ntype and ptype 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 impurityconcentration
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
phosphorusdoped silicon, and about 70% for borondoped 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
firstorder 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 cm3
[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 ntype material) or the valance band
(for ptype material).
The second model assumes that (a) the conductionband and
valanceband 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
quasineutrality 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 lowimpurityconcentration 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 energybandgap narrowing in silicon
in the doping range of 1017 to 1019 cm3, 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 MaxwellBoltzmann 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 cm3 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 silicon0.045 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 cm3 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 cm3), as suggested by
Popovic, and include FermiDirac statistics, we get values of the
opencircuit voltage in silicon solar cells and the commonemitter
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 quasicontinuous
levels in the bandgap at high impurity concentrations (in the range
of 1017 to 1019 m3). 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, energybandgap narrowing since allowed bandlike states
would exist at lower energies (for Ntype 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 Fermilevel saturation go handinhand.
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, singlecrystal 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 freeelectron gas model in metals. These experi
mental observations in heavily doped silicon involve measurements of:
resistivity and Hall effect [4748, 5155], electron spin resonance
(ESR) [53], magnetoresistance [54], electronic specific heat [56],
and nuclear magnetic resonance (NMR) [5758]. A thorough discussion
of these mechanisms in group IV semiconductors is given in [58]. We
consider briefly here the implication of measurements of the electronic
specific heat [56] and nuclear magnetic resonance [5758] 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 phosphorusdoped silicon [5758]. 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 cm3, 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 firstorder
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 energybandgap 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 opencircuit
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 pnjunction devices.
CHAPTER V
HEAVILY DOPED TRANSPARENTEMITTER 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 quasineutral 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 pnjunction
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 Pcells (Ptype substrate) [29] and
Ncells [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 PonN and NonP 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) FermiDirac statistics, (c) builtin field due to the
impurity profile, and (d) a finite surface recombination velocity S.
Detailed numerical studies including these various effects have been
done [32,6263], but they have not treated the case of the transparent
emitter.
A major result of this chapter is the demonstration that the
transparentemitter model can predict experimental values of VOC
observed on N+P thin diffused junction silicon solar cells made on
lowresistivity (0.1 0cm) substrates. Thus, the transparentemitter
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 transparentemitter model
gives VOC 0 600 mV for high values of Sp (Sp > 10 cm/sec) provided
the effects of bandgap narrowing (modified by FermiDirac 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
640650 mV range [6465].
In addition to the development of the theory for the
transparentemitter device, and its application to solar cells, this
chapter will include a test for the selfconsistent validity of the
transparentemitter model. This test compares the calculated transit
time of minority carriers across the emitter with the Augerimpact
minoritycarrier lifetime within the emitter region.
5.2 Derivation
We consider an Ntype heavily doped quasineutral emitter
region; analogous results apply to Ptype emitters. The minority
carrier current density in the Ntype 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 lowlevel 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 FermiDirac 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 quasineutral 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
(transittime limtied), that is, if the minority carrier transit time
Tt is much less than the minority carrier lifetime Tp (for an Ntype
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 minoritycarrier
current in a transparent emitter.
To check the condition, .t << Tp, required for transparancy,
we must determine the steadystate 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 HeavyDoping Effects
In thermal equilibrium, heavydoping concentrations of shallow
level impurities affect the minoritycarrier concentration in a quasi
neutral region by two mechanisms: bandgap narrowing and FermiDirac
statistics. These two mechanisms affect the minoritycarrier
concentration in opposite ways. For any given position of the Fermi
level relative to the band edges, bandgap narrowing tends to increase
the minoritycarrier concentration, while inclusion of FermiDirac
statistics tends to decrease the minoritycarrier concentration below
the value calculated using MaxwellBoltzmann 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 rigidband
approximation; it is discussed in Appendix F.
The effects of bandgap narrowing and FermiDirac statistics
can be lumped into a positiondependent 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 Ntype material, and
n = nV = (EV EF)/kT (5.16)
for Ptype 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 cm3 in
Ntype silicon), is [24]
C(n) W 0.04n + 0.3 (5.17)
The above approximation of C(n) gives values of the FermiDirac
integral of order 1/2 with less than 4% error.
In nonequilibrium conditions, bandgap narrowing increases the
minoritycarrier current by:
1. Increasing the minoritycarrier concentration
2. Decreasing the retarding builtin electric field
acting on the minority carriers
The increase in the minoritycarrier concentration P results
from the increase in n e(x). The decrease of the builtin 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 FermiDirac 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 builtin 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 bandgapnarrovwing reduction factor R(N) versus the electron (majoritycarrier) concentration
N for: (A) LanyonTuft model, (B) Hauser model, and (C) SlotboomDeGraaff model
1.0
0.5
0.0
0.5 
1.0
1017 1018 1019
N (cm3)
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 quasineutral 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 FermiDirac statistics increases the
transit time as can be seen in Figure 5.2 Inclusion of FarmiOirac
statistics shifts the value of the transit time closer to that calcu
lated when heavydoping effects are neglected.
