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Random Walk Simulation of Pn Junctions Random Walk Simulation of p-n Junctions by Jennifer Wang Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Arts in Physics at WELLESLEY COLLEGE May 2020 c Wellesley College 2020. All rights reserved. Author.............................................................. Department of Physics May 6th, 2020 Certified by. Yue Hu Professor of Physics Thesis Supervisor Accepted by . Yue Hu Chair, Department of Physics Random Walk Simulation of p-n Junctions by Jennifer Wang Submitted to the Department of Physics on May 6th, 2020, in partial fulfillment of the requirements for the degree of Bachelor of Arts in Physics Abstract Random walks rely on stochastic sampling instead of analytical calculations to simulate complex processes; thus, these methods are flexible and mathematically simple. In the semiconductor field, random walks are often utilized to model bulk transport of charge carriers. However, junction transport models generally rely on more complex numerical methods, which are computationally unwieldy and opaque to a general audience. To aim for greater accessibility, this thesis applies random walks to simulate carrier transport in p-n junctions. We first show that our random walk methods are consistent with finite-difference methods to model drift-diffusion processes. Next, we apply a basic random walk to model a p-n junction at equilibrium. This demonstrates successful proof-of-concept simulation of the depletion layer, as well as calculation of the the electric field and potential of a typical p-n junction. Additionally, because the basic random walk assumption cannot incorporate different mobility parameters for electrons and holes, we develop an improved compound p- n junction model. We demonstrate that the compound model is consistent with the basic model, and that it is adaptable to realistic material parameters for both electrons and holes. Thesis Supervisor: Yue Hu Title: Professor of Physics 2 Acknowledgments I wish to express my deepest gratitude to my thesis advisor, Professor Yue Hu, for her constant guidance throughout my undergraduate experience. She has been incredibly patient and supportive, helping me with crucial obstacles and offering invaluable life advice. This thesis draws heavily from her ideas and quite literally would not be possible without her. I am also profoundly grateful to Professor Robbie Berg, my major advisor, for setting me on the path to a research career; Professor Rebecca Belisle, for sparking my interest in materials science; and Professor Brian Tjaden, for providing a much- needed computer science perspective. Thank you all for agreeing to serve as my thesis reviewers. Thanks also to Hope Anderson, my thesis mate, for thought-provoking discussions and moral support throughout our senior year. I would also like to thank my friends Jackie Ehrlich, Maddy Buxton and Maya Igarashi for their edits and camaraderie, as well as my parents for their love and care. 3 Contents 1 Introduction 6 1.1 Overview . .7 1.1.1 Semiconductor junction structure . .7 1.1.2 Random walk simulation process . 12 1.2 Research motivation . 14 2 Depletion approximation for p-n junctions 17 2.1 Setup and assumptions of depletion approximation . 17 2.2 Mathematical process . 20 2.2.1 Charge distribution, electric field and electrostatic potential di- agrams . 20 2.2.2 Depletion width calculation . 22 3 Random walks and the diffusion process 25 3.1 The drift-diffusion equation . 26 3.1.1 Derivation of drift-diffusion equation from random walk . 27 3.1.2 Derivation of carrier motion probability . 29 3.2 The finite difference method . 32 3.3 Random walk simulation implementation . 33 3.3.1 Simulation mechanics . 33 3.3.2 Simulation cases . 34 3.4 Comparing random walk and finite difference methods . 34 4 4 Simulation method for p-n junctions 37 4.1 Simulation overview . 37 4.1.1 Finding discretized electric field . 39 4.1.2 Calculating carrier motion probabilities . 41 4.1.3 Running the simulation . 42 4.2 Material parameters for p-n simulations . 44 4.3 Scaling parameter derivations . 44 4.3.1 Fixed physical dimension (∆x) and time scale (∆t)...... 44 4.3.2 Doping concentration: fixed A and variable init ........ 46 4.3.3 Electric field: fixed scaling parameter and E-field unit . 47 4.4 Simulation strategy . 48 4.4.1 Basic p-n junction case . 48 4.4.2 Limitation of basic p-n junction case . 49 4.4.3 Compound p-n junction case . 50 5 Results 51 5.1 Basic p-n junction simulation . 52 5.2 Realistic diffusion coefficient issue . 58 5.3 Compound p-n junction simulation . 59 6 Conclusions and future work 62 A Simulation code for random walks 65 A.1 Basic random walk . 65 A.2 Biased random walk . 68 A.3 Compound random walk . 70 B Simulation code for p-n junctions 74 B.1 Basic p-n junction . 74 B.2 Compound p-n junction . 