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Electromagnetic modeling of plasmonic nanostructures
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Electromagnetic modeling of plasmonic nanostructures
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ELECTROMAGNETIC MODELING OF PLASMONIC NANOSTRUCTURES by Prathamesh Pavaskar A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2013 Copyright 2013 Prathamesh Pavaskar ii EPIGRAPH “It is the man of science, eager to have his every opinion regenerated, his every idea rationalized, by drinking at the fountain of fact, and devoting all the energies of his life to the cult of truth, not as he understands it, but as he does not yet understand it, that ought properly to be called a philosopher.” - Charles Sanders Peirce iii DEDICATION To my parents, for their constant support. iv ACKNOWLEDGEMENTS Firstly, I have to thank my advisor, Prof. Stephen B. Cronin, who taught me how to be a good scientist. I am forever thankful for his guidance and support from day one. I have studied many diverse subjects in the exciting field of nanotechnology and his trust in my abilities, encouragement, ideas, and funding helped me immensely in accomplishing significant academic progress. Nobody finishes a dissertation without fellow researchers. Cronin Group members at B113, I have to thank every single one of you, I could not have done all this work here without you. Adam Bushmaker, Charles Le Pere, Chia-Chi Chang, Chun-Chung Chen, David Valley, Fernando Souto, Guangtong Zeng, I-Kai Hsu, Jesse Theiss, Jing Qiu, Ko- Chun Lin, Mehmet Aykol, Mohammed Amer, Rajay Kumar, Rohan Dhall, Shermin Arab, Shun-Wen Chang, Vaibhav Bora, Wei-Hsuan "Wayne" Hung, Wenbo Hou, Zhen Li, and Zuwei Liu, thank you all for making this journey fun. I want to thank Dr. Kian Kaviani for offering me a teaching assistant position for his class. The 4 years I spent teaching for the class were memorable. He was always encouraging, helpful and considerate, and I appreciate all the trust he put in me, without which I would not have successful at my job. The professors and staff in the Electrical Engineering department were extremely helpful. I appreciate all the work Dr. Donghai Zhu did and continues to do for the Powell clean room at USC. Kim Reid, Angelique Miller, Mona Gordon and Jaime Zelada were extremely helpful in their administrative support and management of funding. v I want to thank Prof. Wei Wu and Prof. Alex Benderskii for being on my defense committee. I would also like to thank Prof. Michelle Povinelli, Prof. John O’Brien, and Prof. Ed Goo for being on my qualifying exam committee. I really appreciate your helpful suggestions. I would especially like to thank my Parents and my sister for supporting me through my PhD despite being so far from home. My parents always supported me in whatever I did in my life, and that kind of support helped me get through my research. My girlfriend Arpita was always there for me whenever I needed her. She stood by me while I worked at USC, and linked her life to mine in so many ways. At last, none of this research would have been possible without funding, which came from ONR Awards No. N00014-08-1-0132 and No. N00014-12-1-0570, AFOSR Award No. FA9550-08-1-0019, ARO Award No. W911NF-09-1-0240, NSF Award No. CBET-0846725, and NASA SURP No. 1346414. Some of our research was supported as part of the Center for Energy Nanoscience, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001013. Some research was partially carried out at the Jet Propulsion Laboratory, California Institute of Technology. vi TABLE OF CONTENTS EPIGRAPH ......................................................................................................................... ii DEDICATION ................................................................................................................... iii ACKNOWLEDGEMENTS ............................................................................................... iv LIST OF FIGURES ............................................................................................................ x LIST OF TABLES ........................................................................................................... xix ABSTRACT ...................................................................................................................... xx CHAPTER 1: BACKGROUND ......................................................................................... 1 1.1 The Finite-Difference Time-Domain (FDTD) Method ............................................. 1 1.2 Plasmonics ............................................................................................................... 10 1.3 Photocatalysis .......................................................................................................... 27 CHAPTER 2: OPTIMIZATION OF PLASMON RESONANT NANOSTRUCTURES 30 2.1 Abstract ................................................................................................................... 30 2.2 Introduction ............................................................................................................. 30 2.3 Simulations details .................................................................................................. 31 2.4 Results and discussion ............................................................................................. 34 2.5 Conclusions ............................................................................................................. 41 CHAPTER 3: A MICROSCOPIC STUDY OF PLASMONIC ISLAND THIN FILMS . 42 3.1 Abstract ................................................................................................................... 42 3.2 Introduction ............................................................................................................. 43 3.3 Experiment and simulations details ......................................................................... 44 3.4 Results and discussion ............................................................................................. 45 vii 3.5 Conclusions ............................................................................................................. 54 CHAPTER 4: PLASMONIC HOT SPOTS: NANOGAP ENHANCEMENT VS. FOCUSING EFFECTS FROM SURROUNDING NANOPARTICLES ........................ 56 4.1 Abstract ................................................................................................................... 56 4.2 Introduction ............................................................................................................. 57 4.3 Simulation details .................................................................................................... 60 4.4 Results and discussion ............................................................................................. 61 4.5 Conclusions ............................................................................................................. 68 CHAPTER 5: PLASMONIC ENHANCEMENT OF PHOTOCATALYTIC CHEMICAL REACTIONS .................................................................................................................... 69 5.1 Abstract ................................................................................................................... 69 5.2 Introduction ............................................................................................................. 70 5.3 Experimental details ................................................................................................ 74 Photocatalytic water splitting ................................................................................ 75 Photocatalytic CO 2 reduction ................................................................................ 78 Photodegradation of methyl orange ...................................................................... 81 5.4 Electromagnetic simulations ................................................................................... 83 5.5 Dye-sensitized solar cell .......................................................................................... 86 5.6 Conclusions ............................................................................................................. 94 CHAPTER 6: PLASMONIC NANOPARTICLE ARRAYS WITH NANOMETER SEPARATION FOR HIGH-PERFORMANCE SERS SUBSTRATES .......................... 96 6.1 Abstract ................................................................................................................... 96 viii 6.2 Introduction ............................................................................................................. 97 6.3 Experimental details ................................................................................................ 99 6.4 Electromagnetic simulations ................................................................................. 105 6.5 Conclusions ........................................................................................................... 107 CHAPTER 7: FUTURE DIRECTIONS ......................................................................... 109 7.1 Optimization of nanoparticle dimer arrays fabricated using the angle evaporation method ......................................................................................................................... 109 Preliminary results .............................................................................................. 110 Proposed work .................................................................................................... 112 7.2 Nanoparticle cluster optimization for four-wave mixing ...................................... 112 Preliminary results .............................................................................................. 117 Proposed work .................................................................................................... 118 7.3 Optimization of nanoparticle clusters with nanoparticle size as a parameter ....... 119 Preliminary results .............................................................................................. 121 Proposed work .................................................................................................... 121 BIBLIOGRAPHY ........................................................................................................... 122 APPENDIX ..................................................................................................................... 147 A.1: Additional FDTD concepts ................................................................................. 147 Perfectly matched layers ..................................................................................... 147 Total Field Scattered Field (TFSF) source.......................................................... 150 Stability ............................................................................................................... 153 A.2: Optimization code ............................................................................................... 153 ix A.3 TiO 2 synthesis ....................................................................................................... 176 A.4 TiO 2 doping .......................................................................................................... 180 x LIST OF FIGURES Figure 1.1: Primary and secondary cell within Yee grid in FDTD representing E- and H-field components in the discretized space. Each node i,j,k within the FDTD grid represents three electric and magnetic field components located around a volume element. .......................................................................... 3 Figure 1.2: Position of the E and H fields inside a Cubic unit cell of the Yee space lattice. ...................................................................................................................... 4 Figure 1.3: (a) Half step updates of the E- and H-fields in space and time. (b) Leapfrog method of calculation. ............................................................................. 5 Figure 1.4: A simplified flowchart describing the FDTD algorithm. ................................. 9 Figure 1.5: A surface plasmon polariton as a collective excitation at a metal– dielectric interface [from reference 35]. ............................................................... 13 Figure 1.6: Dispersion relation of SPPs at the interface between a Drude metal with negligible collision frequency and air (gray curves) and silica (black curves). [from reference 2] ................................................................................... 16 Figure 1.7: Localized surface plasmon excitation in nanoparticles in response to an incident light wave. .......................................................................................... 18 xi Figure 1.8: homogeneous, isotropic metallic sphere of radius , placed in an electrostatic field. .................................................................................................. 19 Figure 1.9: (a) Absolute value and (b) phase of the polarizability of a sub- wavelength metal nanoparticle with respect to the frequency of the driving field. [from reference 2] ........................................................................................ 22 Figure 1.10: (a) Scattering spectra of single silver nanoparticles of different shapes obtained in dark-field configuration [from reference 42]. (b) Electric field enhancement for silver nanoparticles of different diameters. ......... 24 Figure 1.11: Surface plasmon dipole resonance of a silver nanoparticle, when the nanoparticle is placed in (a) air, (b) 1.44 index oil and (c) 1.48 index oil. [from reference 45] ............................................................................................... 25 Figure 1.12: For a dimer of nanoparticles, the local electrical field between the gap becomes enormous due to the coupling of the surface plasmons that gives rise to new resonant modes. As the distance between the nanoparticle decreases, the plasmonic coupling increases which in turn increases the local electric field intensity. ............................................................ 26 Figure 1.13: (a) Traditional p-n junction solar cell, (b) Photoelectrochemical solar cell. (c) Solar fuel generation [from reference 47]. .............................................. 27 xii Figure 1.14: Solar Spectrum superimposed on (a) the absorption spectrum of TiO 2 and (b) the absorption spectrum of a typical plasmonic nanostructure. ............... 29 Figure 2.1: Flowchart illustrating the algorithm used for optimization. ........................... 32 Figure 2.2: (a) Optimization electric field intensity map with respect to spatial coordinates of the third nanoparticle. (b) Movement of the third nanoparticle with the optimization algorithm. The initial position is shown in green and the final position is shown in red. (c) Change in E-field intensity and (d) resonant wavelength with each iteration. ................................... 35 Figure 2.3: Optimization of 10 nanoparticles. .................................................................. 36 Figure 2.4: Optimum electric field intensity enhancement as a function of the number of nanoparticles. ....................................................................................... 37 Figure 2.5: Electric field profiles of optimized geometries of 3- to 20-nanoparticle clusters. Here, the electric field intensity is optimized at (0,0)............................. 38 Figure 2.6: Electric field profiles for the optimized geometry of 20 nanoparticles for (a) the 2D case and (b) the 3D case. Near field enhancement spectra of the optimized geometry for (a) the 2D case and (d) the 3D case. ......................... 40 Figure 3.1: TEM images and electric field profiles of (a,b) a 5nm thick Au island film. The color axis in (b) shows the enhancement of electric field xiii intensity relative to the incident electric field intensity. (c) Experimental and simulated absorption spectra of the 5nm Au island film. ............................... 46 Figure 3.2: TEM images and electric field profiles of (a,b) a 5nm thick Ag island film. The color axis in (b) shows the enhancement of electric field intensity relative to the incident electric field intensity. (c) Experimental and simulated absorption spectra of the 5nm Ag island film. ............................... 47 Figure 3.3: TEM images and electric field profiles of (a,b) a 10nm thick Ag island film. The color axis in (b) shows the enhancement of electric field intensity relative to the incident electric field intensity. (c) Experimental and simulated absorption spectra of the 10nm Ag island film. ............................. 48 Figure 3.4. Electric field distribution of a 5nm Au film irradiated with (a) horizontally and (b) vertically polarized light. (c) The distribution of the electric field intensity enhancements for various gap sizes. The color axes in (a,b) show the enhancement of electric field intensity relative to the incident electric field intensity. ............................................................................. 50 Figure 3.5. (a) TEM image of an annealed 5nm Au film. Experimental and simulated absorption spectra of annealed 5 nm (b) Au and (c) Ag films. ............ 52 Figure 4.1: (a) HRTEM micrograph and (b) electric field profile of the region surrounding a hot spot on a 5nm Au island film. The black squares of area xiv (d x d) in (a) correspond to the different film areas used in the simulations shown in (c). (c) Electric field intensity at the hot spot plotted as a function of area (d x d).......................................................................................... 58 Figure 4.2: (a) HRTEM micrograph and (b) electric field profile of the region surrounding a hot spot on a 5nm Au island film. The black squares of area (d x d) in (a) correspond to the different film areas used in the simulations shown in (c). (c) Electric field intensity at the hot spot plotted as a function of area (d x d).......................................................................................... 59 Figure 4.3: Electric field intensity profiles corresponding to the areas highlighted in Figure 4.1c. ....................................................................................................... 62 Figure 4.4: (a) Electric field intensity profile of a region with 3 hot spots labeled HS 0 , HS 1 and HS 2 . (b) Electric field intensities of HS 0 , HS 1 and HS 2 plotted as a function of the HS 0 gap size. ............................................................. 64 Figure 4.5: Electric field intensity profile for an optimized cluster of 20 nanoparticles. ........................................................................................................ 66 Figure 5.1: Energy band alignment of anatase TiO 2 , Au, and the relevant redox potentials of H 2 O under visible illumination. ....................................................... 71 xv Figure 5.2: Energy band alignment of anatase TiO 2 , Au, and the relevant redox potentials of CO 2 and H 2 O under visible illumination.......................................... 73 Figure 5.3: (a) Schematic diagram of the photo-electrochemical measurement setup. (b) Quadrupole mass spectrometer data for H 2 production. Photocurrent of anodic TiO 2 with and without Au nanoparticles at zero bias voltage irradiated with (c) UV (λ = 254nm) and (d) visible light (λ = 532nm) for 22 seconds. ......................................................................................... 74 Figure 5.4: UV-Vis absorption spectra of TiO 2 with and without gold nanoparticles. ........................................................................................................ 76 Figure 5.5: Photocurrent of anodic TiO 2 with and without Au nanoparticles irradiated with λ = 633nm light for 20 seconds. ................................................... 77 Figure 5.6: Schematic of the experimental setup for CO 2 reduction. ............................... 79 Figure 5.7: Photocatalytic product yields (after 15 h of visible irradiation) on three different catalytic surfaces. ................................................................................... 80 Figure 5.8: UV-Vis spectra of MO aqueous solution before (black) and after (red) 1h UV illumination using (a) TiO 2 and (b) Au nanoparticle/TiO 2 photocatalysts. ....................................................................................................... 82 xvi Figure 5.9: UV-Vis spectra of MO aqueous solution before (black) and after (red) 1h 532nm laser illumination using (a) TiO 2 and (b) Au/TiO 2 as photocatalysts. ....................................................................................................... 83 Figure 5.10: (a) SEM image of 5 nm Au island film on TiO 2 . (b-d) Electric field intensity at the interface of Au–TiO 2 at the resonance calculated using FDTD. ................................................................................................................... 85 Figure 5.11: Schematic diagrams of three different Au nanoparticle/dye/TiO 2 configurations. ...................................................................................................... 88 Figure 5.12: Short-circuit photocurrents of DSSCs with three different Au nanoparticle/dye/TiO 2 configurations. .................................................................. 90 Figure 5.13: Photocurrent spectra of DSSCs with different working electrodes. ............. 92 Figure 5.14: FDTD calculated and experimentally measured enhancement factors plotted as a function of wavelength for embedded nanoparticles. ........................ 93 Figure 6.1: Schematic of the angle evaporation method to produce arrays of nanometer gap dimers. ........................................................................................ 100 Figure 6.2: (a) Low magnification TEM image of Ag nanoparticle arrays deposited using the angle evaporation technique on a SiN membrane. (b) xvii and (c) high magnification TEM images of Ag nanoparticle pairs with nm- sized gaps. ........................................................................................................... 102 Figure 6.3: (a) TEM image of a 5x5 matrix of cells containing various nanoparticle dimer geometries. Spatial mapping of the Raman intensity with a 632.8nm laser exciting the 1576 cm-1 p-ATP Raman peak with (b) parallel and (c) perpendicular polarization. ........................................................ 103 Figure 6.4: (a) TEM image of a silver nanoparticle dimer with 2nm separation. FDTD simulation results of the electric field intensity around the dimer plotted on a logarithmic scale with incident light polarized (b) parallel and (c) perpendicular to the axis of the nanoparticle pair. ........................................ 105 Figure 7.1: Schematic of the angle evaporation scheme................................................. 109 Figure 7.2: Optimization of an array of dimers for maximum absorption...................... 110 Figure 7.3: Constrained optimization of an array of dimers for maximum absorption with 6 parameters. ............................................................................. 111 Figure 7.4: Flowchart illustrating the algorithm used for optimization. ......................... 115 Figure 7.5: Optimization of four-wave mixing using 6 nanoparticles. ........................... 117 Figure 7.6: Setup of the proposed four-wave mixing experiment .................................. 118 xviii Figure 7.7: Optimization of a 7 nanoparticle cluster with diameter as an optimization parameter. ...................................................................................... 120 Figure A.1: Interface of air and PML ............................................................................. 149 Figure A.2: Two-dimensional Total field – scattered field formalism with the source injecting in the positive Y-direction. ....................................................... 152 Figure A.3: Scanning electron microscope (SEM) images of solgel TiO 2 showing the mesoporous structure. ................................................................................... 177 Figure A.4: Scanning electron microscope (SEM) images of anodic TiO 2 , (a) top view and (b) cross sectional view. ...................................................................... 177 Figure A.5: schematic of the atomic layer deposition of TiO 2 using TiCl 4 and H 2 O as the precursors. ................................................................................................. 179 xix LIST OF TABLES Table 7.1: Solution of the parameters of the constrained optimization of an array of dimers for maximum absorption..................................................................... 