5.4 Discussicn
From (5.8), the minGritycarrier 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
10l 1  III
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 (SlotboomDeGraaff
model) and Fermi Dirac statistics (BGN + FD), bandgap
narrowing (SlotbocmDeGraaff 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 pnjunction 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 transparentemitter 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 103 (N/1018)1/2 eV
for nondegenerately doped silicon, and (5.25)
AEG = 162. x 103 (N/1020)1/6 eV
for degenerately doped silicon. A Gaussian impurity profile is assumed
with a surface impurtiy concentration of 1020 cm3 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 LanyonTuft model of bandgap narrowing.
The validity of the transparentemitter model is based on the
condition that the minoritycarrier transit time is much smaller than
the minoritycarrier lifetime: Tt << T To test this condition, T
10II
101
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, FermiDirac statistics and bandgap narrowing included: (A) lanyonTuft,
(B) Hauser, and (C) SlotboomDeGraaff. For low values of Sp (less than about 10' cm/sec) the
selfconsistency test yields Tt > Tp so the emitter current is then due to Auger recombination
and may be larger than values reported above.
108
SA
4
10
1010 ..... 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, FermiDirac statistics and
bandgap narrowing are included: (A) LanyonTuft, (B) Hauser, and (C) SlotboomDeGraaff
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 quasineutral
emitter region width, WE, for Sp = 5 x 105 cm/sec. The recombination
lifetime Tp has an upper bound determined by the Auger bandtoband
recombination at high impurity concentration.
To illustrate the selfconsistency test for transparency, we
assume the surface concentration of a diffused emitter to be 1020 cm3.
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 pnJunction Silicon Solar Cells
In this section, we apply the transparentemitter model to
calculate the opencircuit voltage of silicon pnjunction solar cells
having low substrate resistivity.
The opencircuit 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 ShockleyReadHall 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) LanyonTuft, (B) Hauser, and (C)
FermiDirac
included:
SlotboomDeGraaff
z
0
C3
e
1010
10 1
0.8
I
i t I I
I08
**
where JSC is the shortcircuit 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 minoritycarrier saturation current density
and JNO is the base minoritycarrier saturation current density.
The base saturation current density is
9
qniDN (
NO NALN (5.2
Consider lowresistivity silicon solar cells with base doping concen
tration of NAA = 5 x 1017 cm3. Measurements made on such cells
indicated the minority carrier diffusion length, LN, to be 80 im [66],
corresponding to JNO = 6.2x1014 A/cm2. The general expression for
the transparentemitter 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. FermiDirac statistics and bandgap narrowing are included:
(A LanyonTuft, (B Hauser, and (C) SlotboomDeGraaf. In (D) heavydoping 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 metalcovered surface, region 1, and
the nonmetalcovered surface, region 2, are given by (5.8). In region
3, the flow of minority carriers is twodimensional 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
twodimensional analysis, we make the firstorder 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 (nonmerfi)
oxide
meta I  7 
,(metal) Jp143 2
7~
(b)
Figure 5.7 (a) The structure of a pnjunction 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, thinjunction
pnjunction solar cells made on low resistivity (= 0.1 ncm) material,
the maximum observed opencircuit voltage is about 600 mV [1]. As
one example, if we consider the LanyonTuft 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 cm3. 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, thinjunction solar cells. Note that
Figures 5.4 and 5.5 indicate the selfconsistent validity of the
transparency assumption for this device in the LanyonTuft model
which permits use of (5.20).
Although the preceding discussion has focused on the
transparentemitter model applied to N+P silicon solar cells, the
model can also be applied to P+N cells. It is straightforward to
show that heavydoping effects (bandgap narrowing and FermiDirac
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 Ptype material than in Ntype material [32], if both Ntype and
Ptype regions have the same bandgap narrowing. Thus the net effect
of bandgap narrowing and FermiDirac statistics is to degrade the
Ntype heavily doped region more than the Ptype region with the same
impurity concentration. This may, in part, be responsible for the
high efficiency P+NN+ cells that have been observed experimentally
[6869].
5.6 Perspective
This chapter has dealt with the transparentemitter model of
a solar cell, which is defined by the condition that the minority
carriers in the dark quasineutral 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 pnjunction 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 transparentemitter model, provided the emitter
69
recombination center density is low enough for the Auger process to
dominate over the ShockleyReadHall process.
