27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9
MODELING OF MULTICRYSTALLINE SI SOLAR CELLS BASED ON LIFETIME DISTRIBUTIONS
H. Wagner1,*, M. Müller2, G. Fischer2 and P.P. Altermatt1
Leibniz University of Hannover, Dep. Solar Energy, Appelstr. 2, 30167 Hannover, Germany
2
SolarWorld Innovations GmbH, Berthelsdorfer Str. 111 A, 09599 Freiberg, Germany
1
ABSTRACT: We present a numerical simulation model for multicrystalline Si (mc-Si) solar cells, which predicts cell
behavior from lifetime mappings of mc-Si wafers to a precision of about 3 mV in open-circuit voltage, Voc. From a
scientific point of view, the model allows us to answer commonly discussed questions such as: which is a suitable
method for averaging lifetimes in the quality control of mc-Si wafers, or the role of lower lifetimes in mc-Si cell
performance. It is shown that mathematical averaging procedures of wafer lifetime mappings can predict cell’s Voc
performance only within a rather broad range (30 mV for the arithmetic mean value, 7 mV for the geometric and 4
mV for the harmonic mean value). From an industrial point of view, the simulation model enables cell manufacturers
to predict cell behavior from lifetime images of mc-Si wafers and hence help in deciding on improvement strategies.
Keywords: Electrical properties, Lifetime, Multi-Crystalline, Modeling, Qualification and Testing, Silicon Solar Cell,
Simulation
1
INTRODUCTION
The influence of lifetime inhomogeneity on mc-Si
cell performance has been investigated using various
methods referenced e.g. in Refs. [1] and [2]. For a recent
study, see for example Ref. [3]. In the present paper, we
use the modeling software Sentaurus Device [4] (also
called Dessis or TCAD) to predict mc-Si cell performance from lifetime images. The model is tested on
mc-Si cells made from various ingots and it is used for
investigating various phenomena such as the role of low
lifetime regions in cell performance. We critically assess,
to which precision mathematical averages of the lifetime
images can predict mc-Si cell performance.
2
passivated with Al2O3. Accessible to measurement is
only the effective excess carrier lifetime, τeff(x,y), which
is related to τbulk(x,y) via [5]:
1
𝜏𝑒𝑓𝑓
=
1
𝜏𝑏𝑢𝑙𝑘
+
2𝑆
𝑊
=
1
𝜏𝑆𝑅𝐻
+
1
𝜏𝐴𝑢
+
1
𝜏𝑟𝑎𝑑
+
2𝑆
, (2)
𝑊
where S is the surface recombination velocity, W is the
sample thickness, and τ the Shockley-Read-Hall, Auger,
and radiative lifetimes, respectively. Independent
measurements yield S ≈ 5 cm/s [6], so we can neglect the
term containing S in Eq. (2).
τbulk [µs]
SIMULATION MODEL FOR MC-SI CELLS
To simulate mc-Si solar cells, the following wellknown approach is used. An image of the excess carrier
lifetime in the bulk of the cell, τbulk(x,y), is discretized in
a lifetime histogram having a number n of lifetime values
τsim,i with corresponding areas Ai. The sum of all Ai gives
the total cell area. We then simulate a sequence of n
monocrystalline Si (c-Si) cells, having the homogeneous
lifetime τsim,i, and obtain their I-V curves I(V)mono,i. To
calculate the I-V curve of the mc-Si cell I(V)multi, these
simulated I(V)mono,i curves are connected in parallel in a
numerical spice circuit model, using Ai as area factors.
The circuit approximates the lateral current flow within
both the silicon and the metallization. Alternatively the
I(V)mono,i curves can be analytically calculated using
𝐼(𝑉)𝑚𝑢𝑙𝑡𝑖 = ∑𝑛𝑖=1 𝐴𝑖 ∙ 𝐼(𝑉)𝑚𝑜𝑛𝑜,𝑖 .
(1)
The resistance due to the front metallization, Rs, is
included only as a lumped parameter (0.3 Ωcm2) but can,
if required, be simulated with the spice model from finger
line resistances. In the following, we establish the model
by comparing the simulated with the measured I-V
curves.
