Sophisticated light management is crucial for optimal thin-film solar cell efficiency.^{1} One successful approach, based on nanostructured composite layers and illustrated in Figure 1, incorporates materials with highly optimized optical properties. As might be expected, the design, development, and testing of these new solar cell prototypes is a time-consuming process. For this reason, suitable models and simulation techniques are required for the analysis of optical properties within thin-film solar cells.^{2} These can be widely applied to nanostructure designs and may include ray tracing, optical admittance analysis, a transfer matrix method, or finite difference discretization of Maxwell's equations.

**Figure 1. **Structure of a thin-film solar cell. Onto a rough silver (Ag) surface are deposited, in order, transparent conductive oxide (TCO), microcrystalline and amorphous silicon (μc-Si:H and a-Si:H, respectively), a second layer of TCO, and glass.

The latter leads to the most accurate simulations because it includes optical effects such as interference, optical near-field properties, and plasmon effects. Suitable discretization methods are finite edge elements (FE), the finite integration technique (FIT), and finite difference time domain (FDTD). However, these approaches are all computationally intensive owing to the difficulty of achieving the randomness of rough interfaces between composite layers formed on rough metallic surfaces.

We have developed a simulation tool for calculating quantum efficiency and short-circuit current density of thin-film solar cells. We use the FIT method since it can be used for curvilinear interfaces and is less computationally intensive compared with FE. We optimized the software using the standard message-passing interface (MPI) to harness the power of several thousand processors. The topography of interfaces between different layers used to build our simulation uses data from atomic force microscopy (AFM) scans of real surfaces. For the simulation itself, suitable boundary conditions must be defined to handle the non-periodic structure of the AFM scan data. We modeled sunlight for different polarizations and oblique incident light, and carried out simulations in parallel nodes for a given solar spectrum. The simulation also takes account of the absorption of light in different layers of the solar cell.

To simulate materials with negative permittivity at visible wavelengths, such as the silver used to back the thin film, we used a time harmonic inverse iteration method. Silver and other noble metals metals have negative permittivity because they are opaque to electromagnetic waves at visible wavelengths. But at shorter wavelengths electromagnetic waves are above the resonant frequency of the material and begin to pass through it. We use our time harmonic inverse iteration method^{3} to predict the paths of electromagnetic waves as they pass through the material. ‘Time harmonic’ relates to the fact that we are dealing with the time-dependent partial derivative of Maxwell's equations in a periodic (harmonic) treatment. This solves Maxwell's equation for a given frequency without requiring a ‘Drude’ approximation of the permittivity of the material. In other words, the refractive index and absorption of silver do not have to be approximated as they do for other methods.^{4}

As an example, we simulated a tandem thin-film solar cell consisting of one layer of amorphous silicon (a-Si:H) and one layer of microcrystalline silicon (*μ*c-Si) (see Figure 1), with the aim of analyzing the effect of adding a back transparent conductive oxide (TCO) and silver back contact. It is widely known in the field that plasmonic effects at the silver-backed contact influence the short-circuit current density. The plasmonic effects themselves are in turn influenced by the surface roughness of the silver layer, which we measured using AFM.

**Figure 2. **Absorption, loss, and reflection in a tandem thin-film solar cell. The computational domain was a cube of size 3.5μm in all directions. The solar spectrum was covered by 41 wavelengths from 300–1100nm. QE is quantum efficiency.

To optimize the back contact, we explored a number of different composite architectures, including with and without the TCO back layer, and for smooth and rough surfaces of the silver back contact. Our simulations showed that a TCO back layer is essential for obtaining a high short-circuit current density. However, while the roughness of the silver back contact has a minimal effect on the total short-circuit current density, that of the TCO back layer contact can strongly influence the efficiency of the solar cell. Figure 2 shows absorption, loss, and reflection of a solar cell with a TCO back contact incorporating AFM scans of real interfaces. The numerical simulations were performed on the high-performance Lima Computer Cluster in Erlangen, Germany, using 1536 processors.

Our next steps are to analyze the optimal structure of the TCO back layer compared with the structure of the front TCO layer. This can be done by applying and comparing modified AFM scan topologies, measured from real samples. We will then compare real prototypes of our theoretically optimized solar cells to check the validity of our model.

Christoph Pflaum, Christine Jandl

Friedrich-Alexander University of Erlangen-Nürnberg

Erlangen, Germany

Christoph Pflaum was awarded a technical degree in mathematics and electrical engineering in 1992 at the Technical University of Munich, and a PhD in mathematics at the same university in 1996. From 2003 he has been professor of high-performance computing at the University of Erlangen-Nürnberg.

References:

2. C. Haase, H. Stiebig, Optical properties of thin-film silicon solar cells with grating couplers, *Prog. Photovolt. Res. Appl.* 14(7), p. 629-641, 2006.

3. A. Taflove, S. C. Hagness, *Computational Electrodynamics: The Finite-Difference Time-Domain Method*, Artech House, 2000.

4. C. Pflaum, Z. Rahimi, An iterative solver for the finite-difference frequency-domain (FDFD) method for the simulation of materials with negative permittivity, *Num. Linear Algebra Appl.* 18(4), p. 653-670, 2011.