NUMERICAL ANALYSIS OF DIFFUSION IN CRYSTALLINE AND POLYCRYSTALLINE MATERIALS-APPLICATION TO PHOTOVOLTAICS Anuja Parikh A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2019 Committee: Marco Nardone, Advisor Lewis P. Fulcher Alexey T. Zayak Copyright c May 2019 Anuja Parikh All rights reserved iii ABSTRACT Marco Nardone, Advisor Cadmium telluride (CdTe) and copper indium gallium diselenide (CIGS) based thin-film pho- tovoltaic devices are leading candidates for cost-effective solar electricity. Inter-diffusion of atoms in device components play a critical role in the performance and degradation of a thin-film solar cell. Diffusion studies require special attention because device components are in general poly- crystalline and exhibit grain boundaries. In this work, we review the physics of diffusion kinetics in solids (crystalline and polycrystalline) and develop a general numerical simulation tool for ion drift-diffusion. We validate the model against literature data and analytical expressions for the technologically important cases of copper (Cu) and phosphorus (P) in CdTe. We present empiri- cal and numerical analysis for evaluation of sodium (Na) diffusion in CIGS. The calculations are conducted by the finite element method using COMSOL Multiphysics R software. Our study indi- cates that both Cu and P diffuses in crystalline CdTe using two mechanisms, slow and fast. In the case of Cu in crystalline CdTe, our model confirms the slow diffusivity values of 3.7×10-4 exp(- 0.67eV/kT ) cm2/s and 6.6×10-5 exp(-0.50eV/kT ) cm2/s depending on the experiment, along with the fast diffusivity of 2×10-8 cm2/s. In case of P in crystalline CdTe, the slow and fast diffusivities are confirmed to be 4.86×10-3 exp(-2.03eV/kT ) cm2/s and 2.15×10-5 exp(-1.0eV/kT ) cm2/s, re- spectively. Migration of P in polycrystalline CdTe includes a grain boundary diffusion coefficient of 6.61×102 exp(-1.97eV/kT ) cm2/s. Our analysis for Na in polycrystalline CIGS indicates the grain boundary diffusivity of 5.06×10-9 exp(-0.09eV/kT ) cm2/s. iv To The Nature, On the pillar of my faith in her, I flourish... v ACKNOWLEDGMENTS I owe my deepest gratitude to my advisor, Dr. Marco Nardone. He has offered a wonderful research experience over the last year that taught me the values of good research and inspired me to continue in the field of research. I certainly appreciate his availability for valuable discussions and suggestions. I also acknowledge the financial support from the National Renewable Energy Laboratory (NREL) and U.S Department of Energy that was made available through my advisor. I want to express my gratitude to Dr. Fulcher and Dr. Zayak for their time in reviewing this thesis. I want to take an opportunity to thank them, also for the learning experiences that they have provided during my study. I also appreciate administrative support from Sandra Gardner and friendly conversations with her that helped me move on. I find myself very fortunate that I had a great support from my friends through out my studies. I am indebted to my parents for showering their unconditional love when I needed it the most. Lastly, this thesis would not have been possible without my sweet and affectionate daughters, Maanya and Noopur, and without my loving, supportive, and patient husband, Viral. Their love brings the best out of me. vi TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION . 1 1.1 BACKGROUND . 2 1.1.1 PRINCIPLE OF OPERATION . 2 1.1.2 CASE 1: MIGRATION OF COPPER IN CdTe . 4 1.1.3 CASE 2: MIGRATION OF PHOSPHORUS IN CdTe . 6 1.1.4 CASE 3: MIGRATION OF SODIUM IN CIGS . 6 1.2 THESIS OBJECTIVES AND HYPOTHESIS . 8 CHAPTER 2 FUNDAMENTALS OF DIFFUSION . 9 2.1 FICK’S LAWS . 9 2.2 FISHER’S MODEL . 10 2.3 HARRISON’S MODEL . 14 2.3.1 TYPE A . 14 2.3.2 TYPE B . 15 2.3.3 TYPE C . 17 2.4 GRAIN BOUNDARY SEGREGATION . 17 2.5 DIFFUSION MECHANISMS . 18 2.5.1 INTERSTITIAL MECHANISM . 18 2.5.2 COLLECTIVE MECHANISMS . 19 2.5.3 VACANCY MECHANISM . 20 2.5.4 INTERSTITIAL-SUBSTITUTIONAL EXCHANGE MECHANISM . 21 CHAPTER 3 METHODOLOGY . 23 3.1 SIMULATION TOOL . 23 3.2 DIFFUSION: METHODS AND PARAMETER VALUES . 24 3.3 OUTPUT . 27 vii CHAPTER 4 RESULTS AND DISCUSSION . 29 4.1 DIFFUSION OF CU IN CRYSTALLINE CdTe . 29 4.1.1 SUMMARY OF DIFFUSION EXPERIMENTS . 29 4.1.2 NUMERICAL CALCULATIONS . 30 4.1.3 DISCUSSION . 33 4.2 DIFFUSION OF P IN CdTe . 33 4.2.1 SUMMARY OF DIFFUSION EXPERIMENTS . 33 4.2.2 NUMERICAL CALCULATIONS . 34 4.2.3 DISCUSSION . 36 4.3 DIFFUSION OF Na IN POLYCRYSTALLINE CIGS . 36 4.3.1 SUMMARY OF DIFFUSION EXPERIMENTS . 36 4.3.2 EMPIRICAL ANALYSIS . 37 4.3.3 NUMERICAL CALCULATIONS AND DISCUSSION . 41 CHAPTER 5 CONCLUSIONS . 45 BIBLIOGRAPHY . 