METHODS FOR INVESTIGATING THE PROPERTIES OF
POLYCRYSTALLINE SILICON P-N JUNCTION SOLAR CELLS
BY
JEFFREY ALAN MAZER
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
1981
ACKNOWLEDGEMENTS
The author sincerely thanks the chairman of his supervisory
committee, Dr. Arnost Neugroschel, and the co-chairman,
Dr. Fredrik A. Lindholm, for their help and guidance during the course
of the research presented in this dissertation. The author also thanks
the other members of his supervisory committee, Dr. Jerry G. Fossum,
Dr. Paul H. Holloway, and Dr. Arun K. Varma for their friendly
assistance.
Appreciation is extended to the author's colleagues and friends for
stimulating discussions: Shing C. Pao, J. Ignacio Arreola,
Franklin N. Gonzalez, Dersun Lee, and Phillip E. Russell. Appreciation
is also extended to Raymond Wilfinger, William Axson, Dean Schoenfeld,
Bruce Chovnick, and William Wagner for technical assistance during the
fabrication of devices.
The author gratefully acknowledges the financial support and
technical assistance of the U.S. Department of Energy and the Solar
Energy Research Institute during the course of this research.
Last, but not least, the author thanks his parents for their
frequent support and encouragement throughout his graduate school
career.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGMENTS . . . . . . . . ... .... ii
ABSTRACT . . . . . . . . ... . . ... . vi
CHAPTER
1 INTRODUCTION . . . . . . . . .. . 1
2 DEGRADATION OF SOLAR-CELL PERFORMANCE
BY AREAL INHOMOGENEITY . . . . . . . . 3
2.1 Introduction . . . . . . . .... .. 3
2.2 Type 1 Areal Inhomogeneity . . . . . . 5
2.3 Type 2 Areal Inhomogeneity . . . . . .. 11
2.4 Discussion . . . . . . . .... . 12
3 A METHOD FOR EXPERIMENTAL ASSESSMENT OF THE SHIFTING
APPROXIMATION, WITH APPLICATION TO POLYSILICON SOLAR
CELLS . . . . . . . . ... .. ... 14
3.1 Introduction .... . . . . . . .... 14
3.2 Method for Analyzing the Measured I-V Curves . 15
3.3 Experimental Procedure . . . . . ... 21
3.4 Experimental Results . . . . . . ... 24
3.5 Discussion . . . . . . . .... . 24
4 EFFECTS OF GRAIN BOUNDARIES ON THE CURRENT-VOLTAGE
CHARACTERISTICS OF POLYSILICON SOLAR CELLS . . .. 33
4.1 Introduction . . . . . . . . ... 33
4.2 Fabrication of Devices and Evaluation of Prefer-
ential Grain-Boundary Diffusion . . . ... 36
4.3 Analysis of the I-V Curves . . . . ... 45
4.3.1 Space-Charge Region Current Components
(n+-p diodes) . . . . . ... .. 48
4.3.2 Quasi-Neutral Region Current Components
(n+-p diodes) ..... .. . . . . 53
4.3.2.1 IGB . . . . . . . 55
4.3.2.2 IB and GB Passivation .... 55
QNE
4.3.2.3 IGB . . . . . .. 57
B
PAGE
4.3.3 Illuminated I-V Curves (n -p diodes). . 58
4.3.4 Grain-Boundary Passivation by
Hydrogenation Treatment . . . .. 62
4.3.5 I-V Characteristics (p+-n diodes) . . 65
4.3.6 Grain-Boundary Shunt Resistance
RG . . . . . . . . . 67
Sh
4.4 Comparison of Mesa Diode and Planar Diffused
Diode I-V Curves . . . . . . .. 70
4.5 Discussion . . . . . . . . . 71
5 SMALL-SIGNAL ADMITTANCE METHOD FOR DETERMINING THE
SURFACE-STATE DISTRIBUTION AT THE PREFERENTIALLY
DIFFUSED PART OF THE GRAIN BOUNDARY . . . ... 77
5.1 Introduction . . . . . . . .... 77
5.2 Small-Signal Equivalent Circuit Model of
a Diode with a Preferentially Diffused
Grain Boundary . . . . . . . ... 78
5.3 An Admittance Method for Determining N . . 87
ss
5.4 Inversion along the GB in the p-Type Bulk . .. 92
5.5 Experimental Procedure and Results ...... 92
5.6 Conductance Method for Determining N . .. 101
ss
5.7 Discussion . . . . . . . . . 103
6. DESCRIPTION OF SEVERAL METHODS INTENDED TO SUPPRESS
THE GRAIN-BOUNDARY DARK RECOMBINATION CURRENT . . 108
6.1 Introduction . . . . . . . ... 108
6.2 Low-Temperature-Enhanced Preferential Diffusion
of Phosphorus . . . . . . . ... 108
6.3 Low-Temperature-Enhanced Preferential Diffusion
of Boron . . . . . . . .... ... 110
6.4 Grain-Boundary Passivation by Hydrogen Plasma
Treatment . . . ......... ............ 111
6.5 Preferential Etching of Grain Boundaries to
Enhance Performance . . . . . . ... 112
6.6 Discussion .... . . . . .. . . 113
7 DISCUSSION . . . . . . . . . . 114
APPENDIX
I FORTRAN PROGRAMS FOR SIMULATING THE EFFECT OF AREAL
INHOMOGENEITY IN AN n+-p SILICON SOLAR CELL ... 117
II FORTRAN PROGRAM FOR PROJECTING THE PERFORMANCE OF
A SOLAR CELL GIVEN THE EMPIRICAL PARAMETER VALUES
OF THE SUBCELLS . . . . . . . . ... 121
III DERIVATION OF A SIMPLIFIED EXPRESSION FOR THE SPACE-
CHARGE REGION RECOMBINATION CURRENT . . . .. 124
PAGE
IV GROOVE AND STAIN EXPERIMENT TO DETERMINE THE
EXTENT OF STAINING . . . . . . . ... 126
V STANDARD WAFER CLEANING AND POLISHING PROCEDURES . 128
VI FABRICATION SCHEDULES FOR RUNS 4P4, 6P1, 7P, 8P1,
13P3 and 13P4. . . . . . . . .... .129
VII FABRICATION SCHEDULE FOR RUNS 22P and 25P. . . ... 132
VIII FABRICATION SCHEDULES FOR RUNS 34P, 36P, AND 37P . 133
IX FABRICATION SCHEDULES FOR RUNS 39P AND 40P . . .. .135
X COMMENTARY ON THE RELIABILITY OF GROOVE AND STAIN
RESULTS IN CHAPTERS 4 AND 5. . . . . . ... 137
REFERENCES . . . . . . . . . . . . 139
BIOGRAPHICAL SKETCH . . . . . . . .... .144
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
METHODS FOR INVESTIGATING THE PROPERTIES OF
POLYCRYSTALLINE SILICON P-N JUNCTION SOLAR CELLS
By
Jeffrey Alan Mazer
August 1981
Chairman: Dr. Arnost Neugroschel
Co-Chairman: Dr. Fredrik A. Lindholm
Major Department: Electrical Engineering
Experimental and analytical methods are developed for investi-
gating the properties and performance-degrading mechanisms of poly-
crystalline silicon p-n junction solar cells.
The degrading effects of areal inhomogeneity are demonstrated by
means of a parallel-subcell equivalent circuit model. It is shown that
it is the area of the poor-quality material in a silicon p-n junction
solar cell that dominates in determining the overall cell performance.
An experimental method is developed for assessing the validity of the
shifting approximation for solar cells made from polysilicon and other
material. The experimental data suggest that the shifting approximation
is valid for a variety of polysilicon solar cells in which the intra-
grain base minority carrier diffusion length is smaller than or equal to
the average grain diameter. The current components associated with the
grain boundaries of diffused p-n junction polysilicon solar cells made on
Wacker substrates are analyzed and experimentally identified. The
analysis shows that the dominant current component at small bias
levels (0-300 mV) is the recombination current at the grain boundary
within the p-n junction space-charge region. At higher bias levels
(V = V0C 500-600 mV), both this current component and the current
component due to recombination at that part of the grain boundary
which is adjacent to the quasi-neutral base region are important.
New electrical methods for determining the presence or absence of
preferential diffusion along the grain boundaries and for determining
the average doping density of preferentially diffused regions along
the grain boundaries are described. A small-signal admittance method
is developed for the determination of the grain-boundary surface-state
distribution in the energy gap for that part of a grain boundary which
has been preferentially diffused with phosphorus. Various experimen-
tal attempts at suppressing the grain-boundary dark recombination
current are described. It is shown that the large leakage currents
of small polysilicon p-n junction mesa diodes cause the measured I-V
characteristics of these diodes to be of questionable value in
analyzing the grain boundary component of the current.
vii
CHAPTER 1
INTRODUCTION
Recent attention has been focused on polycrystalline silicon solar
cells because of their potential low cost. The anticipated advantage
of low cost is offset by the fact that polycrystalline solar cells
have displayed efficiencies that are much less than those of the
corresponding single-crystal devices [1,2,3]. This lower efficiency
is caused, to varying degrees, by the presence of dark recombination
currents associated with the grain boundaries, by the degrading effects
of areal inhomogeneity, by a low short-circuit current density, and
by a low shunt resistance. In this dissertation, we develop methods
for investigating the properties and performance-degrading mechanisms
of polysilicon p-n junction solar cells.
In Chapter 2, we demonstrate the degrading effects of areal inhomo-
geneity by means of a parallel-subcell equivalent-circuit model.
Chapter 3 describes an experimental method for assessing the validity
of the shifting approximation. These two chapters are applicable to
single-crystal as well as to polycrystalline solar cells.
Most of the experimental devices used in this research were
fabricated on Wacker polysilicon substrates. Wacker material (both p
and n-type) was chosen because it has large enough grain diameters
(~1 mm) to conveniently enable the fabrication of devices that contain
either zero, or at most a few, grain boundaries. By comparing the data
(e.g., capacitance, current-voltage characteristic) on a device
containing a few grain boundaries with the corresponding data on a
similarly fabricated grain-boundary-free (GBF) device, the grain-
boundary (GB) component of the data could be isolated and accurately
analyzed provided that the surface and edge leakage currents were
adequately suppressed. The fabrication of GBF devices also enabled
the determination of some of the intragrain material parameters,
e.g., the intragrain base minority carrier diffusion length. This
fabrication and measurement strategy was used frequently in the
research described in chapters 4, 5, and 6.
In Chapter 4, the current components associated with the grain
boundaries are analyzed and experimentally identified. New electrical
methods for determining the presence or absence of preferential
diffusion along the grain boundaries and for determining the average
doping density of preferentially diffused regions along the grain
boundaries are described.
Chapter 5 describes a small-signal admittance method for the determina-
tion of the grain-boundary surface-state distribution in the energy
gap for that part of a grain boundary which has been preferentially
diffused with phosphorus.
Chapter 6 describes various experimental attempts at suppressing
the grain-boundary dark recombination current. Many of these experi-
+
ments were done on 50-mil n -p mesa diodes. Such experiments were
inconclusive because of the presence of large surface and edge leakage
currents. The subsequent awareness of this fact motivated the fabrica-
+ +
tion of the 30-mil n -p and p -n dark diodes and solar cells used for
the research described in Chapter 4.
CHAPTER 2
DEGRADATION OF SOLAR-CELL PERFORMANCE
BY AREAL INHOMOGENEITY
2.1 Introduction
Areal inhomogeneity refers to the spatial variation of material
properties across the area of a solar cell. Unavoidable statistical
fluctuations in doping concentration across the area always occur. In
addition to this, fabrication procedures can sometimes result in large
areal fluctuations in recombination rates. For polycrystalline cells,
preferential diffusion of impurities down the grain boundaries and the
non-uniform spatial distribution of these grain boundaries can result in
drastic changes in the recombination rates across the area of the
cell [1-3]. The intent here is to quantitatively indicate the limita-
tions on silicon p-n junction solar-cell performance that can be caused
by areal inhomogeneity.
To deal with the areal inhomogeneity, a solar cell can be modeled
as the parallel combination of as many one-dimensional diodes [4] as are
needed to approximate the spatial distribution of the material proper-
ties. Resistive coupling of the diodes is then used to represent
interdiode paths for the recombination and shunt currents (Fig. 2.1).
The use of one-dimensional diodes in this modeling scheme is a first
approximation for polycrystalline solar cells having columnar grains in
which the grain diameter greatly exceeds the minority carrier diffusion
length in the base. This equivalent circuit representation of an illumi-
nated solar cell assumes the validity of the shifting approximation that
IL,l Sh,l IL,n RSh,n
RSB,1 SB,n
BASE -..
Figure 2.1 Parallel-subcell model of a p-n junction silicon solar cell with areal inhomogeneity.
