Estimating Pyramid Density of a Random-Textured Surface by
Capacitance-Voltage Measurement of c-Si Solar Cells
Xufeng Wang and Muhammad A. Alam
Purdue University, West Lafayette, Indiana, 47907, U.S.A.
Crystalline silicon (c-Si) solar cell dominates today’s PV
market due to its low cost, mature technology, ease of
manufacturing, and many other advantages. Unfortunately, the
indirect bandgap of c-Si makes sunlight absorption difficult.
Effective light trapping is therefore essential for a c-Si solar
cell to have high efficiency.
Surface texturing of c-Si cells allows effective light trapping
through random scattering and total internal reflection. These
textures can be created by etching the silicon substrate with
dilute KOH/NaOH solution [1]. For (100) silicon wafer, which
is commonly used in manufacturing, this etching exposes
intersecting {111} planes, thereby forming a pyramidal shape.
Typical height of the pyramids ranges between 1-6 um. The
randomly textured wafer is then exposed to a phosphorus
diffusion process to form the n-type emitter, such that the
topography of the PN junction conforms to that of the
pyramidal texture.
To characterize the geometry, density, and surface
properties of this randomly textured surface, one generally
resorts to advanced metrological techniques such as SEM
imaging. These techniques are expensive and time consuming,
and thus not suitable for inline measurement, and may not
easily be accessible to medium/small scale manufacturers on a
routine basis. Very often, however, even a rough
characterization on the random texture can be an important
predictor of the overall quality. Thus, we are motivated to
develop a rapid and low cost characterization method for such
tasks.
978-1-4799-3299-3/13/$31.00 ©2013 IEEE
II. A THEORY OF ANOMALOUS CAPACITANCE
Although the randomly formed pyramids have different
shapes and sizes, they are all enclosed by {111} planes. As a
result, the microscopic surface area of the cell ( Atexture ) has a
fixed ratio to the base area ( Aplanar ) of the pyramid,
Atexture / Aplanar ≈ 1.732 , which is a result well known to the cSi PV community [2].
If the capacitance were directly proportional to area, one
might conclude that Ctexture / C planar ≈ 1.732 as well, regardless
meas
)
of the applied voltage. Unfortunately, measured C-V ( Ctexture
meas
shows Ctexture / Cplanar < Atexture / Aplanar ≈ 1.732 (see Fig. 1a).
This non-ideal voltage-dependent distortion reflects the
complexity of capacitance in a dense pyramidal structure with
dimensions comparable to the depletion region, as shown in
Fig. 1c. The depletion region (blue in Fig. 1c), where lowfrequency capacitance arises from, smears out the sharp
features of the pyramid edges, such that the junction-area is no
longer conformal to the surface of the pyramid. As a result,
meas
Ctexture
/ Cplanar < 1.732 , and we call the difference
the “correction capacitance”, a
CC = Ctmeas
exture − 1.732 × C planar
quantity easily measured from typical measurements.
(
)
ï
ï
ï
(a)
1.732*Cplanar
C
meas
Cplanar
ï
ï
I. INTRODUCTION
The paper is organized as follows. In Sec. II, we first
describe our theoretical approach toward transforming the
random pyramid surface into an arrangement defined by two
characteristic topological features. We then describe an
algorithm to obtain pyramid density of commercial c-Si solar
cell samples. In Sec. III, we present the results of pyramid
density of the samples deduced from our purposed method and
characterization. We summarize our findings in Sec. IV.
Abstract — Random surface texturing allows efficient lighttrapping and therefore, is widely used in commercial c-Si PV
technology. The geometry, periodicity, and interface
characteristics of the texture greatly impact the efficiency of the
c-Si solar cells, and thus, the ability to evaluate the texture
‘quality’ in a time-and-cost effective manner without
sophisticated metrology is desired. In this paper, we report a
simple and efficient algorithm to estimate the pyramid density
via standard capacitance-voltage measurement. We demonstrate
that the overall capacitance of a random pyramid surface can be
decomposed into three distinct capacitive components, and this
decomposition can be used to estimate the pyramid density. We
validate the algorithm by sampling the pyramid density of
commercial c-Si solar cells and show that the density obtained
from image analysis is consistent with the estimates suggested by
the purposed model.
Index Terms — surface texture, capacitance-voltage
characteristics, statistical analysis, photovoltaic cells, silicon.