As the excess carrier lifetime in the bulk changes
during the fabrication of the cells, lifetime images of
finished mc-cells are used. To obtain τbulk(x,y), the
metallization, the emitter, and the BSF are etched off
prior to measurement, and the newly formed surface is
Figure 1: Local excess carrier lifetime in the bulk of
industrially fabricated multicrystalline solar cells,
measured with the photoconductance-calibrated photoluminescence technique. The emitter and the BSF were
etched off and the newly formed surface was passivated
with Al2O3 prior to the measurement.
We measure τeff(x,y) with photoconductancecalibrated photoluminescence images [7-9] using the
LIS-R1 system from BT Imaging [10], which contains a
Sinton Instruments photoconductance set-up [11]. The
optical reflectivity, required by the LIS-R1 system, is
derived using a PerkinElmer UV/VIS spectrometer [12],
while the wafer thickness is measured with a Käfer
digital-dial gauge [13]. An example of a τbulk(x,y) image
is shown in Fig. 1.
27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9
For validating the model, six mc-Si cells were taken
from an industrial production line; they were made from
six different ingots. Two groups of three cells were
produced separately, what means that in each case three
of them have an almost identical emitter phosphorus
diffusion profile, Al-induced back-surface-field (BSF),
geometry, etc. Their I-V parameters were measured with
a flash I-V tester from h.a.l.m. [14] such as the shortcircuit current density Jsc, the open-circuit voltage Voc, fill
factor FF and efficiency η. After this, they underwent the
above described procedure so τbulk(x,y) could be
extracted.
The two groups were numerically simulated in two
dimensions using the software Sentaurus Device [4] from
Synopsys to solve the complete set of semiconductor
equations [15]. The physical models of Refs. [16] – [18]
are applied. All dopant profiles were independently
measured with the electrochemical capacitance-voltage
technique (ECV) using the CVP21 profiler [19] and the
procedure described in Ref. [20]. The front surface
recombination velocity Sfront is adjusted such that the
measured saturation current-density J0e of the emitter is
reproduced in the model. J0e is measured with the
procedure of Kan and Swanson [21] on independently
fabricated, textured samples. It turns out that Sfront,
necessary to reproduce J0e, is compatible with the data on
planar wafers in Ref. [18] when taking Ref. [22] for
extrapolating from planar to textured wafers.
The τsim,i values, extracted from the histogram of the
measured τbulk(x,y), are inserted into the simulation as
τSRH in Eq. (2), concretely by choosing equal lifetime
parameters τn and τp in the well-known SRH equation
[23,24] and the defect level energy at midgap (τAu and
τrad are chosen as given in Ref. [16], Table 3 therein). The
only free parameter is the SRH lifetime in the Al-induced
BSF, which seems to be limited by the density of Al-O
complexes [17] and hence depends on processing
conditions. We adjusted it in a separate study which is
compatible with Ref. [17].
With all these parameters independently measured,
our model contains no remaining free parameters. As
described at the beginning of this Section, we include the
I-V curves, simulated with the homogeneous τsim,i values,
into a circuit simulation to obtain e.g. the Voc of the
multicrystalline cells. A comparison between the
experimental and simulated IV parameters is shown in
Fig 2. Overall, the simulation model reproduces the
measurements very well. For example, the model predicts
Voc with a precision of about 3 mV. The fill factor is
reproduced with the least precision because the resistive
losses due to the metallization are included in the model
merely as a lumped value.
3
Experimental Data
Simulated Data
16.8
η [%]
16.5
16.2
78.9
FF [%]
78.6
78.3
34.4
Jsc [mA/cm2]
34.0
33.6
621
618
Voc [mV]
615
612
1
2
3
1. Cell Process
4
5
6
2. Cell Process
Figure 2: Comparison between experimental and
simulated I-V parameters of multicrystalline Si solar cells
from an industrial production line made from six
different ingots and applying two different fabrication
processes.
