Tel. Phenomena
No.: 0049-0711-6857181
Solid State
Vol. 93 (2003) pp 399-404
© (2003)
Trans
Tech
Publications, Switzerland
Fax.
No.:
0049-0711-6857206
doi:10.4028/www.scientific.net/SSP.93.399
e-mail: kurt.taretto@ipe.uni-stuttgart.de
A simple method to extract the diffusion length from the output
parameters of solar cells - application to polycrystalline silicon
K. Taretto, U. Rau, T. A. Wagner, and J. H. Werner
Institut für Physikalische Elektronik, Universität Stuttgart, Pfaffenwaldring 47, 70569 Stuttgart,
Germany
Keywords: solar cell, diffusion length, internal quantum efficiency, grain boundary recombination,
polycrystalline silicon
Abstract. This work presents a simple method to obtain the effective diffusion length Leff of a solar
cell directly from measured values of open circuit voltage, short circuit current density, and the
doping density in the base of the cell. In the second part of this paper, we extract Leff from literature
data of polycrystalline silicon cells, with grain sizes from 10-2 to 104 µm, modeling the extracted Leff
as a function of the grain size g, and the recombination velocity SGB at the grain boundaries. For g >
1 µm, our model predicts 105 < SGB < 107 cm/s. Cells with g < 1 µm, are understood with 101 < SGB
< 103 cm/s. This finding supports the hypothesis that the key to high efficiencies at small grain sizes
is the use of {220}-textured films.
Introduction
The efficiency of solar cells depends strongly on the effective diffusion length Leff of minority
carriers. Internal quantum efficiency (IQE) measurements give Leff, but its determination is based on
an exact knowledge of the absorption and dispersion of light in the cell. This makes the IQEmethod rather complicated, but up to now, no simple substitute to this method was found. This
work presents such a substitute, where Leff is calculated directly from the open circuit voltage, the
short circuit current density, and the doping level in the base of the solar cell. In contrast to the IQEmethod, our approach constitutes a fast and simple method to estimate Leff. To prove the validity of
our model, we compare a large data set of Leff measured by IQE, and compare these values to the
predictions of our model. We show that throughout three orders of magnitude of Leff, our method
allows us to determine Leff within an accuracy of 35 %.
The efficiency of a solar cell generally increases with increasing grain size g [1]. The
explanation for this increase is simple: if g increases, the ratio of grain boundary area to grain
volume decreases, reducing the amount of recombination centers, increasing the minority carrier
diffusion length. In the second part of this paper, we show the increase of Leff with g, extracting Leff
from literature data of polycrystalline silicon cells having 10-2 < g < 104 µm. We model the
extracted Leff-data as a function of g and the recombination velocity SGB at the grain boundaries. At
g > 1 µm, our model predicts values of SGB between 105 and 107 cm/s, while cells with g < 1 µm are
only understood with SGB between 101 and 103 cm/s. This finding supports the hypothesis that the
nanocrystalline silicon cells benefit from {220}-textured films introduced in Refs. [2] and [3].
Model
The double-diode current(J)/voltage(V) characteristics of the pn cell is given by the equation [4]
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400
Polycrystalline Semiconductors VII
J = J 01 exp
V
V
- 1 - J SC ,
- 1 + J 02 exp
Vt
2V t
(1)
where Vt is the thermal voltage kT/q, J01 and J02 are the saturation current densities, and JSC is the
short-circuit current density. The first term of Eq. (1) represents the recombination current in the
base, characterized by an ideality factor nid = 1. The second term belongs to the recombination in
the space-charge region (SCR), and it is modeled considering Shockley-Read-Hall recombination
via a single trap located in the middle of the bandgap, resulting nid = 2 [4]. In a monocrystalline
material, J01 is a function of an effective diffusion length Leff = Leff,mono, which is given by Leff,mono =
Lnf(W,Sb,Ln), where Ln is the diffusion length of electrons in the p-type base, and f is a function of
the base thickness W, the recombination velocity Sb at the back contact, and Ln [5].