112 xx ABSTRACT In this thesis, plasmonic properties of metal nanostructures are investigated by electromagnetic simulations using the finite difference time domain (FDTD) method. Chapter 1 covers the background knowledge required to read this thesis. It talks about the fundamentals of the FDTD method, the physics of plasmonics and a brief description of photocatalysis. In chapters 2 to 6, I am going to present the work our research group has done in the field of plasmonics that has already been published. Chapter 7 contains work that we can derive from our past research. In chapter 2, we perform optimization of plasmonic nanoparticle geometries. An iterative optimization algorithm is used to determine the configuration of the nanoparticles that gives the maximum electric field intensity at the center of the cluster. We observe that the optimum configurations of these clusters have mirror symmetry about the axis of planewave propagation, but are otherwise non-symmetric and non- intuitive. The maximum field intensity is found to increase monotonically with the number nanoparticles in the cluster, producing intensities that are 2500 times larger than the incident electromagnetic field. In chapter 3, I will talk about the optical properties of thin metallic discontinuous films of Au and Ag, which are known to exhibit a strong plasmonic response under visible illumination. In this work, evaporated thin films are imaged with high resolution transmission electron microscopy (HRTEM), to reveal the structure of the semicontinuous metal island film with sub-nm resolution. The electric field distributions xxi and the absorption spectra of these semicontinuous island film geometries are calculated using the finite difference time domain (FDTD) method and compared with the experimentally measured absorption spectra. In addition to that, we calculate the SERS enhancement factors and photocatalytic enhancement factors of these films. We also study the effect of annealing on these films, which results in a large reduction in electric field strength due to increased nanoparticle spacing. In chapter 4, we study the effects of surrounding nanoparticles on a plasmonic hot spot. From our simulations, we show that the surrounding film contributes significantly to the electric field intensity at the hot spot by focusing energy to it. Widening of the gap size causes a decrease in the intensity at the hot spot. However, these island-like nanoparticle hot spots are shown to be robust to gap size than nanoparticle dimer geometries, studied previously. In fact, the main factor in determining the hot spot intensity is the focusing effect of the surrounding nano-islands. In chapter 5, I will talk about electromagnetic simulations of plasmonic enhancement of photocatalytic chemical reactions. By integrating strongly plasmonic Au nanoparticles with strongly catalytic TiO 2 , we demonstrate plasmon-enhanced photocatalytic water splitting, and reduction of CO 2 with H 2 O to form hydrocarbon fuels. Under visible illumination, we observe enhancements of up to 66X in the photocatalytic splitting of water in TiO 2 with the addition of Au nanoparticles. We also perform a systematic study of the mechanisms of Au nanoparticle/TiO 2 -catalyzed photoreduction of CO 2 and water vapor over a wide range of wavelengths. In this case, under visible light illumination, we observe a 24-fold enhancement in the photocatalytic activity due to the xxii intense local electromagnetic fields created by the surface plasmons of the Au nanoparticles. Above the plasmon resonance, under ultraviolet radiation we observe a reduction in the photocatalytic activity. Electromagnetic simulations indicate that the improvement of photocatalytic activity in the visible range is caused by the local electric field enhancement near the TiO 2 surface, rather than by the direct transfer of charge between the two materials. Here, the near-field optical enhancement increases the electron-hole pair generation rate at the surface of the TiO 2 , thus increasing the amount of photogenerated charge contributing to catalysis. This mechanism of enhancement is particularly effective because of the relatively short exciton diffusion length (or minority carrier diffusion length), which otherwise limits the photocatalytic performance. Our results suggest that enhancement factors many times larger than this are possible if this mechanism can be optimized. In chapter 6, I will talk about a method for fabricating arrays of plasmonic nanoparticles with separations on the order of 1nm using an angle evaporation technique. High resolution transmission electron microscopy (HRTEM) is used to resolve the small separations achieved between nanoparticles fabricated on thin SiN membranes. These nearly touching metal nanoparticles produce extremely high electric field intensities when irradiated with laser light. We perform surface enhanced Raman spectroscopy (SERS) a non-resonant dye molecule (p-ATP) deposited on the nanoparticle arrays using confocal micro-Raman spectroscopy. Our results show significant enhancement when the incident laser is polarized parallel to the axis of the nanoparticle pairs, whereas no enhancement is observed for the perpendicular polarization. These results demonstrate xxiii proof-of-principle of this fabrication technique. Finite difference time domain (FDTD) simulations based on HRTEM images predict an electric field intensity enhancement of 82,400 at the center of the nanoparticle pair, and an electromagnetic SERS enhancement factor of 10 9 -10 10 . 1 CHAPTER 1: BACKGROUND In order to establish a fundamental background for the work presented in this thesis, theories about the finite difference time domain (FDTD) algorithm, plasmonics, and photocatalysis are explained in this chapter. 1.1 The Finite-Difference Time-Domain (FDTD) Method After Kane Yee first introduced the FDTD method in 1966 254 , people began to realize its accuracy and flexibility for solving electromagnetic problems. The FDTD method provides a direct time-domain solution of Maxwell's Equations in differential form by discretizing both the spatial and temporal dimensions using a uniform grid. The power of this method lies in the fact that it can solve Maxwell's equations on any scale with almost all kinds of environments in a wide variety of different electromagnetic problems. In this section, Maxwell's equations and Yee's FDTD algorithm will be introduced. The differential form of Maxwell's equations and constitutive relations can be written as: Faraday's law: ⃗ ⃗ ⃗ ⃗ ⃗⃗ (1.1) Ampere's Law: ⃗ ⃗ ⃗ ⃗ ⃗ (1.2) 2 Gauss's Law for electric fields: ⃗ ⃗ (1.3) Gauss's Law for magnetic fields: ⃗ (1.4) Constitutive relations: ⃗ ⃗ ⃗ (1.5) ⃗ ⃗ ⃗ (1.6) From these equations we can get Maxwell's curl equations in linear, isotropic, nondispersive, lossy materials: ⃗ ⃗ ⃗ ⃗ ⃗ ⃗⃗ (1.6) ⃗ ⃗ ⃗ ⃗ (1.7) If we write these equations in rectangular coordinates we will get: (1.8) (1.9) (1.10) (1.11) 3 (1.12) (1.13) The basic idea of Yee's algorithm is to discretize both the physical region and the time interval of the differential-form three-dimensional Maxwell's equations shown above on uniform grids. The algorithm updates electromagnetic field values time step by time step from two parts: the field values calculated in the previous time steps and the field values in the adjacent space cells. This algorithm makes possible the use of modern Figure 1.1: Primary and secondary cell within Yee grid in FDTD representing E- and H-field components in the discretized space. Each node i,j,k within the FDTD grid represents three electric and magnetic field components located around a volume element. 4 computational resources to solve Maxwell's equations, and has created a new era of electromagnetic scientific research. The FDTD algorithm proposed by Yee is based on a description of the coupled system as described in equations (1.6) and (1.7) on the basis of a finite central difference approximation. Figure 1.1 depicts the position of the electric (yellow) and magnetic (green) field components for each primary and secondary Yee cell, respectively, allocated within the cartesian grid. The fields are located such that four ⃗ ⃗ components surround every ⃗ component and vice versa, which leads to a spatially coupled system of field circulations corresponding to the law of Faraday and Ampere. Each three ⃗ and ⃗ ⃗ field components are assigned to a node i, j, k within the three dimensional (3-D) FDTD grid. Figure 1.2: Position of the E and H fields inside a Cubic unit cell of the Yee space lattice. 5 Fig. 1.2 shows the primary (E) grid in detail. x, y, z are the three dimensions of this cube. We use (i, j, k) to denote the point whose real coordinates are (i x, j y, k z) in the model space. The “curl" operation in Maxwell’s equations can be easily understood looking at this figure. For example, the H x component located at point (i, j+1/2, k+1/2) is Figure 1.3: (a) Half step updates of the E- and H-fields in space and time. (b) Leapfrog method of calculation. 6 surrounded by four circulating ⃗ components, two E y components and two E z components, exactly matching to the equation (1.8), which states that the H x component increases directly in response to a curl of ⃗ components in x direction proportional to a constant µ. The constant µ specifies the magnetic permeability of the material at the location of this unit cell. Similarly, the ⃗ components increase directly in response to the curl of ⃗ ⃗ components, with a constant proportional to the electrical permittivity of the material at the current location. The ⃗ and ⃗ ⃗ components are also coupled in time. As illustrated in Figure 1.3, all the ⃗ components are updated at NΔt and all the ⃗ ⃗ components are updated at (N+1/2)Δt. This means that, the previously stored ⃗ ⃗ data at the (N-1/2)-th time step is used to update the ⃗ components in the N-th time step, and then the ⃗ ⃗ components in the (N + 1/2)-th time step are updated and stored in memory also using the ⃗ data which are just calculated. Thus, the electromagnetic field values can be updated at any time interval. To discretize the differential form of Maxwell's equations, we need a difference approximation of the space and time derivatives. Consider the Taylor series expansion of a certain variable at the current space location and time step: [ ] (1.14) This first part of the Taylor series expansion is nothing but the central difference approximation. It is chosen because it is computationally simple and has relatively good second-order accuracy. So the discretized Maxwell's equations can be derived from equations (1.8) to (1.13) using the central difference approximations. The resulting finite- 7 difference equations are solved in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved. For single electromagnetic components in Maxwell's equations, for example H x , the averaging of the two consecutive time steps value is applied in discretized equations. ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] ( ) [ ( ) ( )] ( ) (1.15) ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] ( ) [ ( ) ( )] ( ) (1.16) 8 ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] ( ) [ ( ) ( )] ( ) ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] ( ) [ ( ) ( )] ( ) (1.18) ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] ( ) [ ( ) ( )] ( ) (1.19) (1.17) 9 ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] ( ) [ ( ) ( )] ( ) (1.20) Figure 1.4: A simplified flowchart describing the FDTD algorithm. 10 Apart from being accurate and flexible, the most appealing characteristic of FDTD algorithm is its simplicity of computation and structure. All the operations given in equations 1.15 to 1.20 are additions, subtractions and multiplications, which are easy to compute and computationally inexpensive requiring few resources in a hardware implementation. It should also be noted that the structures of these six equations are almost the same, making it possible to apply various methods to speed up the algorithm. What's more, the algorithm simply iterates the same procedures for each time step in the time domain, making the computation regular. 1.2 Plasmonics In the last few decades, a fast growing research field of plasmonics 150 – named after the electron density waves that propagate along the interface of a metal and a dielectric, has been prominent in scientific journals. We have witnessed a flurry of activity in the fundamental research and development of plasmonic structures and devices. Much before the recent advances in the field of plasmonics, metal nanostructures were employed by artists to generate vibrant colors in glass artifacts and in the staining of church windows. One of the most widely cited examples is the Lycurgus cup dating back to the Byzantine Empire (4th century AD). The first scientific studies involving surface plasmons date back to the beginning of the twentieth century. Plasmonics builds a bridge between two different regimes: bulk regime and molecular regime, by confining light from far field to sub-wavelength near field. Most useful of these plasmonic nanostructures are metallic nanoparticles (MNPs) and colloids, 11 or thin metal films in the case of plasmonic waveguides. Such nanostructures have been utilized to dramatically change the optical properties of various optically active objects of similar or smaller dimensions such as atoms, molecules, and quantum dots 16,34,50,75,197,209,232 . There have been some promising applications of plasmonics, such as Raman spectroscopy 127,241 , fluorescence spectroscopy 142,168 , superfast computer chips 13 , cancer treatments 143 , ultrasensitive molecular detectors 6,80 , cloaking 140,189 , data storage 36,180,260 , optical data processing 25 , quantum optics 3,130 , optoelectronics 64,131 , photovoltaics 67,191 and photocatalysis 146 , non-linear wave mixing 179 . A plasmon is essentially a quantum of plasma oscillation. The plasmon is a quasiparticle resulting from the quantization of plasma oscillations just as photons and phonons are quantizations of electromagnetic and mechanical vibrations, respectively (although the photon is an elementary particle, not a quasiparticle). Plasmons can be described in the classical picture as an oscillation of free electron density with respect to the fixed positive ions in a metal. To visualize a plasma oscillation, imagine a cube of metal placed in an external electric field pointing to the right. Electrons will move to the left side (uncovering positive ions on the right side) until they cancel the field inside the metal. If the electric field is removed, the electrons move to the right, repelled by each other and attracted to the positive ions left bare on the right side. They oscillate back and forth at the plasma frequency until the energy is lost in some kind of resistance or damping. Plasmons are a quantization of this kind of oscillation. Since plasmons are the quantization of classical plasma oscillations, most of their properties can be derived directly from Maxwell's equations. 12 There are several different types of plasmon resonances including bulk or volume plasmons, surface plasmon polaritons and localized surface plasmons. Bulk or volume plasmons are longitudinal oscillations of free electrons within the bulk of a material. Volume plasmons do not couple to transverse electromagnetic waves, due to their longitudinal nature and can only be excited by particle impact. Volume plasmons can decay only via energy transfer to single electrons, by a process known as Landau damping 128 . Their oscillating frequency or plasma frequency, , is given by: (1.21) where n is the density of electrons, m is the electron mass, e is the charge of an electron and is the permittivity of free space. Light of frequency below the plasma frequency is reflected, because the electrons in the metal screen the electric field of the light. Light of frequency above the plasma frequency is transmitted, because the electrons cannot respond fast enough to screen it. In most metals, the plasma frequency is in the ultraviolet, making them shiny (reflective) in the visible range. Volume plasmons are extremely difficult to excite with visible light because of the reasons noted above. Therefore, experimentally the plasma frequency of metals typically has to be determined via electron loss spectroscopy experiments. The plasma frequency for most metals lies in the ultraviolet region, with being on the order of 5 − 15 eV, depending on the band structure of the metal 122 . 13 Surface plasmon polaritons (SPPs), are infrared or visible frequency electromagnetic waves trapped at or guided along metal-dielectric interfaces. These are shorter in wavelength than the incident light (photons). Hence, SPPs can provide a significant reduction in effective wavelength and a corresponding significant increase in spatial confinement and local field intensity. SPPs are combined excitations of free electrons i.e. plasmons and photons and form a propagating mode bound to the metal- dielectric interface 20,22 . For a SPP mode to exist, the materials at the interface must have the of opposite sign. Metals, which inherently have a < 0, satisfy this condition when interfaced to a dielectric material. Upon excitation of the surface plasmon polaritons, maximum electric field intensity is present at the interface. These fields Figure 1.5: A surface plasmon polariton as a collective excitation at a metal– dielectric interface [from reference 35]. 14 exhibit near-field or evanescent behavior with exponential decay away from the interface. The frequency of surface plasmon polaritons, is related to the bulk plasmon frequency of a metal by the following relation: √ (1.22) Consider the differential equations shown in equations 1.8 to 1.13. For propagation along the x-direction ( ) and homogeneity in the y-direction ( ) for a metal dielectric interface as shown in Figure 1.5, the TM solutions of the equations look like: (1.23) (1.24) (1.25) for z > 0, and (1.26) (1.27) (1.28) for z < 0, where is the component of the wave vector perpendicular to the interface and is the propagation constant. Applying the boundary condition enforcing the continuity of and at the interface gives us the following relation: 15 (1.29) The negative sign in the above equation ensures that the surface waves exist only at interfaces between materials with opposite signs of the real part of their dielectric permittivities, i.e. between a metal and a semiconductor. The wave equation for the TM mode is given as, (1.30) The expression for Hy further has to fulfill the above wave equation, yielding (1.31) Combining equations 1.29 and 1.31, we get the dispersion relation for surface plasmon polaritons at a metal-dielectric interface: √ (1.32) Drude model of dielectric function for a metal is given as: (1.33) where is known as the collision frequency which typically of the order of 100THz. For large frequencies close to , the quantity >> 1, such that the damping can be neglected. In this case, the dielectric function can be simplified to: (1.34) 16 Figure 1.6 shows the dispersion relation defined by equation 1.32 using the dielectric function given by equation 1.34 for an air ( = 1) and a fused silica ( = 2.25) interface. The frequency is normalized to the plasma frequency . The real part of the wave vector is shown by continuous curves and the imaginary part is shown by the broken curves. The bound nature of the SPP excitations necessitates that they correspond to the part of the dispersion curves lying to the right of the respective light lines of air and silica. Therefore, they cannot be excited without special phase-matching techniques such as grating or prism coupling 90 . Radiation into the metal occurs in the transparency regime > . Between the regime of the bound and radiative modes, a frequency gap region exists, where the propagation is prohibited due to a purely imaginary . Figure 1.6: Dispersion relation of SPPs at the interface between a Drude metal and air (gray curves) and silica (black curves). [from reference 2] 17 Localized surface plasmons (LSP) are substantially different from both bulk plasmons and surface plasmon polaritons. LSPs are non-propagating collective excitations of free electrons that are bound in a confined solid or void like a spherical nanoparticle. The curved surface of the particle exerts an effective restoring force on the driven electrons, leading to a resonance, and field amplification both inside and in the near-field zone outside the particle. The curved surface also enables excitation of plasmon resonances by direct light illumination, in contrast to propagating SPPs, where the phase-matching techniques have to be employed. Fig. 1.7 illustrates LSPR in a nanoparticle coherently oscillating in response to an incident light wave. The electron cloud is first displaced from the nanoparticle leaving behind positive ions. Columbic attraction between the electrons and the ions gives rise to a restoring force, resulting in an oscillation of the electron cloud about the nanoparticle. The resonances of the electron cloud are discrete frequencies that are dependent on different factors such as the size, shape and material of the nanoparticles 118 . So, the energy of the incident wave is first absorbed to displace the electron cloud about the nanoparticle, and then scattered in the form of a dipolar field. If the size of nanoparticle is much smaller than the wavelength of field, such that , the electrical field can be considered as uniform within the volume of the nanoparticle. This is known as the quasi-static approximation, which allows the Maxwell equations to be reduced to strictly electro- and magneto-static conditions because the electric field incident on the volume of the particle is spatially constant. 18 Let us look at the analytical description of LSP using the electrostatic approach. For this, we will consider a homogeneous, isotropic metallic sphere of radius , being placed in dielectric media with the dielectric constant as shown in Figure 1.8. If there are no external charges, the electric potential under electrostatic approximation should satisfy the Laplace equation: (1.35) Using this Laplace equation, we should be able to calculate the electric field, . The solution for the th eigenmode for azimuthal symmetry can be given as, ( ) for ( ) for (1.36) Figure 1.7: Localized surface plasmon excitation in nanoparticles in response to an incident light wave. Electric field Nanoparticle Electron cloud 19 where are the Legendre polynomials of order , and and are normalization factors. The continuity of the normal component of the electrical displacement , demands that the dielectric function of the metal and the dielectric media must satisfy the following relation: (1.37) where is the resonance frequency of the th mode. If we consider the simple Drude model dielectric function for the metal given in equation 1.33, the mode frequency is given by the following relation: Figure 1.8: homogeneous, isotropic metallic sphere of radius , placed in an electrostatic field. 𝑍 𝑌 𝑋 𝑬 ̅ ̅ ̅ ̅ 20 √ (1.38) So for a dipole solution ( = 1), the dielectric functions must satisfy the relation which is known as the Fröhlich condition: (1.39) As increases, the multipole solutions gradually approach the SPP resonance condition. Equality of the tangential components of the electric field demands that must be continuous. Using the two boundary conditions, we can get the potentials inside and outside the nanoparticle for the dipole mode ( = 1) as, (1.40) (1.41) Here, we can introduce a dipole moment given as, (1.42) Using this, can now be written as, (1.43) 21 Thus, the applied field induces a dipole moment inside the sphere of magnitude that is proportional to | | . Dipole moment can also be written as , where alpha is the polarizability given by (1.44) Figure 1.9 (from ref x) shows the absolute value and phase of α with respect to frequency ω (in energy units) for a Drude dielectric function fitted to the optical constants of silver 112 . As shown earlier by the Fröhlich condition in equation 1.39, the polarizability experiences a resonant enhancement under the condition that | | is a minimum, which for the case of small or slowly-varying [ ] around the resonance simplifies to [ ] . For a sphere consisting of a Drude metal with a dielectric function given by equation 1.33 located in air ( ), the Fröhlich criterion is met at the frequency √ . Now, using the equation , we can find the electric fields inside and outside the nanoparticle as, (1.45) (1.46) 22 Figure 1.9: (a) Absolute value and (b) phase of the polarizability of a sub- wavelength metal nanoparticle with respect to the frequency of the driving field. [from reference 2] (b) 23 As expected, the internal and dipolar fields also show resonant enhancement die to the resonance in . The corresponding cross sections for scattering and absorption and can be calculated via the Poynting-vector determined from the total electric and magnetic fields in the near, intermediate and radiation zones of a dipole 24 to | | (1.47) [ ] (1.48) Here, is the wave vector. Since the efficiency of absorption scales with , it dominates over the scattering efficiency, which scales with , for small particles with . As , it becomes very diff icult to pick out small objects from a background of larger scatterers. From these equations, one can see that the plasmon resonance varies for nanoparticles with different sizes, shapes. Objects other than spheres, such as ellipsoids and stars, have been shown to have different plasmon modes corresponding to the different axes of the object 86,167 . Optical absorption for nanoparticles with different shapes 158 and sizes is shown in Figure 1.10. Several other factors also influence the LSP resonance including the electron mean free path and skin depth. For bulk Ag, the electron mean free path is ~ 50 nm and decreases almost linearly with the particle diameter 199 . For nanoparticles with diameters smaller than ~10nm, electron scattering from the particle's surface becomes significant. This results in dephasing, where a broadening of the resonance peak occurs 133 . 