RSE is series resistance in the emitter, RSB is series resistance in the base, RSh is
shunt resistance, IL is photogenerated current; and, ISCR and IQN are the dark recom-
bination currents in the space-charge region and quasi-neutral region, respectively.
m M
illuminated current is equal to the dark current shifted by the short-
circuit photocurrent [5].
We consider a simple ideal case in which only two subcells are used;
one subcell represents the good-quality material part of the cell, and
the other represents the poor-quality material part of the cell. The
series and shunt resistances are neglected, and the current in each
subcell is assumed to be one-dimensional (Fig. 2.2). For this simple
+
case, two types of areal inhomogeneity for an n -p Si solar cell are
considered. We define AG and Ap to be the areas of the good and poor
regions of the cell, respectively. Then the areal quality factor is
defined to be AQF = AG/A, where A = AG + A is the total solar cell area.
2.2 Type 1 Areal Inhomogeneity
For this type of areal inhomogeneity, the non-illuminated (or dark)
quasi-neutral current densities of the good-quality and poor-quality
diodes, JQNG and JQNP' respectively, are allowed to differ by several
orders of magnitude. For both diodes, the recombination currents are
dominated by recombination in the quasi-neutral regions rather than in
the space-charge regions. Both diodes have the same short-circuit
current density JSC. This type of areal inhomogeneity could occur if the
quasi-neutral emitter dark recombination current density JQNE experiences
drastic areal variations due, for example, to impurity clusters or
variations in the surface recombination velocity in the heavily doped
emitter. For these variations in JQNE to be important, JQNE must be the
dominant component of JQNP. To increase the likelihood of this, we will
17 -3
assume a high base doping concentration: N = 10 cm We assume
further a long base diode for which the short-circuit current density
JSC 25 mAcm2, and the base minority carrier diffusion length
J = 25 mA/cm and the base minority carrier diffusion length
I
+ AI
POOR AREA
Figure 2.2
Two-diode modeling of areal inhomogeneity in a solar cell, assuming the validity of the
shifting approximation, and neglecting the series and shunt resistances.
L n 100 pm. By the shifting approximation, the illuminated current of
the solar cell is
S= ISC DARK
AG[(JSC)G JQNGOexp(qV/kT)] + Ap[(Js)P JQNPOexp(qV/kT)]
= A {JSC -[exp(qV/kT)][(AQF)JQNGO + (1-AQF)JQNPO]}. (2.1)
In (2.1), JQNGO and JQNPO are the dark saturation quasi-neutral current
densities of the good and poor areas of the cell, respectively. In the
good portion of the cell we are assuming that JQNE is negligible compared
G102 -13
toJ Thus, J J qn D/LN 4.4 x A/cm2 at 250 C,
QNB' QNGO QNBO nn
where JQNBO is the dark saturation current density of the quasi-neutral
base as derived in [6], and D is the electron diffusivity in the base.
In the poor portion of the cell we are assuming that JQNB is negligible
compared to JQNE; thus JQNPO JQNE By defining JR = JQNPO QNGO'
(2.1) may be rewritten as
I = A {JSC -[exp(qV/kT)]JQNGO[AQF + (1-AQF)JR]}. (2.2)
From (2.2), we calculate [Appendix I] and plot the open-circuit voltage
VOC, the fill-factor FF, and the power conversion efficiency n as a
function of AQF with JR as a parameter in Figs. 2.3, 2.4, and 2.5,
respectively. These figures show that even a small poor-material area,
for example 5% (corresponding to AQF = 0.95) of the total solar cell
area, can drastically decrease the overall performance of the solar cell.
2 +
The parameter value JR = 10 corresponds to a silicon n -p thin-junction
solar cell with negligible recombination in the space-charge region,
JSCR JQN. The surface of the good-quality subcell is covered by a
SCR UiM
I
I I I I I I I.
1.0
1.0
TYPE 1 .- -
0.9 0.1/=^ L =LR
102= jQNO)POOR/Q NO GOOD =JR
TYPE 2 /
x 0.7 -
> 0.6
> 0.5
0.4 -
0.3
0.2 Z 0.01
0.1
0 II I I I I I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A GOOD /A TOTAL AQF
Figure 2.3 Normalized open-circuit voltage vs. areal quality factor.
0.8-
0.7
0.6
1.0TYPE 1
102 QNO POOR/( N ) D =--- I
0.1 = Lp /LGoo= DLRT 2
POOR GOOD TYPE 2
- -
A- - - - -L
0.5 <
0.01
I I
0.4
0.7
A OOD /AT TOTAL =AQF
Figure 2.4 Normalized fill-factor vs. areal quality factor.
0.4
0.3
0.2
0.1
0-.-
1.0 1.0
TYPE 1
0.9 -
0.8 '
102 =(NO POOR /(NO)GOOD =JR .
0 / 7
TYPE 2
0.1= LPOOR/LGOOD = DLR
/
/
7
7
7
7
0.1
- .-.
-- 0.01
- -- I I I I
0.4
0.7
0.8
GOOD TOTALL =AQF
Figure 2.5 Normalized power conversion efficiency vs. areal quality factor.
0.6-
0.5-
0.3
0.2-
0.9
passivating thermal SiO2, which yields a low surface recombination
velocity; and the surface of the poor-quality subcell is covered by an
ohmic contact.
2.3 Type 2 Areal Inhomogeneity
In this type of areal inhomogeneity, the p-type base lifetime Tn and
diffusion length L vary across the area of the cell for either of two
reasons: (i) as a result of large variations in the bulk recombination-
center density; or (ii) as a result of the presence of grain boundaries,
in which case T is an effective lifetime. The dark current J K comes
n DARK
mainly from the junction space-charge region (SCR) and from the quasi-
neutral base (QNB), whereas the short-circuit current JSC comes mainly
from the QNB. The shifting approximation then gives the illuminated
current density in each of the two subcells as
J = JSC DARK (2.3)
= JSC JQNB[exp(qV/kT) 1] JSCRO[exp(qV/mkT) 1] (l
in which the expression for JDARK is derived in [7]. A simplified form
of (2.3), which follows from the Sah-Noyce-Shockley treatment of the SCR
recombination current [8] is
J = JC JQNBO[exp(qV/kT) 1] JsCRO[exp(qV/2kT) 11, (2.4)
where
JC qn.W Dn/2L2. (2.5)
SCRO i SCR n n
The details are given in Appendix III. The total illuminated current of
the solar cell is then
12
I = AQF {(JS (JQNBO)G[exp(qV/kT) 1] (JSCRO)G[ex(qV/2kT) 1]}
+ (1 AQF) {(JSC)- (JQNBO)[exp(qV/kT) 1]
(JSCRO)[exp(qV/2kT) 1]}. (2.6)
To demonstrate this type of areal inhomogeneity, we let
16 -3
NAA = 10 cm and, for the good portion of the cell, let
2
(L)G = 100 pm and (JSC)G = 25 mA/cm We assume that at any point on
the cell area JSC"log(Ln), where the constant of proportionality is that
found in both experimental [9,10] and numerical [11] studies for
1 pm L L 100 pm. We define the diffusion length ratio to be
DLR = (Ln)p/(Ln)G. From (2.6), we plot VOC, FF, and n as a function of
AQF with DLR as a parameter in Figs. 2.3, 2.4, and 2.5, respectively.
Again, we notice that it is the poor-quality area of the solar cell that
-2
dominates the overall cell performance. The case for DLR = 10- corre-
sponds to empirical observations by Schwuttke [12] that the generation
lifetime of silicon ribbon material (and thus the related defect density)
varies in a random fashion across the area of the ribbon by between four
and five orders of magnitude.
2.4 Discussion
The above treatment assumes several idealities: resistance and
fringing effects are neglected; the good and poor-material is concen-
trated into two single-connected regions; the current flow in each region
is assumed to be one-dimensional. We now consider the effect of series
resistance on the illuminated solar cell.
In a solar cell with an adequate grid geometry, the dominant
component of series resistance will be that contributed by the bulk
material. This bulk component resists the current of majority carriers
that flow in from the contacts to support recombination with minority
carriers. In the poor-region of the cell, the recombination rate is
very high. Consequently, it is in the poor-region of the solar cell
where majority carrier currents, and thus series resistance effects, will
be most important. The series resistance must be included in the
parallel-subcell model if it causes a voltage drop which is a signifi-
cant fraction of the terminal voltage. A two-parallel subcell model
that includes the series resistance has been considered [4,13]. The
significant finding in [4,13] is that the inclusion of the series
resistance has the effect of decoupling the subcells. This is because
the series resistance tends to suppress the currents flowing between
subcells. The result of this is that the degradation of solar-cell
performance is not as severe as that predicted when the series resis-
tances are neglected. Estimation of the series resistance is given
in [4].
The idealization of two single-connected regions can be removed by
extending the two-parallel subcell model into an n-subcell model, where
n is sufficiently large to accurately describe the variation of material
properties across the area. A computer program that predicts the overall
performance of the solar cell when provided with the empirical parameters
of the subcells is given in Appendix II. A discussion of fringing
effects on the two-parallel subcell model is given in [4].
CHAPTER 3
A METHOD FOR EXPERIMENTAL ASSESSMENT OF THE SHIFTING
APPROXIMATION, WITH APPLICATION TO POLYSILICON SOLAR CELLS
3.1 Introduction
The shifting approximation that the illuminated current of a solar
cell is equal to the dark current shifted by the short-circuit photocur-
rent is discussed in detail in [5,14]. We report here the results of an
experimental investigation of the validity of the shifting approximation
for four types of polycrystalline Si solar cells.
A solar cell may be thought of as a system with two inputs and two
outputs. The inputs are the optical generation rate in the base and the
excess minority carrier concentration at the edge of the space-charge
region in the base (if the recombination current in the quasi-neutral
emitter is negligible); the corresponding outputs are the short-circuit
photocurrent and the dark recombination current, respectively. It is
shown in [5] that the shifting approximation is valid if this system is
linear; in [14] it is shown that the shifting approximation may remain
practically valid despite some nonlinearity in the system. In a poly-
silicon solar cell, if the shifting approximation is not valid, it is
most likely because the system has been rendered nonlinear by the depen-
dence of the material properties (e.g., the effective minority carrier
lifetime) on the illumination level or by the existence of a large series
or small shunt resistance. Recent data on the majority-carrier Hall
mobility in polysilicon material [15] indicate that the grain-boundary
potential barrier under illumination vanishes almost completely. This
14
15
suggests one possible origin of a dependence on illumination of the
effective minority carrier lifetime. Such a dependence would tend to
invalidate the shifting approximation.
The equivalent circuit diagram of a solar cell is shown in
Fig. 3.1. The shunt resistance RSh is assumed to be large enough so that
the shunt current may be neglected. The combination of the current
generator and the dark diode constitute the intrinsic system [5] of the
solar cell. Even though the series resistance RS may be large enough so
as to invalidate the shifting approximation for the overall solar cell,
it may nevertheless be the case that the shifting approximation is valid
for the intrinsic system of the cell. The approach taken here is to
measure the dark and illuminated I-V curves and then to compare these
curves after separating out the influence of RS. (Shunt resistance for
these cells was determined to be large.) If, after the correction for
RS, the dark and illuminated I-V curves are identical except for being
shifted or translated by the short-circuit current ISC, then the
shifting approximation is valid for the intrinsic system of the solar
cell. It is this sense of the shifting approximation that we are
concerned with in this chapter. The validity of the shifting approxima-
tion for a given solar cell considerably simplifies the theoretical
understanding of the performance of the cell, because then the dark I-V
characteristic and ISC can be considered separately [5].
3.2 Method for Analyzing the Measured I-V Curves
In Fig. 3.1, the voltage V across the terminals differs from the
voltage across the intrinsic system VIS by a voltage drop across the
series resistance RS. In the dark V = VIS + IDRS, whereas under illumi-
nation V = VS ILRS. In Fig. 3.2, we illustrate qualitatively the
F --- -------I
I I
I I '
I I
I I
II
I I
I I
I I
I I
L___-____i__-St
Intrinsic System
Figure 3.1
Equivalent circuit diagram of a solar cell. The dashed lines denote the intrinsic
system of the solar cell.
Figure 3.2(a)
Schematic representation of the current-voltage depen-
dencies of a solar cell.
(ISC)maxi
7
VOLTAGE (Arbitrary Units)
Figure 3.2(b)
Schematic representation of the current-voltage depen-
dencies of a solar cell with all curves shown in the
same quadrant.
dependencies ID V, IL V, and ISC VC for a solar cell. (IScmax
and (Voc)max are the short-circuit current and open-circuit voltage at
the maximum illumination level of the ISC V dependence. Note that
the ideal (reciprocal slope = 1) ISC V0C dependence is the I-V depen-
dence of the intrinsic cell system, since ISC = IS[exp(qV0C/kT) 1] =
IS[exp(qVIS/kT) i], where IS is the dark saturation current of the
cell. This is valid if RS is small enough so that the externally
measured ISC is equal to the short-circuit current of the current gener-
ator in Fig. 3.1. In general, the current-voltage dependencies will have
the following two properties: (i) the ISC VC dependence will be
negligibly influenced by RS; and (ii) the series resistance will shift
the IL V and ID V curves to the left and right of the ISC VC
curve, respectively.
