Properties of Nanostructures

Dissertation by Prabath Hewageegana Georgia State University
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Georgia State University Digital Archive @ GSU Physics and Astronomy Dissertations Department of Physics and Astronomy 11-18-2008 Theory of Electronic and Optical Properties of Nanostructures Prabath Hewageegana Recommended Citation Hewageegana, Prabath, "Theory of Electronic and Optical Properties of Nanostructures" (2008). Physics and Astronomy Dissertations. Paper 27. http://digitalarchive.gsu.edu/phy_astr_diss/27 This Dissertation is brought to you for free and open access by the Department of Physics and Astronomy at Digital Archive @ GSU. It has been accepted for inclusion in Physics and Astronomy Dissertations by an authorized administrator of Digital Archive @ GSU. For more information, please contact digitalarchive@gsu.edu.

THEORY OF ELECTRONIC AND OPTICAL PROPERTIES OF NANOSTRUCTURES by PRABATH S. HEWAGEEGANA Under the Direction of Dr. Vadym Apalkov ABSTRACT “There is plenty of room at the bottom.” This bold and prophetic statement from Nobel laureate Richard Feynman back in 1950s at Cal Tech launched the Nano Age and predicted, quite accurately, the explosion in nanoscience and nanotechnology. Now this is a fast developing area in both science and technology. Many think this would bring the greatest technological revolution in the history of mankind. To understand electronic and optical properties of nanostructures, the following problems have been studied. In particular, intensity of mid-infrared light transmitted through a metallic diffraction grating has been theoretically studied. It has been shown that for s-polarized light the enhancement of the transmitted light is much stronger than for p-polarized light. By tuning the parameters of the diffraction grating enhancement can be increased by a few orders of magnitude. The spatial distribution of the transmitted light is highly nonuniform with very sharp peaks, which have the spatial widths about 10 nm. Furthermore, under the ultra fast response in nanostruc-

tures, the following two related goals have been proved: (a) the two-photon coherent control allows one to dynamically control electron emission from randomly rough surfaces, which is localized within a few nanometers. (b) the photoelectron emission from metal nanostructures in the strong-field (quasistationary) regime allows coherent control with extremely high contrast, suitable for nanoelectronics applications. To investigate the electron transport properties of two dimensional carbon called graphene, a localization of an electron in a graphene quantum dot with a sharp boundary has been considered. It has been found that if the parameters of the confinement potential satisfy a special condition then the electron can be strongly localized in such quantum dot. Also the energy spectra of an electron in a graphene quantum ring has been analyzed. Furthermore, it has been shown that in a double dot system some energy states becomes strongly localized with an infinite trapping time. Such states are achieved only at one value of the inter-dot separation. Also a periodic array of quantum dots in graphene have been considered. In this case the states with infinitely large trapping time are realized at all values of inter-dot separation smaller than some critical value. INDEX WORDS: Nanoplasmonic, Nanooptics, Nanostructures, Surface plasmon polaritons, Localized surface plasmons, Nanoantennas, Ultra fast nanostructures, Graphene, Graphene quantum dots.

THEORY OF ELECTRONIC AND OPTICAL PROPERTIES OF NANOSTRUCTURES by PRABATH S. HEWAGEEGANA A Dissertation Submitted in Partial Fullfilment of the Requirements for the Degree of Doctor of Philosophy in the College of Arts and Sciences Georgia State University 2008

Copyright by Prabath Sunjeewa Hewageegana 2008

THEORY OF ELECTRONIC AND OPTICAL PROPERTIES OF NANOSTRUCTURES by PRABATH S. HEWAGEEGANA Committee Chair: Dr. Vadym Apalkov Committee: Dr. Unil Perera Dr. Nikolaus Dietz Dr. Ramesh Mani Dr. Douglas Gies Dr. Richard Miller Electronic Version Approved: Office of Graduate Studies College of Arts and Sciences Georgia State University December 2008

iv To Loving... Wife Duleeka, Daughter Nihinsa and My Parents

v Acknowledgments I would like to thank my adviser Dr. Vadym Apalkov for his excellent guidance and great help throughout the process of this research. He has closely followed this work and influenced it with many ideas and recommendations. Discussions have often led to find new perspectives or deeper insight into subject. He is a person who has both knowledge and great sense of humor, which is something we couldn’t see often. My special thanks to Dr. Unil Perera (Associate chair/Physics Graduate Director) for his support through out my graduate studies. Most importantly, I have to thank my family for their great support during the time of my graduate studies in United State of America. Last but not least, I am very thankful to my wife Duleeka and my daughter Nihinsa for their understanding for the little time I was able to spend with them during the past years.

vi Table of Contents Acknowledgments v List of Tables ix List of Figures ix 1 Fundamentals of nanoplasmonics 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Nanolocalization of energy . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Maxwell’s equations and electromagnetics of metal . . . . . . . . . . . 8 1.4 Characteristic lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Nonlocal dielectric constant and Landau damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Local dielectric constant . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6.1 Drude-Sommerfeld theory . . . . . . . . . . . . . . . . . . . . 16 1.6.2 Interband transitions . . . . . . . . . . . . . . . . . . . . . . . 19 1.6.3 Size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Plasmon modes in metal nanoparticles: Particle plasmons 23 2.1 Plasmon resonance of small metal particles . . . . . . . . . . . . . . . 23 2.2 Particle plasmon oscillations: Simple semi-classical model . . . . . . . 26 2.3 Particle plasmons in a metal nanosphere . . . . . . . . . . . . . . . . 28 2.4 Scattering by a small metal sphere . . . . . . . . . . . . . . . . . . . 34 2.5 Beyond the quasi-static approximation . . . . . . . . . . . . . . . . . 37 3 Surface plasmon polaritons in nanostructured systems 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Mathematical description . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Surface plasmon polaritons at a metal-dielectric interface . . . . . . . 45