24 In addition to the mean free path of an electron, it is important to consider the skin depth of incident light on a nanoparticle. The incident field, , penetrating beyond the surface of a nanoparticle by distance is exponentially decaying in nature and is expressed as: | | | | (1.49) where is the skin depth defined as: (1.50) Here is the imaginary part of the refractive index of the metal, also known as the absorption coefficient and is given by: √ √ [ ] [ ] [ ] (1.51) Figure 1.10: (a) Scattering spectra of single silver nanoparticles of different shapes obtained in dark-field configuration [from reference 42]. (b) Electric field enhancement for silver nanoparticles of different diameters. (a) 25 While a typical skin depth for an Ag nanoparticle is calculated to be ~ 30 nm, it increases to ~80 nm at the plasmon resonance. It is intuitive that plasmon resonance occurs when the skin depth and electron mean free path are on the order of the nanoparticles size. So, it is not surprising that the dipolar nature of LSP modes in Ag nanoparticles generally exist for nanoparticles with 20 - 250 nm in diameter. The plasmon resonance of a nanoparticle is also influenced by the local medium surrounding the particle 159 . This dependence is shown in Figure 1.11. Adsorbates on the particle's surface can also cause shift in the plasmon resonance. This happens due to the electronic coupling between plasmon and charge transfer bands of the metal-adsorbate complex 163 . Figure 1.11: Surface plasmon dipole resonance of a silver nanoparticle, when the nanoparticle is placed in (a) air, (b) 1.44 index oil and (c) 1.48 index oil. [from reference 45] 26 Moving from a single nanoparticle, to a dimer of nanoparticles, the peak electric field now appears in the gap between the two nanoparticles, and is much higher in intensity that the electric field intensity in the vicinity of a single nanoparticle. This happens because the individual LSP modes in the nanoparticles now couple to each other and thus reinforce each other, leading to an exponential increase in the electric fields. This can be demonstrated by changing the separation between the 2 nanoparticles and calculating or measuring the electric field intensity. As shown in Figure 1.12, the electric field intensity decays exponentially as the gap between the nanoparticles is increased, due to the decreased coupling between the nanoparticles. Figure 1.12: For a dimer of nanoparticles, the local electrical field between the gap becomes enormous due to the coupling of the surface plasmons that gives rise to new resonant modes. As the distance between the nanoparticle decreases, the plasmonic coupling increases which in turn increases the local electric field intensity. 0 2 4 6 8 10 12 14 16 0 10000 20000 30000 40000 I/I 0 Separation (nm) 27 1.3 Photocatalysis Solar energy can be either converted to electricity directly using a solar cell, or stored in chemical bond of fuels, such as H 2 or CH 4 . Traditional solar cells use pn junctions in single crystal semiconductors as in Figure 1.13a. Typically, a p-type Figure 1.13: (a) Traditional p-n junction solar cell, (b) Photoelectrochemical solar cell. (c) Solar fuel generation [from reference 47]. 3 II (a) (b) (c) 28 semiconductor (base) is doped to n-type on the surface layer (emitter) to form pn junction. The top electrode normally consists of a conductive anti-reflection coating (ARC) and a metal grid as the negative pole of the solar cell. To absorb visible light, the band gap of these semiconductor solar cells are around 1eV. Since, in most cases, one photon only generates one electron-hole pair, a lot of energy is wasted due to either the photon energy being too low to be absorbed or too high, in which case the excess energy is converted to heat (phonon energy). The pn junction solar cell requires complicated semiconductor and doping processing. A more cost-effective solar cell is the photoelectrochemical solar cell using Titanium dioxide (TiO 2 ), as shown in Figure 1.13c. In these cells, electron-hole pair generation is balanced by reversible red-ox reactions (e.g. I/I 3 -- ). Yet another way to utilize solar energy is to store it in the bonds of chemical fuels, such as hydrogen and methane. Photochemical fuel generation 76 , as shown in Figure 1.13d, requires that In photocatalysis, chemical reactions are driven or accelerated by a catalyst under electromagnetic illumination. To be a good photocatalyst, a material should have its conduction band above the energy level of reduction half reaction and the valance band below the energy level of oxidization half-reaction (Figure 1.14). TiO 2 is commonly used as a photocatalyst for solar cell, water splitting, and organic material decomposition, as it satisfies this condition, and also due to its low cost and superior stability. Its two phases commonly used in photocatalysis are anatase and rutile, with 3.2eV and 3.0eV energy band gaps respectively. Pure TiO 2 is a good UV absorber, but transparent to visible light. Even though anatase has a slightly higher band gap, it is 29 generally considered as a better photocatalyst than rutile due to significantly higher surface area and thus higher levels of adsorbed radicals. As mentioned earlier, TiO2 is a great photocatalyst, but it does not absorb visible light. Figure 1.14a shows the TiO2 absorption spectrum together with the solar spectrum, where one can see TiO2’s absorption cutoff at a wavelength around 370nm. Because of this, there are very few solar photons (about 4%) that can be used to drive this photocatalyst. The absorption of the plasmonic nanoparticles shown in Figure 1.14b, on the other hand, can be tuned by varying their size, shape, and separation, and can essentially be tailored to match the solar spectral radiance, as shown here. The rationale for our work is to use plasmonic nanostructures to improve visible light absorption of TiO 2 . Figure 1.14: Solar Spectrum superimposed on (a) the absorption spectrum of TiO 2 and (b) the absorption spectrum of a typical plasmonic nanostructure. (a) (b) 30 CHAPTER 2: OPTIMIZATION OF PLASMON RESONANT NANOSTRUCTURES This chapter is similar to Pavaskar et al. 184 , published in Applied Physics Letters. 2.1 Abstract In this work, we perform finite difference time domain simulations of two- dimensional clusters of metal nanoparticles with incident planewave irradiation. An iterative optimization algorithm is used to determine the configuration of the nanoparticles that gives the maximum electric field intensity at the center of the cluster. The optimum configurations of these clusters have mirror symmetry about the axis of planewave propagation, but are otherwise non-symmetric and non-intuitive. The maximum field intensity is found to increase monotonically with the number nanoparticles in the cluster, producing intensities that are five times larger than linear chains of nanoparticles and 2500 times larger than the incident electromagnetic field. 2.2 Introduction Plasmon resonant nanostructures have become a topic of growing interest, impacting a wide range of fields, including optics 152 , spectroscopy 127,134 , chemistry 101 and medicine 62,212 . Surface plasmon excitations produce very large enhancements in the electromagnetic fields surrounding these nanostructures. This effect is seen especially for wavelengths near the plasmon resonance energy. Jeanmaire and Van Duyne 106 31 demonstrated surface enhanced Raman spectroscopy (SERS) using plasmonic nanoparticles as early as 1977. Kneipp et al. 127 reported SERS enhancement values of 10 14 from colloidal silver clusters. A quantitative theoretical understanding of this electric field enhancement came later with Hao and Schatz 84 and Oubre and Nordlander 174 who calculated electric field enhancements for single nanoparticles of different shapes, as well as dimers and linear chains of nanoparticles. Raman intensity enhancement values as high as 10 8 were found in their calculations of dimers. Experimentally, Atay et al. 11 found the plasmon resonant frequency of nanoparticle dimers to depend strongly on nanoparticle separation. A similar work was carried out both through simulations and experiments by Su et al. 220 confirming these findings. Adaptive algorithms have been used by Gheorma et al. 74 and Seliger et al. 210 to optimize nanoscale dielectric structures to achieve preferential angular scattering profiles. The optimum design showed non-symmetric solutions that could not have been derived by straightforward human logic. An experimental realization of these geometries confirmed the results predicted by these algorithms. 210 2.3 Simulations details In the work presented here, we simulate the electric field response of metal nanoparticle clusters with incident plane wave irradiation. It should be noted that the 2D nature of the simulations imply that these nanoparticles are clusters of infinitely long cylinders rather than spheres. We implement optimization of plasmonic materials considering the 2N-dimensional parameter space (N is the number of nanoparticles), 32 Figure 2.1: Flowchart illustrating the algorithm used for optimization. output ults yes initialize parameters calculate gradients move NPs calculate intensity I and new resonant wavelength I >I max ? I max =I; new optimum configuration i = i + 1 yes no no I max < 1% 33 including non-intuitive configurations. We solve Maxwell’s equations numerically by discretizing the spatial and temporal dimensions of the electromagnetic fields using the finite difference time domain (FDTD) method. 226 A two dimensional spatial grid of 300,000 points is used to represent the computational cell. The grid spacing used for our simulations is 1 nm. To justify the grid spacing used for the simulations, the minimum allowed separation between any two nanoparticles is 2 nm. The diameter of the nanoparticles used in simulations is 20 nm. For boundary conditions, we have implemented perfectly matched layers (PML) with the number of layers being 10. The dielectric function used is based on a fit of the experimental data obtained by Johnson and Christy. to a Lorentz-Drude formula. 112 The temporal grid consists of 10,000 time steps with a spacing of 0.002 fsec. We have used a gradient-based algorithm to optimize the electric field at the center of the nanoparticle cluster with respect to all the nanoparticle positions. Downhill movements are allowed in order to avoid local maxima. Figure 2.1 shows a flowchart illustrating the algorithm used, which is as follows: 1. Run a FDTD simulation with a broad pulse to determine the resonance frequency of the initial nanoparticle configuration. Initialize all parameters, including number of nanoparticles, nanoparticle positions, frequency and bandwidth of incident radiation, and step size. 2. Calculate electric field intensity gradients with respect to nanoparticle positions in the x and y directions. 34 3. Move each nanoparticle in the direction of the electric field intensity gradient with respect to its position, with a step size proportional to the distance of the nearest nanoparticle, provided there is no overlap with other nanoparticles. 4. Determine the resonant wavelength of the new nanoparticle configuration and run a full FDTD simulation of the electric fields over the entire simulation grid. 5. Compare the electric field intensity at the center of the nanoparticle cluster with that of the highest previous value (I max ). a. If this new value exceeds the previous highest value, then the configuration is saved as the new optimum configuration. b. If the electric field intensity is found to be lower than the previous highest value, increase the downhill movement counter j by 1 and proceed to the next iteration. 6. When the maximum field intensity stops changing by more than 1%, the iterative process is stopped, and the highest electric field intensity and corresponding nanoparticle configuration are output. 2.4 Results and discussion Figure 2.2(a) shows an electric field intensity map for the case of 3 nanoparticles. Here, the 2 nanoparticles centered about (0,0) are fixed in position, with a separation of 2 nm. The electric field intensity is calculated at the point of interest (shown by ‘X’) as a function of the positions of the third nanoparticle. This Figure shows that a maximum intensity can be achieved by placing the third nanoparticle at the top or the bottom of the 35 fixed dimer. Figure 2.2(b) shows the stepwise results of the optimization algorithm applied to this third nanoparticle. Figure 2.2: (a) Optimization electric field intensity map with respect to spatial coordinates of the third nanoparticle. (b) Movement of the third nanoparticle with the optimization algorithm. The initial position is shown in green and the final position is shown in red. (c) Change in E-field intensity and (d) resonant wavelength with each iteration. X X (a) (c) (d) (b) X 36 It can be seen from these figures that the algorithm moves the nanoparticle from its initial position (shown in green) to the optimum position (shown in red), as predicted by the intensity map. This optimization was found to reach the convergence criterion after five iterations. Figures 2.2(c) and 2.2(d) show the electric field intensity and the resonant wavelengths obtained after each iteration. The electric field intensity is found to increase monotonically with the number of iterations. The resonant wavelength of the cluster also changes slightly during the iterative process, but only by 1-2%. Such a Figure 2.3: Optimization of 10 nanoparticles. 37 change in the resonant wavelength has been well documented previously in the literature. 11 Using this approach, we add nanoparticles to this cluster, one by one, and optimize the positions of the nanoparticles. Figure 2.4 shows the trend in optimum electric field intensity as a function of the number of nanoparticles together with the electric field intensities of linear chains of nanoparticles with 2 nm separation. A monotonic increase is found in the optimum intensity with increasing number of nanoparticles, whereas the highly symmetric linear chains show a drop in the intensity above four nanoparticles. This is consistent with the previous results in the literature. 246 Figure 2.4: Optimum electric field intensity enhancement as a function of the number of nanoparticles. 38 The electric field intensities of these optimized configurations show a factor of 2000 enhancement over the incident electromagnetic field in the case of 10 nanoparticles Figure 2.5: Electric field profiles of optimized geometries of 3- to 20-nanoparticle clusters. Here, the electric field intensity is optimized at (0,0). 39 and a factor of 2500 enhancement in the case of 20 nanoparticles. The optimized geometric configurations of clusters with 3 to 20 nanoparticles are shown in Figure 2.5. There are several subtleties that facilitate the practical implementation of this iterative algorithm. The allowance of downhill movements in the optimization routine is necessary to avoid local optima. It is also important to choose the correct step size in the algorithm, in order to achieve convergence. Lastly, we maintain a taboo list of nanoparticle configurations in order to avoid computing the same configuration twice. The optimum geometric configurations for clusters of 10 and 20 nanoparticles are plotted in Figures 2.4(a) and 2.4(b), together with the resulting electric field intensity. In the figures, the “ ” symbol indicates the point of optimum electric field intensity. The direction of planewave propagation in both plots is from left to right. These geometries exhibit mirror symmetry about the x-axis. However, their overall configuration is still largely non-symmetric and non-intuitive. We would like to emphasize that these complex geometries are obtained independent of the initial positions of the nanoparticles, before optimization. The FDTD calculations performed so far are two-dimensional. As such, they represent infinitely long cylindrical rods rather than spherical nanoparticles. We also performed three-dimensional simulations, which require significantly more computational resources. In 2D, moving each nanoparticle requires roughly 5 minutes of computation time. However, in 3D, parallel processors are required, and computation times are on the order of hours rather than minutes for each computational iteration. Another distinguishing factor between 2D in 3D is the plane of irradiation. In 2D, we had 40 the source irradiating in the plane of the nanoparticles, whereas for the 3D case, the source is irradiating at normal incidence. Figure 2.6b shows the optimized geometry of 20 nanoparticles, for the case of 3D optimization 187 . This geometry shows much more symmetry than the 2D optimized geometry shown in Figure 2.6a. Figures 2.6c and 2.6d show the near field enhancement spectra of the two optimized geometries. Figure 2.6: Electric field profiles for the optimized geometry of 20 nanoparticles for (a) the 2D case and (b) the 3D case. Near field enhancement spectra of the optimized geometry for (a) the 2D case and (d) the 3D case. 400 450 500 550 600 0.0 5.0x10 2 1.0x10 3 1.5x10 3 2.0x10 3 2.5x10 3 |E| 2 /|E 0 | 2 Wavelength (nm) 400 450 500 550 600 0.0 2.0x10 3 4.0x10 3 6.0x10 3 8.0x10 3 1.0x10 4 |E| 2 /|E 0 | 2 Wavelength (nm) (a) (b) (c) (d) 41 2.5 Conclusions In conclusion, we have implemented an iterative optimization algorithm to determine the geometric configurations of metal nanoparticle clusters that produce the strongest electric field intensity from an incident electromagnetic plane wave. The optimum configurations of these clusters are found to have mirror symmetry about the axis of planewave propagation, but are otherwise non-symmetric and non-intuitive. The optimized electric field intensity increases monotonically with the number nanoparticles in the cluster. The electric field intensity enhancement for 20 nanoparticles is 2500 times larger than the incident electromagnetic field. This enhancement is significantly higher than a linear chain of nanoparticles or the centermost dimer alone. If these optimized nanoparticle configurations can be achieved experimentally, this method could enable new devices based on near-field and non-linear effects. This research was supported in part by ONR Award No. N00014-08-1-0132 and AFOSR Award No. FA9550-08-1-0019. 42 CHAPTER 3: A MICROSCOPIC STUDY OF PLASMONIC ISLAND THIN FILMS This chapter is similar to Pavaskar et al. 186 , published in the Journal of Applied Physics. 3.1 Abstract Thin Au and Ag evaporated films (~5nm) are known to form island-like growth, which exhibit a strong plasmonic response under visible illumination. In this work, evaporated thin films are imaged with high resolution transmission electron microscopy (HRTEM), to reveal the structure of the semicontinuous metal island film with sub-nm resolution. The electric field distributions and the absorption spectra of these semicontinuous island film geometries are then simulated numerically using the finite difference time domain (FDTD) method and compared with the experimentally measured absorption spectra. We find surface enhanced Raman scattering (SERS) enhancement factors as high as 10 8 in the regions of small gaps (≤ 2nm), which dominate the electromagnetic response of these films. The small gap enhancement is further substantiated by a statistical analysis of the electric field intensity as a function of the nanogap size. Areal SERS enhancement factors of 4.2 10 4 are obtained for these films. These plasmonic films can also enhance the performance of photocatalytic and photovoltaic phenomena, through near-field coupling. For TiO 2 photocatalysis, we calculate enhancement factors of 16 and 19 for Au and Ag, respectively. We study the 43 effect of annealing on these films, which results in a large reduction in electric field strength due to increased nanoparticle spacing. 3.2 Introduction Plasmonic phenomena have demonstrated great promise for a variety of photonic device applications including subwavelength waveguides 153 , tunable filters 54 , resonators 31 , and modulators 55 . Plasmonic lasers 45,175 , photocatalysts 92,96,146,231 , and solar cells 14,67,97,165 have also been realized experimentally. A wide range of fabrication techniques exist for producing plasmonic nanostructures including electron beam lithography 230 , colloidal self assembly 137,218 , nanoimprint lithography 251 , nanosphere lithography 38,87 , and block copolymer lithography 105 , just to name a few. Thin films (~5nm) of noble metals evaporated on dielectric substrates present a simple one-step and reproducible way to make stable plasmonic nanostructures. These semicontinuous thin films of Ag or Au have been used in SERS 44,125 , plasmon-enhanced photocatalysis 92,96,146 , and sensing applications 113,114,234 . SERS enhancement factors of 10 5 have been previously reported for Ag films 241 . Van Duyne et al. showed that large electric field enhancements are obtained due to the island-like structure of these films 241 . Regions of strong local electric field enhancements called “hot spots” are observed between the nanoislands 26 . An extensive study of these thin films, has been done by the Purdue group both theoretically 26,29,41 and experimentally 58,77 . These hot spots are generally believed to be formed in regions in which the gaps between neighboring islands are very small, which allows for maximum plasmonic coupling. This is a well-known plasmonic phenomenon 44 in the case of nanoparticle dimers 85,174,178,230 . While these semicontinuous films consist of random geometries of metal islands, their performance is significant and reproducible. We would like to develop an understanding of the microscopic electric field distribution of these substrates, and their relationship to nanoparticle morphology. In previous works, the morphology of these semicontinuous metal island films have been studied by scanning electron microscopy (SEM) 134,201 , transmission electron microscopy (TEM) 207 , scanning tunneling microscopy (STM) 49 , and atomic force microscopy (AFM) 30,241 . Schlegel and Cotton 207 , using TEM, and Dawson et al. 49 , using STM, studied the relationship of the deposition rate of silver films with SERS intensities. Both these works showed that slowly deposited films produce higher SERS intensities than rapidly deposited films. Electromagnetic simulations of semicontinuous metal films have been carried out based on SEM and AFM images by several groups 30,41,206,241 . In the work presented here, FDTD simulations are performed based on HRTEM images, which provide a spatial resolution more than one order of magnitude higher than previous studies. The grid size of the FDTD calculations are also one order of magnitude higher than previous studies, which is important in accurately capturing the behavior of the quickly decaying fields at the surface of these metals. 3.3 Experiment and simulations details Electron-beam evaporation was used to deposit 5nm and 10nm Au and Ag films on 100nm thick SiN membranes, as determined by a quartz crystal oscillator thickness monitor. A JEOL JEM-2100F advanced field emission transmission electron microscope 45 was used to obtain high resolution TEM images of these films with a resolution of <2Å. FDTD simulations were performed on USC’s 0.15 petaflop supercomputing facility, which consists of 14,734 CPUs connected by a high-performance, low-latency Myrinet network. For the simulations, a cell of size 1000nm × 800nm × 500nm is used, which represents an experimentally measurable area, larger than the diffraction limit. The film occupies an area of 450nm × 300nm in the simulation. We use a grid spacing of 2Å in the volume of 500nm × 500nm × 40nm around the film and 10nm elsewhere. A temporal grid spacing of 0.002fsec is used with a total of 100,000 time steps. A planewave source is used, which irradiates the metal film with a Gaussian pulse with a spectrum of wavelengths ranging from 300nm to 800nm. Perfectly matched layers (PML) boundary conditions are used with 25 layers. The dielectric functions of Au and Ag are based on the optical constants given by Palik and Ghosh 176 . 3.4 Results and discussion Figure 3.1a shows a high resolution TEM image of a Au thin films with a nominal thickness of 5nm. The island-like formations are clearly visible in the images, where the dark grey regions are the gold islands and the light grey regions are interstitial space in between. Figure 3.1b shows the electric field distribution of the Au film. Here, the color axis shows the enhancement of electric field intensity relative to the incident electric field intensity. As postulated before, these results show that electric field enhancement is dominated by a few nm-sized hot spots rather than regions extending over large areas. 