High Performance Computing for
the Simulation of Thin-Film Solar Cells
C. Jandl, K. Hertel, W. Dewald, and C. Pflaum
Abstract To optimize the optical efficiency of silicon thin-film solar cells, the absorption and reflection of sunlight in these solar cells has to be simulated. Since
the thickness of the layers of thin-film solar cells is of the size of the wavelength,
a rigorous simulation by solving Maxwell’s equations is important. However, large
geometries of the cells described by atomic force microscope (AFM) data lead to
a large computational domain and a large number of grid points in the resulting
discretization. To meet the computational amount of such simulations, high performance computing (HPC) is needed. In this paper, we compare different high performance implementations of a software for solving Maxwell’s equations on different
HPC machines. Simulation results for calculating the optical efficiency of thin-film
solar cells are presented.
1 Introduction
Silicon thin-film solar cells are an innovative, inexpensive, and environmentally
friendly technique to produce renewable energy. Increasing the efficiency of these
cells is currently an important research topic in the field of photovoltaic. To this
end, the optical path in the solar cell has to be improved to enhance absorption.
Therefore e.g. the light coupling into the cell can be adjusted by structured front
C. Jandl · K. Hertel · C. Pflaum
Erlangen Graduate School in Advanced Optical Technologies (SAOT) and Department of Computer Science 10, University Erlangen-Nuremberg, Cauerstr. 6, 91058 Erlangen, Germany
e-mail: christine.jandl@informatik.uni-erlangen.de
e-mail: kai.hertel@informatik.uni-erlangen.de
e-mail: pflaum@informatik.uni-erlangen.de
W. Dewald
Fraunhofer Institute for Surface Engineering and Thin Films IST, Bienroder Weg 54 E, 38108
Braunschweig, Germany
e-mail: wilma.dewald@ist.fraunhofer.de
S. Wagner et al. (eds.), High Performance Computing in Science and Engineering,
Garching/Munich 2009, DOI 10.1007/978-3-642-13872-0 46,
© Springer-Verlag Berlin Heidelberg 2010
553
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C. Jandl et al.
Fig. 1 Cross-section of a
thin-film solar cell interface
described by AFM scan
contacts. Since the rough morphology of the front TCO is passed to all following films of the cell, light trapping in the cell can be achieved. In order to design
such a solar cell, numerical simulations of Maxwell’s equations are needed (see
[1] and [2]). Widely used methods for solving Maxwell’s equations for high frequencies include the finite difference time domain method (FDTD) and the finite
integration technique (FIT). The computational effort required by these methods is
extremely high in case of thin-film solar cell simulations. Therefore, the use of HPC
is needed. In this paper, we study different implementations of the FDTD iteration
method.
To show the practicability of our simulation tool, we present results of the simulation of a solar cell structure based on AFM scan data. In this simulation, we study
a thin-film solar cell consisting of a layer of glass, a transparent conductive oxide
(TCO) layer, a layer of microcrystalline silicon (μ c-Si:H), a second layer of TCO,
and a layer of silver (Ag) (see Fig. 1). The texture of the first TCO layer influences
the texture of all the other layers. There exist several publications analyzing the optical efficiency of a thin-film solar cell based on this kind of model structure (see
[1, 2, 4–6], and [8]). However, the real texture of TCO in a thin-film solar cell is
a random texture. Therefore, simulations based on model structures can give a hint
on what an optimal texture might look like. In order to compare different TCO textures though, it is important that simulations based on AFM scans are performed
on large computational domains to gain statistical information. Results gained from
these simulations greatly help in optimizing the optical properties of the TCO layer
without the need to manufacture a complete thin-film solar cell.