The histograms used in Sec. 2 are obtained from
lifetime images, containing a certain number of τbulk(x,y)
values, which are binned into n bins. In this section, we
compute the histograms synthetically, while we do not
care about how the τbulk(x,y) values could be distributed
in the synthetic lifetime image. The boundary conditions
for our synthetic histograms are as follows. The number
of τbulk values is kept constant at 106, which is
comparable to the number of pixels in the experimental
lifetime images of Sec. 2. Each histogram has an
adjustable lowest lifetime value τmin, as indicated in Fig.
3. Each bin has the same area A0, except a selectable area
A1 ≥ A0 with lifetime values between selectable limits τ1
and τ2. Additionally, the histograms’ mean lifetime M is
INFLUENCE OF LOW LIFETIME REGIONS ON
CELL PERFORMANCE
It is widely known that the low lifetime regions in
mc-Si limit the cell’s performance. If two mc-Si cells
have equal mean lifetimes but different lifetime
distributions, the wafer having smaller regions of low
lifetimes yields a better cell performance than the other.
In this Section, we synthetically vary the lifetime
distribution while keeping the mean lifetime constant,
and show that Voc may vary up to 30 mV because Voc is
influenced by the varying areas of low lifetimes.
Figure 3: Schematic illustration of synthetic lifetime
distributions. The values τmin, τ1, τ2 and A1 are set as input
parameters. A selectable mean value M is achieved by
variation of τmax and A0.
27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9
(ii) The slope of the Voc lines is influenced by τ2. At the
line with the strongest decrease (lowest curve of blue
squares) the area set by the parameters τ1 and τ2 contains
in this case only the lowest lifetimes. By increasing the
parameter τ2, higher lifetimes become more abundant,
resulting in a smaller slope of Voc. This shows that the
lowest lifetimes limit the Voc values the most.
(iii) The second set of synthetic lifetime distributions
(green circles) starts at higher lifetime, τmin = 10 µs,
which shifts all Voc values up. This shows again that the
lowest lifetimes limit the maximum achievable Voc.
(iv) Note that the spread in Voc is about 22 mV, causing a
spread of cell efficiency η of 1.5 % absolute as is
apparent in Fig. 5. This demonstrates that the arithmetic
mean of wafer lifetimes (which remains constant in Figs.
4 and 5) is a poor indicator for cell performance.
620
Voc [mV]
615
610
τmin
τ2
605
τ2
600
595
First set of synthetic lifetime distributions
Second set of synthetic lifetime distributions
0
10
20
A1 [%]
30
40
Figure 4: Modeled Voc values from 720 synthetic lifetime
histograms having a constant arithmetic mean value of
64 µs but different amounts of lower and higher lifetime
values. The variation in Voc is 22 mV.
16.5
16.2 τmin
τ2
15.9
η [%]
kept constant. To achieve this selected M value, the
parameter τmax must be allowed to be free. And to keep
the number of τbulk values at 106, the value A0 must be
free as well. This procedure allows us to synthetize
different histograms having equal mean values M by
using τmin, τ1, τ 2, and A1 as input parameters.
In the following, we generate histograms that have a
constant arithmetic mean lifetime Mari of 64 µm, as
observed in the sample 5 shown in Fig. 2. The first set of
distributions starts at a very low lifetime τmin = τ1 = 1 µs
with a variation of τ2 from 6 µs to 30 µs in steps of 3 µs.
For all of these distributions, the area A1 of lower
lifetimes between τ1 and τ2 is varied from 1 to 40% of the
total area. With the device model of Sec. 2, Voc is
simulated and shown as blue squares in Fig. 4. A second
set of distributions is generated with a higher minimal
value of τmin = τ1 = 10 µs plus variations in τ2 from 15 µs
to 39 µs in steps of 3 µs and A1 from 1 to 40%. The
resulting Voc values are indicated by the green circles in
Fig. 4. A clear tendency of Voc is observable:
(i) With increasing A1 the number of low lifetime values
set between τ1 and τ2 increases. To keep the mean value
of the distributions constant, the upper limit of high
lifetimes, τmax, increases in these histograms. As a result,
Voc continuously decreases because the higher lifetimes
cannot compensate the influence of the lower lifetimes.