46 Figure 3.1c shows the calculated and measured absorption spectra of the Au film shown in Figure 3.1a. The experimental absorption spectra were measured using a Perkin-Elmer Lambda 950 UV/Vis/NIR spectrometer with an integrating sphere detector. This figure shows good agreement between the experimental and calculated spectra, which confirms the accuracy of our simulations. Figure 3.2a shows a HRTEM image of a 5nm Ag film. The 5nm Ag island nanoparticles are more loosely packed than the Au island film (Figure 3.1a) because the Figure 3.1: TEM images and electric field profiles of (a,b) a 5nm thick Au island film. The color axis in (b) shows the enhancement of electric field intensity relative to the incident electric field intensity. (c) Experimental and simulated absorption spectra of the 5nm Au island film. (b) (a) 50 nm (c) Wavelength (nm) Experimental FDTD Absorption (a.u.) 47 percolation threshold (i.e., the thickness above which the film goes from being insulating to conductive) for Ag films is 10nm 61,233 . Figure 3.3a shows the TEM image of a 10nm thick Ag film. Figures 3.2b and 3.3b show the electric field profiles for the 5nm and 10nm Ag films respectively. The absorption spectra for the 5nm Au film (Figure 3.1c) and 10nm Ag film (Figure 3.3c) show a much broader response than the 5nm Ag film (Figure 3.2c), due to the inhomogeneity of the island shapes and sizes, which gives rise to different plasmon modes 72 . (b) (a) 50 nm (c) Wavelength (nm) Experimental FDTD Absorption (a.u.) Figure 3.2: TEM images and electric field profiles of (a,b) a 5nm thick Ag island film. The color axis in (b) shows the enhancement of electric field intensity relative to the incident electric field intensity. (c) Experimental and simulated absorption spectra of the 5nm Ag island film. 48 The Au film shows higher electric field intensities than Ag, due to the smaller gaps between the islands. As such, these films utilize only a very small fraction of the sample area. For the gold film, the local electric field intensity in each of these hot spot regions is approximately three to four orders of magnitude larger than the incident electric field. Figures 3.4a and 3.4b show the polarization dependence of the electromagnetic response of the 5nm Au film. As can be seen in these figures, gaps oriented vertically are 300 400 500 600 700 800 Experimental FDTD Absorption (a.u.) Wavelength (nm) (c) (a) 50 nm (b) Figure 3.3: TEM images and electric field profiles of (a,b) a 10nm thick Ag island film. The color axis in (b) shows the enhancement of electric field intensity relative to the incident electric field intensity. (c) Experimental and simulated absorption spectra of the 10nm Ag island film. 49 primarily excited by horizontally polarized light and vice versa. This polarization dependence confirms the notion that the hot spots are formed due to strong plasmonic coupling between nearly touching islands. Some of the hot spots can produce SERS intensity enhancements exceeding 10 8 . The average electric field intensity enhancements for various gap sizes is shown in Figure 3.4c, where each data point is a statistical average of several hot spots of the same gap size. There is a monotonic decrease in the enhancement by a factor of 117, as the gap size increases from 2nm to 10nm, further demonstrating that the high electric fields in the gaps are due to strong plasmonic coupling. Such a relation between the electric field and the gap size has been shown previously in the case of dimers 2,85 , but not in the case of island films. It should be noted that sub-10nm gaps cannot be fabricated by standard lithographic techniques. We have also investigated the effect of annealing on nanoparticle morphology and the absorption spectra of these films. Figure 3.5a shows a high resolution TEM image of a 5nm Au film after annealing at 200˚C for 2 hours. The absorption spectra taken of annealed Au and Ag films are shown Figures 3.5b and 3.5c. These spectra exhibit narrow plasmon resonant absorption centered at 540 nm and 474 nm for Au and Ag, respectively, which is consistent with the absorption spectra of isolated nanoparticles. This occurs due to the fact that this random distribution of island shapes is transformed into a fairly uniform distribution of ellipsoidal nanoparticles after annealing. 50 Figure 3.4. Electric field distribution of a 5nm Au film irradiated with (a) horizontally and (b) vertically polarized light. (c) The distribution of the electric field intensity enhancements for various gap sizes. The color axes in (a,b) show the enhancement of electric field intensity relative to the incident electric field intensity. 2 3 4 5 6 7 8 0.0 2.0x10 3 4.0x10 3 6.0x10 3 |E| 2 Enhancement Gap (nm) (c) (a) 50 nm (b) 51 From the FDTD simulations described above, we have calculated the SERS enhancement factors of these films. Since the Raman intensity depends on the electric field to the fourth power (E 4 ), for small phonon energies, each of the hot spot regions is able to produce a SERS enhancement factors exceeding 10 8 . However, this requires that the molecule(s) of interest be located precisely in the 1-2nm hot spot region. Therefore, most of the area of this film does not demonstrate significant SERS enhancement. This makes detecting trace amounts of chemical species difficult and impractical. The areal SERS enhancement is another way to assess the performance of SERS substrates, and is obtained by integrating E 4 over the entire sample area and dividing by the incident electric field to the fourth power (E o 4 ) integrated over the same area, as follows: ∫ | | ∫ | | (3.1) Based on this equation, an areal SERS enhancement factor of 4.2 10 4 was obtained for both the 5nm Au and 10nm Ag films. The 5nm Ag film exhibited a smaller enhancement of 1.3 10 4 because the islands are more loosely packed resulting in a fewer number of small nanogaps. The annealed films produced even smaller enhancement factors of 366 and 45 for the Au and Ag films, respectively. Another reason for the smaller enhancements of the 5nm Ag film and the annealed Au and Ag films is their lack of pointed and sharp edges. It has been shown previously that such sharp features give rise to large plasmonic enhancements due to the so called lightning rod effect 73,151,217 . In addition to SERS enhancement, we have also calculated the photocatalytic enhancement factors for these plasmonic metal island films, 52 Figure 3.5. (a) TEM image of an annealed 5nm Au film. Experimental and simulated absorption spectra of annealed 5 nm (b) Au and (c) Ag films. Experimental FDTD Absorption (a.u.) (a) (b) 300 400 500 600 700 800 Absorption (a.u.) Wavelength (nm) Experimental FDTD (c) 53 as reported previously for nanoparticles deposited on top of a photocatalytic semiconductor. For the photocatalytic enhancement factor, we integrate over E 2 instead of E 4 , since the electron-hole pair generation rate is proportional to the square of the electric field. For the calculation of the photocatalytic enhancement factor, we need to consider absorption in the underlying substrate, since the electron-hole pairs driving the photocatalytic reaction are created in the substrate. Therefore, we also integrate over the z-dimension in our calculation. Since the minority carrier diffusion length in TiO 2 is only about 10nm, we only integrate over that length in the z-direction. Thus, the formula for the photocatalytic enhancement factor is: ∫ ∫ | | ∫ ∫ | | (3.2) where E sub is the electric field intensity in the underlying semiconductor without the plasmonic film. The dielectric function of anatase TiO 2 is used for the substrate 157 . Using this formula, the calculated photocatalytic enhancement factors for 5nm Au and 10nm Ag films are 16 and 19, respectively. For a single hot spot this enhancement factor is 176 in the 5nm Au film. The annealed films barely produce any enhancement with photocatalytic enhancement factors of 2.8 and 2.2 for the Au and Ag films, respectively. Several experimental photocatalytic studies have reported plasmonic enhancement using these 5nm semi-continuous Au films deposited on top of photocatalytic semiconductors. For water splitting, photocatalytic enhancement factors of 4 and 66 were reported at 532nm and 633nm, respectively 146 . For methane formation 54 from the reduction of CO 2 with water, a plasmonic enhancement factor of 24X was reported 92 . While these experimental values span a wide range, the enhancement is of the same order of magnitude as that predicted from the electric field distributions of FDTD simulations for similar nano-island films conducted in this study. 3.5 Conclusions In conclusion, we have simulated semicontinuous thin films of Au and Ag based on high resolution transmission electron microscope images with 2Å resolution using the finite difference time domain method. The island-like structure of these films produces strongly plasmonic behavior due to the presence of “hot spots” located between small nm-sized gaps. Our results verify the notion previously put forth that macroscopically observed SERS is dominated by a few nm-sized “hot spots”, with SERS intensities reaching 10 8 times the incident intensity. We obtained photocatalytic enhancement factors of 16 and 19 for 5nm Au and 10nm Ag films, respectively, over the whole film, but a much higher enhancement factor of 176 for a single hot spot. This indicates that if these films are optimized to have a large number of hot spots using numerical optimization, much higher enhancement factors over the whole film can be obtained. Based on a statistical analysis of electric field enhancements for various gap sizes, we demonstrate that the hot spots arise due to strong plasmonic coupling between nearly touching islands. Thermal annealing of these island films results in significant changes in their morphology and absorption spectra. 55 This research was supported by AFOSR Award No. FA9550-08-1-00190019 and ONR Award No. N00014-12-1-0570. 56 CHAPTER 4: PLASMONIC HOT SPOTS: NANOGAP ENHANCEMENT VS. FOCUSING EFFECTS FROM SURROUNDING NANOPARTICLES This chapter is similar to Pavaskar et al. 187 , published in Optics Express. 4.1 Abstract Thin Au films (~5nm) are known to form island-like structures with small gaps between the islands, which produce intense electric field “hot spots” under visible illumination. This chapter deals with finite difference time domain (FDTD) simulations based on experimentally observed high resolution transmission electron microscope (HRTEM) images of these films in order to study the nature of the “hot spots” in more detail. Specifically, we study the dependence of the electric field intensity in the hot spots on the surrounding film environment and on the size of the nanogaps. From our simulations, we show that the surrounding film contributes significantly to the electric field intensity at the hot spot by focusing energy to it. Widening of the gap size causes a decrease in the intensity at the hot spot. However, these island-like nanoparticle hot spots are far less sensitive to gap size than nanoparticle dimer geometries, studied previously. In fact, the main factor in determining the hot spot intensity is the focusing effect of the surrounding nano-islands. We show that these random Au island films outperform more 57 sophisticated geometries of spherical nanoparticle clusters that have been optimized using an iterative optimization algorithm. 4.2 Introduction It is well known that thin films (~ 5nm) of noble metals such as Au and Ag exhibit a strong plasmonic response under visible illumination due to their discontinuous nature, as discussed in chapter 3. These thin plasmonic films have been used in surface enhanced Raman spectroscopy (SERS) 44,124 , plasmon-enhanced photocatalysis 93,94,144 , and sensing applications 113,114,234 . Van Duyne et al. showed that the island-like structure of these films facilitates large enhancement in electric fields 240 . Royer et al. studied the structure of 5nm thick Au and Ag films using scanning electron microscope (SEM) and approximated the shape of the film islands using oblate spheroids in order to perform theoretical calculations 202 . These island-like structures create localized “hot spot” regions, in which the local electric field intensity can be enhanced by several orders of magnitude 27 . The Purdue group has studied these hot spot thin films both theoretically 27,28,41 and experimentally 58,77 . The general consensus is that these hot spots tend to form in regions where the neighboring islands are nearly touching, allowing for maximum plasmonic coupling. This has been observed in the case of nanoparticle dimers as well 84,174,178,230 . We believe that the electric field enhancement in these hot spots, apart from being dependent on the gap size between the islands, is also influenced by the surrounding nanoparticles in the film. In chapter 3, I talked extensively about the optical properties of 58 these films. In this chapter, I will focus on our study to determine the effect of the surrounding environment on the local electric field at the hot spot. Specifically, we determine how much of the electric field at the hot spot is contributed by the focusing effect of the surrounding film. To achieve this, we performed finite difference time Figure 4.1: (a) HRTEM micrograph and (b) electric field profile of the region surrounding a hot spot on a 5nm Au island film. The black squares of area (d x d) in (a) correspond to the different film areas used in the simulations shown in (c). (c) Electric field intensity at the hot spot plotted as a function of area (d x d). (b) (c) hot spot 0 10,000 20,000 30,000 40,000 0.0 5.0x10 6 1.0x10 7 1.5x10 7 2.0x10 7 2.5x10 7 c d b Intensity Area (nm 2 ) a 50 nm d (a) 59 domain (FDTD) simulations 225 of several hot spots based on experimentally measured high resolution transmission electron microscope (HRTEM) images of 5nm Au films and systematically varied the nano-island and gap size. Figure 4.2: (a) HRTEM micrograph and (b) electric field profile of the region surrounding a hot spot on a 5nm Au island film. The black squares of area (d x d) in (a) correspond to the different film areas used in the simulations shown in (c). (c) Electric field intensity at the hot spot plotted as a function of area (d x d). (b) (c) 0 10,000 20,000 30,000 40,000 0.0 2.0x10 7 4.0x10 7 Intensity Area (nm 2 ) hot spot 50 nm d (a) 60 4.3 Simulation details Au films were deposited on 100nm thick SiN membranes by electron-beam evaporation to achieve a nominal thickness of 5nm, as determined by a quartz crystal oscillator thickness monitor. HRTEM images were taken in a JEOL JEM-2100F advanced field emission transmission electron microscope, which provides high contrast images of the Au island films with <2Å resolution. These images were then used to define the spatial extent of the Au islands in the electromagnetic simulations. The thickness of the Au islands for the simulations was chosen to be 5nm. We neglected the surface roughness of the film, as the significant electric fields are present in the gaps between the islands and not on the surface of the islands. Full three-dimensional FDTD simulations were carried out on the University of Southern California’s 0.15 petaflop supercomputing facility, which consists of 14,734 CPUs connected by a high- performance, low-latency Myrinet network. A simulation cell size of 600nm x 600nm x 500nm was used with a grid spacing of 2Å in the volume of 200 x 200 x 20nm 3 around the film. 100,000 discrete time steps with a temporal grid spacing of 0.002 fsec were used in these calculations. A normal incidence, Gaussian pulse planewave source was used in the simulations with a broad spectrum of wavelengths ranging from 500nm to 1100nm. Perfectly matched layers (PML) were used as boundary conditions with 64 layers. The dielectric function of Au was a Lorentz-Drude model fitted to the optical data given by Palik and Ghosh. A uniform mesh was used over the entire simulation volume 176 . 61 4.4 Results and discussion Figure 4.1a shows a high resolution TEM image of a 5nm nominal Au film centered on a hot spot. Figure 4.1b shows the simulated electric field profile of the selected region. We performed several simulations on this hot spot, selectively removing the metal material around the hot spot and replacing it by vacuum. This scheme is illustrated more clearly in Figure 4.3. The black squares of area (d x d) in Figure 4.1a show the different film regions used in these simulations. We, thus, limit the total size of the film to an area of (d x d), while keeping the simulation area and mesh constant, to avoid any unphysical factors that may influence the results of the simulation. The electric field intensity at the centermost hot spot, I, was calculated by integrating the square of the electric field intensity over a 10 x 10 x 15nm 3 volume centered on the hot spot region and over all wavelengths. The integral in the z-dimension is centered on the Au-SiN interface. Thus, I is calculated as ∫ ∫∫∫ | | (4.1) The electric field intensity integrated over a small volume provides a more reliable value than the intensity at a point, by minimizing the effects that may arise due to sharp features artificially created by the spatial discretization. The hot spot intensity calculated in this manner is plotted as a function of plasmonic film area in Figure 4.1c. In this plot, the intensity increases with area initially, but drops beyond 6400nm 2 (i.e., 80nm x 80nm). The surrounding film is, therefore contributing energy to the local hot spot 62 causing this initially increasing trend. It should be noted that the integrated intensity, I, is different than the electric field intensity shown in Figure 4.1b. Similar results were observed for several such hot spots. Figure 4.2 shows the result for another hot spot. In order to fully understand the drop in intensity for larger Figure 4.3: Electric field intensity profiles corresponding to the areas highlighted in Figure 4.1c. (a) (b) (c) (d) 50 nm 63 areas, we plot the electric field intensity profiles of several different areas in Figure 4.3. The electric field profiles shown in Figures 4.3a to 4.3d correspond to the highlighted data points shown in Figure 4.1c. Figures 4.3a to 4.3c clearly show that the hot spot at the center is the dominant hot spot, and there is a monotonic increase in the hot spot intensity with area. However, the electric field profile shown in Figure 4.3d shows the presence of an additional hot spot that is more dominant than the one at the center. The new dominant hot spot is indicated by the arrow in Figure 4.3d. The intensity drop from 6400nm 2 (80nm x 80nm) to 10,000nm 2 (100nm x 100nm) can, therefore, be explained by the presence of the additional hot spot, which takes energy from the centermost hot spot. We also notice another hot spot appearing and disappearing as the area is increased. This phenomenon can be attributed to the fact that, the addition of the surrounding nanoparticles focuses light to a given hot spot by varying degrees, depending on their specific morphology. We also study the effect of widening the gap of a hot spot on the electric field intensity by artificially removing metal from the hot spot region, thus reducing the gap size. Figure 4.4b shows the relation between the hot spot intensity and gap size. As expected, the intensity at the centermost hot spot (labeled HS 0 in Figure 4.4a) decreases with gap size, due to the decreased coupling between plasmons across the gap. This effect has been reported both theoretically and experimentally in the case of plasmonic nanoparticle pairs 12,84,219 . The intensities of the other two hot spots (HS 1 and HS 2 ), however, remain mostly unaffected as the gap size of the central hot spot HS 0 is changed. 64 Figure 4.4: (a) Electric field intensity profile of a region with 3 hot spots labeled HS 0 , HS 1 and HS 2 . (b) Electric field intensities of HS 0 , HS 1 and HS 2 plotted as a function of the HS 0 gap size. 2 3 4 5 6 7 8 9 10 4.0x10 6 6.0x10 6 8.0x10 6 1.0x10 7 1.2x10 7 1.4x10 7 1.6x10 7 HS 0 HS 1 HS 2 Intensity Gap size (nm) (b) HS 0 HS 1 HS 2 (a) 50 nm 65 The monotonic decrease in the hot spot intensity as the gap size is increased shows that there is plasmonic coupling between the islands across the gap. However, the rate of decrease in the intensity is not as high as in the case of nanoparticle dimers 2,84 , which show an exponential decrease in the electric field intensity when the gap size is increased. Hao and Schatz reported a 10-fold decrease in the electric field intensity with only a 2nm increase in the gap size 84 . Aizpurua et al. showed a 25-30 drop in the electric field intensity when the gap size is changed from 2nm to 10nm 2 . However, in our case, there is only a 2.6X decrease in the intensity when the gap is increased from 2nm to 10nm. Thus, these island-like nanoparticle hot spots are more robust to changes in the gap size than the corresponding spherical or cylindrical nanoparticle geometries. This is due to the significant contribution made by the surrounding metal islands to the hot spot intensity, as shown in Figure 4.1 and Figure 4.2. This has important implications on the future design of nanoparticle arrays for optimized plasmonic hot spot applications. It should be noted that widening the gap of a hot spot only changes the plasmonic coupling locally and thus should not affect the surrounding hot spots in any way, if the area of the film (d x d) is held constant, as seen in Fig. 4.4b. In chapter 2, the geometries of nanoparticle clusters giving the maximum electric field intensity at the center of the cluster was determined using an iterative optimization algorithm 185 . Here, the surrounding nanoparticles in the cluster focus light on the central hot spot. This previous work was carried out using 2D FDTD simulations for light incident in the plane of the nanoparticle cluster. More recently, we have carried out full 3D FDTD simulations of nanoparticle clusters under normal incidence illumination. 66 Figure 4.5 shows the electric field intensity plot for the optimum configuration of a 20-nanoparticle cluster. This nanoparticle configuration was the result of an iterative optimization scheme with respect to 40 dimensions (i.e., x and y positions of 20 Figure 4.5: Electric field intensity profile for an optimized cluster of 20 nanoparticles. 67 nanoparticles). In order to obtain this geometry, we started with a cluster of two nanoparticles and performed the optimization routine for the position of a third nanoparticle. The position of each additional nanoparticle was optimized using this scheme up to a total of 20 nanoparticles each having a diameter of 40nm. Adding more nanoparticles would have increased the computational cost while not providing any further insight into plasmonic optimization, since the resulting geometries were highly symmetric. Additionally, the overall size of this cluster geometry is of the same order as that of the largest film area we used in our simulations. These calculations were performed using 16 processors in parallel, and altogether they took approximately 1700 hours. Despite the relative computational intensity of this calculation, the resulting optimized nanoparticle cluster only produces an integrated electric field intensity, I, of 10 6 at the centermost hot spot. This is more than an order of magnitude smaller than the integrated electric field intensity observed in the randomly-shaped Au island films of comparable size. The main reason for the relatively poor performance of the optimized nanoparticle cluster is that it contains many hot spots, which all compete for energy from the same incident electromagnetic flux. This will be an important consideration in future efforts to design and optimize plasmonic hot spot nanostructures, which has not yet been taken into account. It is likely that more intense hot spots that make more efficient use of the available surface area are possible by considering the focusing effects associated with arbitrary shaped geometries. Again, we would like to point out that the quantity plotted in figure 4.5 is the electric field intensity, which is different than the integrated intensity, I. 68 4.5 Conclusions In conclusion, we have studied the dependence of nanogap “hot spots” in thin plasmonic films on the presence of surrounding nano-islands. FDTD simulations demonstrate that the surrounding film environment contribute significantly to the intensity at the hot spot by focusing energy to it. However, the presence of other dominant hot spots in the surrounding film can take energy away from the central hot spot. We have shown that widening of the gap size causes a decrease in the intensity at the hot spot, while the surrounding hot spots remain unaffected. However, these island- like nanoparticle hot spots are far less sensitive to gap size than nanoparticle dimer geometries. In fact, the main factor in determining the hot spot intensity is the focusing effect of the surrounding nano-islands. In the future design of plasmonic nanostructures, this focusing effect will be an important consideration in producing optimized plasmonic films. This research was supported by AFOSR Award No. FA9550-08-1-00190019 and ONR Award No. N00014-12-1-0570. 69 CHAPTER 5: PLASMONIC ENHANCEMENT OF PHOTOCATALYTIC CHEMICAL REACTIONS This chapter is similar to Liu et al. 146 published in Nano Letters, Hou et al. 92 published in ACS Catalysis, Hou et al. 96 published in Journal of Catalysis, and Hou et al. 97 published in Energy & Environmental Science. 