It follows from the above discussion that the shifting approximation
will be valid if the series resistance RS is independent of the illumina-
tion level, and if the IL V and ID V curves are symmetrically shifted
with respect to the ISC VC curve. By symmetrically shifted we mean
that at any given current I (ISC)max, the distance AV1 between the
ID V and ISC VC curves when the ISC VOC curve is shifted into the
first quadrant is the same as the distance AV2 between the IL V and
ISC VOC curves when these two curves are rotated about the V-axis into
the first quadrant. This test for symmetry is shown in Fig. 3.3. Note
that the ISC V0C curves cross at (ISC)max/2 and, if symmetry holds,
then AV1 = AV2 = AV at (ISC)max/2. The series resistance of the solar
cell can then be calculated [16] as
RS = AV/[(IS)max/2] = 2AV/[(IS)max].
I
S about V axis AV AV
2 7 -
/D v
6
Z (1 )/2- -I V rotated AV--AV
r 5 5 about V axis
3 -\ \
4-
/ v
3-,
2-
0 1 2 3 4 5 6 (Voc)max
VOLTAGE (Arbitrary Units)
Figure 3.3 Illustration of the test for symmetry in the measured
I-V curves. The ISC-VOC curves cross at (ISC)max/2.
If the I-V curves are symmetric, then AV =AV2 at any
given current I < (ISC)max and AV, = AV2 = AV at
I= (IS) max/2.
Sc max
To determine whether RS is independent of the illumination level, we must
perform a second test which involves an independent determination of RS
in the dark. This is done by a method employing small-signal admittance
measurements [17] at a frequency of 4 MHz with zero de-bias. We then
compare this measured value of RS with the value of RS obtained from the
displacement of the curves. If the calculated and measured values of RS
agree, then this would indicate that (i) RS is independent of the
illumination level, and (ii) RS is the only reason for the voltage
displacement of the curves in Fig. 3.3.
3.3 Experimental Procedure
The ID V, IL V, and ISC VOC dependencies at 25.0C were
measured for the seven solar cells listed in Table 3.1. The illuminated
curves were obtained as tracings on an X-Y recorder by continuously
varying (with a helipot) the external load across the solar cell while
the cell was being illuminated by an Oriel solar simulator equipped with
an AMO filter. The level of illumination was varied from exactly 1 sun
intensity, which produced (ISC)max, to about 1/3 sun intensity. Calibra-
tion was accomplished by using a single-crystal standard solar cell
(device No. 7) calibrated at the NASA Lewis Research Center. The dark
I-V curves at 25.00C were obtained by varying the dark diode current with
a digital current source and measuring the corresponding voltage across
the diode terminals. The experimental setup for both the dark and
illuminated measurements (Fig. 3.4) used a four-point probe technique
so as to eliminate the effects of the contact resistance between the cell
and the measuring probes on the top, and between the cell and the vacuum
chuck on the bottom.
Table 3.1 The seven experimental solar cells for which the I-V dependencies were measured.
Device No.
Description
Total Area
(cm2)
Grain Diameter
milss)
4
(25P4)
5
(36P#28-L2)
6
(36P#3-L1)
+
n -p 50 pm thick epi-layer grown on Dow
Corning grade 2P polysilicon substrate.
SnO2 on n-type Wacker polysilicon
substrate.
Indium-Tin-Oxide on 0.1 0.3 Q-cm p-type
polysilicon substrate.
Phosphorus diffused on 5 0-cm Wacker
polysilicon p-type substrate.
Phosphorus diffused on 5 --cm Wacker
silicon p-type substrate. One grain bound-
ary goes through the cell. This solar cell
is 30 mils in diameter and has an MOS guard
ring.
Same as No. 5, but with five grain bound-
aries.
4.45
4.0
10 150
20 70
20 50
20 70
RCA
Exxon
Colorado
State Univ.
UF
4.6 x 10-3
4.6 x 10-3
Diffused n -p single-crystal control. 2.0
Diffused n -p single-crystal control. 2.0
Source
Sandia
hv from solar simulator
Digital
Current Source
Figure 3.4 Four-point probe experimental setup for investigating the
validity of the shifting approximation.
(a) Setup for measuring IL-V and ISC-VOC
(b) Setup for measuring ID-V.
In both setups, X and Y are the voltage and current axes on
an X-Y recorder. The chuck is connected to a circulating
water bath. The temperature of the chuck is controlled by
the water bath and is measured by a thermocouple inside of
the chuck.
The small-signal admittance measurement for determining the value of
RS at zero de-bias in the dark was accomplished with an HP 4275A LCR
meter. The ac-signal was 10 mV. The values of the series resistance
determined by the small-signal admittance method and by the method of
comparing the dark and illuminated I-V curves are shown in Table 3.2.
3.4 Experimental Results
The symmetries exhibited in the I-V curves, Figs. 3.5 3.9, along
with the corresponding data on the series resistance, suggest that the
shifting approximation is valid for all of the devices measured. From
Table 3.2, note that the values of RS for devices No. 5 and 6 are much
higher than those for the other devices. This is primarily because
devices No. 5 and 6 are much smaller in area than the other devices. If
we approximate the current flow as one-dimensional, then the value of RS
2
for these two devices normalized to an area of 2 cm would be about
0.07 Q.
3.5 Discussion
As mentioned in Section 3.1, it is possible in polysilicon material
that the effective minority carrier lifetime will depend on the illumina-
tion level. The likelihood of this dependency is increased if (i) the
illumination-induced lowering of the grain-boundary potential barrier
significantly changes the effective surface recombination velocity at the
grain boundary, and (ii) the intragrain base minority carrier diffusion
length L is greater than or equal to the average grain diameter dG.
Thus, it is particularly interesting to investigate the validity of the
shifting approximation in polysilicon solar cells where L dG. All six
of the polysilicon devices in Table 3.1 have relatively large grain
Values of series resistance by the small-signal admittance
method and by the method of comparing the dark and illumi-
nated I-V curves.
RS by small-signal
admittance method
(Q)
0.43
0.42
by comparison
the I-V curves
(2)
0.51
0.29
0.30
0.55
0.56
0.05
Table 3.2
Device No.
0.05
E 100
I-
z
w
80
60-
IL -V
40 -
20-
0 I I i i
250 300 350 400 450
VOLTAGE (mV)
Figure 3.5 I-V curves for device No. 1.
140
I
120 T = 25.00C
A = 4.0 cm2
Isc141mA 1 SunAMO
Voc = 539mV I
100 \ -
\ l
I VI
40 I- -
280
z
I.I
l ID.v
60
40 IL ---- \ ---- --V--- l --- ----
I
Iv-
40 ~II
I
20 I
I
0
300 350 400 450 500 550 600
VOLTAGE (mV)
Figure 3.6 I-V curves for device No. 2.
375 400 425
VOLTAGE (mV)
Figure 3.7 I-V curves for device No. 3.
< 20
I-
w
cc
o 15
10
450
5-
0 -
350
Z 20
E
\ /
w
- 15-
/
I-v // \
10 I I
5
450 475 500
VOLTAGE (mV)
Figure 3.8 I-V curves for device No. 4.
Figure 3.9
Curve tracer photographs of the dark and illuminated I-V
curves for device No. 6. The symmetry of these curves is
displayed in this fashion rather than in the fashion of
Fig. 3.3 because the voltage displacement of the curves
is very small. ISC = 60 pA, V0C = 467 mV.
(a) Dark I-V curve.
(b) Illuminated I-V curve.
diameters except device No. 6 (see Fig. 3.10) for which L = dG. L was
determined by applying the method of [7] to the dark I-V dependence of a
small grain-boundary-free device and was found to be about 130 Um. For
the small grain-boundary-free device, ISC = 62 pA and VC = 497 mV;
whereas, for device No. 6, ISC = 60 pA and VOC = 467 mV. The 30 mV
decrease in V0C resulting from the presence of the grain boundaries
corresponds to approximately a threefold increase in the dark recombi-
nation current. These data show the degrading effect that the grain
boundaries have on the performance of device No. 6. We note, however,
that the shifting approximation remains valid for device No. 6 in spite
of the very strong influence of the grain boundaries on VOC.
Though no general conclusions can be drawn from these experimental
results, they suggest that the nonlinearities introduced by illumination
levels of one sun are insufficient to invalidate the shifting approxima-
tion in polysilicon solar cells. If this is true, then the simplifica-
tions afforded to solar cell theory by the shifting approximation remain
intact for polysilicon. We emphasize that the experimental method
described here provides a general technique for assessing the validity
of the shifting approximation for solar cells made from polysilicon and
other material, including single crystal, polycrystalline, and highly
disordered semiconductors.
Figure 3.10 Microphotograph showing a top view of device No. 6. The
device is 30 mils in diameter and has five grain bound-
aries going through it. The five small white circles
are the top ohmic contacts, and the white annulus is an
MOS guard-ring gate that overlaps the p-n junction dif-
fusion edge.
CHAPTER 4
EFFECTS OF GRAIN BOUNDARIES ON THE CURRENT-VOLTAGE
CHARACTERISTICS OF POLYSILICON SOLAR CELLS
4.1 Introduction
The performance of polycrystalline solar cells is limited by the
effects of the grain boundaries (GB's) on the current-voltage (I-V)
characteristics. The I-V characteristics determine the efficiency of a
solar cell. Several theoretical models of the GB's and their influence
on the recombination currents have been published [18-23]. The purpose
here is to experimentally investigate the effects of the GB's on the
dark and illuminated I-V characteristics of polysilicon p-n junction
solar cells. The analysis of the experimental I-V characteristics, with
the help of the theoretical models [18-23], will lead to a determination
of the parameters which govern the recombination at the GB's. It will
also lead to a determination of the dominant GB current components.
Most experimental studies on polycrystalline solar cells have been
done on large area (~ 1 cm2) devices which, in general, contain hundreds
of GB's. Due to nonuniform distribution of the GB's and nonuniform grain
geometries, it is difficult to obtain reproducible results on such cells;
thus, an evaluation of the effects of the GB's and different fabrication
procedures on the performance of the cells is difficult. Moreover, poly-
crystalline solar cells usually exhibit large leakage currents which can
mask the contribution of the GB's.
To overcome these difficulties, most of the work reported here was
done on small-area (30 mil in diameter) diodes which contain at most a
few GB's. On such devices the total length and the area of the GB's can
be found, and comparison among the devices can be easily made. The
length and area of the GB's have to be exactly known in order to derive
quantitative results concerning their parameters from the data. Large
area devices containing many GB's can be used mostly for qualitative
studies only.
To assure that the surface and edge currents are suppressed, the
diodes were fabricated with a thermally-grown silicon dioxide on the top
surface and an MOS guard-ring gate overlapping the edge of the diffused
layer, as shown in Fig. 4.1. The surface recombination current compo-
nents, including the surface inversion channel current and the recombina-
tion current through the surface states at the periphery and at the GB's
intersecting the edge of the diode, are suppressed by applying a suit-
able gate voltage to the MOS guard-ring. The remaining current compo-
nents are then only the bulk intragrain and grain-boundary components.
An additional and very important advantage of using small-size diodes is
the possibility of placing diodes inside of grains. This grain-boundary-
free (GBF) diode will then allow us to measure the properties of the
bulk of the grain and will serve as a reference diode for comparison
with a nearby diode containing GB's.
Diffusion of impurities into the polycrystalline material is
expected to proceed preferentially down the GB's [24-26]. The width and
average doping density of this enhanced grain-boundary diffusion region
is determined for the case of phosphorus dopant. Also, the current
components associated with the enhanced grain-boundary diffusion will be
discussed and analyzed here for thefirst time.
Gate
SiO2
Figure 4.1
+
(a) Cross section of the 30 mil in diameter n -p solar cell
with MOS guard ring showing one columnar grain boundary
crossing through the middle.
(b) Top view showing five metal contact circles (5 mil in
diameter) and one grain boundary crossing the cell. The
total metal coverage is 20%.
xi Jn-
Grain Boundary
P type
- -1 1
4.2 Fabrication of Devices and Evaluation of Preferential Grain-Boundary
Diffusion
+ + +
Both n -p and p -n diodes were studied. The n -p diodes were
fabricated on 5 0-cm p-type Wacker polycrystalline silicon substrates.