The current density J02 depends on Ln, not including contact recombination [4]. In order to
simplify the analysis, we make an assumption that allows to use the same diffusion length to
calculate J01 and J02: the value of Ln does not depend on the position in the cell (SCR or bulk), and
the recombination of carriers at the back contact does not affect strongly Leff. This assumption
imposes Ln < W, since, Leff,mono equals Ln within an error smaller than 20 % provided Ln/W < 1.
Assuming Ln/W < 1, both current densities J01 and J02 become a function of an unique diffusion
length Leff. In the p-type base, the saturation current density J01 is given by [5]
qD n n i2 1
J 01 =
,
(2)
N A L eff
where Dn is the diffusion constant of electrons, ni the intrinsic carrier concentration, and NA the
doping density in the base of the cell. In the SCR, the saturation current density J02 is given by [6]
q pD n n i V t 1
.
J 02 =
(3)
Fmax
L 2eff
If the doping profiles are step-like, the maximum electric field Fmax is given by Fmax =(2qNAVbi/eS)1/2
[6], being Vbi the built-in voltage, and eS the semiconductor’s absolute dielectric constant.
Replacing Eqs. (2) and (3) in Eq. (1), the whole J/V-curve is written as a function of Leff. Thus,
we express Leff as a function of the cell’s open circuit voltage VOC and JSC. At J = 0, we have V =
VOC, and using the definitions of J01, J02, and Fmax, Eq. (1) is rewritten as
es
qD n n i2 1
V
V
1
J SC =
exp OC + q pD n n i V t
exp OC .
(4)
2
N A L eff
Vt
2 qN A V bi L eff
2V t
Solving this equation for Leff, we obtain
z + z 2 + 2 pV t J SC
L eff =
2J
2 Dn e S
V bi
1/ 2
1/2
z1/2
(5)
,
SC
where z is given by
z = qn i D n exp
V OC
NA
- ln
Vt
ni
.
(6)
Figure 1 shows the increase of Leff with [VOC - Vt ln(NA/ni)] , given by Eqs. (5) and (6) with Dn =
10 cm2/s, and Vbi = 0.8 V. This estimate of Vbi meets commonly found values in silicon cells.
Equation (5) yields values of Leff differing less than 1 % for the range 0.5 < Vbi < 1.0 V. Figure 1
indicates that a material with low recombination (high Leff) is required to obtain solar cells with high
values of VOC. The plot shows two regions: the region for low Leff, where the recombination in the
SCR determines VOC, and nid = 2; and the region of higher Leff, where VOC is limited by bulk
recombination, resulting nid = 1.
The curves in Figure 1 suggest that, by calculating the ideality factor nid at VOC from the slope of
the condition Leff/W < 1. Figure 2 shows that the
values of Leff obtained with the present model, agree
with the IQE values over three orders of magnitude
of Leff. The circles in Figure 2 belong to silicon
epitaxial cells prepared with the ion-assisted
deposition method [7], while the triangles belong to
multicrystalline silicon cells [8]. The solid line in
Figure 2 gives the identity Leff (modeled) = Leff
(measured by IQE). The dashed lines represent the
least-square standard deviation of the data from the
identity line, indicating that the present model
predicts Leff with an error of 35 % (assuming that
the IQE values are exact).
Effective diffusion length in polycrystalline cells
401
nid = 1
2
10
2
JSC = 2 mA/cm
1
10
0
10
5
10
nid = 2
20
30
40
-1
10
-0.2
-0.1
0.0
0.1
VOC-Vtln(NA/ni) [V]
0.2
Figure 1: The effective diffusion length Leff
obtained by the present model, as a function
of the open circuit voltage VOC, and the short
circuit current density JSC. At low Leff the
curves are described by the recombination in
the space-charge region (ideality nid = 2). At
high Leff, the recombination in the base (nid =
1) limits VOC.
2
Leff [µm], modeled
a measured J/V curve, one can determine where the
highest recombination takes place: in the SCR, or in
the bulk. Cells with a small diffusion length, will
show nid = 2, and will have VOC limited by the
recombination in the SCR. By increasing Leff, the
generated electron-hole pairs will not recombine in
the SCR but mainly in the bulk, showing nid = 1.