80 5 Chapter 1 Introduction The p-n junction is an interface structure between semiconductor materials, which forms the basis for electronic devices such as diodes, transistors, and solar cells. This junction occurs at the boundary between a p-type material, with excess positive charge carriers, and a n-type material, with excess negative charge carriers. This allows electrical current to flow through the junction in only one direction. Depending on external controls, such as a directional voltage bias, it is possible to achieve precise control of the current flow, thus creating an `on' and ‘off' state and enabling the p-n junction to function as a diode. Due to their ubiquitous use in technology infrastructure, p-n junctions and semi- conductors in general are well-studied; many commercial software packages exist to model junction behavior. However, most of these models are highly complex, in- volving multiple coupled differential equations and inherent assumptions that remain opaque to a general audience. This thesis will lay out a simple method of simulating p-n junctions using the Monte Carlo random walk. Through this investigation, I will endeavor to demonstrate three points: firstly, the random walk method achieves a level of accuracy comparable to that of analytical methods, and results in a more realistic solution than a conventional approximation treatment. Secondly, investigat- ing junction structures using a random walks enables semiconductor simulations to achieve a high degree of flexibility when incorporating external conditions. Thirdly, the random walk method is easily understandable by a broader audience, which bodes 6 well for technology outreach in the computing and clean energy industries. In this first chapter, I will provide necessary background and elaborate on the mo- tivation for using random walks to simulate p-n junctions. Chapter 2 will delineate the theoretical analysis of p-n junctions through the depletion approximation. Chap- ter 3 will justify how random walks can be used to model drift-diffusion processes, as well as how to simulate random walks using Monte Carlo methods. Next, Chapter 4 will introduce the setup and parameter analysis for p-n junctions simulations. In Chapter 5, we present simulation results for several different p-n junction models, to evaluate the accuracy, flexibility and applicability of our methods to the p-n junction system. Finally, Chapter 6 will conclude the thesis with a discussion of limitations and potential future explorations. 1.1 Overview 1.1.1 Semiconductor junction structure Electrical conductivity is a measure of how easily a material conducts electricity, often calculated as the ratio of current density to electric field within the material.[1] It is a fundamental material property and can be qualitatively understood with concepts from energy band theory. Every material has a distinctive band gap determined by internal structure. The size of the band gap is defined as the energy difference between the lowest energy of electrons in the conduction band, in which electrons are free to move around, and the highest energy of electrons in the valance band, in which electrons are bound close to the nuclei in the atoms. We can characterize material conductivity based on the size of this band gap. For example, as shown in Figure. 1-1, metals are conductors because the conduction band (blue) and valance band (red) overlap, which means electrons with that energy level are free to flow away from their original nuclei. Insulators, like rubber, have a large band gap, which indicates that electron flow in the material is not easy to achieve; in order to escape their bindings from the nuclei, electrons in the valance band have to absorb a large 7 amount of energy to reach the conduction band. Semiconductors like monocrystalline silicon have a small band gap, which means they have a conductivity between that of metals and insulators. We note that the Fermi energy (indicated with dashed line) is the highest energy of electrons if the material is kept at a temperature of 0 Kelvin; it provides a benchmark for the general energy region in which the conduction and valance band reside. Figure 1-1: Metals are conductors, since they have overlapping conduction and valance bands (zero band gap). Insulators have large band gaps (commonly greater than 9 eV). Semiconductors have smaller, easily tunable band gaps, which indicates that they have a conductivity between that of metals and insulators. Semiconductor band gaps are small enough (usually around 1.1 eV for silicon) that a tiny shift in energy level can have a large effect on electronic properties within the material.[2] Thus, the key advantage of using semiconductors to engineer devices is that their conductivity can be tuned by external factors, such as doping, which is the introduction of impurities into the material structure. In Figure. 1-2, we take a closer look at both the positive and negative doping of silicon, which lends the p-n junction its name.
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