The phosphorus emitter was predeposited at 900C for 30 minutes followed
by a drive-in diffusion at 1050C for 40 minutes. Wet oxide about
0
3000 A thick was grown on the top surface during the drive-in step. The
junction depth xj was 1.8 pm and the sheet resistance was 8 n/square.
+
The p -n diodes used 0.3 Q-cm n-type Wacker substrates. The boron was
predeposited at 900C for 25 minutes. The drive-in was done at 10000C
for 120 minutes resulting in a junction depth of 0.8 pm and sheet
resistance of 800 0/square. The top surface was passivated by an Si02
layer, about 3000 X thick, grown during the drive-in step.
In Fig. 4.2(a), a grooved and stained section [27,28] of an n -p
diode shows a preferential diffusion of phosphorus down the GB's [24-26].
A copper staining solution was used. The preferential diffusion spike is
about 6 pm deep and is uniformly about 2.3 pm wide. (We refer to a
longitudinal cross-section of a preferentially diffused grain-boundary
region as a diffusion spike.) It was determined by an investigation of
the cross-sections of the 15 mil thick substrate wafers that the GB's go
all the way through the material at an angle between 200 and 450 to the
normal [29]. This angle is referred to as a in Fig. 4.2(b). The GB
also cuts the groove at another angle which is referred to as @ in
Fig. 4.2(b). Both of these angles were considered in calculating the
depth of the n-spike. No phosphorus preferential diffusion spikes were
observed after the 30 minute 900C predeposition step; however, we
measured 5-10 pm spikes after 48 hours heating at 600C after the
predeposition. For the p -n diodes, no preferential diffusion spikes of
4
I -
'A.
II
'Ir
U
N
*,4V~
Figure 4.2(a)
Grooved and stained section of the n -p junction showing preferential
phosphorus diffusion along the grain boundary. A copper staining solu-
tion was used. The junction depth is about 1.8 pm; the depth of the
preferential diffusion is about 6 pm. A commentary on the reliability
of groove and stain results appears in Appendix X.
Figure 4.2(b)
Diagram of a groove and stain sample showing the two
angles, a and a, that define the orientation of the
GB-plane with respect to the substrate wafer. The spike
depth d = d/cos a, where d is the depth for the plane
with a = 0; the angle 8 does not influence d .
n
boron were observed for various diffusion schedules employing 900-1000C
for 20-120 minutes. In order to remove the uncertainties in delineating
the very narrow diffusion spikes by the groove and stain method, we have
developed an electrical measurement procedure which can positively
identify the presence or absence of a preferential diffusion down the
GB's. This electrical measurement will also yield an average doping
density within the preferentially diffused region.
A test structure for determining the presence of preferential GB
+
diffusion is demonstrated for the case of n -p mesa diodes made from a
wafer that received a 30 minute 900C phosphorus predeposition followed
by a 40 minute 10500C drive-in. As stated previously, for this diffusion
GB
schedule x. = 3x. in the bulk. The mesa structures are formed by
J J
masking small dots on the top of the diffused wafer with wax or photo-
resist and then etching off about a 2 pm layer of silicon. The etch
depth is just slightly in excess of x.. This leaves about a 4 pm depth
J
of preferentially diffused GB's around the mesa diodes. Figure 4.3 shows
four mesa diodes, the top two of which are GBF, and the bottom two of
which are connected by a GB.
Two tests can be made on these mesa diodes. The first test, alluded
to above, is the measurement of conductance between the top two diodes
and between the bottom two diodes. The top two diodes represent back-to-
+
back n -p junctions, and very little current will flow between them when
a bias is applied. The bottom two diodes, which are connected by the
diffused GB, constitute a structure similar to a JFET and will show a
current which is dependent on the conductance of the diffused GB channel.
For the second test, reverse-biased capacitance is measured between the
n -diffusion layer and the p-type substrate. The capacitance measured
Figure 4.3 Test structures for determining the presence of a prefer-
ential diffusion in the GB. The top two mesa diodes are
GBF; the bottom two mesa diodes are connected by a prefer-
entially diffused GB conducting channel.
between one of the top two diodes and the substrate is merely the
capacitance of the GBF diode. The capacitance measured between the small
diode in the bottom pair and the substrate will be equal to the sum of
the capacitances of the two diodes on the bottom plus the capacitance
contributed by the diffused GB channel. Figure 4.4 shows the capacitance
of a 10-mil GBF diode and the capacitance of a 10-mil GB diode which is
connected by a diffused GB channel to a much larger GB diode.
Two conclusions can be made based on Fig. 4.4. First, the capaci-
tance of the 10-mil GB diode is much larger than the capacitance of a
10-mil GBF diode at VR = 0 V. This indicates that the diffusion sched-
ule used in the fabrication of these devices (30 minute 900C predeposi-
tion followed by a 40 minute 10500C drive-in) resulted in a preferential
diffusion of phosphorus down the GB's. Such devices that share a common
GB will be electrically connected by that diffused GB. The 6 pm deep
preferential diffusion will increase the p-n junction area of the 10-mil
GB diode by only a few percent. This slight increase in the area cannot
account for the large value of capacitance at VR = 0 V, which is due
mainly to the contribution from the larger GB diode. Secondly, at
VR 3 V, the capacitance of the 10-mil GB diode drops to the level of
the 10-mil GBF diode, and the two curves are identical for VR Z 3 V.
The reason for this capacitance dependence on reverse bias is a widening
of the depletion layer in the channel by the reverse bias until the
channel is completely depleted. The GB-connected diodes thus become
electrically disconnected due to a depleted nonconducting channel.
+
This capacitance experiment also was done on n -p devices made by a
30 minute diffusion at 900C and on the p -n diodes. No capacitance
difference comparable to that exhibited in Fig. 4.4 was observed. This
0 1 2
REVERSE BIAS VOLTAGE (V)
Figure 4.4
Capacitance versus reverse bias measured on the test struc-
tures of Fig. 4.3. The insert shows the measured conduc-
tance of the JFET-like channel between the two bottom GB
diodes of Fig. 4.3 as a function of voltage between the
diodes.
indicates that no inversion layer was created at the intersection of
the GB's with the silicon surface or along the GB's in the p-type bulk;
such an inversion layer could also lead to results shown in Fig. 4.4.
The stained width of the diffusion spike, Fig. 4.2(a) is about
2.3 pm. Based on a series of experiments (see Appendix IV) on both
+ +
n -p and p -n junctions, we concluded that the stained region using the
copper stain solution includes the p-n junction space-charge-region
(SCR) from both sides of the diffusion spike. Inside the spike, a GB
potential barrier is created due to the GB surface states [18,19]. The
reverse bias, VR, required to deplete the channel of width W (Fig. 4.5)
n
will have to push the edge of the p-n junction SCR to the edge of the
grain-boundary SCR. By using a linearly-graded junction approximation
for the p-n junction, we estimated that the average channel doping den-
16 -3
sity is approximately NDD 1 x 10 cm and W = 0.5 pm. We assume
as a first approximation, that the width of the grain-boundary SCR is
independent of the reverse bias VR.
The conductance test described above using the simple structure
created by black wax masking, which results in large spacing between
the GB diodes ( 10 mil), will yield rather small channel current. It
was difficult to separate this channel current from the reverse leakage
16 -3
current of the mesa diodes for the device with NDD 10 cm The
conductance method is demonstrated here for another, more heavily doped,
structure diffused at 1050C for one hour which resulted in a preferen-
tially diffused region 11 pm deep; the stained width of the n-spike was
about 3.6 pm and W = 1 pm. The insert of Fig. 4.4 shows the measured
n
channel current of this JFET-like structure versus the applied voltage.
The existence of a large conductance confirms the occurrence of a
Z &e
Figure 4.5
WSCR
+
Section of an n -p diode with a columnar GB and a preferen-
tially diffused n-spike with depth d The broken lines
indicate the edges of the p-n junction SCR and the edges of
the GB potential barrier within the n-spike. For the polar-
GB GB
cities indicated, I IB and IB are negative currents.
QNB 1B QNB
The p-n junction along the preferentially diffused GB is
modeled, as a first approximation, as having a square-well
shape with a one-sided step junction at the bottom of the
square well. A more accurate model for the diffusion pro-
file along the GB is one in which the p-n junction appears
wedge-shaped with phosphorus dopant extending below the a-
pex of the wedge. In such a model, the phosphorus concentra-
tion below the apex gradually diminishes to zero.
preferential diffusion. In addition, we can calculate the value of NDD
from the linear portion of the channel current-voltage dependence and
geometry [30]:
NDD = GPGB/qA = 2 x 1017 cm-
-3
where G = 1 x 10-3 mhos is the conductance of the linear portion of the
-4
channel I-V dependence, GB = 23 x 10 cm is the length of the channel,
P = 400 cm2/V-sec is the estimated electron mobility in the channel, and
-7 2
A = 2.2 x 10 cm is the cross-sectional area of the channel. The gate
voltage between the p-type substrate and one of the GB diodes (source)
can be used to modulate the channel conductance and thus to study the
diffusion profile within the preferentially diffused region. NDD also
can be estimated from the reverse gate bias required to deplete the
channel and decrease the channel current to zero; this was not possible
in this device because of the large NDD and the wide channel.
4.3 Analysis of the I-V Curves
A solar cell under illumination is forward biased; the external
current is given by the photogenerated short-circuit current ISC minus
the dark current ID, providing that the superposition principle is
valid [5]. In the following discussion we will concentrate on the open-
circuit voltage VOC of the cell, and also on the dark current ID, since
for many cells, VOC is degraded much more by the GB's than is ISC [31,32].
The illuminated I-V characteristics will be considered in more detail in
Section 4.3.3.
In order to analyze the I-V characteristics of GB diodes, we first
define and describe all current components. Figure 4.5 shows a cross-
section of a portion of the n-p diode showing one columnar grain
section of a portion of the n -p diode showing one columnar grain
boundary with the n-diffusion spike. We assume that the GB is perpen-
dicular to the top surface. The presence of the GB and the diffusion
spike will result in dark current components in addition to those present
in the GBF device. All these additional components, shown in Fig. 4.5,
are designated by a superscript "GB"'. The total dark current is equal to
the sum of all current components:
GB GB GB GB GB GB GB
D QNB + QNE +SCR SCR +SCR' +B QNB + QN + h. (4.1)
The dark current of the GBF diode is given by
ID = IQNB + QNE + ISCR + V/RSh. (4.2)
The current components in (4.1) and (4.2) are defined as follows: IQNB
and IQNE are the recombination currents within the quasi-neutral base
+
and emitter, respectively, and orginate from the lateral n -p junction.
GB
I and ISG are the recombination components due to the carrier
SCR SCR
recombination in the bulk space-charge region (SCR) adjacent to the
lateral n -p junction, and adjacent to the n-diffusion spike, respec-
tively. IGB is the recombination current at the GB within the
SCR
SCR [23]; 1GB is the recombination current at the grain boundary adjacent
G
to the quasi-neutral base (QNB) region; INB is the current component
QNB
injected from the diffusion spike and recombining within the QNB region;
IGB is the current injected from the substrate into the emitter diffu-
QNE
sion spike and recombining within the spike and at the GB surface; and
GB
RS and RGB are the shunt resistances of the GBF and GB diodes, respec-
Sh Sh
tively. The components IQNB IQNE and ICR in (4.1) and (4.2) are
nearly equal in a special case. Their equality requires that the
fundamental kinetic parameters (recombination-center density, capture
cross-sections, etc.) describing recombination in the bulk of the GBF
diode are the same as those relating to the bulk (as opposed to the GB
surface) of the GB diode under study. Additionally, (a) for ISCR
equality, the volume of the SCR straddling the lateral n -p metallurgical
junction must be nearly the same for the GB and the GBF diodes; (b) for
IQNB equality, the electron diffusion length in the quasi-neutral base
of a GB diode containing one GB must be much smaller than the diode
diameter d; and (c) for IQNE equality, the effective hole diffusion
length (which includes the influence of drift) in the quasi-neutral
emitter of a diode containing one GB must be much smaller than the diode
diameter.
The presence of current components associated with the GB will
result in a complicated two-dimensional current flow inside the diode.
The problem can be greatly simplified using an empirical relationship for
the measured I-V curve [7] which expresses the measured I-V dependence of
the GB diode as a sum of three terms:
GB GB GB GB GB
ID = I[exp(qV/mX kT) 1] + IQN[exp(qV/kT) 1] + V/RSh, (4.3)
where IB is the lumped SCR saturation current component and ~B is the
GB
reciprocal slope factor of that component. IQN is the lumped saturation
QNO
current of all quasi-neutral current components, which have a reciprocal
GB
slope factor m = 1.0. The IB is generally a function of injection level
in low injection [23], and its reciprocal slope factor m can be different
from m = 1.0. However, for very large surface recombination velocity
SGB at the GB, which is the case for our devices as will be shown later,
GB
m = 1.0, and the IB can then be lumped together with the other QN
components. Equation (4.3) is strictly valid only if one of the SCR
components in (4.1) is dominant or if all SCR components in (4.1) have
the same reciprocal slope factor.
