FDTD Algorithm for Plasmonic Nanoparticles with Spatial Dispersion THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Li Zhang, B.S. Graduate Program in Electrical and Computer Engineering The Ohio State University 2016 Master's Examination Committee: Prof. Fernando Teixeira, Advisor Prof. Betty Lise Anderson Copyright by Li Zhang 2016 ABSTRACT We develop a finite-difference time-domain (FDTD) algorithm incorporating the hydrodynamic Drude model for the permittivity of noble metals. This model in- corporates spatial dispersion (nonlocal) effects on the permittivity. The resulting FDTD algorithm fully includes eddy current effects and is employed to study the extinction cross section of different metallic nanoparticles: nanospheres, nanoshells, nanospheroids, nanodisks, and nanorings, in the visible spectrum. It is observed that spatial dispersion can have significant effects on the extinction cross section of nanoparticles with characteristic sizes around 20 nm or below. For larger particles, the effect is mostly negligible. It is determined that inclusion of spatial dispersion yields two general trends on the behavior of the extinction cross section versus frequency as the particle size is reduced. The first is a blue-shift on the extinction spectrum and the second is an overall decrease on the extinction cross section as well as on the sharpness of resonance peaks. Also we find the strength of spatial dispersion effects do not have significant dependence on the orientation of nanoparticles. ii ACKNOWLEDGMENTS Hereby, I want to give my sincere thanks to Prof. Fernando L. Teixeira, who gave me a lot of help in my academic studies, and who has been alway patient even when I went through hard times. I always present a graceful heart to my parents. They are the backbone of my achievement. iii VITA 2011 . .B.S. Electrical Engineering 2011-present . .M.S. Electrical and Computer Engi- neering FIELDS OF STUDY Major Field: Electrical and Computer Engineering iv TABLE OF CONTENTS Page Abstract . ii Acknowledgments . iii Vita . iv List of Tables . vii List of Figures . viii Chapters: 1. Introduction . 1 1.1 Introduction . 1 2. Theoretical Background . 3 2.1 Drude model . 3 2.2 Nonlocal Permittivity . 4 2.3 Hydrodynamic Plasma Equation . 5 2.4 Hydrodynamic Drude model for permittivity . 9 3. Formulation . 13 3.1 Permittivity Data . 13 3.2 Nonlocal FDTD Algorithm . 16 3.3 Additional Boundary Conditions . 19 3.4 Numerical Setup . 20 3.4.1 General Aspects . 20 3.4.2 Source Excitation . 21 v 3.4.3 Extinction Cross Section Calculation . 21 3.4.4 Near-to-Far Field Transformation . 22 3.4.5 Two-stage PML Formulation . 23 3.4.6 Qualitative Illustration . 24 4. Results . 26 4.1 Au Nanospheres . 26 4.2 Au Nanoshells . 30 4.3 Au Nanospheroids . 34 4.4 Au Nanodisks . 42 4.5 Au Nanorings . 45 5. Conclusion . 47 Bibliography . 48 vi LIST OF TABLES Table Page 3.1 Additional boundary condition for nonlocal models at metal-dielectric interfaces. 20 vii LIST OF FIGURES Figure Page 3.1 Real part of the permittivity function function (solid line) for Au com- pared to experiment data (circle) [1, 2]. 15 3.2 Imaginary part of the permittivity function function (solid line) for Au compared to experiment data (circle) [1, 2]. 16 3.3 Qlatitative illustration of the field distribution around Au nanopar- ticles excited by a plane wave. The top-left plot represent the field around a 4 nm Au nanosphere modeled by the hydrodynamic Drude mode. The top-right plot represent the field around the same 4 nm Au nanosphere modeled by the standard Drude mode. The bottom- left plot represent the field around a Au nanospheroid modeled by the hydrodynamic Drude mode. The bottom-right plot represent the field around the same Au nanospheroid modeled by the standard Drude mode. 25 4.1 Comparison between local (solid red) and nonlocal (dashed blue) ex- tinction cross section for a 30 nm diameter Au nanosphere embedded in silica glass substrate. 27 4.2 Comparison between local (solid red) and nonlocal (dashed blue) ex- tinction cross section for a 20 nm diameter Au nanosphere embedded in silica glass substrate. 28 4.3 Comparison between local (solid red) and nonlocal (dashed blue) ex- tinction cross section for a 10 nm diameter Au nanosphere embedded in silica glass substrate. 29 4.4 Comparison between local (solid red) and nonlocal (dashed blue) ex- tinction cross section for a 5 nm diameter Au nanosphere embedded in silica glass substrate. 30 viii 4.5 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for a 60 nm radius Au nanoshell for different values of wall thicknesses embedded in a silica glass substrate. 32 4.6 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for a 30 nm radius Au nanoshell for different values of wall thicknesses embedded in a silica glass substrate. 33 4.7 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for a 10 nm radius Au nanoshell for different values of wall thicknesses embedded in a silica glass substrate. 33 4.8 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for a 5 nm radius Au nanoshell for different values of wall thicknesses embedded in a silica glass substrate. 34 4.9 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in x direction for different values of short/long axial ratio. The spheroids are embedded in silica glass substrate. 36 4.10 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in y direction for different values of short/long axial ratio. The spheroids are embedded in silica glass substrate. 37 4.11 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in z direction for different values of short/long axial ratio. The spheroids are embedded in silica glass substrate. 38 4.12 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in x direction for different values of short/long axial ratio. The spheroids are embedded in silica glass substrate. 38 4.13 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in y direction for different values of short/long axial ratio. The spheroids are embedded in silica glass substrate. 39 ix 4.14 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in z direction for different values of short/long axial ratio. The spheroids are embedded in silica glass substrate. 40 4.15 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 16 nm and long/short axial ratio equal to 2 for different long axis orientation. The spheroids are embedded in silica glass substrate. 41 4.16 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au spheroids with short axis a = 16 nm and long/short axial ratio equal to 1/2 for different long axis orientation. The spheroids are embedded in silica glass substrate. 42 4.17 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au nanodisks with radius r = 5 nm for different heights as indicated, with the disk plane facing the y direction. The nanodisks are embedded in silica glass substrate. 43 4.18 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au nanodisks with radius r = 5 nm for different heights as indicated, with the disk plane facing the z direction. The nanodisks are embedded in silica glass substrate. 44 4.19 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au nanodisks with radius r = 10 nm for different heights as indicated, with the disk plane facing the y direction. The nanodisks are embedded in silica glass substrate. 44 4.20 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for Au nanodisks with radius r = 10 nm for different heights as indicated, with the disk plane facing the z direction. The nanodisks are embedded in silica glass substrate. 45 4.21 Comparison between local (solid red) and nonlocal (dashed blue) nor- malized extinction cross sections for an Au nanoring with its ring shape laid down on the x−y plane and with its wall height in the z direction. The nanoring is embedded in silica glass substrate. 46 x CHAPTER 1 Introduction 1.1 Introduction Nanoscale metallic structures are of interest for a number of applications in optics. By harnessing plasmon resonances, they can provide subwavelength field confinement at levels otherwise not possible with other types of structures. Nonlocal electromag- netic effects such as spatial dispersion can have important effects in the response of some nanoscale devices made of noble metals [3] such as Au and Ag. In particular, spatial dispersion can play a significant role in affecting the plasmon resonances of metallic nanoparticles such as spheres, disks, and rings.