Equation (5) is extremely useful for the
experimentalist who wants to estimate Leff, because
JSC, VOC, and NA are easy to measure. The standard
technique to determine Leff is much more
complicated, since it is based on internal quantumefficiency (IQE) measurements, which require an
exact knowledge of the absorption constant of the
material [5], making a determination of Leff rather
intricate.
Now we prove that Eq. (5) gives the correct
value of Leff, using literature data of silicon solar
cells where Leff was obtained from IQE
measurements, and compare them to the values of
Leff predicted by Eq. (5). The selected data satisfies
effective diff. length Leff [µm]
Solid State Phenomena Vol. 93
10
1
10
0
10
0
1
2
10
10
10
Leff [µm], from IQE measurements
Figure 2: The effective diffusion length Leff of
silicon solar cells determined by our model
agrees with the values of Leff obtained by
internal quantum efficiency measurements,
over three orders of magnitude of Leff. The
circles belong to silicon epitaxial cells [7],
while
the
triangles
correspond
to
polycrystalline silicon cells [8].
In a polycrystalline material, we must consider
a diffusion length Leff,poly, which contains the
recombination SGB velocity at the grain boundaries
(GBs). In Ref. [10], the diffusion length Leff,poly
was calculated considering the diffusion and
recombination of minority carriers in the base of a
three-dimensional np cell, with no GBs
perpendicular to carrier flow. With the recombination of carriers inside the grains described by
Leff,mono, the diffusion length Leff,poly is given by the equation
L eff ,mono
L eff , poly =
,
(7)
2 S GB L 2eff , mono
1+
Dn g
402
Polycrystalline Semiconductors VII
Table 1: Experimental polycrystalline silicon solar cell parameters extracted from the literature. The
geometrical quantities given are the area A, the cell thickness W, and the grain size g. The doping
density NA corresponds to the p-type base of the cells. The electrical parameters, given under
AM1.5 illumination conditions, are the efficiency h, the open circuit voltage VOC, the fill factor FF,
and the short circuit current density JSC. Cells denoted as pnn-type have low doped and n-type
middle-layers.
cell cell
Ref.
A
W
g
NA
VOC
FF
JSC
h
2
-3
2
type
[cm ] [mm]
[cm ]
[%] [mV] [%] [mA/cm ]
[mm]
a
np
[11]
4
60
104
2x1016 (a) 16.5 608
77
35.1
3
b
np
[12]
1
100
10
2x1016 (a) 16.6 608
82
33.5
3
16 (b)
c
np
[13]
1
72
10
2x10
9.3
567
76
21.6
d
np
[13]
1
30
103
2x1016 (b)
11
570
76
25.6
16 (b)
e
np
[15]
1
300
500
2x10
9.95 517
7.2
27.1
f
np
[15]
1
300
500
2x1016 (b) 11.1 538
72.4
28.5
16 (b)
g
np
[14]
1
500
250
2x10
10.7 527
69
31.1
h
np
[17]
1
49
200
2x1017
8.2
525
66
23.8
16
i
np
[16]
1.3
30
150
3x10
8.3
561
74
20.1
j
np
[18]
?
330
20
2x1016 (b) 4.3
430
64
16.7
k
np
[19]
0.01
4.2
10
4.3x1017
6.5
480
53
25.5
17
l
np
[21]
1
15
7
1x10
5.2
461
64
17.5
m
np
[17]
1
15
5
2x1017
2.8
368
59
12.8
17
n
np
[20]
0.17
15
1
1x10
5.3
400
58
23
o
np
[22]
1
20
1-3
2x1017
2.0
340
59
10.1
77
24.35
p
[23]
?1
2
2x1016 (b) 10.1 539
pnn
»0.5
16 (b)
q
[24]
1
5.2
1
2x10
9.2
553
66
25
pnn
16 (b)
r
pin
[26]
0.7
2
0.05
2x10
7.5
499
68.7
22
s
pin
[25]
0.25
2.1
0.042
2x1016 (b) 9.5
500
68
28
16 (b)
t
pin
[27]
0.25
2.5
8.6
500
66
26.2
»0.01 [28] 2x10
16 (b)
u
pin [29,30] 0.33
2
2x10
8.5
531
70
22.9
»0.01
a) this value an estimate that corresponds to commonly utilized doping levels.