Similarly for the GBF diode:
ID = IX[exp(qV/mkT) 1] + IQN[exp(qV/kT) 1] + V/RSh. (4.4)
All I-V curves of diodes investigated in this work are described by (4.3)
or (4.4).
4.3.1 Space-Charge Region Current Components (n -p diodes)
Figure 4.6 shows the measured dark I-V characteristics for five
+
representative n -p diodes chosen from over 100 devices containing
either zero or a few grain boundaries. A summary including the total
length of the GB's and the parameters defined in (4.3) and (4.4) for
each of these diodes is shown in Table 4.1. A comparison of the data
for the GB diodes with the data for the GBF diode No. 1 shows the strong
effect of the GB's on the I-V characteristics. This is true even for
device No. 3 which had only one GB. The ratio of currents at 160 mV for
diodes No. 3 and No. 1 is about 40.
+
We now analyze the current components in the n -p GB diodes by
comparing the I-V curves of these diodes with the I-V curve of a GBF
diode, Fig. 4.6. The GB component of IB can be obtained by subtracting
the measured current of the GBF diode from the measured current of the
GB
GB diode, i.e., I ID. However, we observe that the GB components
dominate the current at small biases, below about 300 mV. In this range
GB
mX m 1.8; therefore we can write:
GB GB GB GB
I (100 300 mV) = ISCR + ISCR' + V/R h. (4.5)
D SCR SCR' Sh*
0 .1 .2 .3 .4 .5
VOLTAGE (V)
+
Figure 4.6 Measured dark I-V curves for five n -p solar cells: a GBF
cell (No. 1), a cell containing twins only (No. 2), and
three GB cells (No. 3, 4, and 5).
Parameter values for devices No. 1-8.
T = 25.0C, A = 4.6 x 10-3 cm2
Device No.
Type
Description
kGB
milss)
GB
I I0
XO' XO
(A)
GB GB
mX, mX QNO' QNO
(A)
1
(36P#13-L2)
2
(36P#4-L1)
3
(36P#28-L2)
4
(36P#25-L1)
5
(36P#3-L1)
6
(34P#4-L2)
7
(34P#1-D1)
8
(34P#1-L1)
+
n -p
+
n -p
+
n -p
+
n -p
+
n -p
+
p -n
+
p -n
+
p -n
GBF
twins only
one GB
several GB's
several GB's
GBF
twins only
several GB's
Table 4.1
SGB
(cm/sec)
2.1
4.8
4.0
9.9
5.7
1.1
1.8
7
x 1012
x 10-12
x 10-0
x-10
x 1010
x 109
x 1010
x 0-10
x 10
1.18
1.29
1.83
1.84
1.76
2.0
1.88
1.94
10-14
10-13
10-13
10-13
10-13
1-14
10-14
10-14
65
3.4 x 104
2.5 x 10
8.2 x 103
9.7
1.6
2.5
2.4
3.4
5.1
2.0
3
GB
The recombination in the SCR adjacent to the n-diffusion spike I was
SCR
analyzed by Sah-Noyce-Shockley [8] and is, in fact, just an extension of
the ISCR of the GBF diode. The area of the n-spike is A = 2kGBdn where
GB is the total length of the GB's in a diode and d is the GB
GB n
preferential diffusion depth. For the GB diodes No. 2-5 in Table 4.1,
2 -3 2
An << A, where A = rd /4 = 4.6 x 10-3 cm is the top area of a 30-mil
GB
diode, and thus ISCR' can be neglected. This component could be impor-
tant in solar or metallurgical grade material where the impurities can
be segregated at the GB's resulting in a very short lifetime in the SCR
GB
adjacent to the n-spike and a large ISC'. The recombination current at
SCR,.
the GB within the SCR can be expressed as [20,23]
GB GB GB (4.6)
SCR ASCR qiSGBexpqVm kT), (4.6)
where SGB is the GB surface recombination velocity at that part of the
GB adjacent to the bulk, Bm 2.0, and ASG 2 WSCR is the approxi-
mate area over which the GB recombination current is described by
GB
(4.6) [23]. WSR is the SCR width of the GB barrier [18,19]. This cur-
rent component is proportional to n.. Such a current will have an acti-
GB
vation energy of one-half of the bandgap (EG/2). ID was measured for
the GB devices in the temperature range from 2220K to 2860K. The slope
factor mX was almost constant in this temperature range. Figure 4.7
GB
shows the ID versus 1/T plot yielding activation energy
Ea= 0.59 eV = E/2. This result very strongly suggests that IGB is
the dominant GB recombination component at small biases below about
300 mV. The theoretical analysis [23] of IS R, based on idealizations,
SCR,
predicts mB = 2.0. Our data give mX mB = 1.76 1.84 at 250C with
GB
most of the devices having mX 1.8. By using (4.6) and an estimate of
10-9 -
Ea = 0.59 eV
10-11'
10-1r I I I I
3.4
3.6 3.8
1000/T (K-')
Figure 4.7
The dark current of a GB diode measured at 150 mV versus
1/T showing an activation energy Ea = 0.59 eV EG/2.
a. G
10-1o
z
w
C-)
1 I I
WsGB = 0.5 pm, we can calculate the approximate values of SGB. The
results in Table 4.1 indicate that electrically active GB's do not have
a uniform SG. By using the values for IGB = IRGB where
S GB X SCRO
ISRO = ASCR qnSGB, we can calculate the average current per unit
GB -13
length of GB, ISCRG 4.6 x 10 A/pm, with the average slope factor
GB
S mS = 1.81. These two average values can then be used to predict
the behavior of the I-V curves for different grain sizes. The average
GB recombination velocity for devices No. 3-5 is SGB = 2.2 x 104 cm/sec.
The device No. 2 which contains only twin boundaries is almost identical
to the GBF device.
4.3.2. Quasi-Neutral Region Current Components (n -p diodes)
GB
Each value of IQNO, Table 4.1, which includes the QN components
GB
plus the IB from (4.1), is an increasing function of kGB' as expected.
IQNO for the GBF diode is a function of only the doping density in the
p-type substrate and the minority electron diffusion length. The IQNE
can be neglected because of the low doping density in the base
NAA = 3 x 101 cm-3 [7]. The diffusion length L = 130 pm in the GBF
AA n
diode was obtained from the QNB component of the dark current [7]. In
the small area devices, such as the 30 mil diodes used in this work, the
two-dimensional spreading effects will be important if the radius d/2 is
comparable to the diffusion length L [331. For the GBF diode, the
radius of the diode is d/2 = 380 Pm > Ln, which will assure almost one-
dimensional current flow; but for the 30 mil diodes with several GB's,
the grain size could be comparable to the electron diffusion length and
the injected electrons will combine within the grain and at the GB's.
The recombination current in the device can then be described by
Shockley's filament theory [34]:
54
IQN= (AGqn /NAA)[Dn/T(eff)] exp(qV/kT), (4.7)
where AG is the grain area and Tn(eff) is the effective electron life-
time [34] which includes recombination within the grain and at the GB's.
The effective lifetime can be calculated exactly from (4.7) for columnar
grains with rectangular or cylindrical geometry [34]. This is not the
case for the Wacker polycrystalline silicon material. We can, however,
GB
estimate an average effective electron diffusion length L in the poly-
n
crystalline material by measuring the X-ray induced current [35] on a
large area polycrystalline cell and then comparing this current to the
response of a single-crystal cell with known diffusion length. The
2 +
experiment used a 2 cm polycrystalline n -p cell with average grain
GB
size of 500-1000 pm and yielded L = 70 pm which is smaller than
L = 100-130 pm for the GBF diode. This result demonstrates that the
n
+
effect of the GB's on the electrons injected from the top n -p junction
in diodes No. 2-5 cannot be neglected. In addition, the GB's will
contribute to the I QN because of the preferentially diffused n-region
which increases the total area of the p-n junction and contributes
additional current components, as was discussed in Section 4.3.1. The
GB
separation of IQN0 as defined in (4.3) into components for the general
case requires solution of a two-dimensional boundary value problem which
is beyond the scope of this work. We can, however, approximately
accomplish this for our devices by considering the components separately
GB
and identifying the dominant one. Neglecting RSh, there are three quasi-
GB
neutral current components associated with the grain boundary: IQNB,
GB GB
IQNE, and I We now consider each of these components.
GB
4.3.2.1 I
QNB
GB
The IQNB component is due to the electrons injected from the prefer-
entially diffused vertical GB region with an area A into the base.
n
This current component, coupled with the electron current injected from
the top lateral junction with an area A, will result in a two-dimensional
current flow in the base. An accurate solution of such a two-dimentional
problem is not presently available. If, however, A << A, then IGB can
n QNB
be neglected. Considering our n -p devices from Fig. 4.6 and Table 4.1,
the ratio An/A is largest for device No. 5 and is only about 0.1. Thus,
as a first approximation, we will neglect IQB in further analysis of
QNB
GB
our devices. IQ will be important for devices with large depths of
preferential diffusion in the GB.
GB
4.3.2.2. I and GB Passivation
QNE
GB
INE is due to the holes injected from the p-type substrate into the
preferentially diffused region of the GB and recombining inside that
region. The current transferred through the narrow n-spike, Fig. 4.5,
can be treated in a way similar to the current transferred through a
narrow emitter region of a solar cell. A detailed investigation of this
problem was done by Shibib et al. [36]. The recombination current in
the n-spike will be a function of the hole lifetime T in the bulk of
P
the spike, the hole transit time Tt through the spike, and the surface
recombination velocity SGB at the GB inside the spike. The transit time
can be expressed as [36]
T (W2/2D ) + (W /SGB (4.8)
t n p n p(eff)
where D is the average diffusion coefficient corresponding to the
average doping density NDD in the preferentially diffused GB region, W
'D n
GB
is the width of the QN region of the n-spike, and S Gf is the effec-
p(eff) i
tive recombination velocity for holes at the edge of the GB SCR in the
n-spike [18,23]. We will assume now that T << T and check later for
t P
self-consistency. For Tt << Tp, the n-spike will be transparent to the
injected minority holes and the saturation current of the holes
recombining at the GB surface is [36]
GB 2- GB
IQNEO = (Aqni /NDD) [/[(/Sp(eff) ) + (Wnp) (4.9)
This current has to be separated from two other IQN components:
GB GB
IQNB and IB We approach this problem by calculating IN from (4.9).
GB n
This requires a knowledge of SGB Let us assume that S inside of
p(eff)" GB
the n-spike is equal to the surface recombination velocity at that part
GB
of the GB adjacent to the bulk, SB. SG is obtained from IX in
GB GB XO
Table (4.1). From [18,23]:
GB n
Sp(eff) SGB exp(q/kT), (4.10)
where cB is the barrier height of the GB-SCR. By using the estimate
**
GB 6**
S p( ) = 1 x 10 cm/sec. By using this value in (4.9), along with the
p(eff)
16 -3 2
values NDD 1 x 10 cm D 11 cm /sec, and W = 0.4 pm, we obtain
GB -13
IQNO = 2 x 10 A for the device No. 5. This value is close to the
QNEO
GB
measured value of I (see Table 4.1), which implies that I can
QNO QNE
dominate the dark current of the GB diodes if the preferentially
diffused GB regions are lightly doped and SGB is large.
GB
57
GB GB
Note, however, that I EGO exp(qV/kT) is a one-dimensional
QNE QNEO
current linearly dependent on the total length of the GB's in the diode,
GB GB +
since A = 2k d .Therefore, if I dominates the I of the n -p
n GB n QNEO QNO
GB
devices in Fig. 4.6 and Table 4.1, then I should be a linear function
QNO
GB
of kGB. The measured IQNO is, however, much less than linearly depen-
dent on GB; compare, for example, diodes No. 3 and No. 4 in Table 4.1.
GB GB
This implies that the IQN is dominated by IQNB and I We can thus
QN QNB B
make a reasonable estimate that IGB 0.1 IGB Then for device
QNEO QNO.
No. 5, (4.9) implies that S GB ) 4 x 10 cm/sec, and (4.10) implies
p(eff)
that SGB 400 cm/sec. This suggests that the recombination velocity of
the GB was lowered by the diffusion of phosphorus into the GB. Similar
conclusions are also valid for other devices investigated.
GB
We now use the above calculated value for S in (4.8) and
p(eff)
obtain T < 0.1 psec. Thus, our assumption that Tt << T is reasonable
t t p
16 -3
for a relatively low doped n-spike, such as NDD 10 cm found in our
devices.