15.6
τ2
15.3
15.0
First set of synthetic lifetime distributions
Second set of synthetic lifetime distributions
0
10
20
A [%]
30
40
1
Figure 5: Modeled η values from the same simulations
as shown in Fig. 4. The maximum variation in η is 1.5 %
absolute.
More synthetic distributions can be produced by
moving the area between τ1 and τ2 from the lowest
lifetimes up to the highest, while holding the mean value
constant at 64 µs. The maximum variation in Voc then
becomes 30 mV.
Besides the arithmetic mean value, the harmonic and
geometric mean values have been analyzed in the same
way. A maximum range in Voc of 7 mV for the geometric
and 4 mV for the harmonic mean is observable. The
arithmetic mean value has the highest variation in Voc
because it weights the lowest lifetimes most, the
harmonic mean weights the lowest lifetimes least and is
most stable by changing the synthetic distributions. The
geometric mean weights all lifetime values equal.
4
MEAN VALUES FOR PREDICTING
CELL PERFORMANCE
In the following, we show that no global mean value
exists for predicting cell performance from lifetime
images with certainty.
We broaden our scope with the generalized theory of
mean values, as was done in Ref. [2]. The generalized
mean value of x1,…,xn positive real numbers is defined
as:
𝑝 1/𝑝
𝑀𝑝 (𝑥1 , … , 𝑥𝑛 ) = �𝑛1 ∑𝑛𝑖=1 𝑥𝑖 �
,
(3)
where the exponent p is a real number and characterizes
the mode of mean. For example, p = 1 is the arithmetic
mean, p = –1 the harmonic mean and p = 0 the geometric
mean. In the following, we show that there is no global p
value for predicting cell performance (with a given
fabrication process). Or in other words, each wafer has its
individual p value for predicting cell performance, which
can only be found by physical modeling, not by pure
mathematical means.
Each simulated or experimental Voc value of a mc-Si
cell can be reproduced with the simulation model
described in Sec. 2 using a c-Si cell and a single, welldefined τsim,i value; it acts as a kind of average value,
which we now call τVoc,av. Hence, to each τVoc,av value
corresponds a p value in Eq. (3) with τVoc,av = Mp. For
example, the sample 3 shown in Fig. 2 has p = –0.69 and
27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9
τVoc,av = 54.97 µs, whereas sample 5 in Fig. 2 has
p = –0.58 and τVoc,av = 43.70 µs. We have investigated a
[8]
number of mc-Si cells and found that each waver has its
individual p value which, on top of this, may depend on
the fabrication process.
Hence, whatever kind of mathematical averaging
procedure we use for assessing the quality of mc-Si
wafers, we cannot predict cell performance with
certainty. This makes sense because, if a mathematical
averaging procedure existed to predict cell performance,
no physical models would be necessary to describe solar
cells – it could be done on purely mathematical grounds.
[12]
5
EXCLUSION PROCEDURE FOR LOW-QUALITY
WAFERS BEFRORE FABRICATION
[13]
This still leaves the question of which averaging
procedure predicts cell performance most precisely so it
can be used to exclude low-quality wafers from
fabrication.
A possible way for obtaining a procedure to exclude
low-quality mc-Si wafers from cell fabrication may be as
follows:
[14]
1. Measure τeff(x,y) images of wafers before production
and compute their mean values. Gettering and
thermal treatments during cell fabrication will change
τeff(x,y), but to some extent it is still possible to
2. correlate these initial mean values with the resulting
Voc values after cell fabrication.
3. Based on a rather broad range of data, establish a
lowest mean value below which a wafer should be
excluded. Sec. 3 gives the reasons why this
correlation is fuzzy to, say, 7 mV in Voc. Hence, any
exclusion criterion is only able to exclude rather few
wafers from cell fabrication without running the risk
that too good wafers are excluded as well.
ACKNOWLEDGEMENTS
Funding was provided by the German Ministry for
the Environment, Nature Conservation and Nuclear
Safety (BMU). The Al2O3 passivation was carried out at
the Institute of Solar Energy Research Hamelin (ISFH),
Germany.
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