5.1 Abstract In this chapter, I will talk about electromagnetic simulations of plasmonic enhancement of photocatalytic chemical reactions. By integrating strongly plasmonic Au nanoparticles with strongly catalytic TiO 2 , we demonstrated plasmon-enhanced photocatalytic water splitting, and reduction of CO 2 with H 2 O to form hydrocarbon fuels. Under visible illumination, we observe enhancements of up to 66 in the photocatalytic splitting of water in TiO 2 with the addition of Au nanoparticles. We also perform a systematic study of the mechanisms of Au nanoparticle/TiO 2 -catalyzed photoreduction of CO 2 and water vapor over a wide range of wavelengths. In this case, under visible light illumination, we observe a 24-fold enhancement in the photocatalytic activity due to the intense local electromagnetic fields created by the surface plasmons of the Au nanoparticles. Above the plasmon resonance, under ultraviolet radiation we observe a reduction in the photocatalytic activity. Electromagnetic simulations indicate that the improvement of photocatalytic activity in the visible range is caused by the local electric 70 field enhancement near the TiO2 surface, rather than by the direct transfer of charge between the two materials. Here, the near-field optical enhancement increases the electron-hole pair generation rate at the surface of the TiO2, thus increasing the amount of photogenerated charge contributing to catalysis. This mechanism of enhancement is particularly effective because of the relatively short exciton diffusion length (or minority carrier diffusion length), which otherwise limits the photocatalytic performance. Our results suggest that enhancement factors many times larger than this are possible if this mechanism can be optimized. 5.2 Introduction Solar energy is a promising source of renewable energy, which can be used to supplement and possibly replace fossil fuels. The amount of energy striking the Earth from sunlight in one hour (4.3 10 20 J) is more than the total energy consumed on this planet in one year (4.1 10 20 J). In the wake of the global energy crisis, photocatalysis, which uses photonic energy instead of thermal energy to drive chemical reactions, has attracted much attention. The direct conversion of solar energy to electricity is problematic because of our inability to store large amounts of electricity. Photocatalysis circumvents this problem by converting solar energy directly into chemical bonds, which can be stored indefinitely for later use. Semiconductor photocatalysts (e.g., TiO 2 , ZnO, SnO, In 2 O 3 ) have been used effectively as catalysts in many chemical reactions, including decomposition of pollutants 121,229,245 , water splitting 155,198,223 , CO oxidation 46,250 , and reduction of aqueous CO 2 1,7,82,103 . TiO 2 , which is one of the most commonly used 71 semiconductor catalysts, does not absorb light in the visible region of the electromagnetic spectrum. With a bandgap of 3.2eV the cutoff wavelength of TiO 2 is around 380nm. There have been several attempts to extend the cutoff wavelength of this catalyst, including doping 204,213,245 and defect creation 227,228 . These efforts, however, have only been able to extend the absorption edge of TiO 2 to approximately 420nm 204,213,245,253 . Therefore, most of the solar spectrum is still unable to drive this photocatalyst. -8 -7 -6 -5 -4 -3 3.2 eV Anatase TiO 2 H 2 O/O 2 H 2 O/H 2 Vacuum (eV) NHE (Volts) 3 2 1 0 -1 -2 Fermi level of Au By depositing plasmonic metal nanoparticles on top of anatase TiO 2 , several research groups have observed enhanced photocatalytic water splitting under visible illumination 57,144,147,198,215,235 . In the photocatalytic splitting of water 69 , the oxidation and reduction half-reactions are given by the following chemical equations: Figure 5.1: Energy band alignment of anatase TiO 2 , Au, and the relevant redox potentials of H 2 O under visible illumination. 72 2H 2 O + 4h + O 2 + 4H + (1) 2H 2 O + 2e - H 2 + 2OH - (2) The photocurrents were measured using a potentiostat for this reaction, and the H 2 formed was monitored using gas chromotography and/or mass spectrometry. While the plasmonic enhancement factors reported above are relatively large, the overall photoconversion efficiencies are still quite low particularly in the visible wavelength range. The photoconversion efficiency for water splitting is reported to be around 1% under an overpotential of 0.6 Volts 120 . In Figure 5.1, the energy band diagram shows the conduction band and valence bands of TiO 2 together with the oxidation and the reduction potentials for the water splitting chemical reaction. The conduction band edge of TiO 2 is just barely high enough to carry out the water reduction and oxidation. Plasmonic enhancement has also been demonstrated in photocatalytic methane formation by the reduction of CO 2 with H 2 O 98 . The oxidation and reduction half- reactions are given by the following chemical equations: 2H 2 O + 4h + O 2 + 4H + (1) CO 2 + 8H + + 8e - → CH 4 + 2H 2 O (2) Prior to plasmon-enhanced studies, photocatalytic methane formation has been studied by Grimes’ group 66,200,242 and several others 1,81,82,103,243 . The products formed have been identified qualitatively and quantitatively by gas chromotography. This reaction involving 8 photons, is slightly more complicated process than water splitting reaction. In addition to methane, methanol, formaldehyde are other possible byproducts. In Figure 5.2, the energy band diagram shows the conduction band and valence bands of 73 TiO 2 together with the reduction potentials for several possible products: CH 4 , HCHO, and CH 3 OH. The conduction band edge of TiO 2 is just barely high enough to carry out direct CO 2 reduction and water oxidation. Because of this, an additional CO 2 reduction catalyst is typically needed in order to accomplish reasonable CO 2 reduction directly on a semiconductor surface. -8 -7 -6 -5 -4 -3 3.2 eV CO 2 /CH 3 OH -0.32 V CO 2 /HCHO -0.40 V Anatase TiO 2 H 2 O/O 2 0.82 V CO 2 /CH 4 -0.244 V Vacuum (eV) NHE (Volts) 3 2 1 0 -1 -2 Fermi level of Au A vast number of the publications in the literature discussing plasmon-enhanced photocatalysis have focused on reactions involving photocatalytic decomposition of organic compounds 5,8,15,39,51,63,65,68,79,99,102,107,110,116,117,129,132,136,148,149,154,160,161,214,216,236,248,252,255- 257,259 . These are typically dye molecules such as methyl orange whose decomposition rates can be easily observed by optical means 94 . In photocatalytic degradation, Figure 5.2: Energy band alignment of anatase TiO 2 , Au, and the relevant redox potentials of CO 2 and H 2 O under visible illumination. 74 photogenerated electrons reduce adsorbed oxygen (O 2 ) to form superoxide (HO 2 ·) radicals and photogenerated holes react with H 2 O to form OH· radicals 192 . Hydroxyl radicals subsequently oxidize the organic pollutants resulting in mineralization and complete degradation 23,156,192 . These pollutants are broken down into small molecules such as CO 2 , H 2 O, and NH 3 , which are not particularly toxic. 5.3 Experimental details Figure 5.3: (a) Schematic diagram of the photo-electrochemical measurement setup. (b) Quadrupole mass spectrometer data for H 2 production. Photocurrent of anodic TiO 2 with and without Au nanoparticles at zero bias voltage irradiated with (c) UV (λ = 254nm) and (d) visible light (λ = 532nm) for 22 seconds. 0 10 20 30 40 50 60 0 20 40 60 80 without Au with Au Short circuit current ( A) Time (sec) UV 0 10 20 30 40 50 60 0 5 10 15 20 25 Short circuit current ( A) Time (sec) without Au with Au Vis 0 100 200 300 1E-14 1E-13 1E-12 1E-11 H 2 QMS Ion Current (A) Time (seconds) (b) (a) UV/Vi s quartz window TiO 2 Ar gas in gas out to mass spec. ref. A V UV on graphite H 2 potentiostat (c) (d) 4x 5x 75 Photocatalytic water splitting: We prepared TiO 2 in the anatase crystalline phase by electrochemically oxidizing titanium foils in an ethylene glycol electrolyte containing 0.25 wt% NH 4 F and 2% wt H 2 O at an anodization potential of 30V for two hours, using a graphite rod as the cathode 78 . A gold film with a nominal thickness of 5nm was then evaporated on the surface of the TiO 2 . This thin gold film as described in chapter 3, is known to form island-like growth that is strongly plasmonic and serves as a good substrate for surface enhanced Raman spectroscopy (SERS) 59,134 . Absorption spectra of the bare TiO 2 and Au nanoparticle/TiO 2 films were recorded on a Perkin-Elmer Lambda 950 UV/Vis/NIR with an integrating sphere detector. Figure 5.3a shows a schematic diagram of our experimental setup. Here, the photocatalytic reaction rates of TiO 2 with and without Au nanoparticles are measured in a 1M KOH solution using a three-terminal potentiostat, with the TiO 2 film, a Ag/AgCl electrode, and a graphite electrode functioning as the working, reference, and counter electrodes, respectively. The TiO 2 film is irradiated in a sealed quartz flask, while the generated gas is monitored by mass spectrometry, verifying the production of H 2 under illumination, as shown in Figure 5.3b. Photocurrents of anodic TiO 2 with and without Au nanoparticles irradiated with ultraviolet (20mW/cm 2 @ 254nm) and visible light (7W/cm 2 @ 532nm) for 22 seconds are shown in figures 5.3c and 5.3d. Under UV illumination (λ = 254nm) (Figure 5.3c), the addition of gold nanoparticles results in a 4- fold reduction in the photocurrent. This reduction is due to the presence of the gold nanoparticles, which significantly reduce the TiO 2 surface area in direct contact with the aqueous solution. Under visible irradiation (λ = 532nm) (Figure 5.3d), however, the 76 addition of gold nanoparticles results in a 5-fold increase in the photocurrent due to the large plasmonic enhancement of the incident electromagnetic fields. Considering the 25% reduction in active TiO 2 surface area, based on the 4-fold reduction in Figure 5.3c, this plasmonic enhancement corresponds to a factor of 20 increase in the photocatalytic performance of the exposed TiO 2 . The transient decay observed in Figure 5.3d is the result of trapped surface charge that is released upon irradiation 164 . 300 400 500 600 700 800 Optical Absorption Wavelength (nm) Solgel TiO2 thin film Anodic TiO2 Anodic TiO2 with Au Figure 5.4 shows the UV-Vis absorption spectra of TiO 2 with and without gold nanoparticles. The spectrum taken for an undoped TiO 2 film prepared by the solgel Figure 5.4: UV-Vis absorption spectra of TiO 2 with and without gold nanoparticles. 77 method (solid black curve) shows transparency for wavelengths above 370nm 21,211 . The anodic TiO 2 film (red curve), however, shows significant absorption in the visible range (at longer wavelengths), likely due to N- and F-impurities produced during the anodization process, which produce defect states in the bandgap 100 . The absorption spectrum taken from anodic TiO 2 with gold nanoparticles (dashed blue curve) exhibits a slight increase in the absorption. The broad absorption of the Au film is a result of the inhomogeneity of these plasmonic nanoparticles described in chapter 3. 15 30 45 60 0.0 0.1 0.2 0.3 0.4 Current ( A) Time (sec) Laser 633 no Au Laser 633 with Au We also measured photocatalytic enhancement when irradiated with 633nm light (Figure 5.5), which is significantly below the bandgap of TiO 2 . Unlike 532nm light, no detectible photocurrent can be observed for bare TiO 2 . However, we do see significant enhancement in the photocurrent for samples with plasmonic nanoparticles deposited on Figure 5.5: Photocurrent of anodic TiO 2 with and without Au nanoparticles irradiated with λ = 633nm light for 20 seconds. 78 top of TiO 2 , resulting in a photocurrent on the order of 1μA. We can estimate a lower limit for the photocatalytic enhancement factor at 633nm by taking the unenhanced photocurrent to equal the noise in the photocurrent (4.5nA). This gives a lower limit for the enhancement factor of 66 . Photocatalytic CO 2 reduction: In this work, anatase titania thin films were prepared by the sol-gel process, following the general recipe of acid catalyzing dilute titanium ethoxide in ethanol 4 . The solution was then mixed with surfactant (P123) and stirred for several hours until a sol forms. Substrates of glass or quartz were spin-coated to achieve the desired film thickness of 400nm. The substrates were then positioned horizontally and dried at room temperature in air for 24h, thereby allowing most of the solvents and hydrochloric acid to evaporate and the surfactant to self-organize. The dried films were then annealed at 400 o C in air for 4h to improve their crystallinity and drive off any remaining solvents and surfactant. A thin film of gold was deposited on the TiO 2 surface in vacuum using electron beam evaporation, while the film thickness is monitored with a crystal oscillator. A 5nm deposition of gold produces an island-like morphology as described earlier in chapter 3. Subsequent annealing of this island-like film was performed at 400 o C in air for 1h in order to obtain more spherical Au nanoparticles. In order to make bare Au nanoparticles on an inactive support, a gold film with a nominal thickness of 5nm was evaporated on the surface of the glass. The photocatalytic reduction of CO 2 and H 2 O was carried out in a sealed 51.6ml stainless steel reactor with a quartz window. The photocatalytic films were placed on the catalyst holder, which is on the bottom of the reactor. A schematic diagram of the 79 experimental setup is shown in Figure 5.6. The reactor was first purged with CO 2 saturated water vapor for 1h before closing the system. The reactor was then illuminated with either UV light (254nm 20mW/cm 2 or 365nm 20mW/cm 2 UV lamp) or visible light (532nm 350mW/cm 2 green laser) for 15h at 75 o C. The irradiated surface area was limited by the surface area of the photocatalysts (10cm 2 ). Reaction products were analyzed using a Varian gas chromatograph (GC) equipped with TCD (with a detection limit of 100 ng for CO 2 ) and FID (with a detection limit of 50 pg for small organic molecules) detectors. The GC was calibrated by a series of gas samples with known amounts of CH 4 , CH 3 OH, HCHO, and C 2 H 6 . 300µL of gas (products and unreacted reagents) was sampled after 15h of illumination for each reaction. Since only 300µL of unreacted reagents and products were sampled and tested using GC, the yields were calculated by normalizing to the full volume of the reactor (51.6ml). Figure 5.7 shows the product yields of the photoreduction of aqueous CO 2 expressed per 1m 2 of catalyst surface area after 15h visible (532nm laser) illumination. Figure 5.6: Schematic of the experimental setup for CO 2 reduction. Catalyst UV or visible light MF C Quartz window Sampling gases using a syringe for GC measurement Catalyst holder CO 2 Water vapor bubbler 80 Here, methane is the only product detected by the GC for the three basic sample types, bare sol-gel TiO 2 , Au nanoparticle/TiO 2 , and bare Au nanoparticles. 0 5 10 15 20 25 =532 nm Product Yield ( mol/m 2 -cat.) CH 4 TiO 2 Au/TiO 2 Au Since the conduction band of TiO 2 lies above the reduction potential of CO 2 /CH 4 , 123 it is energetically favorable for electrons from the conduction band of TiO 2 to transfer to CO 2 to initiate the reduction of CO 2 with H 2 O producing CH 4 10 . Since the reduction potentials of CO 2 /HCHO and CO 2 /CH 3 OH lie above the conduction band of TiO 2 , methane is the only favorable product 88,103 . For the bare TiO 2 -catalytzed reduction, only a small amount of methane is detected by GC since the energy of the 532nm Figure 5.7: Photocatalytic product yields (after 15 h of visible irradiation) on three different catalytic surfaces. 81 wavelength light (2.41eV) is significantly lower than the bandgap of TiO 2 (3.2eV). Because of electronic transitions excited to and from defect states in the bandgap of TiO 2 , the yield is finite, yet small (0.93µmol/m 2 -cat.). On the other hand, the yield of Au nanoparticle/TiO 2 -catalyzed reduction is 22.4µmol/m 2 -cat., a 24-fold enhancement over the bare constituent materials. This enhancement in sub-bandgap absorption/photocatalysis is consistent with our previous work 95,145 , wherein the intense local fields produced by the plasmonic nanoparticles couple light very effectively from the far-field to the near-field, short-lived defect states at the TiO 2 surface. Bare Au nanoparticles without TiO 2 were also tested as a control experiment, and found to exhibit a negligible photocatalytic yield (Figure 5.7), indicating the importance of the TiO 2 surface in this catalytic process. Under visible illumination, electron-hole pairs are generated by the sub-band transitions in TiO 2 , instead of in Au. Plasmon-excited electrons in Au nanoparticles are not be able to transfer to the either TiO 2 or the reagents. Photodegradation of methyl orange: Anodic TiO 2 was fabricated using a method discussed earlier in this section. A gold film with a nominal thickness of 5nm was then evaporated on the surface of the ATO. The photocatalytic activity was tested using methyl orange (MO) photodegradation as the model reaction. The decay in absorbance of the MO aqueous solution at 460 nm was monitored by Varian Cary 50 UV-Vis spectrophotometer after 1h exposure to UV (365nm, mercury lamp with a bandpass filter centered near 365nm, 0.02W) or green laser (532nm, 0.2W) irradiation. Figure 5.8 shows the photocatalytic degradation of MO achieved under UV irradiation. Here, the absorption spectra taken before and after irradiating with UV light 82 (365nm) were used to quantify the relative MO concentration and, hence, the photocatalytic decomposition rate. After 1h of UV illumination, the absorbance of the MO aqueous solution and, hence, concentration, dropped by 23% for bare TiO 2 (Figure 5.8a), but only by 10% for the Au nanoparticle/TiO 2 sample (Figure 5.8b). Therefore, the addition of gold nanoparticles resulted in more than a 2-fold reduction in the photodecomposition rate due to the reduction in the active TiO 2 surface area, as the gold nanoparticle film covers a significant fraction of the TiO 2 surface, preventing it from coming into direct contact with the aqueous solution to be photocatalyzed. In this photochemical process, the photogenerated electrons and holes react with H 2 O and O 2 in the MO aqueous solution to produce highly active oxidizing species, which in turn result in the photodecomposition of MO into inorganic final products (SO 4 2- , NO 3 - , NH 4 + , CO 2 and H 2 O) 8,132,229,249,255 . Figure 5.9 shows the MO absorption spectra taken before and after irradiating anodic TiO 2 with and without Au nanoparticles with visible light (532nm laser). For bare Figure 5.8: UV-Vis spectra of MO aqueous solution before (black) and after (red) 1h UV illumination using (a) TiO 2 and (b) Au nanoparticle/TiO 2 photocatalysts. 300 400 500 600 0h 1h Absorbance Wavelength (nm) TiO 2 @ UV 300 400 500 600 Au/TiO 2 @ UV Absorbance Wavelength (nm) 0h 1h 83 TiO 2 (Figure 5.9a), the absorbance (or concentration) of the MO solution is only observed to decrease by 1.4% after 1h of illumination. However, with the addition of gold nanoparticles (Figure 5.9b), a 13% reduction in the MO absorbance can be seen due to the plasmon-enhanced photocatalytic decomposition mechanism, as described below. This corresponds to a more than 9-fold enhancement in the photocatalytic activity. Agin this improvement in photocatalytic activity is mainly due to the local near-field enhancement, which is not reflected in the bulk UV-vis absorption spectra. In addition, the bulk UV-vis spectra contain absorption processes that do not contribute to photocatalysis, such as recombination centers due to impurities. 5.4 Electromagnetic simulations In order to understand the underlying mechanism of catalytic enhancement, we performed numerical simulations of the electromagnetic response of these plasmonic Au nanoparticles using the finite-difference time-domain (FDTD) method, carried out on Figure 5.9: UV-Vis spectra of MO aqueous solution before (black) and after (red) 1h 532nm laser illumination using (a) TiO 2 and (b) Au/TiO 2 as photocatalysts. 300 400 500 600 0h 1h TiO 2 @ Vis Absorbance Wavelength (nm) 300 400 500 600 0h 1h Au/TiO 2 @ Vis Absorbance Wavelength (nm) 84 USC’s 0.15 petaflop supercomputing facility. Figure 5.10a shows a scanning electron microscope (SEM) image of a gold nanoparticle-island film deposited on top of anodic TiO 2 . The light grey regions correspond to gold islands and the dark regions correspond to spaces in between. The electromagnetic response of this film, shown in Figures 5.10 (b-d), is dominated by local “hot spots” (bright yellow) that can be seen between nearly touching Au nanoparticles. This well-known phenomenon is described in more detail in chapters 3 and 4 83,108,173 . Figure 5.10d shows a cross-sectional plot of the electric field distribution of one of these hot-spot regions in the z-dimension. In this hot-spot region, the electric field intensity at the TiO 2 surface reaches 1000 times that of the incident electric field intensity. This means that the photoabsorption (and hence electron-hole pair generation) rate is 1000 times higher than that of the incident electromagnetic radiation. This is particularly advantageous considering the small crystal grain sizes and high defect concentrations in anodic TiO 2 , which limit the minority carrier diffusion length to ~10nm 9,139,205 . As a result, only photons absorbed within 10nm of the TiO 2 surface will contribute to the photocatalytic splitting of water. Here, the plasmonic nanoparticles couple light very effectively from the far-field to the near-field at the TiO 2 surface. Consequently, most of the photogenerated charge created by the plasmon excitation will contribute to the surface catalysis (water splitting). 85 Figure 5.10: (a) SEM image of 5 nm Au island film on TiO 2 . (b-d) Electric field intensity at the interface of Au–TiO 2 at the resonance calculated using FDTD. (c) (d) (a) (b) 100 nm 100 nm 86 We have calculated the photocatalytic enhancement factor (EF) based on the results of the FDTD simulation, as shown in chapter 3. Since the photon absorption rate is proportional to the electric field squared (|E| 2 ), we integrate |E| 2 over the whole film and divide by the integral of the incident electromagnetic field squared (|E o | 2 ), as shown in the equation: ∫ ∫ | | ∫ ∫ | | (5.1) In the z-dimension, we integrate only from the TiO 2 surface (z = 0) to one minority carrier diffusion length below the surface (z = -10nm). The value for the EF when integrating over the whole simulation area (400nm 300nm) is 12 , which is consistent with the values observed experimentally. It should be noted, however, that this value is for a random distribution of gold islands that is not optimized geometrically. If, instead, we integrate only over the area of one hot spot, as shown in Figure 5.10c, an EF of 190 would result. We, therefore, believe that enhancement factors many times larger than this are possible if this plasmonic film can be optimized to make more efficient use of the total surface area. 5.5 Dye-sensitized solar cell Ever since the first dye-sensitized solar cell (DSSC) was demonstrated by O'Regan and Grätzel in 1991 171 , scientists have been working on improving its performance. A variety of methods, which include novel sensitizers 37,70,71,123,194,244 , electrolytes 18,35,104 , and semiconductors 43,91,258 , have been utilized to enhance the power 87 conversion efficiency of DSSCs during the past two decades. Several attempts have been made to employ noble metal nanoparticles to increase the efficiency of DSSCs 40,42,135,188,221 . In these previous studies, the enhanced efficiency of the DSSC was attributed to the Schottky barrier formed at the metal-semiconductor interface 40,42,135,221 . In other works, a decreased efficiency was observed with the addition of metal nanoparticles, and attributed to the decreased surface area of the underlying semiconductor in direct contact with the absorbing dye molecules 188 . To date, metal nanoparticle plasmon enhanced DSSCs are is still open study. In this work, three basic working electrode configurations with different geometric structures were fabricated and characterized: #1) a monolayer of Ru dye N719 deposited on top of Au nanoparticles embedded in a TiO 2 film, #2) a monolayer of Ru dye N719 on top of an evaporated 5 nm Au-island thin film deposited on the TiO 2 layer, #3) an evaporated 5 nm Au-island thin film deposited on top of the dye monolayer and the TiO 2 layer. Schematic illustrations of these three sample configurations are given in Figure 5.11. The conventional DSSC with a monolayer of Ru dye N719 on top of TiO 2 was also prepared as a control. Anatase titania films were prepared in our lab by the sol- gel process described in section 5.3. For Sample #1, Au nanoparticles were embedded in the TiO 2 films using a modified recipe described by Li et al. 141 In this recipe, P 123, titanium ethoxide, and HAuCl 4 were mixed in the ethanol at room temperature and spin- coated on ITO substrates. After aging at 40 o C for 24h and 100 o C for 12h, the gels were calcined at 350 o C for 4h in air using a heating rate of 0.5 o C/min. In order to prepare Sample #2, a gold film with a nominal thickness of 5nm was evaporated on the top of the 88 sol-gel TiO 2 film. A second annealing process was carried out by placing the sample into a furnace and calcining in air at 400 o C for 1h, which makes the Au islands more Figure 5.11: Schematic diagrams of three different Au nanoparticle/dye/TiO 2 configurations. TiO 2 ; Au nanoparticles; Dye N719 #1 #2 #3 89 spherical, as described in chapter 3. The conventional DSSC and Samples #1 and #2 were then sensitized by immersing into a 0.3mM solution of N719 dye in ethanol. All samples were sensitized for 48 h. In order to make Sample #3, a gold film with a nominal thickness of 5nm was evaporated on top of the dye surface of the conventional DSSC. For the counter electrode, a 10nm film of Pt was deposited by electron-beam evaporation onto ITO. The working electrode and the counter electrode were separated by a polypropylene spacer approximately 20µm thick and bonded using binder clips. The internal space of the cells was filled with electrolyte (Iodolyte AN-50 from Solaronix) and sealed with hot-melt sealing foil (Meltonix 1170-25 from Solaronix). The open- circuit photovoltage, short-circuit photocurrent, and I-V characteristics of these solar cells were measured using a digital multimeter (Keithley 2400). A fixed wavelength green laser (60 mW, 532 nm) was employed to illuminate the solar cells. In addition, the spectral response of the photocurrent was measured using a Fianium supercontinuum white light laser in conjunction with a Princeton Instruments double grating monochromator. Figure 5.12 shows the short-circuit photocurrent during a 28-second laser exposure at a wavelength of 532nm. Under illumination, the highest photocurrent was produced by working electrode #1 (dye monolayer deposited on top of embedded Au nanoparticles/TiO 2 ). Comparing this with the photocurrent produced by the conventional DSSC, the photocurrent increased by a factor of 2. This enhancement is due to the intense local electromagnetic fields produced by the plasmonic nanoparticles, which couple light very effectively from the far-field to the near-field of the dye molecule monolayer. Under 90 these intense local fields, the exciton generation rate in the dye molecule monolayer increases significantly, thereby improving the photocurrent. The photocurrents produced by #2 and #3, however, were lower than that of the conventional DSSC, even with the addition of Au nanoparticles. The decrease in photocurrent, here, is attributed to the following facts. For #2, the 5nm Au thin film is deposited between the TiO 2 film and the dye monolayer. The Au thin film, therefore, decreases the active surface area of TiO 2 in 20 40 60 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Photocurrent (mA) Time (s) #1-Embedded Au NPs Conventional DSSC #3- Au NPs on top #2-Au NPs in between direct contact with the dye molecules. This decrease also demonstrates that charge is not transferring from the dye to the Au or from the Au to the TiO 2 . That is, in order for this solar cell to function properly, the charge must be created in the dye layer and then transferred to the TiO 2 and surrounding solution. Even though the plasmon enhancement Figure 5.12: Short-circuit photocurrents of DSSCs with three different Au nanoparticle/dye/TiO 2 configurations. 