GB
4.3.2.3. IB
B
IB is the electron current recombining at the GB adjacent to the
QN base. This component will not be linearly proportional to GB due to
the two-dimensional nature of the electron current flow injected from
GB
the top junction. The importance of IB will depend on the grain size
dG, the electron diffusion length Ln, and the surface recombination
velocity SGB. Its influence on the total quasi-neutral current, IQN,
as defined in (4.3) and (4.7) can best be demonstrated by the dependence
of Tn(eff) on these parameters. For our devices with large
SGB ~ 10 cm/sec, we can write [23,37]:
/nef) = (/)[ + 2r 2(L /d )]. (4.11)
n(eff) n n G
For dG >> L Tn(eff) T and IQNB dominates; for small dG, Tn(eff) < T
G n n(eff) n QNB G n(eff) n
and I becomes important. For device No. 5 with diameter d = 760 pm
and five GB's, the approximate grain size is dG = 150 pm and
Tn(eff) 0.9 psec. This value is to be compared to Tn = 6 psec,
GB
corresponding to L = 130 pm for the GBF diode No. 1. For IB to be
n B
important, the intragrain base electron diffusion length, Ln, has to be
larger than the preferential diffusion depth d = 4 pm. From our
GB
experiment we found L = 130 pm >> d The effect of IB will obviously
n n B
increase with decreasing d .
n
4.3.3. Illuminated I-V Curves (n -p diodes)
Table 4.2 shows the summary of results of measurements on illumi-
nated diodes. The short circuit current ISC is almost constant except
for device No. 5, but VOC decreases slightly with increasing RGB. This
is consistent with previous results [31,32] and also with a recently
proposed model [38] for devices with grain size dG > L The decrease
G n
in VO is due to increased ISCR which is directly proportional to kGB'
GB
and also due to increased I The slight decrease in IS for device
No. 5 is because, in this device with five GB's within the 760 pm diode,
the average grain size is comparable to L = 130 pm; thus, some of the
light-generated electrons will recombine at the GB's and will not
contribute to the external measured ISC.
The preferentially diffused n-regions can contribute to the ISC if
d ~ L For the devices studied, d << L and no increase in ISC is
n n n n SC
observed for the GB diodes. The fill factor decreases with kGB as
expected due to the increasing importance of IX with mX > 1.0.
Table 4.2 Parameter values
illumination. T
Device No.
for devices No. 1-8 while under 1 Sun AMO
-3 2
250C, A = 4.6 x 10 cm
ISC
(PA)
VOC
(mV)
497
497
489
485
467
494
480
496
0.81
0.80
0.78
0.75
0.77
0.78
0.78
0.78
Another important consideration for the preferentially diffused
n-regions is the possibility of pinching-off the narrow-n-channel due to
the current passing through it. The detailed description of current flow
through the n-region, both in the dark and under illumination, is very
complicated; but we can roughly estimate the current required for pinch-
off. As a first approximation we will assume that the current leaves or
+
enters the n-region at a distance d /2 from the top n -p junction, i.e.,
n
in the middle of the n-region. We can then treat this n-region as the
channel of a JFET with floating drain and calculate the saturation
current of the n-region channel for our geometry and parameters [30].
This approximation gives Isat = 2mA. The available ISC for 1-sun AMO
conditions for the 30 mil diameter device is only about 0.14 mA [31].
This indicates that the preferentially diffused n-region will not be
pinched-off at 1-sun, even if the entire photogenerated current is
collected by this region. The n-region, however, could be pinched-off
at high concentrations of illumination or in devices which have narrower
Wn or lower NDD Isat also can be directly measured by using the
conductance method on suitable structures shown in Fig. 4.3. This
measurement is demonstrated in the insert of Fig. 4.4.
The device No. 2 which contains only twins has about the same VC
and ISC as the GBF diode.
+
The n -p diodes in Tables 4.1 and 4.2 are representative illumi-
nated diodes (solar cells) from Run 36P. Parameter values for additional
diodes in Run 36P are presented in Tables 4.3 and 4.6. The fabrication
schedule for Run 36P is presented in Appendix VIII.
I
Table 4.3
Description
Parameter values for additional diodes in Run 36P
GB GB -3 2
ICRO IX. T = 25.0C, A = 4.6 x 10 cm
GB GB GB
GB XO' X mX,' X QNO' QNO
milss) (A) (A)
GBF
twins only
one GB
several GB's
several GB's
several GB's
one GB
several GB's
Averages for GB diodes including
the GB diodes in Table 4.1:
64 5.2 x 10 0 1.76
Diode
SGB
(cm/sec)
36P#10-D1
36P#4-D1
36P#14-D2
36P#9-D1
36P#1-D2
36P#34-D1
36P#8-L1
36P#9-L2
8.2
4.7
1.9
2.4
2.6
7.4
6.1
1.4
4.1
1.9
2.4
1.9
10-13
10-12
10-
10-10
10-10
10-10
1010
10-10
10-14
10-13
10-13
10-13
10-13
10-13
1.17
1.32
1.64
1.71
1.70
1.80
1.73
1.81
8.4
1.0
9.2
8.8
2.4
x 101
x 104
x 103
x 103
x 104
1.4 x 1013
2.2 x 10-13
2.6 x 104
3.0 x 10
10-13 4
2.5 x 10 2.0 x 10
4.3.4 Grain-Boundary Passivation by Hydrogenation Treatment
+
Tables 4.4 and 4.6 present parameter values for the n -p diodes in
Run 37P. The Run 37P diodes underwent the same fabrication schedule as
the Run 36P diodes, except for an additional processing step [39] that
was intended to tie up dangling bonds along the grain boundaries with
monoatomic hydrogen [39,40]. After the 37P wafer had undergone the
drive-in step, the oxide grown during the drive-in was covered with a
6000 X layer of vacuum-evaporated aluminum. The wafer was then sintered
at 450C for 12 hours in dry N2. The aluminum was then removed by
chemical etchant and the processing of the wafer was continued as in
Run 36P. This additional processing step is known to tie up dangling
bonds at the Si-SiO2 interface in MOS devices by the generation of some
form of active hydrogen at the Si02-Al interface [41]. By following
this procedure, we were able to use the diodes of Run 36P as a control
group for determining the effect of the hydrogenation step on the diodes
of Run 37P. The 37P wafer was not sintered after the ohmic contacts were
formed so as to avoid out-diffusion of the hydrogen from the GB's [39].
A comparison of the values of SGB for Run 37P in Table 4.4 with
the values of SGB for Run 36P in Table 4.3 shows that the hydrogenation
step has a negligible effect on SGB. By applying both the method of [7]
and the method of [35] to several GBF diodes in Run 37P, it was deter-
mined that the hydrogenation step lowers the intragrain base minority
carrier diffusion length L from about 130 pm to about 90 ilm. This
lowering of Ln is reflected in the I-V characteristics shown in Fig. 4.8.
In agreement with the data reported in [39], Table 4.6 indicates that
the hydrogenation treatment slightly increases VOC. This conclusion is
not firm though, because the increase observed averaged only 11 mV and
the spread of values of VOC was large.
Table 4.4 Parameter values for diodes in Run 37P. T = 25.0C, A =
Description
GB(
milss)
GB
XO' XO
(A)
GB
mx, m"
GB
SCRO
(A)
-3 2
4.6 x 10 cm.
GB GB
SCR QNO' QNO
(A)
37P#11-D2
37P#12-D1
37P#2-D2
37P#1-D1
37P#5-D2
37P#9-D2
37P#10-D1
37P#2-L1
37P#6-L1
37P#8-L1
37P#11-L1
1.3 x 10-10
2.5 x 10-10
5.8 x 10-10
8.5 x 10-11
1.8 x 10-10
6.9 x 10-11
1.9 x 10-9
3.6 x 109
3.9 x 10-10
79 7.4 x 1010 1.78 8.0 x 1010
1.85 3.8 x 1013 2.2 x 104
Diode
GBF
GBF
SGB
(cm/sec)
several
several
several
several
one GB
one GB
several
several
several
4.0
7.5
1.3
2.4
5.4
8.0
1.7
6.7
1.8
3.5
3.6
GB's
GB's
GB's
GB's
GB's
GB's
GB's
x 10-12
x 10-12
x 10-10
x 10-11
x 10-10
x 10-11
x 10-10
10-11
10-9
10-9
10-10
140
75
35
90
20
80
68
150
50
1.19
1.30
1.63
1.69
1.94
1.59
1.61
1.57
2.03
2.01
1.93
x 103
x 103
x 104
x 103
x 104
1.3
8.1
4.1
4.3
2.9
2.3
4.0
4.8
3.1
3.8
4.8
x 10-13
x 10-14
x 10-13
x 10-13
x 10-13
x 10-13
x 10-13
x 10-13
x 10-13
x 10-13
x 10-13
1.67
1.75
2.02
1.71
1.67
1.69
2.08
2.05
2.03
2.0
7.2
3.6
2.0
1.9
1.8
6.0
5.2
1.7
Averages for GB diodes:
.1 .2 .3 .4 .5 .B
VOLTAGE (V)
Figure 4.8
Measured dark I-V curves of two GBF n -p solar cells
showing the effect of the hydrogenation treatment on the
intragrain base minority carrier diffusion length, L .
n
The solar cells had identical fabrication schedules ex-
cept that the cell represented by the upper curve under-
went the hydrogenation treatment. This treatment lowered
L from about 130 pm to about 90 pm.
n
1E-04
1E-05
1E-06
IE-07
1E-08
1E-09
1E-10
1E-11
Figure 4.8 also displays the effect that the absence of sintering
of the ohmic contacts had on the series resistance of the 37P diodes:
the extra series resistance of the 37P diode causes the I-V curve of
that diode to bend to the right and cross the I-V curve of the 36P
diode.
4.3.5 .I-V Characteristics (p -n diodes)
Figure 4.9 shows the measured I-V curves for two representative
p -n diodes; device No. 6 is a GBF diode; device No. 8 is a GB diode. A
summary of parameters for these two diodes and for diode No. 7 which
+
contains only twins is in Tables 4.1 and 4.2. In contrast to the n -p
diodes, the effect of the GB's on diode No. 8 is very small. This is
evident both from Fig. 4.9 and Tables 4.1 and 4.2. The intragrain hole
diffusion length for the GBF p -n diode No. 6, obtained from the dark
I-V curve [7], is Lp 35 pm. Thus, for the GB diode No. 8, dG >> L ,
GB
and the effect of the GB's on I QN is negligible. At small bias levels
QNO
the SCR recombination currents are dominated by the recombination in
+ GB
the SCR adjacent to the top p -n junction, IR, and ISCR is negligible.
SCR' SCR
With several ideal assumptions (Appendix III), the Sah-Noyce-Shockley
theory [8] allows ISCR to be written as
SCR
ISCR [AqWSCRn /(2TSCR)]exp(qV/mkT), (4.12)
where TSCR is the time constant controlling the recombination in the
SCR at small bias levels, and m = 2.0. Using (4.12) we find
+
TSCR 0.2 psec for the p -n GBF device No. 6 and TS 100 psec for
+
the n -p GBF device No. 1. This comparison of values for TSCR and also
a comparison of L with L explains the insensitivity of parameters in
p n
1 E-04
0 .1 .2 .3 .4 .5
VOLTAGE (V)
Figure 4.9
+
Measured dark I-V curves for two p -n solar cells: No. 6
is a GBF cell, No. 8 is a GB cell.
GB +
Tables 4.1 and 4.2 (including IXO and V C) to the GB's in the p -n
diodes. The grain size of p -n diodes would have to be comparable to
L = 35 pm in order to observe the effects of the GB's on the I-V curves.
p
+ +
The other important difference between the n -p and p -n diodes is
that no preferential diffusion of dopant impurity (boron) was observed
+
in the p -n diodes. The differences in the values for IX and IN0 for
XO QNO
+
the p -n devices in Table 4.1 are due mainly to the measured variations
16 -3 16 -3
in the base doping density (NDD = 1.3 x 1016 2.6 x 106 cm-3).
Notice also that the device No. 7 containing only twins is very similar
to the other two p -n diodes. Values for ISC are slightly higher for
+ +
the p -n diodes than for the n -p diodes because of the shallower
p -junction depth of 0.8 pm compared to 1.8 pm for the n -junction depth.
+
The p -n diodes in Tables 4.1 and 4.2 are representative diodes
from Run 34P. Parameter values for additional diodes in Run 34P are
presented in Tables 4.5 and 4.6. The relative scattering of values for
these diodes (as opposed to the Run 36P n -p diodes) is attributed mainly
to the measured variation in base doping density.