(b)
ï
ï
ï
ï
Cmeas - 1.732*Cplanar
ï
1465
3499
ï
(c)
(a)
Decreased
(b)
Virtual cut
1
2
Same
4
3
B
Enhanced
Virtual cut
A
Fig. 1. (a) The open circles denote the measured C-V
characteristics for a c-Si solar cell with base doping 1x1016 /cm3 (
meas
). For comparison, capacitances for a planar cell ( C planar ) and
Ctexture
1.732 × C planar are also shown. (b) ‘Capacitance correction’ defines
the gap between the measured capacitance and 1.732* C planar . (c)
Simulated cross-section of a single pyramid, displaying depletion
region (blue). The nonconformity of the depletion edge, with respect
to the surface of the pyramid, interprets the capacitance gap.
Although the smearing of the junction makes the calculation
of capacitance difficult, we can actually take advantage of the
feature to characterize the surface! Microscopically, each
unique type of the ‘rough’ features due to intersecting {111}
planes, which occurs N i times over the surface, contributes a
capacitance correction Ci per occurrence to the overall CC (as
in Fig. 1c):
CC = ∑ N i Ci = Cmeasure − 1.732C planar
(1)
Fortunately, the topological configurations restrict Ci to
only few types, which we will discuss next.
A. The Decomposition of Capacitance
The random surface is composed of 4 {111} planes (labeled
as 1, 2, 3, and 4 in Fig. 2a) that can intersect with each other in
only 3 different ways. These are (angles mentioned are
obtained by the cross product of the direction vectors
associated with the pair of planes):
• 1 to 2 / 1 to 4 (orange, dash-dot): intra-pyramidal
109.47O ridge is formed.
• 1 to 2 / 1 to 4 (green, solid): inter-pyramidal 109.47O
shoulder is formed.
• 1 to 3 (red, dash): inter-pyramidal 70.53O valley is
formed.
In addition, the pyramid tops formed by all 4 {111} planes
introduce another major correction factor:
• 1+2+3+4 (blue dot): pyramidal peaks.
(c)
Fig. 2. Upper left: (a) Ridge (orange), shoulder (green), and valley
(red). Upper right: (b) Virtual cuts draw across a SEM image. Lower:
(c) The cross section of a virtual cut showing A and B features. The
corresponding features are marked as white lines in (a). Feature A
consists of two ‘109.47 degree ridges’ separated by a ‘70.53 degree’
valley, while Feature B consists of a ‘109.47O shoulder’ connected to
a ‘109.47O ridge’.
The randomly formed pyramids are tiny features covering
the entire cell surface. If we cut along either [010] or [001]
direction (yellow dash lines in Fig. 2b), we will obtain a long
cross section with various combinations of three mountainous
“sub-features” (i.e., ridges, shoulders, and valleys), see Fig.
2c. A closer examination reveals that these sub-features can be
further grouped into two composite features (A and B in Fig.
2c). Feature A consists of two ‘109.47O ridges’ separated by a
‘70.53O valley’, while Feature B consists of a ‘109.47O
shoulder’ connected to a ‘109.47O ridge’.
Since each virtual cut is performed over a large cell area, we
expect on average each cut intersects the same amount of
Feature A and Feature B. Eq. (1) can be written as
CC L2 = (N AC A + N BC B )L + N P CP ,
(2)
where N A ( N B ) is the number of Feature A (B) encountered
along a virtual cut (in unit of counts/cm), L is the width of the
2
sample assuming the sample is a square with area L , and N P
is the pyramid total over the cell area (in unit of counts). If the
sample is not a square, one can always normalize the surface
to an equivalent square and find L .
Note the ends of Feature A and Feature B are “flat”, and
this makes any combination of the AB chain a valid profile for
the cross-section. The capacitance correction factors for
Feature A and Feature B are
C A = Cvalley + 2Cridge
C B = Cridge + Cshoulder
,
(3)
(a)
(b)
doped side, K S is the relative dielectric constant, and ε0 is
the permittivity of free space.
ï
ï
ï
shoulder
ridge
ï
ï
valley
ï
with Cvalley , Cridge , and Cshoulder being the capacitance
correction factors for valley, ridge, and shoulders,
respectively. We can find the capacitance correction factors
for each of these sub-features and combine them to find C A
and C B . Fig. 3a shows 3D SentaurusTM [3] simulation
structure for the extraction of the capacitance correction
factors.
ï
peaks
ï
(a)
ï
ï
ï
(b)
ï
ï
Fig. 4. (a) Capacitance correction density from each of the subfeatures: valley, shoulder, and ridge. (b) Capacitance correction for
each pyramid peak.
Fig. 3. (a) Simulation domain for a 4 μm high pyramid. Notice the
rounded emitter regions (blue) at the peak and base. (b) Cross-section
of a valley sub-feature showing the integration box used during the
charge integration method.