2 Simulation of Thin-Film Solar Cells
To calculate the short-circuit current density of a solar cell, it is necessary to take
into account the intensity of light with respect to the wavelength. Fig. 2 depicts the
corresponding solar spectral irradiance AM1.5. Using these data, the short-circuit
current density can be calculated by
High Performance Computing for the Simulation of Thin-Film Solar Cells
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Fig. 2 Intensity of the solar spectrum AM1.5 which
ranges from λ = 0.3 μm to
λ = 1.1 μm
JSC = ∑ QE(λ )PλAM1.5 Δ λ
λ
eλ
,
hc
(1)
where PλAM1.5 is the intensity of light with linewidth Δ λ , e is the elementary charge,
h the Planck constant, and c the velocity of light in vacuum. Furthermore, the quantum efficiency is defined by the quotient of the absorbed power in silicon over input
power:
Pabs,λ
.
(2)
QE(λ ) :=
Pin,λ
The main difficulty is to calculate the absorbed power Pabs,λ for an incoming wave
of power Pin,λ . Since the relevant structures of the thin-film solar cell are of the order
of magnitude of one single wavelength, it is important that Maxwell’s equations are
solved by an accurate simulation method. A common approach is to apply the finite
integration technique.
2.1 Modeling Maxwell’s Equations
Maxwell’s equations describe the propagation of electromagnetic waves in a very
broad manner. Based on some model assumptions about the materials involved in
the production of solar cells (see [3, 9]), we can reduce them to the following set of
equations:
μ ∂t H = −∇ × E − σ H ,
ε ∂t E = ∇ × H − σ E ,
0,
÷ (ε E) =
÷H =
0.
(3)
(4)
(5)
(6)
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In order to be able to solve these equations numerically, a discretization scheme
has to be employed. In the following, finite differences will be shown to be a
practicable method for this kind of problem. Our implementation relies on rectangular staggered grids for spatial discretization. It is well known that such a staggered grid discretization satisfies (5) and (6) in a discrete sense, if these conditions
hold for the initial value of the iteration. So, with two out of four equations being
dealt with implicitly, the finite difference discretization allows for Maxwell’s equations to be solved by a relatively simple system of equations. In order to be able
to model interfaces of curvilinear geometry more accurately, the finite integration
technique is applied. Resulting stencils in the time-harmonic case are equivalent
to the ones produced by the FDTD method, while integrals are hidden in the coefficients. So, for reasons of simplicity we limit the following discussion to the
FDTD and derived finite difference frequency domain (FDFD) cases, which turns
out to be equivalent to the FIT method where coefficients are constant. Due to interdependencies between Eqs. (3) and (4), temporal decoupling is an important step
towards developing an iterative solver. It turns out that a leap frog scheme can
be used to this end. This leads to the following update steps for the time evolution:
τσ τ
τσ t+ τ2
t− τ2
t− τ
t
HC = HC −
grad ×EC −
(1 − ρ ) HC 2 and (7)
1+ρ
μ
μ
μ
h
τ
τ
τσ
τσ t+τ
t+
EC = ECt +
grad ×HC 2 −
1+ρ
(1 − ρ ) ECt ,
(8)
ε
ε
ε
h
where ρ ∈ [0; 1] is a steering parameter for interpolating the given quantities as
needed. Note that the subscript C simply identifies the node in the mesh relative
to the adjacent ones while iterating over the vector field (refer to Fig. 3 for further
details). In order to allow for a linear iterative solver to converge towards a solution
of this evolution scheme, we exploit the fact that electromagnetic waves expose
periodic behavior, and thus solutions can also be expected to be periodic. This leads
to the related stationary problem in the frequency domain:
Fig. 3 A node in the finite
difference mesh and its neighbors
High Performance Computing for the Simulation of Thin-Film Solar Cells
1+ρ
τσ
ε
n+ 21 iωτ
HC
e
n− 21
= HC
τ
− μτ gradh ×ECn eiω 2
τσ n+1 iωτ
1+ρ
EC e
ε
(9)
n− 1
− τσμ (1 − ρ ) HC 2 and
τ
n+ 1
= ECn + τε gradh ×HC 2 eiω 2
557
(10)
n
− τσ
ε (1 − ρ ) EC ,
where t and τ have been dropped in favor of n as a label for iterative updates to
the E and H fields, since this scheme does not represent a time evolution in the
strict sense anymore. In order to meet stability requirements for materials of negative permittivity, especially silver in the case of thin-film solar cells, this iterative
scheme can be slightly modified as has been shown in [7]. To gain better control
over spurious reflections at the computational boundaries, perfectly matching layers
are deployed. To this end, a split field approach is helpful, effectively turning the
above two equations into four. Take, for example, the x component of the electric
field. In its original form, the equation reads:
τ
τσ n+1 iωτ
n+ 21
n+ 21
n
−1
ECx e = ECx +
hy
HNz − HSz
1+ρ
ε
ε
τ
n+ 21
n+ 21
−1
HTy − HBy
ei ω 2
−hz
after resolving the discrete curl operator in the finite difference formulation. Note
that h denotes the spatial granularity of the mesh. After splitting up this equation
subject to the constraint of ECx = ECxy + ECxz being satisfied for all steps n of the
iteration, we get:
τσ n+1 iωτ
τ
n+ 21
n+ 21
n+ 21
n+ 21
1+ρ
ECx e = ECnx − τε h−1
H
eiω 2 and
+
H
−
H
−
H
z
Tyx
Tyz
Byx
Byz
y
y
ε
τσ n+1 iωτ
τ
n+ 21
n+ 21
n+ 21
n+ 21
1+ρ
ECx e = ECnx + τε h−1
H
ei ω 2 .
+
H
−
H
−
H
y
Nzx
Nzy
Szx
Szy
z
z
ε
For reasons of reducing computational complexity, coefficients can be absorbed into
separate vectors, as they are not subject to change after the beginning of the iterative
process:
ĉExy =
1
1+ρ τσ
ε
e−iωτ ,
cExy = − 1+ρ1 τσ e− 2 iωτ τε h−1
z ,
1
ε
ĉExz =
1
1+ρ τσ
ε
e−iωτ ,
cExz = + 1+ρ1 τσ e− 2 iωτ ετ h−1
y .
1
ε
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Note that ε and σ generally are not constant throughout the computational domain,
and as a consequence neither are the coefficients above. This gives rise to the following formulation of the split-field approach:
n+ 21
n+ 21
n+ 21
n+ 21
n+1
n
,
(11)
ECx = ĉECxy ECx + cECxy HTyx + HTyz − HByx − HByz
y
y
n+ 21
n+ 21
n+ 21
n+ 21
n +c
H
.
(12)
=
ĉ
E
+
H
−
H
−
H
ECn+1
E
E
N
N
C
C
C
S
S
x
x
zx
zy
x
x
z
z
z
y
z
z
x
y
So, the resulting iteration scheme effectively solves for a vector of approximately 6
components times the number of grid points in the computational domain.
2.2 Solar Cell Model
In order to demonstrate the efficiency of our simulation technique, let us study a
thin-film solar cell model with one layer of microcrystalline silicon. Fig. 1 depicts
the cross-section of a simulated solar cell with the interface being described by
an AFM scan. The size of the complete simulated solar cell is 2.4 μm in x and y
directions, and 2.38 μm in z direction. The thickness of the glass layer is 0.2 μm,
the size of the upper TCO layer is 0.6 μm, and that of the lower TCO layer is
0.08 μm. The height of the microcrystalline silicon is 1 μm, and the silver layer
has a thickness of 0.5 μm. A real thin-film solar cell consists of a glass layer of a
thickness larger than 0.5 cm. Simulating such a thick layer of glass by using the
FDTD method is computationally extremely intensive. However, light waves approximately behave like plane waves inside the glass layer. Therefore it is sufficient
to start with an incoming plane wave at the top of a thin layer of glass (thickness of
0.2 μm). The AFM scans of the TCO layer are courtesy of Fraunhofer IST Braunschweig.
To calculate the short-current density, we apply the finite integration technique.