GB
4.3.6 Grain-boundary Shunt Resistance RSh
The shunt resistance effects on the I-V curve will be most effective
at very small biases. An analysis of the I-V curves in the voltage
+ +
range of about 0-300 mV for both the n -p and p -n GBF and GB diodes
shows that the measured curves can be described by Icexp[(qV/nmkT) 1],
i.e., they follow exactly an exponential dependence for a certain
constant slope factor mX. This indicates that the V/RG term in (4.3)
GB GB
and (4.5) can be neglected and GSh = 1/RSh = 0. This is an important
conclusion regarding the shunt resistance. The effects of RSh on I-V
curves can be confused with the effects of IGB or the edge effects.
SCR
Table 4.5 Parameter values for additional diodes in Run 34P. T = 25.0C, A = 4.6 x 10-3 cm2 .
Description
GB
milss)
GB
IXO IX
(A)
GB
x, "mx
GB
QNO' QNO
(A)
GBF
GBF
twins only
one GB
several GB's
several GB's
several GB's
several GB's
several GB's
several GB's
Averages for GB diodes including
the GB diode in Table 4.1:
5.3
5.5
70 1.3
90 1.2
40 2.8
50 5.7
55 3.0
30 3.0
40 7.3
34P#4-D1
34P#9-D1
34P#1-D1
34P#11-D2
34P#1-D2
34P#3-D1
34P#3-D2
34P#5-D1
34P#6-D2
34P#1-L2
1.91 3.4 x 1014
Diode
2.9 x 10-10
10-11
10-10
10-10
10-10
10-10
10-10
10-10
10-10
10-10
1.58
1.93
1.88
1.74
2.06
1.72
1.90
1.79
2.19
1.97
10-15
10-14
10-14
10-14
10-14
10-14
10-14
10-14
10-14
1.4 x 10-14
50 4.1 x 10 10
Summary of IS and V for
T = 25.0C, A = 4.6 x 10-3
Number of
Diodes
Measured
VOC spread
(mV)
all 30-mil diameter
2
cm 1 Sun AMO.
VOC average
(mV)
solar cells.
ISC spread
(PA)
ISC average
(1A)
471 522
451 558
478 504
466 491
494
480
496
495
496
481
73 76
66 84
61 75
59 74
64 82 70
Table 4.6
Type
34P
36P
37P
GBF
Twins
GB
GBF
Twins
GB
GBF
Twins
GB
+
p -n
+
n -p
+
n -p
468 502 492
However, the use of a structure with an MOS guard ring and the careful
analysis described above shows that the GB's in diffused p-n junction
polycrystalline solar cells made on Wacker material do not cause notice-
able leakage effects due to the GB shunt resistance. This conclusion
is also supported by measuring the I-V curve in the reverse direction
which shows negligible current between zero and about IV of reverse
bias.
4.4 Comparison of Mesa Diode and Planar Diffused Diode I-V Curves
Prior to the fabrication of the 30-mil n -p and p -n planar diffused
diodes described in this chapter, we fabricated (Appendix VI) 50-mil n -p
dark mesa diodes and attempted to isolate the GB-component of IB by
the method of subtracting I from IGB. The I-V measurements on these
mesa diodes tended to be instable and unreliable because of the surface
and edge leakage currents described in Section 4.1. In particular,
the leakage currents caused the measured values for the current densi-
GB
ties JD and JD in the low voltage range (0-300 mV) to be as much as
three orders of magnitude higher for the mesa diodes than for the planar
diffused diodes. The mesa diodes also tended to have higher measured
GB
values of mX and mX (often approximately 2.0) than did the planar
diffused diodes. Since, for the 50-mil mesa diodes, the measured current
in the low voltage range (0-300 mV) was predominately leakage current,
it was not possible to isolate and accurately analyze the GB-component
of IB for these diodes. As described in Section 4.1, the surface and
edge leakage currents are suppressed in the planar diffused diodes by
the use of an oxide on the top surface and an MOS guard-ring gate over-
lapping the edge of the diffused layer.
Figures 4.10 and 4.11 show the effect of leakage current on the J-V
+
characteristics of 50-mil n -p mesa diodes. The leakage current of the
GB
GB mesa diode in Fig. 4.11 causes the measured value of IX,G and conse-
quently, the calculated value for SGB, to be erroneously high. By the
method of Section 4.3.1, SGB is calculated to be 1.5 x 10 cm/sec. This
+
value exceeds the corresponding values for all of the n -p diodes in
Tables 4.1 and 4.3 by more than a factor of four. Though SGB can vary
significantly from diode to diode (Tables 4.1 and 4.3), the above
comparison demonstrates the questionable value of I-V measurements on
GB
small mesa diodes in analyzing the GB-component of ID Some researchers
have failed to notice this point [42]. In Fig. 9 of [42], for example,
GB
the I-V curves of 50-mil mesa diodes display values of mB > 2.0, which
indicates that the current at low bias levels is dominated by recombina-
tion at the surface around the perifphery of the diode [43]; and,
consequently, these I-V data are of little analytical use.
Surface leakage current is proportional to the circumference, and
thus to the radius r of the diode; whereas, the current of a leakage-
2
free diode is proportional to the area of the diode, and thus to r
Consequently, in a mesa diode, the relative contribution of the surface
leakage current to the total measured current will increase as r
decreases. As seen in Fig. 4.10, for a mesa diode with r 25 mils,
the leakage current can dominate the total measured current at low bias
levels (0-300 mV).
4.5 Discussion
This chapter has described and analyzed the effects of GB's on the
performance of polysilicon p-n junction solar cells.
72 -
VOLTAGE (V)
Figure 4.10
Measured dark J-V characteristics of GBF mesa and planar
diffused diodes showing the effect of leakage current on
the low-voltage current density and on the reciprocal
slope mX of the mesa diode. Diode 4P2-1,3,f is a mesa
diode; diode 36P#10-D1 is a planar diffused diode with
an MOS guard ring.
1E-03
1E-04
1E-05
IE-06
1E-07
1E-0e
1E-09
1E-01
1E-02
1E-03
1E-04
1E-05
1E-07l /
0
Figure 4.11
VOLTAGE (V)
Measured dark J-V characteristics of GB mesa and planar
diffused diodes showing the effect of leakage current
on the low-voltage current density and on the reciprocal
GB
slope mX of the mesa diode. Diode 4P2-4,8,f is a mesa
diode with 300 mils of GB's; diode 36P#1-D2 is a planar
diffused diode with an MOS guard ring and has 65 mils of
GB's. The leakage current of the mesa diode causes the
GB
measured value of IXO and consequently, the calculated
value for S to be erroneously high.
IxD
In order to obtain quantitative results about the GB-recombination
currents in the bulk of the cell, care must be exercised to eliminate
surface and edge-leakage currents. In our devices (30-mils in diameter)
this was accomplished by fabricating the diodes with a thermally-grown
SiO2 on the top surface and an MOS guard-ring gate overlapping the edge
of the diffused layer.
Diffusion of phosphorus into polysilicon material results in a
preferential diffusion along the GB's. The preferential diffusion can
be directly observed by using a groove and stain procedure. This proce-
dure was found to be unreliable, though, because the time and conditions
required to clearly delineate the very narrow preferentially diffused GB
region (spike) varied with the stain solution used and the doping
concentration in the spike. In Section 4.2 we described two new
electrical methods which can positively identify the presence or absence
of a preferentially diffused region. Once the preferentially diffused
region is identified, we may use a suitable groove and stain procedure
to measure the depth of the preferential diffusion and use electrical
methods to determine the average doping concentration in the region.
Our results show that a phosphorus predeposition at 9000 for 30 minutes
followed by a drive-in at 1050C for 40 minutes will preferentially
diffuse the GB's in p-type Wacker material to a depth d of about 4 pm
and will yield an average doping concentration in the diffused GB
16 -3
regions of NDD = 1 x 10 cm The substrate doping was
15 -3
NAA = 3 x 10 cm A predeposition at 1050C for 30 minutes followed
by a drive-in at 1050C for 30 minutes yields dn = 9 m, and
- 2 17 -3
DD 2 x 107 cm The quasi-neutral width W of the preferentially
diffused n-region is about 0.5 pm for the first case and about 1.0 pm
for the second case. On the other hand, a diffusion at 900C for
30 minutes did not result in the creation of a p-n junction along the
GB's.
No preferential diffusion of boron was observed either by the groove
and stain method or by the electrical methods for various diffusions at
900 1000C for 20 120 minutes in n-type Wacker substrate with
16 -3
N D 2 x 101 cm3
DD
Both the GB and the preferentially diffused region along the GB
will have recombination-current components associated with them. These
current components were identified by comparing the I-V characteristics
of the GB devices with the I-V characteristics of devices that were
grain-boundary-free (GBF). A first order analysis of the I-V curves of
the GB diodes, and separation of the total measured current into compo-
nents, was done in Section 4.3. This analysis has shown that the
dominant current component in the GB diodes at small bias levels
(0-300 mV) is the recombination current at the GB within the p-n junction
GB GB
SCR, ISCR. At higher bias levels (V = VC = 500-600 mV), both I and
the recombination current at that part of the GB which is adjacent to
GB
the quasi-neutral base region, IB are important. Both of these
GB
components cause degradation in V and I ; ISR also degrades FF. It
OC SC SCR
was also observed that twin GB's have very little effect on the I-V
characteristics of solar cells.
The preferentially diffused region along the GB increases the area
of the p-n junction. This will result in additional carriers injected
GB
from this region into the base, IQNB, and also in additional carrier
GB
injection from the base into the region, IQNE. A first order analysis
of these currents was done here for the first time.
This analysis suggests some important conclusions regarding the
effects of the preferential GB diffusion on the solar cell I-V curves.
(i) The surface recombination velocity at the GB within the preferen-
tially diffused n-region is about two orders of magnitude smaller than
outside this region. This suggests that a phosphorus diffusion into the
GB's does indeed passivate the GB's. (ii) The preferentially diffused
1018 -3 GB
regions should be heavily doped with NDD = 10 cm to suppress IQNE
n GB
even for large SGB and large d (iii) If IQNE can be suppressed, then
GB n' QNE
the depth of the preferential diffusion should be made comparable to the
GB
base diffusion length, i.e., d = L so as to minimize I (iv) The
n n B
preferentially diffused regions will aid in collection of the photogen-
erated carriers, thus increasing ISC. (v) Increased area of the p-n
GB
junction will increase the IQNB dark current component. However, the
QNB
total dark current will not increase proportionally to the total junction
area because of a two-dimensional coupling between IQNB injected from the
GB
QNB
currents tend to oppose each other.
Another important conclusion of this chapter is that the GB's do not
cause any measurable shunt resistance effects in diffused p-n junction
cells made on Wacker polysilicon material.
CHAPTER 5
SMALL-SIGNAL ADMITTANCE METHOD FOR DETERMINING
THE SURFACE-STATE DISTRIBUTION AT THE PREFERENTIALLY
DIFFUSED PART OF THE GRAIN BOUNDARY
5.1 Introduction
Lattice mismatch at the grain boundaries (GB's) of polysilicon
solar cells causes the formation of energy levels (surface states) in
the energy gap that serve as recombination centers. Surface states
can be either donor-type or acceptor-type. Those surface states that
are near the center of the energy gap will afford the highest recombina-
tion rates [44] and, thus, will be the most efficient recombination
centers.
For an n -p polysilicon diode, methods for calculating the sur-
face recombination velocity SGB at that part of a GB which is adjacent
to the quasi-neutral p-type bulk and for estimating the surface recom-
bination velocity SGB at that part of a GB which is in a preferential-
ly diffused n-region have been demonstrated in sections 4.3.1 and
4.3.2.2, respectively. With the assumption of a uniform distribution
of surface states in the energy gap, the GB surface-state density N
SS
can be calculated [45] for both the diffused and undiffused sections
of the GB:
N s= S/ vh (5.1)
ss th '
where, S, c, and vth are the surface recombination velocity, capture
cross-section, and thermal velocity for minority carriers at the GB,
1
respectively. The above assumption is generally invalid and, con-
sequently, (5.1) at best gives an estimate of N In order to de-
SS
termine the feasibility of preferential diffusion as a means of GB
surface-state passivation and, subsequently, as a means of lowering
the dark recombination current and increasing VOC and n, a more ac-
curate method for determining Nss must be developed. In this chapter,
ss
we develop a small-signal admittance method that enables the determina-
tion of Nss in the energy gap for that part of the GB which is in a
ss
preferentially diffused n region. The method, in principle, can
yield the determination of all of the fundamental kinetic parameters
in the energy gap at the GB surface: Nss(E), ET, Cn(ET), c (ET) en(ET),
and e (E ). In addition, the method can yield the GB barrier height,
and the doping and mobility in the preferentially diffused region.