To better resemble a typical solar cell, the doping profile is
rounded along sharp edges as suggested in [2]. In order to
separate the effect of each feature, a charge integration method
is used to evaluate the capacitances, as follows. For example,
consider the calculation of the valley capacitance, based the
structure is defined in Fig. 3b. First, we define the overall
capacitance for valley sub-feature ( Cvalley
′ )
Cvalley
′ =
Qvalley (V + dV ) − Qvalley (V )
dV
where Qvalley (V ) is the integrated charge for feature i for a
bias voltage V . The charge integration box is set up in a way
that exclusively and fully encloses the valley region, and all
the box edges are sufficiently apart from the valley, thus
cutting through regions not being affected by the capacitance
distortion from valley. The correction capacitance density is
′
subsequently obtained by subtracting Cvalley
from planar
geometrical capacitance
Cvalley = 2(ld C planar − Cvalley
′ )
where the planar capacitance C planar can be deduced
analytically, and ld is the geometrical length of the cell
surface enclosed by the integration box. Finally, a
multiplication of 2 is used since simulation only captures one
half of the full valley structure. For one-sided abrupt junction,
C planar =
qN B K Sε0
2(Vbi − VA )
where Vbi is the built-in voltage of the diode, VA is the
applied voltage, N B is the doping concentration of the lighter
Once the three elementary capacitances due to shoulder,
ridge, and valley are calculated from numerical simulation,
C A and C B can then be obtained from (3). Note that C p is
also determined from the same numerical simulation that
determined Cvalley , Cridge , and Cshoulder . For a given sample
with an emitter doping of 1× 1019 /cm3 and base doping of
1× 1016 /cm3, each of the sub-feature correction values are
shown in Fig. 4a. These doping values are typical in
commercial c-Si solar cell.
To find the capacitance correction factor for pyramid peaks
CP , we simulate an individual pyramid with 4 μm height,
same as shown in Fig. 3a, in 3D and find its overall
capacitance CP′ . CP can be separated out by
CP = CP′ − AsurfaceC planar − 4lridgeCridge − 2lvalleyCvalley
(4)
where Asurface is the surface are of the pyramid, C planar
planar 1D capacitance, lridge is the length of each of
ridges, and lvalley is the length of the pyramid base. The
2 for valley is because each pyramid encloses only half
total 4 valleys formed at the base.
is the
the 4
factor
of the
B. Geometrical Properties of a Textured Surface
With the capacitance correction factors ( Cridge , Cshoulder ,
Cvalley , and CP ) for all features obtained, we now find the rest
of the unknowns in (2), namely N A and N B . If we can relate
N A and N B to the pyramid population N P , equation (2)
becomes solvable.
To estimate the number of each feature, N A and N B ,
empirically, consider the following: First, the actual number
of pyramids is always bounded by the two limits: a surface
composed of purely of A or of B. Second, a count of N A and
N B along the virtual cuts in an SEM image shows
N A / N B ≈ 2 / 5 . Although this ratio is obtained by sampling
the SEM image, it can be justified intuitively, as follows. The
virtual cut extends from one side of the wafer to the other and
travels over many pyramids. If one wants to get from one
pyramid to another (e.g., from face 4 of one pyramid to face 4
of another pyramid), one has to traverse surfaces such as 4-12-4 or 4-1-3-4. Such a transition always consists of two type B
transitions (i.e., 4-1/1-2 and 1-2/2-4, see 2c) and a single type
A transition (4-1/1-3/3-4), thus a ratio N A / N B = 1/ 2 = 0.5 .
In practice, the distribution of the pyramid sizes and shapes
[4] makes this ratio slightly smaller: empirically, we find
N A / N B ~ 2 / 5 = 0.4.
In addition to the ratio N A / N B , along each virtual cut, we
need to relate the number of each geometrical type ( N A and
N B ), on average, to the number of pyramids ( N P′ )
intersecting the virtual cut. One obvious way is to find the
number of ridges the virtual cut has crossed. Note that a pair
of ridges must always meet and form a pyramid peak.
Therefore,
N P′ = N ridge / 2
(5)
Notice that N P′ (#/cm) is different from the total number of
pyramids N P (#). If, for example, a virtual cut line along x
direction intersects 1000 pyramids, N P′ = 1000 . Since the
pyramids are fine structures randomly distributed over a large
solar cell area, we expect that, if we do a virtual cut along y
direction, the line will as well intersect 1000 pyramids on
average. Therefore, the total number of the pyramids over the
entire surface
N P = ( N P′ )2
(6)
significant. Cvalley is about 1/3 of Cridge , while Cshoulder is
insignificant. One can then approximate (9) as
CC L2 = 1.55Cridge N P L + N P CP
(11)
Alternatively, in terms of the pyramid density DP = N P / L2 ,
CC = 1.55Cridge DP + DP CP
(12)
III. RESULT VALIDATION AGAINST SEM IMAGING
In order to verify our theoretical approach, we separately
developed a characterization routine to obtain the pyramid
density from commercial solar cells. Special software is
developed to process SEM images taken from these solar cells
in order to locate the pyramidal peaks. Fig. 5 shows the result
of peak finding for one of the samples. Furthermore, knowing
the precise location of these peaks, we can reconstruct the 3D
texture surface from 2D SEM images by interpolating along
{111} planes. Fig. 5b displays the result for 3D reconstruction
with the relative attitudes for each pyramid labeled. This suite
of software allows us to investigate pyramid density, heights,
distribution, and other useful statistics to estimate the pyramid
density more precisely.