Although the transparentemitter model may describe many
conventional shallow pnjunction silicon solar cells, the high value
of the surface recombination velocity S necessary to validate the
transparentemitter 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 FIRSTORDER ENGINEERINGDESIGN 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, FermiDirac 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 firstorder engineeringdesign
model that includes all the mechanisms of the transparent emitter model
except that we relax, in this treatment, the transparency assumption
and include a positiondependent Auger lifetime. Thus we provide a
general firstorder model that includes all the fundamental heavy
doping effects. This model can be easily used to characterize the
minoritycarrier current in the heavily doped emitter.
There is no analytic solution for the minoritycarrier con
tinuity equation when a positiondependent 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 firstord.r moJel.
Second, it verifies the accuracy of the firstorder model.
6.2 Modeling Approach
The key approximation of the firstorder model is the assump
tion of a negligible net drift component in the minoritycarrier 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 FermiDirac 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 LanscnTuft 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 firstorder model, therefore, the minority carriers flow
primarily by diffusion. We characterize the positiondependent 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'
ntype Si /,
/
/
/"
[ ~//
I.
2 11111111 I 11111111
10 1
1019
I I 1 I I I
/
I ii
1020
1021
NDD (cm3)
Figure 6.1 The effective doping density NDn versus the actual doping density NDD fur Ntype silicon.
LanyonTuft model of energyban ga narrowing and FermiDirac statistics a;'e included.
CII~PI~PIIU~I~UUI~II~~LIEI LU l~~r~yRn~g~_IIULLS~
1017
20
ro 19 / /
F"
10  f/ 
p iype Si 
a. / /
, /
o 10
17
1017 10' 1019 1020 102'
NAA (cm3)
Figure b.2 The Offective doping density NAAeff versus the actual doping density NAA for Ptype silicon.
LanyonTuft model of encrj.ybandgap narrowirng and FermiDirac 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 minoritycarrier saturation current
(JPo in an NFemitter, 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 quasineutral 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 minoritycarrier 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 1fp + 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 FirstOrder Model's Parameters
In this section, we provide means for characterizing the
parameters of the firstorder 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 quasineutral region width 'c,
and the surface recombination velocity S. The parameters of the
firstorder 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 Ntype and NAAeff and NAA for Ptype,
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: nondegenerate 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 Ntype and Ptype silicon, respectively. These characteriza
tions are derived from the rigorous computeraided numerical solution.
6.5 Verification of FirstOrder Model
by Computer Analysis
A computer program has been developed that solves the
minoritycarrier continuity equation witn heavydoping mechanisms
such as energybandgap narowing, FermiDirac statistics, and a
posiciondependent lifetime included. This program (see Appendix D
for a listing) is used to verify the accuracy of our firstorder 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 ( cm3)
Figure 6.3 The average actual doping density NDD and the average ef
fective doping density NDDeff versus the surface impurity
concentration NS for Ntype silicon. LanyonTuft model of
energy bandgap harrowing and FermiDirac 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 Ptype silicon. Lar.yonTaft model
of energybar,dgap narrowing and FermiDirac statistics
are included.
107
0
108
io0 l . ,, I , L ,r
IO19 1020 1021
N (Cm3)
Figure 6.5 The average Auger lifetime T (calculated by computer
aided analysis) versus N, for Ntype silicon
80
W, = 0.25 j.m
(p region)
IO7
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 Ptvoe silicon
as a function of the surface impurity concentrsticn. Sample results
of the computeraided analysis aie presented and discussed in Append;i
E.
In Figures 5.7 and 6.8, we plot the minoritycarrier 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 cm3 and Wr = 0.25 un. The curves are derived from our
computeraided analysis and the points are derived from calculations
using the firstorder model as described earlier in this chapter.
Note the good agreement between the results of the firstorder model
and those of the computeraided analysis.
In Tables III, IV. and V, we compare values of the minority
carrier saturation current obtained from the firstorder model and the
numerical analysis for N+ and P+emitters having NS = 1019 m3
(Table III), 1020 cm3 (Table IV), and 4 x !020 cm3 (Table V). Note
that good agreement is obtained for NS = 101 crm and 1022 cm3 in
both N+ and P+emitters. For Ns = 4 x 1020 cm3, fair agreement is
obtained for the N+emitter, but poor agreement is obtained for the
P emitter. The inadequacy of the firstorder 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 cr3, whereas in Figure 6.2
(P+emitter) the flatness of NAAeff extends up to only about NA =
2 x 1020 en3. Hence, at about NS = 4 x 1020 c3, the firstorder
model does not give accurate results i'or PLregions, 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 firstorder model,
SN (cm/sec)
10I
1013
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 firstorder model.
