Simulating Nanoscale Optics in Photovoltaics with the S-Matrix Method Dalton Chaffee, Xufeng Wang, and Peter Bermel Purdue University

Introduction – Solar cell physics – Scope of the research Methodology – Transfer matrix method Results – Comparison to existing simulations – Future utility Overview 2

Pushing solar cell efficiency limits with increasingly exotic materials and designs requires advanced modeling Introduction 3

Solar cells make electricity from solar photons Photovoltaic effect converts solar photons into positive and negative charges Putting two different materials together creates an internal electric field to extract charges as electricity Introduction Solar Cell Basics + - Solar photon electron current (p-n junction) Solar cell 4

Thin-film cells (~1-40 µm thick) exhibit two important properties in absorbing photons: Introduction Thin-Film Cells 5

Thin-film cells (~1-40 µm thick) exhibit two important properties in absorbing photons: 1.Light Trapping – Increased effective optical thickness of cell – Greater chance of absorption Introduction Light Trapping + - Photon Transmission Reflection electron current (p-n junction) 6

Thin-film cells (~1-40 µm thick) exhibit two important properties in absorbing photons: 2.Photon Recycling – Absorbed photons may be reemitted before their energy can be used – Photon recycling reuses these otherwise wasted photons Introduction Photon Recycling electron current (p-n junction) Photon 7

ADEPT 2.0 is an existing solar cell device simulator, available on nanoHUB.org Introduction ADEPT 2.0 8

Problem: ADEPT (and other simulations) does not precisely capture the reflection, transmission, absorption, and recycling of photons Solution: Build a self-consistent layered wave optics module to be incorporated into ADEPT Will facilitate analysis and improvement of thin-film solar cells Introduction Research Goal 9

Introduction – Solar cell physics – Scope of the research Methodology – Transfer matrix method Results – Comparison to existing simulations – Future utility Overview 10

A general mathematical model commonly used to solve problems of waves in layered media Specific techniques: T-matrix and S-matrix methods Methodology Transfer Matrix Method Inputs Sunlight spectrum Amplitudes Wavelengths Optical Properties Refractive indices Thicknesses Outputs Reflection Transmission Total absorption Carrier generation versus depth 11

Assigns a matrix to each layer based on optical properties Structure matrix formed from product of the layer matrices: Methodology T-Matrix Method Advantages: Intuitive, relatively easy Disadvantages: Lossy materials have exponentially large matrix elements  instability L. Li, JOSA A 13, 1024 (1996) 12

S-matrix method recursively forms a structure matrix that solves directly for transmission and reflection Inversion of exponential terms  greater stability Methodology S-Matrix Method L. Li, JOSA A 13, 1024 (1996) 13

Introduction – Solar cell physics – Scope of the research Methodology – Transfer matrix method Results – Comparison to existing simulations – Future utility Overview 14

Because reflection is ignored in each case, the difference is minimal Results Silicon Wafer 15 Figure 1 Optical Generation in a 200 Micron Si Wafer with Reflection = 0 Depth (microns) Optical Generation (photons absorbed/cm^3/s) Max Current: TMM: 37.6 mA/cm 2 ADEPT: 37.2 mA/cm 2

Results Thin-Film Silicon Cell The TMM module’s calculated reflection values (Figure 2) lead to a more accurate generation profile (Figure 3) 16 Figure 3Figure 2 Depth (microns) Optical Generation in a 2 Micron Thin- Film Si Cell with Reflection > 0 Optical Generation (photons absorbed/cm^3/s) Max Current: TMM: 11.4 mA/cm 2 ADEPT: 16.2 mA/cm 2

Anti-reflection coatings (ARCs) are thin coatings placed on top of solar cells to minimize reflection New module allows for easy simulation of any ARC Previously, ADEPT could only estimate these effects Results Anti-Reflection Coatings Without ARCWith ARC 17

We were able to construct a module that solves layered wave optics using the S-matrix method This module was integrated into the existing ADEPT 2.0 software Solar cells can be more accurately modeled with ADEPT Conclusions 18

The module will be upgraded to calculate the effects of optical generation at any point in the device This will allow for the inclusion of photon recycling The effects of photon recycling in high-performance photovoltaics are a topic of current research Future Work Photon Recycling X. Wang et al., OE 22, A344 (2014) 19

Studies will be conducted to investigate the optimum design parameters for thin-film solar cells – Parameters to be optimized include material, thickness, backside mirror reflectivity, anti-reflection coating, and doping strength The module will be incorporated into the ADEPT 2.0 graphical user interface on nanoHUB.org for global access and ease of use Future Work 20

Acknowledgments 21

"PVeff(rev140627)" by Greg Wilson and Keith Emery - National Renewable Energy Laboratory (NREL), Golden, CO. L. Li, “Formulation and comparison of two recursive matrix algorithms for modelling layered diffraction gratings,” J. Opt. Soc. Am. A., vol. 13, no. 5, pp , X. Wang et al., “Design of GaAs Solar Cells Operating Close to the Shockley–Queisser Limit,” IEEE Journal of Photovoltaics, vol. 3, no. 2, pp , G. Lush and M. Lundstrom, “Thin film approaches for high efficiency III-V cells,” Solar Cells, vol. 30, pp. 337–344, Jeff Gray; Xufeng Wang; Xingshu Sun; John Robert Wilcox (2013), "ADEPT 2.0," L. Pettersson et. al., “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” Journal of Applied Physics, vol. 86, no. 1, pp , X. Wang et al., “Performance-limiting factors for GaAs-based single nanowire photovoltaics,” Optics Express, vol. 22, no. S2, pp. A344-A358, Aigner, F., “New Material Promises Better Solar Cells,” Vienna University of Technology, References 22

Questions?

Backup Slides T-Matrix Method L. Li, JOSA A 13, 1024 (1996) 24

Backup Slides S-Matrix Method L. Li, JOSA A 13, 1024 (1996) 25

Backup Slides Generation at a Point L. Pettersson et al., JAP 86, 487 (1999) 26