91 partially compensates for the decrease in photocurrent caused by the reduced contact area between the dye and TiO 2 , the photocurrent produced by #2 is still lower than that of the conventional DSSC. For #3, the 5nm Au thin film lies on the top of the dye monolayer. For this geometric structure, less dye can absorb the light since the Au thin film covers about 40% surface area of the dye. Therefore, while Samples #1, #2, and #3 all consist of Au nanoparticles/dye/TiO 2 working electrodes, the effectiveness of the plasmon- enhancement depends on the specific geometric configuration of these three constituent materials. Anatase TiO 2 prepared by the acid catalyzing sol-gel process has a hexagonal close packed mesoporous structure 4 . In the plasmon enhanced configuration (#1), the Au nanoparticles sit in the pores of the TiO 2 . These embedded Au nanoparticles not only create intense local fields, but also maintain most of the contact area between the dye and the TiO 2 . The photocurrents of bare TiO 2 and Au nanoparticles embedded in TiO 2 are also measured as a control. No photocurrents are observed for these two samples, as expected, since no dye molecule is deposited. This result further confirms that the plasmon excited electrons are not able to transfer directly from the Au nanoparticles to the TiO 2 . In contrast to photocatalytic reactions described in section 5.3, doping and defects in the TiO 2 do not affect the DSSC performance, since the electron-hole pairs are generated in the dye rather than the TiO 2 95,145 . Figure 5.13 shows the photocurrent spectra of DSSCs with the four different working electrodes. Sample #1 shows a large enhancement in photocurrent with respect to the conventional DSSC over most of the visible wavelength range. The inset in Figure 92 5.13 shows the photocurrent enhancement ratio of Sample #1 with respect to the conventional DSSC. We observe enhancement from 460nm all the way out to the near infrared (730nm). This enhancement ratio peaks at 6.5 at wavelengths of 613nm. The peak photocurrents for #1 (red dashed curve), #2 (blue dotted curve), and #3 (dark cyan dash dotted curve) are redshifted with respect to the conventional DSSC (dark solid curve). This redshift is caused by the plasmonic enhancement, which has a maximum around 566nm. The overall photocurrents of samples #2 and #3 are lower than the Figure 5.13: Photocurrent spectra of DSSCs with different working electrodes. 450 500 550 600 650 700 750 800 0 5 10 15 20 25 Conventional #1-embedded Au NPs #2-Au NPs in between #3-Au NPs on top Photocurrent ( A) Wavelength (nm) 450 500 550 600 650 700 750 800 0 1 2 3 4 5 6 7 PC Enhancement Ratio Wavelength (nm) 93 conventional DSSC for the same reasons given above, namely the decreased effective area of the dye molecule monolayer. Figure 5.14 shows the calculated enhancement factor due to the embedded Au nanoparticles. These results were calculated using full three-dimensional finite difference time domain (FDTD) simulations of the different Au nanoparticle geometries studied experimentally. The enhancement factors were calculated as ratios of the absorption 450 500 550 600 650 700 750 800 0 1 2 3 4 5 6 7 8 9 10 Experimental FDTD PC Enhancement Ratio Wavelength (nm) cross-sections in the sol-gel TiO 2 with and without Au nanoparticles. The dielectric function used for Au was derived from the data by Palik and Ghosh 177 . The dielectric constant of the sol-gel TiO 2 was calculated using the Maxwell Garnett effective medium Figure 5.14: FDTD calculated and experimentally measured enhancement factors plotted as a function of wavelength for embedded nanoparticles. 94 theory due to its mesoporous geometry 170 . A random ensemble of nanoparticles of diameters between 10-20nm embedded inside the block of TiO 2 was used for the simulations. The enhancement factor for the embedded Au nanoparticles in Figure 5.14 shows a well defined peak with an enhancement of 9, which is of the order of the experimental value of 6.5. 5.6 Conclusions We demonstrated enhancement in the photocatalytic reaction rates in visible region of the electromagnetic spectrum by exploiting the surface plasmon resonance of gold nanoparticles. The intense local fields produced by the surface plasmons couple light efficiently to the surface of the TiO 2 . This enhancement mechanism is particularly effective because of anodic TiO 2 ’s short minority carrier diffusion length, which would otherwise limit its photocatalytic activity. For water splitting, enhancements in the photocatalytic activity of 5 and 66 are observed at wavelengths of 532nm and 633nm, respectively. We also observed 24-fold plasmonic enhancement of the photocatalytic reduction of CO 2 with H 2 O under visible illumination when Au nanoparticles are deposited on top of TiO 2 . In the case of methyl orange, a 9-fold improvement in the photocatalytic decomposition rate was observed. Electromagnetic simulations of this process suggest that enhancement factors many times larger than this are possible if this mechanism can be optimized. The experimental data and fundamental understanding described here provide a path toward resolving the photon absorption/electron diffusion length mismatch that has made 95 photovoltaics and direct photocatalysts far too expensive to find broad applicability in our energy infrastructure. For photocatalysis, this area is especially exciting because it presents a possible route to direct solar to fuels production. A 2.4-fold plasmonic enhancement in the overall conversion efficiency of DSSCs under visible illumination is observed when Au nanoparticles are embedded in TiO 2 . Embedded Au nanoparticles show an enhancement in the photocurrent over the wavelength range from 460nm to 730nm. Conversely, depositing Au nanoparticles on top of the TiO 2 results in a decreased efficiency by reducing the effective area of TiO 2 in contact with dye or by reducing the dye surface area exposed to the light. Finally, electromagnetic simulations agree well with the experimental observations, further corroborating the mechanism of plasmonic enhancement in these DSSCs. This research was supported in part by AFOSR Award No. FA9550-08-1-0019, ARO Award No. W911NF-09-1-0240, and NSF Award No. CBET-0846725. W.H. was supported as part of the Center for Energy Nanoscience, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001013. 96 CHAPTER 6: PLASMONIC NANOPARTICLE ARRAYS WITH NANOMETER SEPARATION FOR HIGH-PERFORMANCE SERS SUBSTRATES This chapter is similar to Theiss et al. 230 , published in Nano Letters. 6.1 Abstract This chapter demonstrates a method for fabricating arrays of plasmonic nanoparticles with separations on the order of 1nm using an angle evaporation technique. High resolution transmission electron microscopy (HRTEM) is used to resolve the small separations achieved between nanoparticles fabricated on thin SiN membranes. These nearly touching metal nanoparticles produce extremely high electric field intensities when irradiated with laser light, which result in surface enhanced Raman spectroscopy (SERS) signals. These enhancements are quantified by depositing a para- aminothiophenol (p-ATP) dye molecule on the nanoparticle arrays and spatially mapping their Raman intensities using confocal micro-Raman spectroscopy. Our results show significant enhancement when the incident laser is polarized parallel to the axis of the nanoparticle pairs, whereas no enhancement is observed for the perpendicular polarization. These results demonstrate proof-of-principle of this fabrication technique. Finite difference time domain (FDTD) simulations based on HRTEM images predict an 97 electric field intensity enhancement of 82,400 at the center of the nanoparticle pair, and an electromagnetic SERS enhancement factor of 10 9 -10 10 . 6.2 Introduction Raman spectroscopy is an invaluable tool for many applications, as it provides a unique signature for chemical identification and differentiation. Its utility is limited, however, due to the small Raman scattering cross-sections characteristic of most molecules, typically 10 6 times smaller than Rayleigh scattering. Surface-enhanced Raman spectroscopy (SERS) can be used to greatly improve the low Raman intensities. Since its discovery in 1977, hundreds of papers on the subject of SERS have been published 239 . SERS enhancement factors up to fourteen orders of magnitude have been reported in the literature 126,169 . Since most of the previous work in this field has involved roughened metal surfaces and nanoparticles in solution, it was impossible to image the exact geometry of the nanoparticle complex 32 . Furthermore, the number of molecules contributing to a SERS enhanced signal is generally unknown and can only be ascertained through statistical analysis 126,169 and/or a discriminating selection of analyte molecules 52,53 . Therefore, several unexplored experimental factors remain, including the separation between the nanoparticles, the number of nanoparticles within the focal volume, the number of molecules on each nanoparticle, and the extent to which nanoparticles couple plasmonically to each other. Methods for providing reliable SERS substrates, whose parameters can be controlled precisely on the 1nm-scale are much 98 needed, and will help bring forth a more complete understanding of SERS and enable new applications of Raman spectroscopy achievable with handheld spectrometers. The calculations of Schatz et al., using an interacting dipole model, showed an electric field intensity enhancement factor over 10 6 for the plasmonic resonance between two nearly touching nanoparticles 261 . The electric field enhancement increases sharply for nanoparticle separations below 2nm. Jiang et al. calculated that two 60nm diameter spherical Ag nanoparticles separated by 9nm, 3nm, and 1nm produced Raman enhancement factors of 1.5 x 10 4 , 1.7 x 10 6 , and 5.5 x 10 9 , respectively 109 . Numerical finite difference time domain (FDTD) simulations have also shown that coupled nanoparticles produce large electric field enhancements, with SERS enhancement factors up to 10 10 for a nanoparticle separation of 1nm 173 . A concurrent and independent chemical mechanism may also contribute to the SERS enhancement due to chemisorption of the analyte molecule to the metal surface. This interaction may produce an intermediate charge-transfer complex with increased Raman scattering cross-section or quantum-mechanically alter the electronic states of the analyte molecule 33,126,172 . However, it is generally accepted that this contribution is limited to a few orders of magnitude 56,115 . Based on this work, it is now well established that the SERS enhancement, enabling single molecule detection, is dominated by the high electric fields that occur in the small gaps between metal nanoparticles 89,109,138,183,261 . Several research groups have performed experimental studies of metal nanoparticle dimers fabricated by standard electron beam lithography. However, the nanoparticle separations in these works were limited by the lithographic techniques to 10-20nm 12,219 , which is one order of 99 magnitude larger than that predicted theoretically to produce significant electric field enhancement. Shadow evaporation is a well-known method for making sub-10-nm gaps between metal structures that has been extensively used for making electrodes, although the feature sizes are generally larger than those of plasmonic nanodevices 17,190,222 . Angle evaporation was used in conjunction with nanosphere lithography to produce triangular nanoparticles with angle-controlled gaps measured in the range of 4-25nm 87 . This method limits the shape and size of the particles, which are determined by the hexagonally- packed nanosphere mask. The use of a self-assembled monolayer and electron beam lithography can also produce a high yield of 2nm gaps for nanodevices but contaminates the metal surfaces with a thiolated molecular layer 166 . Other successful methods that can produce molecular-sized gaps include break junctions 195 , electroplating 162 , and electromigration 181,182 . The yield of electromigrated nanogaps has improved significantly, making it a good candidate for the study of SERS phenomena 247 , however, the density is ultimately limited by the electrode geometry. Hence, new methods are desired for controllably creating nanogaps in the range 0.5 to 2nm for providing reliable SERS substrates. 6.3 Experimental details In this work, we fabricated arrays of metal nanoparticles with separations on the order of 1nm using electron beam lithography combined with an angle evaporation technique. This work builds on previous research using controlled angled deposition to fabricate nano-scale tunnel junctions used to form single electron transistors 208 and 100 spintronic devices 238 . Figure 6.1 shows a schematic diagram of the angle evaporation technique. In this scheme, a layer of ZEP-520 electron beam resist was spun on top of a layer of methyl methacrylate (MMA) resist. Since the MMA is more chemically sensitive to electron radiation, electron exposure results in a large undercut by over-exposing the Figure 6.1: Schematic of the angle evaporation method to produce arrays of nanometer gap dimers. 101 more sensitive MMA layer. This leaves a free-standing ZEP mask, as shown schematically in Figure 6.1. In patterning the nanogaps, a thin layer of metal was first deposited at normal incidence. The sample was then tilted by a small angle (5-10°), and a second layer of metal was deposited. The size of the nanogap is determined by the angle of the second evaporation, θ, and the thickness of the first evaporation, t 1 , and is given by t 1 tanθ. Therefore, an angle of θ = 10° and a thickness of t 1 = 10nm would yield a gap of 1.8nm. By decreasing the angle of evaporation to θ = 5°, the gap size is reduced to 0.87nm. We patterned these nanogap structures on thin SiN membranes, which were then imaged using a JEOL JEM-2100F high resolution transmission electron microscopy (HRTEM). HRTEM enables the nanoparticle geometry and gap size to be determined with less than 0.2nm resolution. In this work, a JEOL 9300FSZ electron beam lithography system was used to write 25 sets of nanoparticle arrays, as shown in Figure 6.2a. Each of these sets covers an approximately 6µm x 6µm area and contains a slightly different nanoparticle geometry (i.e., size, shape, separation). A TEM image of one of these arrays is shown in Figure 6.2b. Figure 6.2c shows a high magnification TEM image of one nanoparticle pair with a gap size of 2nm. The second evaporation results in a second nanoparticle that is smaller than that of the first evaporation. This occurs for two reasons. First, the deposition of material tends to close the holes in the lithographic mask. And second, the holes’ effective cross-sections are decreased at oblique incidence. After fabrication, these silver nanoparticle samples were coated with a non- Raman-resonant dye molecule, para-aminothiophenol (p-ATP). The thiol group of the p- 102 ATP molecule has a high affinity for gold and silver surfaces and forms monolayers in dilute solutions followed by rinsing to remove the unbonded molecules. Samples were incubated at room temperature in a 1mM solution of p-ATP in ethanol for 24 hours, rinsed repeatedly with ethanol and DI water, and then dried with a gentle stream of nitrogen gas. Raman spectra were measured in a Renishaw inVia micro-Raman spectrometer. A 632.8nm HeNe laser spot was accurately positioned using a high precision Prior ProScan II microscope stage, which enabled spatial mapping of the Raman spectra. The laser was focused to a small Gaussian spot (~0.5µm diameter) Figure 6.2: (a) Low magnification TEM image of Ag nanoparticle arrays deposited using the angle evaporation technique on a SiN membrane. (b) and (c) high magnification TEM images of Ag nanoparticle pairs with nm-sized gaps. (b) (a) 250 nm (c) 25 µm 103 through a 100X objective lens with a numerical aperture of 0.9. A cylindrical lens was inserted in the beam path before the objective lens to spread the laser into an elliptical spot, enabling faster mapping of the Raman intensity across the sample. Figure 6.3: (a) TEM image of a 5x5 matrix of cells containing various nanoparticle dimer geometries. Spatial mapping of the Raman intensity with a 632.8nm laser exciting the 1576 cm-1 p-ATP Raman peak with (b) parallel and (c) perpendicular polarization. (a) (b) (c) 104 We measured the SERS response of these p-ATP/Ag nanoparticle arrays by spatially mapping their Raman intensities. Being proportional to the fourth power of the electric field |E| 4 for small phonon energies, Raman intensity serves as a good measure of the electric field enhancement and relative plasmonic strength. Figures 6.3b and 6.3c show the Raman intensity spatial maps of the 5x5 matrix shown in Figure 6.3a coated with p-ATP, taken with the incident laser polarization oriented perpendicular (Figure 6.3b) and parallel (Figure 6.3c) to the axis of the nanoparticle pairs 19,109 . Several nanoparticle cells show a significant increase in the Raman intensity when the polarization is matched to the angle-evaporated nanometer size gaps, demonstrating the electric field enhancement of the plasmonically coupled nanoparticles. The lack of uniformity in the SERS intensity over these cells is likely due to the inherent sensitivity of the electromagnetic response to the size, shape and separation of these nearly touching nanostructures. Even a small variation in the particle or gap size can cause a large shift in the resonant frequency 262 , pushing the particle off-resonance for a fixed excitation frequency. By comparing the SERS-enhanced cells of Figure 6.3c with the TEM image of Figure 6.3a, it is clear that the SERS enhancement does not simply correspond to the metal filling factor of these cells. This further demonstrates the plasmonic nature of this SERS enhancement mechanism, which relies intimately on the plasmonic coupling between adjacent nanoparticles. 105 6.4 Electromagnetic simulations Based on the TEM image shown in Figure 6.4a, we simulated the electromagnetic response of this Ag nanoparticle pair by defining the spatial extent of the metal nanoparticles from this high resolution TEM image. The x and y dimensions of the particles were determined from the TEM image, while evaporation thicknesses of 15nm and 30nm were used to specify the height of the left and right particle, respectively. FDTD simulations 226 were performed in USC’s 0.15 petaflop supercomputing facility. Here, full three-dimensional simulations were performed using a grid of 14 million points Figure 6.4: (a) TEM image of a silver nanoparticle dimer with 2nm separation. FDTD simulation results of the electric field intensity around the dimer plotted on a logarithmic scale with incident light polarized (b) parallel and (c) perpendicular to the axis of the nanoparticle pair. (a) 10 nm (b) (c) 106 to discretize the spatial extent of the electric and magnetic fields with up to 2.5Å resolution in the gap region, and carrying out 200,000 time steps. The dielectric function used is based on a fit of the experimental data obtained by Johnson and Christy to a Lorentz-Drude formula 111 . Figures 6.4b and 6.4c show the electric field intensity distributions of this silver nanoparticle dimer integrated over the z-dimension, irradiated at normal incidence at the plasmon resonance frequency. For incident light polarized along the axis of the nanoparticle dimer (Figure 6.4c), the maximum electric field intensity lies in the gap between the nanoparticles, with a value 82,400 times that of the incident field intensity at the calculated plasmon resonance occurring at 552nm. For this nanoparticle pair, the SERS enhancement factor at the most intense point is given by the square of this electric field intensity enhancement factor, giving a value of 6.9 x 10 9 . With polarization perpendicular to the nanoparticle axis, the SERS enhancement factor at the most intense point is calculated to be 6.4 x 10 4 at this same frequency. Integrating over the area shown in Figure 6.4c, we find a total SERS EF of 1.4 x 10 11 for parallel polarization. We can also estimate the areal SERS enhancement factor over the area inside the focal volume. Based on the electric field distribution shown in Figure 6.4b, we calculate the expected areal SERS enhancement as defined in chapter 3 by the following equation: ∫ | | ∫ | | (6.1) Following this procedure, we estimate an areal electromagnetic SERS enhancement factor from our FDTD simulation of 5.8 x 10 3 integrated over a 0.4µm x 0.4µm area. 107 The reason for the low areal EM enhancement factors of our samples is sub-optimal nanoparticle coverage densities, as seen in Figure 6.2. Based on the self-assembled monolayer coverage density and the dimensions of our nanoparticles, the experimental SERS EF for this particular nanoparticle geometry is calculated to be 1.1 x 10 6 . This experimental EF is over three orders of magnitude smaller than the 6.9x10 9 value from our simulations, which may result from exciting the nanoparticle pair slightly off- resonance and overestimating the number of molecules contributing to the measured Raman intensity. 6.5 Conclusions In conclusion, we have demonstrated a method for fabricating nanoparticle pairs with nanometer separations. A significant increase in the Raman intensity is observed when the polarization is matched to the angle-evaporated nanometer size gaps, demonstrating the electric field enhancement of the plasmonically coupled nanoparticles. Numerical simulations of the electromagnetic response of these nanoparticles show significant enhancements in the calculated electric field and SERS signal, which also depend strongly on the polarization of the incident light. Based on the 10 9 -10 10 SERS enhancement factor, these substrates could be used in devices approaching chemical detection at the single molecule level. This research was supported in part by ONR award No. N00014-08-1-0132, AFOSR award No. FA9550-08-1-0019, ARO award No. W911NF-09-1-0240, NSF award No. CBET-0854118, and NASA SURP No. 1346414. Electron beam lithography 108 was performed by Richard E. Muller. This research was partially carried out at the Jet Propulsion Laboratory, California Institute of Technology. 