47 viii LIST OF FIGURES Figure Page 1.1 Schematic diagram of (a) CdTe and (b) CIGS single junction solar cell. 3 1.2 Best cell efficiencies reported by Dean Levi. 4 2.1 Schematic diagram of polycrystalline material. 11 2.2 Interpretation of type A, B and C kinetics by Mehrer. 14 2.3 Schematic diagram of change in the shape of concentration contour as β changes (Philibert, 1991). 17 2.4 Schematics of various diffusion mechanisms. 19 2.5 Schematic diagram of interstitial-substitutional mechanism. 22 3.1 Rectangular and cylindrically symmetric geometry used for the case of P in poly- crystalline CdTe and Na in CIGS. 25 3.2 Illustration of a concentration profile for a typical case of Na diffusion in polycrys- talline CIGS. 28 4.1 Depth profile of Cu diffused in crystalline CdTe at 300 ◦C and 208 ◦C. 31 4.2 Depth profile of P diffused in (a) crystalline CdTe and (b) polycrystalline CdTe for various annealing parameters. 34 4.3 A 2D view of P distribution in (a) Crystalline CdTe and (b) Polycrystalline CdTe , for T =633 ◦C and time = 7 hr. 35 4.4 Sodium concentration profiles for seven CIGS thin-film samples at various anneal- ing temperatures and times (Laemmle, Wuerz, Schwarz, Cojocaru-Mirdin, Choi, and Powalla, 2014). 38 4.5 Log normalized concentration versus depth6=5..................... 40 4.6 Na concentration normalized to surface value versus depth for fine grain samples. 41 4.7 Sodium concentration profiles for six of the CIGS thin-film samples at the anneal- ing temperatures shown. 42 ix 4.8 Sodium concentration profiles for the T = 157 ◦C sample with model fits at the three grain sizes shown and the same diffusivity used in Figure 4.7. 43 4.9 Arrhenius plot from the model fits in Figure 4.7. 44 x LIST OF TABLES Table Page 3.1 Input parameters used in numerical analysis for a typical case of Na diffusion in polycrystalline CIGS. 27 4.1 Critical parameters used for simulations of Cu and P in CdTe. 32 4.2 List of samples and their critical diffusion parameters. The annealing time, temper- ature grain size and diffusivity (D) are collected from Laemmle et al. (Laemmle et al., 2014). 39 4.3 List of samples and their critical diffusion parameters used for the model fit. 43 1 CHAPTER 1 INTRODUCTION The usage of renewable resources has been increasing and the trend is projected to continue by an average rate of 2.8 % every year until 2040 (U. S. Energy Information Administration, 2017). According to the 2015 renewable energy data book, solar photovoltaics (PV) was the fastest grow- ing renewable electricity technology in 2014 and 2015 with 28% increase in global capacity each year (Philipp and Tian, 2015). Even though global annual capacity additions have been highest for PV with a growth rate of 45% between 2005-2015, it only accounted for 1.2% of total electricity generation in 2015 (Philipp and Tian, 2015). On the other hand, carbon fossil fuels such as natural gas, coal, and oil, are the largest supplier of energy production and capture more than 80% of the market. A 1.6% of increase in carbon emissions due to energy consumption has been reported for the year of 2017 (British Petrolium, 2018). Basic issues involved with non-renewable resources such as sustainability, waste disposal, and release of green house gases suggest a need for reduc- tion in consumption of fossil fuels and nuclear energy through a dramatic growth in electricity generation by renewable resources. Electricity generation through renewable resources is mainly achieved by hydro-power, geother- mal, biomass, wind, and solar. While all renewable resources leave low environmental impact, solar PV is a critical renewable resource due to it’s simplicity, reliability, and versatility com- pared to other renewable resources (U.S. Department of Energy, EPRI, 1997). The most efficient photovoltaic devices are commonly developed using monocrystalline silicon wafers. Producing a monocrystalline silicon wafer is a complex and expensive procedure. Thin-film solar cells en- tered the market owing to their simplicity and cost reduction in device fabrication. The leading candidates among thin-film devices are amorphous silicon (a-Si:H), cadmium telluride (CdTe) and copper indium gallium diselenide (CIGS) based polycrystalline devices. However, CdTe and CIGS are almost not used in the fastest growing electronics industry, which means that there is lack of expertise in fabrication of such devices compared to crystalline silicon or a-Si:H (Archer and Hill, 2001). 2 Some of the critical research issues in thin-film PV revolves around process development, back- contact and module stability related to performance changes over time (Ullal, 2008; Kazmerski, 2006).