b) value estimated from the resistivity values between 1-2 Wcm (p-type material), given in the paper
corresponding to each cell.
which shows that at high grain sizes g, Leff,poly approaches the limit given by the monocrystalline
value Leff,mono. Figure 3 shows the increase of Leff with g, where the values of Leff were obtained
using the data of Table 1 using Eq. (5). All the circles belong to np-type cells, and all triangles to
pin- or pnn- cells. The solid lines give Leff,poly from Eq.(7), assuming Dn = 10 cm2/s, Leff,mono = 102
mm, and Vbi = 0.8 V.
Associating the data points to the solid lines in Figure , we distinguish two groups of data with
different ranges of SGB:
i) cells with g > 1 mm, show values of SGB in the range 105 < SGB < 107 cm/s,
ii) cells made from nano- and microcrystalline material, where g < 1 mm, are only understood
with 101 < SGB < 103 cm/s.
Despite the fact that the present model assumes np junctions and not pin junctions, the difference
of SGB between the two regions is large. Are the low SGB values found for the pin cells misleading
because the model does not strictly apply to them? The answer to this question is given by the
numerical simulations of pin cells given in Ref. [32]. The simulations show that SGB indeed must
have values between 300 to 1100 cm/s (at grain sizes around g = 1 mm) in those pin cells.
Solid State Phenomena Vol. 93
403
diffusion length Leff,poly [µm]
An explanation for such low grain
2
boundary recombination velocities is given in
1
limit Leff,mono=10 µm
S
=
10
cm/s
GB
2
Refs. [2] and [3], where it is argued that the
10
B
A
low SGB comes from structural differences
D
3
C
between the cells with g < 1 mm and cells
P
10
I
1
Q
F
10
G
showing g > 1 mm. The pattern found to make
5
U
E
10
R
that estimation is that all cells with g < 1 mm
S
H
T
L
0
J
of Table 1, were reported to have a {220}
7
10
10
N
M K
surface texture. With that information, the
O
low SGB is explained as follows: “The
-1
10
measured {220}-texture implies a (110)oriented surface for most of the grains. A
-2
-1
0
1
2
3
4
large number of the columnar grains must
10 10 10 10 10 10 10
therefore be separated by [110] tilt grain
grain size g [µm]
boundaries. Symmetrical grain boundaries of
this type are electrically inactive because they Figure 3: Data points give the diffusion length Leff
extracted from the data of Table 1 using Eq. (5).
contain no broken bonds” [3].
Circles correspond to np cells, and triangles to pin
cells. The overall increase of Leff with the grain
Discussion and Conclusions
size g, indicates that the recombination velocity
SGB at the grain boundaries determines Leff The
In the first part of this contribution we solid lines model L considering a recombination
eff
present a method to extract the effective velocity S at the grain boundaries.
GB
diffusion length in pn-type solar cells from
I/V measurements. Our model considers recombination in the bulk as well as in the space-charge
region of the cell. We prove that over three orders of magnitude of the effective diffusion length,
our method agrees very well with values obtained from internal quantum-efficiency measurements.
The method to extract the effective diffusion length presented here requires only the short-circuit
current of the cell, the open-circuit voltage, and the doping density in the base of the solar cell,
making it simple compared to quantum-efficiency measurements.
The second part of this contribution extracts the diffusion length of polycrystalline silicon solar
cells with grain sizes between 10-2 and 104 mm. The increase of diffusion length with grain size is
then modeled with an equation that considers the recombination velocity at the grain boundaries.
We find that the reported nanocrystalline and microcrystalline cells reaching efficiencies up to 10 %
are only explained with low grain boundary recombination velocities between 101 and 103 cm/s.
Such low recombination velocities support earlier predictions ascribing low defect densities to the
GBs in those small grained cells.
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Polycrystalline Semiconductors VII
10.4028/www.scientific.net/SSP.93
A Simple Method to Extract the Diffusion Length from the Output Parameters of Solar Cells Application to Polycrystalline Silicon
10.4028/www.scientific.net/SSP.93.399
DOI References
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