5.2 Small-Signal Equivalent Circuit Model of a Diode with a Prefer-
entially Diffused Grain Boundary
+
We consider again the n -p diode with a preferentially diffused
n-region along the GB shown in Fig. 4.5. (The preferentially diffused
n-region of the GB, seen from the perspective of Fig. 4.5, is sometimes
referred to as the n-spike.) The equilibrium band diagram for that
part of the diode that is in the proximity of the preferentially
diffused GB is shown in Fig. 5.1. We assume that the region immediate-
ly next to the GB is depleted due to the presence of the surface
states at the GB [18,19]. The band bending due to those surface states
GB
is q#GB. The narrow depletion region of width WSCR on both sides of
the GB is similar to the surface channel of an MOS transistor in the
depletion regime; and the GB with its surface states is similar to the
Si-Si02 interface of that surface channel. Consequently, the simplified
Figure 5.1
+
(a) Cross-section of an n -p polysilicon diode showing the
preferentially diffused n-region of a GB.
(b) The corresponding thermal equilibrium band diagram. The
GB is located at x = 0. The width of the GB space-charge
region is WD; the width of the quasi-neutral n-region
in the n-spike is Wn; the band-bending at the GB is
94GB*
equilibrium small-signal transmission-line equivalent circuit model
of the preferentially diffused n-region shown in Fig. 4.5 will be
similar to the circuit model of the surface channel of an MOS tran-
sistor [46]. This simplified model is shown in Fig. 5.2. By consider-
ing the two-dimensional current flowing through the preferentially
diffused n-region, this model can be derived from the general small-
signal transmission-line equivalent circuit model developed by Sah [47]
for any two-terminal one-dimensional semiconductor device.
The circuit parameters in Fig. 5.2 are defined as follows:
pn is the per-unit-length resistance of the quasi-neutral n-region
in C/cm;
C is the capacitance of the p-n junction space-charge region
along the n-spike in F/cm2
CD is the capacitance of the GB space-charge region in the n-spike
in F/cm2;
G is the electron capture conductance of the GB surface states
ns
2
in mhos/cm ;
2
C is the storage capacitance of the GB surface states in F/cm ;
ss
C GBF is the capacitance of the lateral p-n junction space-charge
GBF
region in F/cm2; and,
Z is the length (into the plane of the paper) of the preferen-
tially diffused n-region.
In Fig. 5.2., we have neglected the electron and hole storage
capacitances in the quasi-neutral regions, and we have assumed the
simple case of a single level of surface states at the GB. The
capacitive coupling between the n-spike and the substrate at the bot-
tom of the spike is neglected on the grounds that the volume of the
Figure 5.2 Simplified equilibrium small-signal transmission-line
+
equivalent circuit model of a polysilicon n -p diode with
a preferentially diffused GB.
space-charge region at the bottom of the spike is small compared to the
volume of other regions that contribute capacitance. The resistive
coupling between the n-spike and the substrate through the GB can also
be neglected since it was determined in Section 4.3.6 that G s = 1/RG
Sh Sh
0. The Shockley-Read-Hall recombination through the deep trap levels
in the quasi-neutral bulk is also neglected. The equivalent circuit
model obtained by lumping the circuit elements from both sides of the
GB is shown in Fig. 5.3(a). The transmission line is open-circuited be-
GB +
cause GS 0. This model is also applicable to an n -p diode contain-
Sh
ing more than one GB, because the GB's are all electrically connected
to each other by the lateral n layer on the top.
We now assume, for the circuit shown in Fig. 5.3(a), that small-
signal admittance measurements will be made at frequencies that are
much less than the transmission-line characteristic frequency f [46].
With this assumption, the pn circuit elements can be shorted. The re-
sult is the circuit shown in Fig. 5.3(b). The corresponding simple one-
lump circuit model is shown in Fig. 5.3(c). This model is valid for the
depletion and weak inversion regimes, i.e., for EF(0) 3 Ei(0), where
the GB is located at x = 0.
The input admittance for the circuit of Fig. 5.3(c) is
Yin = Gin + Cin' (5.2)
where
Gin = 2T CssC/[W 2 T (CD + C)2 + (Css + CD + CW) 21 (5.3)
Cin = [wr2C CD (CD + C ) + CW (Css + CD)(Css + CD + CW)]/
[2 2(CD + C W)2 + (Css + CD + CW)2] + CGBF, (5.4)
Pndy
c -----
CD ZAyzy
-: lCoZ,' !
CGBF C ZA
T ~tCW Zdy
0 --- i ------- i ---------------------o
Figure 5.3(a)
The equivalent circuit model obtained from Fig. 5.2
by lumping the circuit elements from both sides of
the GB.
Figure 5.3(b) The equivalent circuit model obtained by shorting the
Pn circuit elements in Fig. 5.3(a).
No in
Figure 5.3(c)
The one-lump equivalent circuit model obtained from
Fig. 5.3(b). Here CD, CW, Css, and CGBF are in farads,
and G is in mhos.
ns
T =/W = C /G (5.5)
ss ss ns
Both (5.3) and (5.4) are dependent on C ; therefore, we can use
ss
either Gi or Ci to obtain information about the surface states.
Experimentally, it is found that the surface states at an Si-Si02
interface [48], and also the surface states at silicon GB's [22], are
continuously distributed in the energy gap rather than existing at a
single energy level. These continuously distributed surface states
can be represented in the circuit model of Fig. 5.3(c) as a parallel
array of Gns Css circuit elements, each element representing surface
ns ss
states at a certain energy level. The capacitance of a G C cir-
ns ss
cuit element is proportional to Nss at the given energy level, and Gs
ss ns
reflects the time constant of the surface states at the given energy
level. The problem of calculating the surface-state capacitance C
ss
for an MOS capacitor with a continuous distribution of surface states
has been considered by Sah [49]. In [49] it is shown that, for the
case of an MOS capacitor with a small-signal current flow, the net C
ss
for distributed states can be obtained by summing the contribution to
Css from all the energy levels between EV and EC referenced to a single
Fermi level. The net Css is determined by those states within about
kT of the Fermi level at the Si surface. Since the equivalent circuit
of a preferentially diffused GB at zero-bias is similar to that of an
MOS capacitor in the depletion regime, the results of [49] may be ex-
tended to include the preferentially diffused GB's of our devices.
Consequently, the circuit model of Fig. 5.3(c) is applicable to the
realistic case of a preferentially diffused GB with a continuous dis-
tribution of surface states.
5.3 An Admittance Method for Determining N
ss
We now describe a small-signal admittance method in which we
use Ci to obtain information about the surface states at that part
of a GB which has been preferentially diffused with phosphorus.
Figure 5.3(c) shows that the zero-bias small-signal capacitance
associated with the preferentially diffused region of the GB's in a
GB diode, C, is equal to the measured terminal capacitance C. of the
in
GB diode minus the measured terminal capacitance of the corresponding
GBF diode:
C = C CGBF (5.6)
The capacitance GGBF is frequency-independent until f ~ 1 GHz. For
sufficiently low frequency, f << f = G /21TC the G can be
ss ns ss ns
short-circuited, and the equivalent circuit will reduce to that shown
in Fig. 5.3(d). From (5.4) and (5.6), the low-frequency limit for C
is then
CLF = Cw(Css + CD)/(Css + CD + CW) (5.7)
For sufficiently high frequencies, f >> f the surface states can-
ss
not follow the signal and the C can be short-circuited. With the
ss
provision that the frequency f is less than the characteristic fre-
quency fo, the equivalent circuit model shown in Fig. 5.3(c) then
reduces to that shown in Fig. 5.3(e); and (5.4) and (5.6) yield the
high-frequency limit for C:
CHF = CWCD/(C + CD). (5.8)
From (5.7) and (5.8), we obtain an expression for Css in terms of CLF
and CHF [50]:
Css = Cw[(CLF/C )/( CLF/CW) (CHF/C)/(1 CHF/CW)]. (5.9)
Css
Figure 5.3(d)
CGBF
----Yin
The equivalent circuit model obtained from Fig. 5.3(c)
for the low-frequency case, i.e., for f << f .
ss
-CGBF
iC w
T_
Figure 5.3(e)
The equivalent circuit model obtained from Fig. 5.3(c)
for the high-frequency case, i.e., for f << f << f .
ss o
----Yin
The surface-state density along the preferentially diffused part of
the GB's is then given by
N = Css/qAGB (5.10)
where AGB= d Z is the total area of the preferentially diffused part
of the GB's.
We assume as a first approximation that the preferentially dif-
fused n-region is uniformly doped. In Fig. 5.1(b), it is seen that
the position of the Fermi level for the preferentially diffused part
of the GB's is given by
E (0) E.(0) = E (0)/2 qn q GB (5.11)
F g n GB
where the GB is located at x = 0. In order to determine the position
of the Fermi level for the preferentially diffused part of the GB's,
qpn and q4GB must be calculated.
With the use of the Boltzmann relation [51] for a nondegenerate
semiconductor in thermal equilibrium, the band diagram in Fig. 5.1(b)
shows that
q n = E (0)/2 kT In (N /ni). (5.12)
The values of E (0) and n. as a function of the temperature T are
g 1
found in [52].
The band bending q4GB which is due to the surface states at the
GB can be approximately determined by the method of [46]. In [18], it
is assumed that Nss is uniformly distributed in the energy gap and
ss
that there exists a "neutral level" qqo at approximately E (0)/3 such
that, for EF (0) = q#0, the net charge in the GB surface states is
zero. The assumption concerning the energy gap position of qco is
supported by [53]. With these assumptions, the requirement of overall
charge-neutrality in the n-region of the GB, and the band diagram in
Fig. 5.1(b), yield the following system of equations:
qG = q2Nss 2 [EF(0) q] 1/8EoNDD (5.13)
^GB ss F o o DD (5.13)
EF(00)= E (0 q qGB (5.14)
where po = EV(0)/q + E (0)/3q, and EV(0) is arbitrarily taken to be
zero. This system of equations can be solved for q GB to yield:
qGB =[(23 + 1) 1 (4aB + 1) 1 /2a, (5.15)
where a = qN 2 /8KS N and 3 = 2E (0)/3 [kT In(N /N )].
ss o DD g C DD
In establishing (5.15), it is assumed that N is uniformly dis-
ss
tribute in the energy gap. This generally invalid assumption can
lead to an error in the calculation of q GB. With the assumption of
a one-sided depletion region at the GB, a more accurate expression
for the band bending at the GB is given by
qGB = q () s(E)dE) 2/8 DD (5.16)
Jq
o0
The determination of q GB by (5.16) requires a knowledge of the distri-
bution of Nss between qdo and EF(0) and is only obtainable by means
of iterative calculations. To avoid the laborious nature of the cal-
culations associated with (5.16) while at the same time improving
upon the accuracy that would be afforded by (5.15) if one value of
N were used in (5.15), we have chosen to calculate qGB by using an
ss
average value for Nss in (5.15).
ss
With the calculation of qPn and q GB' EF(0) may be calculated
for various temperatures from (5.11). Thus, by measuring the small-
signal admittance at various temperatures, we can use (5.10) and
(5.11) to determine Nss vs. [E (0) E.(0)].
ss F 1
5.4 Inversion along the GB in the p-Type Bulk
We now consider that part of the GB which is adjacent to the
p-type bulk. Equation (5.13) shows that, with the assumption of a
uniform distribution of surface states in the energy gap, the band
2
bending at the GB is proportional to N For sufficiently large
ss
N E (0) > E (0) where E. is the intrinsic Fermi level, and the
SS F I
region next to the GB becomes inverted. It is of interest to deter-
mine the value of Nss necessary for the formation of an inversion lay-
ss
er along that part of the GB which is adjacent to the p-type bulk.
In Fig. 5.4, it is seen that the onset of inversion occurs when
EF(0) E (0) = q GB = E (0)/2. By substituting these relationships
into (5.13) with NDD replaced by NAA and with o = E (0)/3q, we may
solve for the value of Ns necessary for the onset of inversion:
(N )v = (12/q)[K NAA /E (0)]1/2 (5.17)
ssny o AA g
Since Nss is, in fact, not uniformly distributed in the energy gap,
(5.17) provides only an estimate of the value of Nss necessary for the
SS
onset of inversion. In Section 4.2, an argument based on experimental
data was presented to show that there does not exist an inversion layer
about the GB in the bulk. Consequently, (N ss)in represents an esti-
mate of the upper limit of Nss along that part of the GB which is ad-
ss
jacent to the p-type bulk.
5.5 Experimental Procedure and Results
To demonstrate the above method for determining Nss, two runs
+
(39P and 40P) of n -p mesa diodes were fabricated on 5 --cm Wacker
polysilicon p-type substrates (Appendix IX). The mesa diodes were
made by the method of wax-masking a small dot on a chip that had been
p-type bulk
( 4.B
S Ev
--GRAIN BOUNDARY
x=0
Figure 5.4 Thermal equilibrium band diagram for that part of a GB
+
which is adjacent to the p-type bulk of an n -p poly-
silicon diode.
Y
**
** |