(b)
(a)
We also know the ratio
(7)
Therefore, on average, each occurrence of Feature B is
accompanied by 2 / 5 occurrence of Feature A. Also, each
Feature B contains a single ridge while each Feature A
contains two ridges. With (5), N P and N B can be related as
N P′ = N ridge / 2 = (N B + 2N A ) / 2
4
= (N B + N B ) / 2 = 0.9N B
5
N P = (0.9N B )2 .
(8)
C. Calculation of the Overall Capacitance
The pyramid population N P can therefore be solved by
combining (2), (7), and (8)
CC L = (0.44C A + 1.11C B ) N P L + N P CP
2
(9)
If we define the pyramid density DP = N P / L2 , we obtain
CC = (0.44C A + 1.11C B ) DP + DP CP
(10)
Equation (9) can be further simplified by noticing the
relative magnitude of each correction factor from Fig. 4.
Evidently, the corrections due to ridge and peaks are the most
Fig. 5. (a) Processed SEM image of a sample c-Si solar cell
showing peaks found (red dots). (b) Reconstructed 3D surface of the
SEM image in (a) with peak attitudes.
As already mentioned, based on the anomalous capacitance
information in Fig. 1b, the calculated capacitance corrections
in Fig. 4, and the observed statistical ratio N A / N B ≈ 2 / 5 , we
can solve (9) to find N P (green line Fig. 6b). This result
matches closely with the pyramid count from the SEM
images, as summarized in Fig. 6a and plotted in Fig. 6b (the
range is within two dash lines labeled as “SEM”).
SEM
#1
#2
#3
Image
Area
451.19 1260.9 892.31
(μm2)
296
187
# Peaks 92
Density
0.20 0.24 0.21
(/μm2)
(a)
Fig. 6.
NA 2
= .
NB 5
statistical
SEM
SEM average
approx.
(b)
ï ï ï ï
ï
(a) Processed data examples from 3 SEM images taken on
the same sample. (b) Estimated of pyramid density overlaid with
measured density range from SEM images. Green line labeled
“statistical” is obtained from (9) with a statistical ratio between NA
and NB. Red circles labeled “approx.” is obtained from the simplified
and approximate formula (12).
This reconstruction of the surface also allows us to build a
3D surface resembling the texture quality of the measured cell
as shown in Fig. 5b. This 3D surface can then be used to
determine other useful quantities such as pyramid heights and
to be used as an input for optical simulations [5].
V. CONCLUSION
By transforming the random pyramidal surface into a
composition of three distinct features, we have demonstrated
the method to estimate the pyramid texture density from
capacitance-voltage measurement. Pyramid density is also
derived by peak-finding from SEM images and shown to be
consistent with our proposed method.
ACKNOWLEDGEMENT
This work is supported by SRC-ERI Network for
Photovoltaic Technology (NPT). The authors would like to
thank Mr. Shanbin Wang of Joy SunPower for arranging
commercial c-Si cell samples, Dr. Kyle Montgomery of UC
Davis for SEM images, Dr. Jeremy Schroeder of Purdue for
providing wet etching station, and Professor Mark Lundstrom
and Professor Peter Bermel of Purdue for helpful discussion.
REFERENCES
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reflectance reduction in multicrystalline silicon solar cells,” J.
Electrochem. Soc., vol. 151, no. 6, pp. G408, May 2004.
[2] D. Hinken, et. al., “Determination of the base-dopant
concentration of large-area crystalline silicon solar cells”, IEEE
Trans. on Electron Devices, vol. 57, no. 11, Nov. 2010.
[3] Sentaurus™ Device, www.synopsys.com. Synopsys® Inc. 2013.
[4] D. Stoyan, W. S. Kendall and J. Meche, Stochastic Geometry
and Its Applications, Wiley, 1995.
[5] McIntosh, K.R.; Baker-Finch, S.C., "OPAL 2: Rapid optical
simulation of silicon solar cells," Photovoltaic Specialists
Conference (PVSC), 2012 38th IEEE, pp.000265, 000271, 3-8
June 2012