The mesh size of the grid is calculated as follows. We compute the local wavelength in every grid point p, which is defined by λ (p) := nλr , where nr is the
real part of the refractive index of the local material (see Fig. 4). Then this local wavelength is used for discretizing the domain with a fixed number of grid
points per wavelength. For the AFM based simulations presented in this paper,
we choose 20 grid points per wavelength. This calculation leads to grids of varying mesh size and a fairly large number of discrete grid points. We calculate
the quantum efficiency for the most relevant part of the spectrum, ranging from
0.55 μm to 0.995 μm with a linewidth Δ λ = 5nm. The number of grid points
varies between 6 and 46 millions depending on the wavelength of the incoming
wave.
High Performance Computing for the Simulation of Thin-Film Solar Cells
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Fig. 4 The real part of the
refractive indices of silicon,
TCO, and silver dependent on
the wavelength
2.3 Integration of AFM Scans
By standard simulation programs for solving Maxwell’s equations, it is difficult to
simulate the real roughness of the solar cell layers. Hence, imitated structures (e.g.
a pyramidal structure) are used to calculate the short circuit current density of a thin
film solar cell. To overcome this restriction, we developed a simulation tool which
is able to integrate AFM scan data. As an example we present simulation results
with AFM scan data provided by Fraunhofer IST Braunschweig (see Fig. 5 and Table 1). We need periodic boundary conditions in x and y directions to compute the
efficiency of the scanned thin-film solar cells. Hence, we have to mirror the scan.
This leads to a domain of size 10 μm × 10 μm. If we wanted to compute the effi-
Table 1 Samples done by
Fraunhofer IST Braunschweig
Fig. 5 AFM scan “Sample
A” from Fraunhofer IST
Braunschweig sized 5 μm ×
5 μm with a resolution of
512 × 512 points
rms-roughness
O2-partial pressure
specific resistance
layer thickness
Sample A
Sample B
32 nm
6 mPa
315 μΩ cm
1.08 μm
53 nm
12 mPa
410 μΩ cm
1 μm
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Fig. 6 AFM Scan: Black framed part is the part of the original scan
ciency for a region of this size, we would need more than 109 grid points for the
resulting discretization. In order to decrease the size of the computational domain
and, with it, the number of grid points, we cut the AFM domain into smaller pieces
(see Fig. 6). There are two ways of picking the part of the scan to be used. One is
to choose a fixed part of the AFM scan, which is then simulated for every wavelength. Another is to choose a different part of the AFM scan for each wavelength.
In this case, the chosen part is computed reproducibly from pseudo random numbers. Numerical simulations have shown that choosing different random pieces for
each wavelength lead to more accurate results. Another aspect when dealing with
AFM scans in simulations is that the spatial resolution of the scan generally does
not match the one of the computational domain, so the positions of the resulting
grid points do not necessarily coincide. To resolve this issue, AFM data are linearly
interpolated.
2.4 Parallelization
Since the spatial resolution of the computational domain needs to be relatively high
in order to obtain accurate results, parallelization and the use of high performance
compute clusters are essential. This is especially true where simulations are based
on AFM data, since a minimum of 20 nodes per wavelength are required on the grid
in order to obtain useful results that can yield information about the efficiency of
applied nanostructures. Therefore, the implementation effort has strongly focused
on the solver being MPI-capable throughout the development process. The scaling
behavior turns out to be near-linear as long as the ratio problem size vs. number of
nodes is considered to be same (see Fig. 7, 8).
For parallel runs, a box-shaped computational domain is decomposed into rectangular sub-boxes. This is done in such a way that a maximum of available compute
nodes is assigned a computational partition, while at the same time interfaces between partitions are kept to a minimum.
High Performance Computing for the Simulation of Thin-Film Solar Cells
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Fig. 7 Speed-up of the solver
relative to the 64 compute
node configuration. Simulation of a synthetic 3dimensional moth structure
consisting of roughly 12.28
million grid points. Comparative run on 64, 96, 128, 192,
256, 384 and 512 nodes of the
LRZ Itanium2 Cluster
Fig. 8 Runtime against the
resolution of the computational domain on a fixed
number of 64 compute nodes.