109 CHAPTER 7: FUTURE DIRECTIONS 7.1 Optimization of nanoparticle dimer arrays fabricated using the angle evaporation method In chapter 6, we describe a method to fabricate arrays of metal nanoparticles with separations on the order of 1 nm using electron beam lithography combined with an angle evaporation technique. This work deals with the optimization of different parameters involved in the fabrication of the nanoparticle dimer arrays. Taking a look at the schematics of this fabrication technique shown in Figure 7.1, it can be seen that the gap size is g = t.tanθ and the diameter of the second nanoparticle is D 2 = d.tanθ – g = (d – t).tanθ. So, by changing the values of d, t and θ, we can change the shape and the size of the dimer. In addition to this, we can also vary the x and y pitches of the dimer array and the diameter of the first nanoparticle, D 1 . Figure 7.1: Schematic of the angle evaporation scheme θ d t 110 So, we can use our optimization scheme to get an optimum set of these 6 parameters that would give us the maximum electric field intensity. Preliminary results: Our first step towards this optimization is to use a TEM image of a dimer created by this technique shown in Figure 6.4a and only vary 2 of the 6 parameters which will eventually be used for optimization, x and y pitch. We use a simple gradient descent based algorithm, similar to the one used in chapter 2. Periodic boundary conditions are used to simulate an array of these dimers. The electric fields intensity is measured at the center of the dimer averaged over a small volume (5nm x 2nm x 5nm) and normalized with the simulation area. The polarization of the source is along the axis of the dimer. The result of this optimization is shown in Figure 7.2. After Figure 7.2: Optimization of an array of dimers for maximum absorption. 0 10 20 30 40 50 60 70 80 0 50 100 150 200 250 300 350 400 450 500 550 Intensity X pitch Y pitch Iterations E-field (a.u.) 50 100 150 200 250 Pitch (nm) 111 85 iterations the optimization seems to be converging to an array with the X pitch = 40 nm and Y pitch = 30 nm. 0 25 50 75 100 0 500 1000 1500 E-field (a.u.) Iterations The next step was to do the same for an idealized angle evaporation dimer that is not based on a TEM image. The limitations of the fabrication technique were taken into account as the constraints of the optimization. The first of these is the minimum pitch that can be achieved before the PMMA/MMA layer loses stability because of the undercut. The other problem with the fabrication, is that the second nanoparticle is usually smaller than the ideal size. This is because during the first evaporation, the window in the PMMA/MMA layer is partially clogged by the evaporated Au. Considering these two limitations could give us results that are different than the unconstrained optimization. Figure 7.3: Constrained optimization of an array of dimers for maximum absorption with 6 parameters. 112 One of the preliminary results of such a constrained optimization is shown in Figure 7.3. Table 7.1 shows the optimum set of parameters obtained at the end of the optimization. 1 st evap thickness = 39.29nm 2 nd evap thickness = 11.3nm Evap angle = 5.32˚ MMA thickness = 123.75nm ZEP thickness = 70.9nm Diameter of NP = 46.65nm Proposed work: Ultimately, we would like to fabricate the optimized geometry and measure its UV-Vis absorption spectrum experimentally. The experimental spectra will then be compared with the simulated spectra. We will also optimize the arrays to give us the maximum intensity at the wavelength of the HeNe laser (632.8 nm). The experimental realizations of these arrays will then be coated with a non-Raman-resonant dye molecule, p-aminothiophenol (p-ATP). We will the characterize the samples using Raman spectroscopy to get maps similar to the ones shown in Figures 13(b, c). We hope to get a more uniform map of Raman intensity using the optimized array. 7.2 Nanoparticle cluster optimization for four-wave mixing The high intensities produced by our optimized plasmonic nanostructures can be used to enhance nonlinear effects. Wave mixing typically requires the use of high intensity lasers in nonlinear media. However, by utilizing the high intensities generated Table 7.1: Solution of the parameters of the constrained optimization of an array of dimers for maximum absorption. 113 by these plasmonic nanostructures, we can significantly enhance nonlinear wave mixing. This general idea of this project is to shine two frequencies of light (ω 1 and ω 2 ) on an array of plasmonic nanoparticles. The strong electric field intensities at the center of the cluster will mix these signals efficiently. For gold, we expect the third order nonlinear coefficient (ω 3 ) to result in four-wave mixing (e.g., ω 3 =2 ω 1 - ω 2 ). Danckwerts and Novotny observed a four order of magnitude enhancement in the intensity of four-wave mixing between two nearly touching gold nanoparticles separated by < 1nm 48 . We believe we can improve this significantly by optimizing the geometries of these plasmonic nanoparticle clusters. The short length scale of the nonlinear optical interaction will mitigate any problems associated with phase mismatch. Since the third order nonlinear coefficient of gold and silver are relatively small 193 , we will also simulate nanoparticle clusters embedded in and on top of materials with a high nonlinear coefficients 224 , for example polymers 47 and semiconductors such as GaInAs 237 , GaAs 203 , and amorphous silicon 196 . An optimization algorithm, similar to the one used in chapter 2, will be used to determine the most effective nanoparticle geometry for maximizing the enhancement of the four-wave mixed intensity. The four-wave mixing term arises due to the polarizability term 179 : For four-wave mixing discussed here, we will consider the single degenerate case where, Thus, we get a signal at the frequency The non-linear polarization for this particular case will thus be, and 114 | | | | Hence the output intensity | | is proportional to | | and . So, the total enhancement here is a combination of 3 enhancement factors: a) enhancement of I(ω 1 ), b) enhancement of I(ω 2 ) and c) enhancement of I(ω 3 ). Therefore, the total enhancement is given as, where, for i = 1,2,3; I is the intensity with the nanoparticles and I 0 is the intensity without them. For our optimization problem, we have chosen the frequencies of He-Ne and Ti- Sapph lasers, giving us ω 1 = 473.6 THz, ω 2 = 562.67 THz and ω 3 = 383.69 THz. We simulate the electric field response of metal nanoparticle clusters with incident plane wave irradiation. A two dimensional spatial grid of 300,000 points is used to represent the computational cell. The grid spacing used for our simulations is 1 nm. To justify the grid spacing used for the simulations, the minimum allowed separation between any two nanoparticles is 2 nm. The diameter of the nanoparticles used in simulations is 40 nm. For boundary conditions, we have implemented perfectly matched layers (PML) with the number of layers being 16. The dielectric function used is based on a fit of the optical constants given by Palik and Ghosh 176 . The temporal grid consists of 10,000 time steps with a spacing of 0.002 fsec. 115 output results initialize parameters calculate enhancements for each particle I >local I max ? I max =I; new local optimum yes no Figure 7.4: Flowchart illustrating the algorithm used for optimization. I >global I max ? I max =I; new global optimum calculate the new velocities and positions last particle? next iteration yes no yes no next particle 116 We are using particle swarm optimization (PSO) algorithm for this work 119 . The main advantage of this algorithm is that it makes few or no assumptions and can search very large spaces of candidate solutions 60 . It is particularly useful in problems that are irregular and noisy. PSO algorithm starts out with a population (a swarm) of candidate solutions (particles). These particles are moved around in the search-space according to a few simple formulas. The movements of the particles are guided by a) their current position, b) their own best known position in the search-space and c) the entire swarm's best known position. When improved positions are being discovered these will then come to guide the movements of the swarm. The process is repeated and by doing so, ultimately, a satisfactory solution is reached. A flowchart describing the algorithm is shown in Figure 7.4. The enhancements values, α i , are calculated with 2 simulations each, at the respective wavelengths, one with the nanoparticles and one without nanoparticles. The positions and velocities for particle p and iteration i are updated using the following formulas: ( ) where w, C 1 and C 2 are constants the values of which are determined mostly by trial and error. pmax and gmax are the local optimum position set of the particle p and the global optimum position set respectively. 117 Preliminary results: Optimization of four-wave mixing of a cluster of 6 Au nanoparticles is shown in Figure 7.5a. In this example, we have fixed 4 nanoparticles in Figure 7.5: Optimization of four-wave mixing using 6 nanoparticles. (b) (a) -150 -100 -50 0 50 100 150 -200 -160 -120 -80 -40 0 40 80 120 160 200 Y (nm) X (nm) 0 5 10 15 20 1.5x10 9 2.0x10 9 2.5x10 9 3.0x10 9 4WM enhancement Iterations 118 the middle and we are varying the 5 th and the 6 th nanoparticle positions. Starting with three distinct candidate solutions, indicated by green, blue and cyan colors, we obtain a single solution indicated by the red colored nanoparticles. Thus, we are getting a very good convergence. The change in four-wave mixing enhancement with each iteration of the optimization is shown in Figure 7.5b. Proposed work: We intend to carry out this optimization for a cluster with up to 20 nanoparticles. While, our initial simulations deal only with 2D nanoparticles, we will extend this to 3D nanoparticles as well, for which, we will be utilizing USC’s 0.15 petaflop supercomputing facility. Finally, we would like to do this optimization to get the maximum four-wave mixing electric field intensity from experimentally realizable structures, such as the nanoparticle dimer arrays fabricated using the angle evaporation technique discussed in chapter 6. 633 nm 532 nm PhotoDiode Dichroic mirror 1 Nanoparticle array Dichroic mirror 2 Focusing objective Figure 7.6: Setup of the proposed four-wave mixing experiment 119 Experimentally, we will then measure the four-wave mixing intensity using the setup shown in Figure 7.6 179 . Here, the first dichroic mirror has a cut-off wavelength such that, it will transmit the 633nm laser wavelength and reflect the 532nm laser wavelength. These 2 wavelengths are then focused with the objective on the nanoparticle array. The scattered light from the nanoparticles then passes through the objective on to the second dichroic mirror, which has a cut-off wavelength such that it reflects 532nm and 633nm wavelengths but transmits 781nm (the four-wave mixed wavelength). The intensity of the transmitted signal will then be measured with the help of a photodiode. 7.3 Optimization of nanoparticle clusters with nanoparticle size as a parameter In chapter 2, I explained an iterative optimization scheme, with which we performed optimization of clusters with up to 20 nanoparticles. In the optimization, we considered the positions of the nanoparticle as the only optimization parameters. We intend to expand our parameter space by including the diameter of a nanoparticle as one of the optimization parameters. Varying size makes this problem a little bit tricky, since we are not allowing any overlap of nanoparticles. This means that, if we make a nanoparticle bigger and as a result, if it overlaps another nanoparticle, then we also have to move it to avoid the overlap. It should be noted that this movement to avoid overlap with a nanoparticle, can make it overlap with another nanoparticle in turn. So, there has to be a routine which moves all the nanoparticles until all the overlap conflicts are solved. 120 Figure 7.7: Optimization of a 7 nanoparticle cluster with diameter as an optimization parameter. 0 5 10 15 20 25 1500 2000 2500 3000 3500 4000 4500 E-Field Intensity (a.u.) Iterations 0 5 10 15 20 25 35 40 45 50 55 60 65 70 NP 3 and 4 NP 5 NP 6 and 7 Radius (nm) Iterations 40 nm10 nm (a) (a) (b) (b) (c) 121 Preliminary results: Figure 7.7 shows one of our preliminary results of such an optimization for the case of 7 nanoparticles. Figure 7.7a shows the changes in the positions of the diameters of nanoparticles during the optimization process. 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If the computational domain is extended sufficiently far beyond all sources and scatterers all waves will be outgoing at the boundary. A suitably formulated boundary condition will absorb majority of the incident wave energy, allowing only a small amount to reflect back into the computational domain. This type of boundary condition is known as the Absorbing Boundary Condition (ABC). Realization of an ABC to terminate the outer boundary of the space lattice in an absorbing material medium is analogous to the physical treatment of the walls of an anechoic chamber. Ideally, the absorbing medium is a only a few cells thick, does not reflect all impinging waves over their fill frequency spectrum, highly absorbing and effective in near field of a source or a scatterer. A new fervor has been created in this area by J.P Berenger's introduction of a highly effective absorbing-material ABC designated as the Perfectiy Matched Layer, or PML (ref). The motivation of Berenger's PML is that plane waves of arbitrary incidence, 148 polarization, and frequency are matched at the boundary and subsequently absorbed in the PML layer. The PML formulation introduced by Berenger (split-field PML) is a hypothetical medium based on a mathematical model. In the split-field PML if such a medium were to exist it must be an anisotropic medium. For a single interface, the anisotropic medium is uniaxial and is composed of both electric and magnetic permittivity tensors. Maxwell’s equation in the PML region are given as: [ ̅ ] ⃗ (A.1) [ ̅ ] ⃗ ⃗ (A.2) ⃗ ([ ̅] ⃗ ⃗ ) [ ] ⃗ ⃗ (A.3) ⃗ ⃗ ([ ̅] ⃗ ) [ ] ⃗ (A.4) Effective permittivity [ ̅ ] and permeability [ ̅ ] are given as: [ ̅ ] ( ) (A.5) [ ̅ ] ( ) (A.5) 149 This type of a medium is called a diagonal anisotropic medium. Now let a plane wave be incident on the boundary between free space and this PML layer as shown in Figure A.1. For convenience let us choose the tensors as follows: [ ̅] [ ̅] [ ] ( ) (A.6) A plane wave incident on the interface can be decomposed into TE and TM modes. Let us consider TE modes first. For the direction of propagation as shown in the figure, the incident, reflected and transmitted fields can be derived as follows: ⃗ ̂ ( ) (A.7) ⃗ ̂ ( ) (A.8) ⃗ ̂ ( (√ √ )) (A.9) Figure A.1: Interface of air and PML 150 where and are the reflection and transmission coefficients for TE mode respectively. For this interface, the following properties hold true. The electric field continuity implies (A.10) Phase continuity implies √ (A.11) and continuity of the x-component of the magnetic field leads to √ (A.12) Solving for using (1.30) to (1.32) √ √ (A.13) Similarly, we can find √ √ (A.14) The goal of the formulation is to obtain a reflectionless interface. Therefore using equation A.11, we can choose bc = 1. A.13 and A.14 together imply that a = b for zero reflection, because of our choice of b and c, leading to a = b = 1/c = a - jb . The resulting material is lossy and uniaxial. The wavelength in the material is determined by a and the attenuation is determined by b. Total Field Scattered Field (TFSF) source: This section describes the Total Field Scattered Field (TFSF) plane wave source. The TFSF source is often used to study scattering from small particles. It is also useful when plane wave illumination of non- periodic devices is required. The TFSF source can be used for 151 Particles in a homogeneous medium (which may be lossy or anisotropic) Non-periodic structures in a multi-layer substrate, which may be lossy or anisotropic. Periodic structures in a multi-layer substrate, when used in conjunction with Periodic or Bloch boundary conditions. The TFSF source allows you to separate the computation region into two distinct regions: Total Field region – which includes the sum of the incident field wave plus the scattered field, and the Scattered Field region - includes only the scattered field. There are two primary reasons for doing this: (1) The propagating plane wave should not interact with the absorbing boundary conditions; (2) the load on the absorbing boundary conditions should be minimized. These boundary conditions are not perfect, i.e. a certain portion of the impinging wave is reflected back into the problem space. By subtracting the incident field, the amount of the radiating field hitting the boundary is minimized, thereby reducing the amount of error. Both regions are visible in Figure A.2. It should be noted that the physical field is the total field and the separation into an incident and a scattered field requires careful interpretation. When adding a TFSF source to a simulation, the most visible difference between the TFSF source and all other sources is that it appears as a 3D box, rather than a 2D plane. The source will inject a plane wave from this side of the source. The slightly confusing aspect is that the source will then 'subtract' the same plane wave when it arrives 152 at one of the boundaries of the source. The fields are injected at one edge, and are then Figure A.2: Two-dimensional Total field – scattered field formalism with the source injecting in the positive Y-direction. Nanoparticle 153 'subtracted' at the other edge. This means that the fields are zero in the 'Scattered field' region, if there are no scattering objects inside the source. Stability: In order to guarantee numerical stability for the common FDTD approach, the upper limit of the time step must be bounded by a criterion which restricts an update cycle’s EM fields propagation from cell to cell being faster than allowed by the phase velocity √ within any material ( , ). In FDTD, this condition is mathematically expressed by the Courant condition which determines the suitable size range of and , respectively the space and time discretization steps: √ (A.15) In the last equation, is the dimension of the simulation. The Courant condition must be respected for the simulation to be physically accurate and to avoid unwanted numerical instabilities. In practice, a Courant number “S” is defined to explicitly determine the relation between and : (A.16) A.2: Optimization code While working in Dr. Cronin’s lab, I wrote a substantial amount for the analysis and optimization of plasmonic nanostructures. In this appendix, I have included the code for the optimization of a 4 nanoparticle cluster with a normal incidence. This code is written in the Lumerical scripting language. 154 #A PROGRAM TO PERFORM OPTIMIZATION OF A 4 NANOPARTICLE CLUSTER USING A GRADIENT DESCENT BASED ALGORITHM #SET THE POSITION OF NPs no=4; n=20; #NUMBER OF ITERATIONS NPx=matrix(n,no); #NP POSITION MATRICES NPy=matrix(n,no); NPx0=matrix(1,no); NPy0=matrix(1,no); NPx(1,1)=0; NPy(1,1)=11e-9; NPx(1,2)=0; NPy(1,2)=-11e-9; NPx(1,3)=0; NPy(1,3)=43e-9; NPx(1,4)=0; NPy(1,4)=-43e-9; px=matrix(1,7); py=matrix(1,3); for(j=1:7){ px(j)=-3e-9+(j-1)*1e-9; } for(j=1:3){ py(j)=0; } s=matrix(1,no); elex=matrix(1,no); eley=matrix(1,no); for(j=1:no){ s(j)=8e-9; #STEP SIZE MATRIX } m=2e-9; #MINIMUM SEPARATION c=1; #COUNTER f=0; dontmove=matrix(1,no); #FLAG FOR MOVEMENT elmax0=0; #MAXIMUM INTENSITY elmaxp=0; #INTENSITY OF THE PREVIOUS ITERATION r=matrix(no,no); for(j=1:no){ for(k=1:no){ r(j,k)=4e-9; #SEPARATION MATRIX } } r0=matrix(1,no); for(j=1:no){ r0(j)=2e-9; } #LOAD THE SIMULATION FILE load("fournp.fsp"); 155 #INITIALIZING THE POSITION OF THE NPs switchtolayout; structures; select("NP1"); set("x",NPx(1,1)); set("y",NPy(1,1)); select("NP2"); set("x",NPx(1,2)); set("y",NPy(1,2)); select("NP3"); set("x",NPx(1,3)); set("y",NPy(1,3)); select("NP4"); set("x",NPx(1,4)); set("y",NPy(1,4)); #INITIALIZE THE ELECTRIC FIELD ARRAY el=matrix(9,3,500); if(dontmove==0){ runparallel; #RUNNING THE SIMULATION frq=getdata("twod","freqDFT"); el=pinch(getelectric("twod")); elmax=max(sum(sum(el,2),1)/27); elavg=matrix(500,1); elavg=sum(sum(el,2),1)/27; for(k=1:500){ if(elavg(k)==elmax){ nmax=k; } } for(k=1:no){ for(j=1:no){ r(k,j)=sqrt((NPx(1,j)-NPx(1,k))^2+(NPy(1,j)-NPy(1,k))^2)- 20e-9; r(j,k)=r(k,j); } } NPxmax=matrix(1,no); NPymax=matrix(1,no); dontmove=matrix(1,no); delex=matrix(1,no); deley=matrix(1,no); elex=matrix(1,no); eley=elex; } #'FOR' LOOP for(x=0;x<2;x=x){ if(elmax<=elmaxp){ f=f+1; }else{ 156 if(f>0){ f=0; } } if(x<2){ elmaxp=elmax; if(mod(c,2)==0){ sign=-1; }else{ sign=-1; } if(elmax>elmax0){ elmax0=elmax; for(i=1:no){ NPxmax(i)=NPx(c,i); NPymax(i)=NPy(c,i); } } #MOVEMENT MATRIX for(j=3:no){ mins=1; for(k=1:no){ if(mins>abs(r(j,k))){ mins=abs(r(j,k)); } } if(f==2){ s(j)=-round(min([mins,abs(r0(j))])*0.5*1e9)*1e-9; if(s(j)==0){ s(j)=-0.5e-9; } g(j)=-sign*s(j); }else{ s(j)=round(min([mins,abs(r0(j))])*0.5*1e9)*1e-9; if(s(j)==0){ s(j)=0.5e-9; } g(j)=sign*s(j); } } for(j=1:no){ NPx0(j)=NPx(c,j); NPy0(j)=NPy(c,j); } np=3; dontmove(np)=0; NPx(c,np)=NPx0(np)+g(np); for(j=1:no){ 157 r(np,j)=sqrt((NPx0(j)-NPx(c,np))^2+(NPy0(j)-NPy0(np))^2)-20e-9; r(j,np)=r(np,j); if(r(np,j)<m){if(j!=np){ dontmove(np)=1; } } } r0(np)=1; for(j=1:9){ for(k=1:3){ rmm=sqrt((NPx(c,np)-px(j))^2+(NPy0(np)-py(k))^2)-10e-9; if(rmm<r0(np)){ r0(np)=rmm; } } } if(r0(np)<m/2){ dontmove(np)=1; } #SET THE POSITION OF THE NPs if(dontmove(np)<1){ switchtolayout; structures; select("NP3"); set("x",NPx(c,np)); runparallel; #RUNNING THE SIMULATION el=pinch(getelectric("twod")); elex(np)=max(sum(sum(el,2),1)/27); #MAXIMUM ELECTRIC FIELD FOR THE CURRENT ITERATION switchtolayout; structures; select("NP3"); set("x",NPx0(np)); } np=3; dontmove(np)=0; NPy(c,np)=NPy0(np)-g(np); for(j=1:no){ r(np,j)=sqrt((NPx0(j)-NPx0(np))^2+(NPy0(j)-NPy(c,np))^2)-20e-9; r(j,np)=r(np,j); if(r(np,j)<m){if(j!=np){ dontmove(np)=1; } } } 158 r0(np)=1; for(j=1:9){ for(k=1:3){ rmm=sqrt((NPx0(np)-px(j))^2+(NPy(c,np)-py(k))^2)-10e-9; if(rmm<r0(np)){ r0(np)=rmm; } } } if(r0(np)<m/2){ dontmove(np)=1; } #SET THE POSITION OF THE NPs if(dontmove(np)<1){ switchtolayout; structures; select("NP3"); set("y",NPy(c,np)); runparallel; #RUNNING THE SIMULATION el=pinch(getelectric("twod")); eley(np)=max(sum(sum(el,2),1)/27); #MAXIMUM ELECTRIC FIELD FOR THE CURRENT ITERATION switchtolayout; structures; select("NP3"); set("y",NPy0(np)); } np=4; dontmove(np)=0; NPx(c,np)=NPx0(np)-g(np); for(j=1:no){ r(np,j)=sqrt((NPx0(j)-NPx(c,np))^2+(NPy0(j)-NPy0(np))^2)-20e-9; r(j,np)=r(np,j); if(r(np,j)<m){if(j!=np){ dontmove(np)=1; } } } r0(np)=1; for(j=1:9){ for(k=1:3){ rmm=sqrt((NPx(c,np)-px(j))^2+(NPy0(np)-py(k))^2)-10e-9; if(rmm<r0(np)){ r0(np)=rmm; } } 159 } if(r0(np)<m/2){ dontmove(np)=1; } #SET THE POSITION OF THE NPs if(dontmove(np)<1){ switchtolayout; structures; select("NP4"); set("x",NPx(c,np)); runparallel; #RUNNING THE SIMULATION el=pinch(getelectric("twod")); elex(np)=max(sum(sum(el,2),1)/27); #MAXIMUM ELECTRIC FIELD FOR THE CURRENT ITERATION switchtolayout; structures; select("NP4"); set("x",NPx0(np)); } np=4; dontmove(np)=0; NPy(c,np)=NPy0(np)-g(np); for(j=1:no){ r(np,j)=sqrt((NPx0(j)-NPx0(np))^2+(NPy0(j)-NPy(c,np))^2)-20e-9; r(j,np)=r(np,j); if(r(np,j)<m){if(j!=np){ dontmove(np)=1; } } } r0(np)=1; for(j=1:9){ for(k=1:3){ rmm=sqrt((NPx0(np)-px(j))^2+(NPy(c,np)-py(k))^2)-10e-9; if(rmm<r0(np)){ r0(np)=rmm; } } } if(r0(np)<m/2){ dontmove(np)=1; } #SET THE POSITION OF THE NPs if(dontmove(np)<1){ switchtolayout; structures; 160 select("NP4"); set("y",NPy(c,np)); runparallel; #RUNNING THE SIMULATION el=pinch(getelectric("twod")); eley(np)=max(sum(sum(el,2),1)/27); #MAXIMUM ELECTRIC FIELD FOR THE CURRENT ITERATION switchtolayout; structures; select("NP4"); set("y",NPy0(np)); } for(j=3:no){ delex(j)=elex(j)-elmax; deley(j)=eley(j)-elmax; } for(j=3:no){ stepx=sign*delex(j)/sqrt((delex(j))^2+(deley(j))^2)*s(j); stepy=sign*deley(j)/sqrt((delex(j))^2+(deley(j))^2)*s(j); if(abs(delex(j))<1e-6){if(abs(deley(j))<1e-6){ NPx(c,j)=NPx0(j); NPy(c,j)=NPy0(j); } }else{ if(j==3){ NPx(c,j)=round((NPx0(j)+stepx)*1e9)*1e-9; NPy(c,j)=round((NPy0(j)+stepy)*1e9)*1e-9; }else{ if(j==4){ NPx(c,j)=round((NPx0(j)-stepx)*1e9)*1e-9; NPy(c,j)=round((NPy0(j)+stepy)*1e9)*1e-9; } } } } for(k=3:no){ dontmove(k)=0; for(j=1:no){ r(k,j)=sqrt((NPx(c,k)-NPx(c,j))^2+(NPy(c,k)-NPy(c,j))^2)- 20e-9; if(r(k,j)<m){if(k!