10 "
1012
IO13
i0
1014 L
102
Sp (cm/sec)
TABLE III
COMPARISON OF EMITTER SATURATION CURRENT
CALCULATED FROM FIRSTORDER MODEL AND COMPUTERAIDED
ANALYSIS FOR NS = 1019 cm3EXCELLENT AGREEMENT
IS OBTAINED FOR BOTH N+P AND P+N DEVICES.
WE= 0.25 m NS = 1019 cm3
N+P: JpoA/cm2 P+N: JNOA/cm2
S Numerical FirstOrder Numerical FirstOrder
(cm/sec) Solution Model Solution Model
102 1.9 x 1014 1.9 x 1014 8.2 x 1015 8.6 x 1015
103 6.3 x 1014 6.7 x 1014 4.4 x 1014 5.0 x 1014
104 4.5 x 1013 4.7 x 1013 3.8 x 1013 4.1 x 1013
105 2.3 x 1012 2.3 x 1012 2.6 x 1012 2.9 x 1012
106 3.8 x 1012 3.4 x 1012 6.1 x 1012 6.3 x 1012
107 4.1 x 1012 4.0 x 1012 7.0 x 1012 7.2 x 1012
TABLE IV
COMPARISON OF EMITTER SATURATION CURRENT
CALCULATED FROM FIRSTORDER MODEL AND COMPUTERAIDED
ANALYSIS FOR NS = 1020 cm3GOOD AGREEMENT
IS OBTAINED FOR BOTH N+P AND F+N DEVICES.
WE = 0.25 pm NS = 1020 cm3
N+P: JpOA/cm2 P+N: JNOA/cm2
S Numerical First Order Numerical FirstOrder
(cm/sec) Solution Model Solution Model
102 1.2 x 1012 9.9 x 1013 1.2 x 1013 8.2 x 1014
103 1.2 x 1012 1.0 x 1012 1.3 x 1013 1.0 x 1013
104 1.6 x 1012 1.4 x 1012 2.0 x 1013 3.1 x 1013
105 3.2 x 1012 3.0 x 1012 8.0 x 1013 1.6 x 1012
106 5.1 x 1012 4.5 x 1012 2.7 x 1012 3.4 x 1012
107 5.4 x 1012 4.4 x 1012 3.8 x 1012 3.8 x 1012
TABLE V
COMPARISON OF EMITTER SATURATION CURRENT
FROM FIRSTORDER MODEL AND COMPUTERAIDED
NUMERICAL ANALYSIS FOR NS = 4 x 1020FOR 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 cm3
N+P: JPoA/cm2 P+N: JNOA/cm2
S Numerical FirstOrder Numerical FirstOrder
(cm/sec) Solution Model Solution Model
102 1.9 x 1012 6.5 x 1013 6.9 x 1014 1.9 x 1016
103 1.9 x 1012 6.5 x 1013 6.9 x 1014 2.4 x 1016
104 2.0 x 1012 2.3 x 1013 6.9 x 1014 7.2 x 1016
105 2.0 x 1012 1.0 x 1012 6.8 x 1014 3.9 x 1015
106 2.2 x 1012 1.3 x 1012 6.8 x 1014 8.4 x 1015
107 2.3 x 1012 1.3 x 1012 6.8 x 1014 9.6 x 1015
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 opencircuit voltage. The
emitter currents predicted by the firstorder model, and verified
by the computeraided 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 firstorder engineering
design model for heavily doped silicon devices and solar cells
that accounts for fundamental heavydoping mechanisms. After
developing the model and discussing its assumptions, we related the
emitter design parameters to the parameters of the firstorder model.
We then verified the accuracy of the firstorder model for both N+
and P emitters by using results of a rigorous computeraided analy
sis. We found that the firstorder mcdel is valid for surface con
centrations up to about 2 x 1020 cm3 for P+emitters and up to about
90
4 x 1020 cm3 for N+emitters. The difference in the range of
validity of the firstorder 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 firstorder 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
COMPUTERAIDED 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 heavydoping mechanisms:
bandgap narrowing, FermiDirac statistics, and Auger recombination.
We also include a finite surface recombination velocity at the emitter
surface. We assume lowlevel 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 opencircuit voltage VOC is given by
kT n JSC (7.1)
VOC q JBO JEO
where JSC is the shortcircuit 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 computeraided 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 1014 A/cm2 for an N+P cell and
3.1 x 1014 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 cm3 for Ntype and Ptype 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 cm3 and
NS = 1020 cm3, 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 cm3. 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,