Simulation of the same synthetic 3-dimensional moth
structure with varying numbers of grid points
Fig. 9 Rectangular partitioning scheme balancing the
computational load between
compute nodes
Generally, parallelization is a relatively straight-forward matter where rectangular grids are concerned. Communication across local boundaries need to be performed between iteration steps in order to ensure that neighboring stencils are correctly fed current data. The only detail that needs special attention when exchanging
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Fig. 10 Comparative runs on different architectures, both with the original implementation and
current progress in optimization. Serial simulation runs of a synthetic 3-dimensional moth structure. Fixed number of 1000 iteration steps, 1.5 million grid points
data between compute nodes is the fact that dual grids are in place for different aspects of the electromagnetic field, leading to a slight index shift.
Besides parallelization, a number of optimizations steps are currently being explored in order to allow for the Maxwell code to run faster and, as a consequence,
reduce workloads and expensive runtime on high performance machines. For once,
a relatively simple cache-blocking technique has been implemented lately. As optimization is highly dependent on the architectures involved, results seem rather
ambivalent at the time and are currently under investigation. Also, we are currently
pursuing to reduce application complexity in critical sections and introduce vectorization where available (see Fig. 10 for a preliminary analysis of the current state of
affairs).
3 Simulation Results
The aim of our simulation in regard to physics is to compare quantum efficiencies
and short-circuit current densities dependent on the geometry and the simulated
parts of AFM scans.
To this end, we compare quantum efficiencies of two different AFM scans “Sample A” and “Sample B” provided by Fraunhaufer IST Braunschweig. The geometry
of “Sample A” is depicted in Fig. 5. “Sample B” is of the same size and spatial resolution as “Sample A”, resulting in the same amount of discretized grid points, but it
has a rougher structure. The scan domains are chosen by pseudo random numbers.
Fig. 11 shows the quantum efficiency (QE) of both solar cells computed by Eq. (2).
High Performance Computing for the Simulation of Thin-Film Solar Cells
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Fig. 11 Quantum efficiency
of thin-film solar cells with
different rough layers
Table 2 Short-circuit current
density
JSC
mA/cm2
Sample A
Sample B
8.9821
10.2116
The short-circuit current density JSC is obtained by using Eq. (1) (see Table 2). It
can be seen that rougher surface structures lead to an increase in short-circuit current
density.
The simulations are performed on the HLRBII of LRZ Munich with up to
512 processors. Depending on the speed of convergence the runtime varies between
41/2 and 7 hours per solar cell simulation. The number of discretization points varies
from 6 million for a wavelength of 0.95 μm to 46 millions for 0.55 μm.
The quantum efficiency and short-circuit current density were computed for each
solar cell structure with three different sets of pseudo random numbers. A maximum
deviation of 2.4% of the short-circuit current density is observed for “Sample A” and
1.1% for “Sample B” respectively. Therefore it is necessary to expand the computational domain to obtain more accurate results.
4 Conclusion
Computationally intensive simulations allow a detailed analysis of light trapping in
thin-film silicon solar cells. In this paper, we present first simulaton results for a
complete thin film solar cell structure using AFM scan data. Our aim is to extend
these simulations to a tandem solar cell which consists of amorphous (a-Si:H) and
microcrystalline (μ c-Si:H) silicon and has a thickness of 3.3 μm. To gain more accurate results a larger part of the AFM scan has to be simulated. This will increase
the number of discretization points and computing time. In addition, simulations for
oblique incident waves have to be done, because real outdoor conditions often lead
to oblique incident sunlight.
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Acknowledgements The authors gratefully acknowledge funding from the Erlangen Graduate
School of Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG) in
the framework of the German Excellence Initiative, the Bavarian Competence Network for Technical and Scientific High Performance Computing (KONWIHR) and funding in the frame of the
joint project LiMa by the Federal Ministry for the Environment, Nature Conservation and Nuclear
Safety BMU under contract No. 0327693A.
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