=j){ dontmove(max([k,j]))=1; NPx(c,max([k,j]))=NPx0(max([k,j])); NPy(c,max([k,j]))=NPy0(max([k,j])); } } } r(k,k)=1; 161 r0(k)=1; for(p=1:9){ for(q=1:3){ rmm=sqrt((NPx(c,k)-px(p))^2+(NPy(c,k)-py(q))^2)-10e- 9; if(rmm<r0(k)){ r0(k)=rmm; } } } if(r0(k)<m/2){ dontmove(k)=1; NPx(c,k)=NPx0(k); NPy(c,k)=NPy0(k); } } #SET THE POSITION OF THE NPs switchtolayout; structures; if(dontmove(3)<1){ select("NP3"); set("x",NPx(c,3)); set("y",NPy(c,3)); } if(dontmove(4)<1){ select("NP4"); set("x",NPx(c,4)); set("y",NPy(c,4)); } runparallel; #RUNNING THE SIMULATION frq=getdata("twod","freqDFT"); el=pinch(getelectric("twod")); elmax=max(sum(sum(el,2),1)/27); elavg=matrix(500,1); elavg=sum(sum(el,2),1)/27; for(k=1:500){ if(elavg(k)==elmax){ nmax=k; } } maxwave=2.9979e8/frq(nmax); } #UPDATE THE ITERATION COUNTER c=c+1; if(c==n){ 162 x=2; #CONDITION TO EXIT THE LOOP } for(i=1:no){ NPx(c,i)=NPx(c-1,i); NPy(c,i)=NPy(c-1,i); } #DISPLAY AND EXPORT TO A TEXT FILE ?"Iterations remaining: "+num2str((n-c+1)); ?elmax; write("4np3dfixed.txt",num2str(c-1)+" "+num2str(elmax)+" "+num2str(NPx(c-1,3))+" "+num2str(NPy(c-1,3))+" "+num2str(NPx(c-1,4))+" "+num2str(NPy(c-1,4))+" "+num2str(maxwave*1e9)+" "+num2str(s*1e9)); } #END OF THE 'FOR' LOOP ?"The maximum intensity = "+num2str(elmax0); #END OF THE PROGRAM 163 # A PROGRAM FOR THE OPTIMIZATION OF PLASMONICALLY ENHANCED FOUR-WAVE MIXED SIGNAL USING A PARTICLE SWARM OPTIMIZATION (PSO) ALGORITHM n=100; #number of iterations ns=3; #number of particles (solutions) nnp=4; #number of nanoparticles nsf=matrix(ns,1); #flags for invisible boundary conditions X=matrix(n,2*ns,nnp); #position matrix V=matrix(n,2*ns,nnp); #velocity matrix lam=matrix(3,1); #wavelength matrix lam(1)=632.8e-9; #wavelength of E1 (m) lam(2)=532e-9; #wavelength of E2 (m) lam(3)=lam(1)*lam(2)/(2*lam(2)-lam(1)); #wavelength of E4wm (m) I0=matrix(3,1); #intensity matrix (no NPs) I=matrix(n,ns,3); #intensity matrix (with NPs) enh=matrix(n,ns); #enhancement matrix (with NPs) #elnp=matrix(n,ns); #Kerr with NPs intensity matrix #el0=matrix(n,ns); #Kerr intensity matrix enhmax=matrix(n,ns); #local maximum enhancement matrix Xmax=matrix(n,2*ns,nnp); #local maximum position matrix enhmaxglobal=matrix(n,1); #global maximum enhancement matrix Xmaxglobal=matrix(n,2,nnp); #global maximum position matrix r=matrix(nnp,nnp); #Separation Matrix rad=20e-9; #Radius of the nanoparticles dmin=2e-9; #minimum separation allowed c=1; #counter dt=1; #time step w0=0.8; #inertial weight w=w0-0.1*round(c/8); const1=0.6; #local maximum scaling factor const2=0.4; #global maximum scaling factor Vmax=5e-9; #maximum velocity allowed #Solution Space: 100 nm x 100 nm #Initialization of particles for(i=1:n){ 164 for(j=1:ns){ X(i,2*j-1,1)=0; #Initial X(i,2*j,1)=21e-9; #positions X(i,2*j-1,2)=0; #for X(i,2*j,2)=-21e-9; #NP 1,2 V(i,2*j-1,1)=0; V(i,2*j,1)=0; V(i,2*j-1,2)=0; V(i,2*j,2)=0; } } #Initial positions for NPs 3 onwards X(1,1,3)=0e-9; X(1,2,3)=120e-9; X(1,3,3)=-80e-9; X(1,4,3)=80e-9; X(1,5,3)=80e-9; X(1,6,3)=80e-9; X(1,1,4)=0e-9; X(1,2,4)=-120e-9; X(1,3,4)=-80e-9; X(1,4,4)=-80e-9; X(1,5,4)=80e-9; X(1,6,4)=-80e-9; V(1,1,3)=1e-9; V(1,2,3)=-2e-9; V(1,3,3)=2e-9; V(1,4,3)=-1e-9; V(1,5,3)=-2e-9; V(1,6,3)=-1e-9; V(1,1,4)=1e-9; V(1,2,4)=-2e-9; V(1,3,4)=2e-9; V(1,4,4)=-1e-9; V(1,5,4)=-2e-9; V(1,6,4)=-1e-9; for(j=1:nnp){ for(k=1:nnp){ r(j,k)=4e-9; #SEPARATION MATRIX } } for(j=1:3){ load("4wmnonp.fsp"); #Lumerical simulation file switchtolayout; 165 sources; select("tfsf"); set("wavelength start",lam(j)); set("wavelength stop",lam(j)); run(2); I0(j)=pinch(getelectric("pt")); } load("4wm4np.fsp"); #Main for loop for(y=0;y<2;y=y){ #INITIALIZING THE POSITIONs OF THE NPs for(s=1:ns){ if(nsf(s)==0){ switchtolayout; structures; for(np=1:nnp){ select("NP"+num2str(np)); #Set positions set("x",X(c,2*s-1,np)); # of the NPs set("y",X(c,2*s,np)); } for(jj=1:3){ switchtolayout; sources; select("tfsf"); set("wavelength start",lam(jj)); set("wavelength stop",lam(jj)); run(2); I(c,s,jj)=pinch(getelectric("pt")); } enh(c,s)=((I(c,s,1))^2*I(c,s,2)*I(c,s,3))/((I0(1))^2*I0(2)*I0(3)) ; if(enhmax(c,s)<enh(c,s)){ #Set local maximum conditions enhmax(c,s)=enh(c,s); Xmax(c,2*s-1,1:nnp)=X(c,2*s-1,1:nnp); Xmax(c,2*s,1:nnp)=X(c,2*s,1:nnp); } } } maxglobal=pinch(max(enhmax(c,1:ns))); for(jj=1:ns){ if(maxglobal==enhmax(c,jj)){ nsmax=jj; 166 } } if(enhmaxglobal(c)<maxglobal){ enhmaxglobal(c)=maxglobal; Xmaxglobal(c,1:2,1:nnp)=Xmax(c,(2*nsmax-1):2*nsmax,1:nnp); } if(w>0.1){ w=w0-0.1*round(c/10); }else{ w=0.1; } for(i=1:2*ns){ for(j=3:nnp){ X(c+1,i,j)=X(c,i,j)+V(c,i,j); V(c+1,i,j)=round((w*V(c,i,j)+const1*(Xmax(c,i,j)- X(c,i,j))+const2*(Xmaxglobal(c,mod(i+1,2)+1,j)-X(c,i,j)))*1e9)*1e-9; if(V(c+1,i,j)>0){ signV=1; }else{ signV=-1; } if(abs(V(c+1,i,j))>Vmax){ V(c+1,i,j)=signV*Vmax; } } } for(s=1:ns){ for(k=1:nnp){ for(j=3:nnp){ r(k,j)=abs(sqrt((X(c+1,2*s-1,k)-X(c+1,2*s- 1,j))^2+(X(c+1,2*s,k)-X(c+1,2*s,j))^2))-2*rad; r(j,k)=r(k,j); if(r(k,j)<dmin){ if(k!=j){ o=1; xdiff=X(c+1,2*s-1,max([k,j]))- X(c+1,2*s-1,min([k,j])); ydiff=X(c+1,2*s,max([k,j]))- X(c+1,2*s,min([k,j])); if(xdiff>0){ signxdiff=1; }else{ signxdiff=-1; } if(ydiff>0){ signydiff=1; }else{ signydiff=-1; } 167 X(c+1,2*s-1,max([k,j]))=X(c+1,2*s- 1,max([k,j]))+round((signxdiff*abs(xdiff)/(abs(xdiff)+abs(ydiff))*(2*ra d+dmin-r(k,j)))*1e9)*1e-9; X(c+1,2*s,max([k,j]))=X(c+1,2*s,max([k,j]))+round((signydiff*abs( ydiff)/(abs(xdiff)+abs(ydiff))*(2*rad+dmin-r(k,j)))*1e9)*1e-9; } } } } } enhmax(c+1,1:ns)=enhmax(c,1:ns); Xmax(c+1,1:ns,1:nnp)=Xmax(c,1:ns,1:nnp); enhmaxglobal(c+1)=enhmaxglobal(c); Xmaxglobal(c+1,1:2,1:nnp)=Xmaxglobal(c,1:2,1:nnp); #UPDATE THE ITERATION COUNTER c=c+1; if(c==n){ y=2; #CONDITION TO EXIT THE LOOP } #DISPLAY AND EXPORT TO A FILE matlabsave("opt4wm4np",enhmaxglobal,enhmax,Xmax,Xmaxglobal); ?"Iterations remaining: "+num2str((n-c+1)); ?"Max enhancement = "+num2str(enhmaxglobal(c)); #?num2str(o); } 168 #A PROGRAM TO PERFORM OPTIMIZATION OF ANGLE EVAPORATION DIMERS USING GRADIENT DESCENT ALGORTIHM #PARAMETERS: ANGLE OF TILT th, THICKNESS OF THE FIRST EVAPORATION t1, THICKNESS OF THE SECOND EVAPORATION t2, THICKNESS OF THE MMA LAYER d1, THICKNESS OF THE ZEP LAYER d2, DIAMETER OF THE FIRST PARTICLE D1. #THE STEP SIZE IS CHOSEN AS A FUNCTION OF dI/I. n=500; #number of iterations np=6; #number of parameters (theta, t1, t2, d1, d2 and D1) fpts=500; #frequency points th=matrix(n,1); #th matrix t1=matrix(n,1); #t1 matrix t2=matrix(n,1); #t2 matrix d1=matrix(n,1); #d1 matrix d2=matrix(n,1); #d2 matrix D1=matrix(n,1); #D1 matrix X=matrix(n,np); #position matrix (dx and dy) X0=matrix(np,1); #current positions I_enh=matrix(n,np,fpts); #intensity enhancement matrix Imax_enh=matrix(n,np); #peak intensity enhancement matrix I_enh0=matrix(n,fpts); #intensity enhancement matrix Imax_enh0=matrix(n,1); #peak intensity enhancement matrix dI=matrix(n,np); #gradient intensity matrix I_enh_maxglobal=matrix(n,fpts); #global maximum matrix Imax_enh_maxglobal=matrix(n,1); #peak global maximum matrix Xmaxglobal=matrix(n,np); #global maximum position matrix c=1; #counter g=matrix(np,1); #STEP SIZE MATRIX step=matrix(np,1); #Gradient weighted step size Vmax=0.1; #maximum velocity allowed thmin=5; #minimum th thmax=25; #maximum th t1min=10e-9; #minimum t1 t1max=40e-9; #maximum t1 t2min=10e-9; #minimum t2 t2max=40e-9; #maximum t2 169 d1min=75e-9; #minimum d1 d1max=200e-9; #maximum d1 d2min=40e-9; #minimum d2 d2max=200e-9; #maximum d2 D1min=20e-9; #minimum D1 D1max=100e-9; #maximum D1 #dx and dy lie within [0,1] #Initialization of cell #D=Dmin+(Dmax-Dmin)*d; X(1,1)=0.5; X(1,2)=0.5; X(1,3)=0.5; X(1,4)=0.5; X(1,5)=0.5; X(1,6)=0.5; t1(1)=t1min+(t1max-t1min)*X(1,2); thmin=180/pi*atan(1e-9/t1(1)); th(1)=thmin+(thmax-thmin)*X(1,1); #Initial t2(1)=t2min+(t2max-t2min)*X(1,3); d1min=max([t1(1),t2(1)])*3; d1(1)=d1min+(d1max-d1min)*X(1,4); D1(1)=D1min+(D1max-D1min)*X(1,6); d2max=D1(1)/(2*tan(pi/180*th(1))); d2(1)=d2min+(d2max-d2min)*X(1,5); if(d2(1)<0){ d2(1)=0; } load("ae_dimer_new.fsp"); #Lumerical simulation file switchtolayout; select("FDTD"); #the simulation size (pitch of the array) should be at least 100 nm bigger than the window size set("x span",D1(1)+50e-9); set("y span",D1(1)+50e-9); select("planewave"); #the source should be at least 100 nm bigger than the window size set("x span",D1(1)+50e-9); set("y span",D1(1)+50e-9); 170 select("under"); #the monitor should be at least 100 nm bigger than the window size set("x span",D1(1)+50e-9); set("y span",D1(1)+50e-9); select("over"); #the monitor should be at least 100 nm bigger than the window size set("x span",D1(1)+50e-9); set("y span",D1(1)+50e-9); sep=t1(1)*tan(pi/180*th(1)); #gap between the NPs D2=D1(1)-d2(1)*tan(pi/180*th(1)); #diameter of the 2nd NP shift=d1(1)*tan(pi/180*th(1)); #shift of the 2nd NP select("mesh"); set("x span",D1(1)+5e-9); set("y min",-D1(1)/2-5e-9); set("y max",D1(1)/2+D2/2+shift+5e-9); set("z max",t1(1)+t2(1)+5e-9); select("NP1"); set("radius",D1(1)/2); set("z max",t1(1)); select("gap"); set("radius",D1(1)/2); set("y",sep); set("z max",t2(1)); select("NP2"); set("radius",D1(1)/2); set("radius 2",D2/2); set("y",shift); set("z max",t2(1)); select("NP2_top"); set("radius",D1(1)/2); set("radius 2",D2/2); set("y",shift); set("z min",t1(1)); set("z max",t1(1)+t2(1)); select("ring"); set("inner radius",D1(1)/2); set("z min",t1(1)); set("z max",t1(1)+t2(1)); select("pt"); set("y",D1(1)/2+sep/2); 171 runparallel; #Run the simulation cwnorm; frq=getdata("pt","f"); lam0=2.9979e8/frq; I_enh0(1,1:fpts)=pinch(getelectric("pt")); Imax_enh_maxglobal(1)=max(I_enh0); Imax_enh0(1)=max(I_enh0); #Main for loop for(y=0;y<2;y=y){ if(mod(c,2)==0){ sign=1; }else{ sign=1; } #INITIALIZING THE POSITIONs OF THE NPs for(jj=1:np){ if(2==2){ if(X(c,jj)>0.5){ g(jj)=(1-X(c,jj))/20; }else{ g(jj)=X(c,jj)/20; } }else{ if(X(c,jj)>0.5){ g(jj)=-(1-X(c,jj))/20; }else{ g(jj)=-X(c,jj)/20; } } #CALCULATING THE GRADIENTS X0(jj)=X(c,jj); X(c,jj)=X0(jj)+g(jj); #SET THE POSITION OF THE NPs t1(c)=t1min+(t1max-t1min)*X(c,2); thmin=180/pi*atan(1e-9/t1(c)); th(c)=thmin+(thmax-thmin)*X(c,1); #Initial t2(c)=t2min+(t2max-t2min)*X(c,3); d1min=max([t1(c),t2(c)])*3; d1(c)=d1min+(d1max-d1min)*X(c,4); D1(c)=D1min+(D1max-D1min)*X(c,6); 172 d2max=D1(c)/(2*tan(pi/180*th(c))); d2(c)=d2min+(d2max-d2min)*X(c,5); if(d2(c)<0){ d2(c)=0; } load("ae_dimer_new.fsp"); #Lumerical simulation file switchtolayout; select("FDTD"); #the simulation size (pitch of the array) should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); select("planewave"); #the source should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); select("under"); #the monitor should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); select("over"); #the monitor should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); sep=t1(c)*tan(pi/180*th(c)); #gap between the NPs D2=D1(c)-d2(c)*tan(pi/180*th(c)); #diameter of the 2nd NP shift=d1(c)*tan(pi/180*th(c)); #shift of the 2nd NP select("mesh"); set("x span",D1(c)+5e-9); set("y min",-D1(c)/2-5e-9); set("y max",D1(c)/2+D2/2+shift+5e-9); set("z max",t1(c)+t2(c)+5e-9); select("NP1"); set("radius",D1(c)/2); set("z max",t1(c)); select("gap"); set("radius",D1(c)/2); set("y",sep); 173 set("z max",t2(c)); select("NP2"); set("radius",D1(c)/2); set("radius 2",D2/2); set("y",shift); set("z max",t2(c)); select("NP2_top"); set("radius",D1(c)/2); set("radius 2",D2/2); set("y",shift); set("z min",t1(c)); set("z max",t1(c)+t2(c)); select("ring"); set("inner radius",D1(c)/2); set("z min",t1(c)); set("z max",t1(c)+t2(c)); select("pt"); set("y",D1(c)/2+sep/2); runparallel; #Run the simulation cwnorm; frq=getdata("pt","f"); lam0=2.9979e8/frq; I_enh(c,jj,1:fpts)=pinch(getelectric("pt")); Imax_enh(c,jj)=max(I_enh(c,jj,1:fpts)); dI(c,jj)=Imax_enh(c,jj)-Imax_enh0(c); #Calculate gradients write("ae_gr_absopt.txt","X"+num2str(jj)+" = "+num2str(X(c,jj))+", X0"+num2str(jj)+" = "+num2str(X0(jj))+", Imax"+num2str(jj)+" = "+num2str(Imax_enh(c,jj))+", dI"+num2str(jj)+" = "+num2str(dI(c,jj))); X(c,jj)=X0(jj); } for(jj=1:np){ step(jj)=0.5*sign*dI(c,jj)/Imax_enh(c,jj); if (round(c/10)>0.5){ step(jj)=step(jj)*(1-0.1*round(c/10)); } else { step(jj)=0.5*step(jj); } X(c,jj)=X0(jj)*(1+step(jj)); if (X(c,jj)>0.99){ 174 X(c,jj)=0.9; }else{ if(X(c,jj)<0.01){ X(c,jj)=0.1; } } } t1(c)=t1min+(t1max-t1min)*X(c,2); thmin=180/pi*atan(1e-9/t1(c)); th(c)=thmin+(thmax-thmin)*X(c,1); #Initial t2(c)=t2min+(t2max-t2min)*X(c,3); d1min=max([t1(c),t2(c)])*3; d1(c)=d1min+(d1max-d1min)*X(c,4); D1(c)=D1min+(D1max-D1min)*X(c,6); d2max=D1(c)/(2*tan(pi/180*th(c))); d2(c)=d2min+(d2max-d2min)*X(c,5); if(d2(c)<0){ d2(c)=0; } load("ae_dimer_new.fsp"); #Lumerical simulation file switchtolayout; #the simulation size (pitch of the array) should be at least 100 nm bigger than the window size select("FDTD"); set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); select("planewave"); #the source should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); select("under"); #the monitor should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); select("over"); #the monitor should be at least 100 nm bigger than the window size set("x span",D1(c)+50e-9); set("y span",D1(c)+50e-9); sep=t1(c)*tan(pi/180*th(c)); #gap between the NPs D2=D1(c)-d2(c)*tan(pi/180*th(c)); #diameter of the 2nd NP 175 shift=d1(c)*tan(pi/180*th(c)); #shift of the 2nd NP select("mesh"); set("x span",D1(c)+5e-9); set("y min",-D1(c)/2-5e-9); set("y max",D1(c)/2+D2/2+shift+5e-9); set("z max",t1(c)+t2(c)+5e-9); select("NP1"); set("radius",D1(c)/2); set("z max",t1(c)); select("gap"); set("radius",D1(c)/2); set("y",sep); set("z max",t2(c)); select("NP2"); set("radius",D1(c)/2); set("radius 2",D2/2); set("y",shift); set("z max",t2(c)); select("NP2_top"); set("radius",D1(c)/2); set("radius 2",D2/2); set("y",shift); set("z min",t1(c)); set("z max",t1(c)+t2(c)); select("ring"); set("inner radius",D1(c)/2); set("z min",t1(c)); set("z max",t1(c)+t2(c)); select("pt"); set("y",D1(c)/2+sep/2); runparallel; #Run the simulation cwnorm; frq=getdata("pt","f"); lam0=2.9979e8/frq; I_enh0(c,1:fpts)=pinch(getelectric("pt")); Imax_enh0(c)=max(I_enh0(c,1:fpts)); for(jj=1:np){ write("ae_gr_absopt.txt","X"+num2str(jj)+" = "+num2str(X(c,jj))+", Imax = "+num2str(Imax_enh0(c))+", step"+num2str(jj)+" = "+num2str(step(jj))); } 176 if(Imax_enh_maxglobal(c)<Imax_enh0(c)){ Imax_enh_maxglobal(c)=Imax_enh0(c); Xmaxglobal(c,1:np)=X(c,1:np); I_enh_maxglobal(c,1:fpts)=I_enh0(c,1:fpts); } Imax_enh_maxglobal(c+1)=Imax_enh_maxglobal(c); Imax_enh0(c+1)=Imax_enh0(c); I_enh_maxglobal(c+1,1:fpts)=I_enh_maxglobal(c,1:fpts); Xmaxglobal(c+1,1:np)=Xmaxglobal(c,1:np); X(c+1,1:np)=X(c,1:np); #UPDATE THE ITERATION COUNTER c=c+1; if(c==n){ y=2; #CONDITION TO EXIT THE LOOP } #DISPLAY AND EXPORT TO A FILE ?"Iterations remaining: "+num2str((n-c+1)); ?Imax_enh_maxglobal(c); matlabsave("ae_gr_absopt_par",X,Xmaxglobal,lam0,Imax_enh0,I_enh_m axglobal,Imax_enh_maxglobal,dI,X0,step); } A.3 TiO 2 synthesis There are various techniques for synthesizing TiO2, including the sol-gel method, sputter deposition, chemical vapor deposition (CVD), atomic layer deposition (ALD), thermal oxidization, and anodization. Cubic or hexagonal mesoporous TiO 2 can be synthesized by the sol-gel process. In this process, Pluronic P123/ethanol (1g/12g for cubic, 2.3g/12g for hexagonal) and titanium ehoxide/hydrochloric acid (4.2g/3.2g) solution are prepared separately under flowing Ar or N 2 . The two mixtures are then mixed together and stirred for 3 hours to form the sample solution. The solution is then dip-coated or spin-coated onto substrates and dried at room temperature in air for 24h to form a gel. The dried film is then heated 177 in air (400 °C for cubic, 250 °C for hexagonal) for 4 h before cooling. The advantage of the solgel method is that it produces mesoporous structures with greatly increased surface area for light absorption or dye-sensitizing. The other advantage of the solgel method is that it can be incorporated with plasmonic nanoparticles, such as silver or gold. Electrochemical anodization is another method for producing TiO 2 . In this method, titanium foil is first cleaned in 10% HF for 10 seconds to smooth the surface and remove surface oxide and/or residues. The titanium foil is then connected to the anode of Figure A.4: Scanning electron microscope (SEM) images of anodic TiO 2 , (a) top view and (b) cross sectional view. (a) (b) Figure A.3: Scanning electron microscope (SEM) images of solgel TiO 2 showing the mesoporous structure. 178 a DC power supply and submerged into an electrolyte consisting of 0.25%wt NH 4 F and 2%wt water in an ethylene glycol solution with graphite as the counter electrode. Under a 30V anodization voltage, TiO 2 is formed with an average pore diameter around 40nm at a rate of approximately 0.5μm/1h, as shown in Figure A.4. In order to obtain well-ordered hexagonal arrays of nano-pore structures, the first anodization TiO 2 layer is removed by tape or sonication, and a second anodization process is applied. After the growth, annealing at 400ºC in air for 4 hours is necessary to form the anatase crystalline phase. The applied voltage determines the separation between pores. The advantage of this anodic TiO 2 is their nanotube structure, which can be controlled by anodization voltage (diameter and thickness) and time (length). The other advantage is that these TiO 2 tubes are natively doped with N and F, which also increases their light absorption in visible range slightly. Atomic layer deposition (ALD) is very similar to CVD. In CVD, two precursors are mixed in the chamber at the same time. ALD, however, breaks the CVD reaction into two half-reactions, and keeps the precursor materials separate during each half-reaction. A typical ALD cycle includes (1) exposure of the first precursor; (2) purge of the reaction chamber to remove the previous non-reacted precursors and the gaseous reaction byproducts; (3) exposure of the second precursor (or another treatment to activate the surface again for the reaction of the first precursor); (4) purge or evacuation of the reaction chamber again. Each half reaction only happens on a non-deposited surface, so after each precursor exposure, only a single layer is deposited. ALD is used to synthesize single crystalline pure anatase phase TiO 2 . 179 Figure A.5 schematically shows the ALD process for the deposition of TiO 2 . The precursors used in this deposition are TiCl 4 and H 2 O. The chemical reactions involved in this reaction are: -OH* + TiCl 4 → -O n TiCl 4-n * + HCl -TiCl* + H 2 O → -TiOH* + HCl * indicates a surface species, n can be 1,2,3. Figure A.5: schematic of the atomic layer deposition of TiO 2 using TiCl 4 and H 2 O as the precursors. Substrate Substrate Substrate : Ti : Cl : H : O 180 This process is performed using Cambridge NanoTech Savannah S100 atomic layer deposition (ALD) system. The ALD chamber is heated to 250˚C, while the TiCl 4 and H 2 O reservoirs are kept at room temperature. The ALD cycles used for the deposition are an exposure to TiCl 4 , followed by a purging cycle, followed by an exposure to H 2 O, followed by yet another purging cycle with the cycle times being 0.1s, 10s, 0.02s and 10s respectively. The amorphous TiO 2 films thus obtained are annealed in a furnace in air in order to change the TiO 2 to the anatase crystalline form which is favorable for photoctalysis. A.4 TiO 2 doping Doping of TiO 2 extends the absorption response of TiO 2 to the visible region. This happens due to the introduction of defect and impurity states in the bandgap of TiO 2 . There are various methods of doping TiO 2 to make it either p-type or n-type. Here are a few of these methods that can be executed using a simple tube furnace. To dope TiO 2 with carbon, TiO 2 can be annealed at 500°C to 800°C in a tube furnace, under a flow of dry O 2 and CO, with a heating rate of 5°C/min and a dwelling time of 3h. The temperature determines the degree of doping. Carbon doping makes the TiO 2 n-type. To perform hydrogen doping of TiO 2 , the TiO 2 sample is placed in a quartz tube inserted into an electric tubular furnace and heated at 650°C for 1h. Lowering the temperature requires increasing the time to get the same degree of doping. Hydrogen 181 annealing is carried out in hydrogen gas stream of 300ml/min. Hydrogen doping also results in n-type TiO 2 . Nitrogen doping of TiO 2 can be performed by annealing anatase TiO 2 under flowing NH 3 at 550 to 650°C for 3h. The amount of doping is proportional to the temperature used in doping. Nitrogen doped TiO 2 is p-type.
Abstract (if available)
Abstract
In this thesis, plasmonic properties of metal nanostructures are investigated by electromagnetic simulations using the finite difference time domain (FDTD) method. Chapter 1 covers the background knowledge required to read this thesis. It talks about the fundamentals of the FDTD method, the physics of plasmonics and a brief description of photocatalysis. In chapters 2 to 6, I am going to present the work our research group has done in the field of plasmonics that has already been published. Chapter 7 contains work that we can derive from our past research. ❧ In chapter 2, we perform optimization of plasmonic nanoparticle geometries. An iterative optimization algorithm is used to determine the configuration of the nanoparticles that gives the maximum electric field intensity at the center of the cluster. We observe that the optimum configurations of these clusters have mirror symmetry about the axis of planewave propagation, but are otherwise non-symmetric and non-intuitive. The maximum field intensity is found to increase monotonically with the number nanoparticles in the cluster, producing intensities that are 2500 times larger than the incident electromagnetic field. ❧ In chapter 3, I will talk about the optical properties of thin metallic discontinuous films of Au and Ag, which are known to exhibit a strong plasmonic response under visible illumination. In this work, evaporated thin films are imaged with high resolution transmission electron microscopy (HRTEM), to reveal the structure of the semicontinuous metal island film with sub-nm resolution. The electric field distributions and the absorption spectra of these semicontinuous island film geometries are calculated using the finite difference time domain (FDTD) method and compared with the experimentally measured absorption spectra. In addition to that, we calculate the SERS enhancement factors and photocatalytic enhancement factors of these films. We also study the effect of annealing on these films, which results in a large reduction in electric field strength due to increased nanoparticle spacing. ❧ In chapter 4, we study the effects of surrounding nanoparticles on a plasmonic hot spot. From our simulations, we show that the surrounding film contributes significantly to the electric field intensity at the hot spot by focusing energy to it. Widening of the gap size causes a decrease in the intensity at the hot spot. However, these island-like nanoparticle hot spots are shown to be robust to gap size than nanoparticle dimer geometries, studied previously. In fact, the main factor in determining the hot spot intensity is the focusing effect of the surrounding nano-islands. ❧ In chapter 5, I will talk about electromagnetic simulations of plasmonic enhancement of photocatalytic chemical reactions. By integrating strongly plasmonic Au nanoparticles with strongly catalytic TiO₂, we demonstrate plasmon-enhanced photocatalytic water splitting, and reduction of CO₂ with H₂O to form hydrocarbon fuels. Under visible illumination, we observe enhancements of up to 66X in the photocatalytic splitting of water in TiO₂ with the addition of Au nanoparticles. We also perform a systematic study of the mechanisms of Au nanoparticle/TiO₂-catalyzed photoreduction of CO₂ and water vapor over a wide range of wavelengths. In this case, under visible light illumination, we observe a 24-fold enhancement in the photocatalytic activity due to the intense local electromagnetic fields created by the surface plasmons of the Au nanoparticles. Above the plasmon resonance, under ultraviolet radiation we observe a reduction in the photocatalytic activity. Electromagnetic simulations indicate that the improvement of photocatalytic activity in the visible range is caused by the local electric field enhancement near the TiO₂ surface, rather than by the direct transfer of charge between the two materials. Here, the near-field optical enhancement increases the electron-hole pair generation rate at the surface of the TiO₂, thus increasing the amount of photogenerated charge contributing to catalysis. This mechanism of enhancement is particularly effective because of the relatively short exciton diffusion length (or minority carrier diffusion length), which otherwise limits the photocatalytic performance. Our results suggest that enhancement factors many times larger than this are possible if this mechanism can be optimized. ❧ In chapter 6, I will talk about a method for fabricating arrays of plasmonic nanoparticles with separations on the order of 1nm using an angle evaporation technique. High resolution transmission electron microscopy (HRTEM) is used to resolve the small separations achieved between nanoparticles fabricated on thin SiN membranes. These nearly touching metal nanoparticles produce extremely high electric field intensities when irradiated with laser light. We perform surface enhanced Raman spectroscopy (SERS) a non-resonant dye molecule (p-ATP) deposited on the nanoparticle arrays using confocal micro-Raman spectroscopy. Our results show significant enhancement when the incident laser is polarized parallel to the axis of the nanoparticle pairs, whereas no enhancement is observed for the perpendicular polarization. These results demonstrate proof-of-principle of this fabrication technique. Finite difference time domain (FDTD) simulations based on HRTEM images predict an electric field intensity enhancement of 82,400 at the center of the nanoparticle pair, and an electromagnetic SERS enhancement factor of 10⁹-10¹⁰.
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Creator
Pavaskar, Prathamesh
(author)
Core Title
Electromagnetic modeling of plasmonic nanostructures
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
04/16/2013
Defense Date
03/22/2013
Publisher
University of Southern California
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University of Southern California. Libraries
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electromagnetic simulations,numerical optimization,OAI-PMH Harvest,photocatalysis,plasmonics,SERS
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English
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Cronin, Stephen B. (
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), Benderskii, Alexander V. (
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), Wu, Wei (
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pavaskar@usc.edu,pratham83@gmail.com
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https://doi.org/10.25549/usctheses-c3-238815
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Tags
electromagnetic simulations
numerical optimization
photocatalysis
plasmonics
SERS