vii 3.3.1 Equations and solutions . . . . . . . . . . . . . . . . . . . . . 45 3.4 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Surface plasmon polaritons at a metal-vacuum interface . . . . . . . . 54 3.6 Propagation length, skin depth and plasmon lifetime . . . . . . . . . 57 4 Enhancement of optical sensitivity of photodetectors 63 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Main system of equations . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Intensity distribution within the active region of photodetector . . . . 75 4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.1 Average intensity . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.2 Intensity distribution and the modes of the diffraction grating 90 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5 Plasmonic enhancing nanoantennas for photodetection 99 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 Nanoantennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Chain of metal nanospheres . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 Tapered nanoplasmonic waveguide . . . . . . . . . . . . . . . . . . . . 106 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6 Ultrafast optical responses of metal nanostructures 111 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Strong-field electron emission . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7 Nanolocalized nonlinear electron photoemission under coherent con- trol 124 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.2 Two photon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8 Electronic properties of graphene 135 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2 Bonding in Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3 Electronic band structure: Nearest-neighbor tight-binding model . . . 140 8.4 Effective mass description: k-p model . . . . . . . . . . . . . . . . . . 145

viii 9 Electron localization in graphene quantum dots 149 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.2 Main equations: single quantum dot . . . . . . . . . . . . . . . . . . . 157 9.3 Strongly localized states in quantum dots . . . . . . . . . . . . . . . . 161 9.4 Main equations: Quantum ring . . . . . . . . . . . . . . . . . . . . . 169 9.5 Quantum ring: Fine tuning of the trapping time . . . . . . . . . . . . 172 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10 Trapping of an electron in coupled quantum dots in graphene 179 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 10.2 Double quantum dot system: Main equations . . . . . . . . . . . . . 181 10.3 Double quantum dot system: Results and discussion . . . . . . . . . . 185 10.4 Array of quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Bibliography 198

ix List of Tables 9.1 The heights of the confinement potential, V0 , at which the electron can be strongly localized, is shown for a few lowest values of n and i. The potential satisfies Eq. (9.11). The potential strength is in units of meV. The radius of the quantum dot is R = 50 nm. . . . . . . . . . . . . . 163

x List of Figures 1.1 Real part (a) and imaginary part (b) of dielectric permittivity for silver as function of excitation frequency ω. The dashed curves represent a fit to Drude formula Eq. (1.21) with parameters ωp = 9.24 eV, Γ = 0.02 eV and εh = 1. The solid curves display the experimental data from [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 (color) Real part (a) and imaginary part (b) of dielectric permittivity for silver as a function of excitation frequency ω for different radii as indicated in colors. Here, the size effect has been considered and εa = 1 and A=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 (color) The first nanotechnologists, Ancient stained-glass makers knew that by varying tiny amount of gold and silver in the glass, they could produce the different colors in stained-glass windows. Middle panel shows the size and shape effect to the different colors. From [21]. . . . 24 2.2 (color) Ancient Roman Lycurgus cup (4th century AD, now at the British Museum in Landon). The colors originates from metal nanopar- ticles embedded in the glass. The Lycurgus cup illuminated by a light source from behind [panel (a)]. Light absorption by the embedded gold or silver nanoparticles leads to a red color of the transmitted light whereas scattering by the particles yields greenish colors [panel (b)]. From [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Schematic of metal nanosphere subjected to electric field of an external electromagnetic wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 The metal nanosphere with dielectric permittivity εm (ω) is illumi- nated by z-polarized plane wave. The dielectric constant of embedding medium is εa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xi 2.5 Local electric field distributions around a silver nanosphere with a 10nm radius. The polarization of the excitation radiation is linear in the z direction. Panels (a)-(d): For frequencies indicated, magnitudes |E| of the local electric field, calculated from Eqs. (2.11) and (2.12), are displayed as density plots. The corresponding density scales are shown in the right hand side of the each panel. . . . . . . . . . . . . . 32 2.6 (color) Scattering cross section, σsca , of a silver nanosphere in vacuum for different radii (as indicated). Here the σsca is normalized by a6 . . . 35 2.7 (color) Absorption cross section, σabs , of a silver nanosphere in vacuum for different radii (as indicated). Here the σabs is normalized to volume, V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8 Schematic of radiative (a) and non-radiative (c) decay of localized sur- face plasmons in noble metal nanoparticles. The non-radiative decay occurs via excitation of electron-hole pairs either within the conduc- tion band (intraband excitation) or between the d band and the sp conduction band (interband excitation). . . . . . . . . . . . . . . . . . 38 3.1 The geometry, used to specify the propagation of surface polaritons at an interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 (a) Interface between metal and dielectric media with dielectric func- tions εm and εd . The interface is defined by z = 0 in a Cartesian coordinate system, and (b) shows the penetration into each medium (skin depth). A complete discussion of skin depth will be given at the end of this chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 The relationship [see Eq. (3.19)] between ω vs. ky for silver (εm ) and a dielectric (εd ) interface. Where εd = 1 for the solid curve and εd = 10 for the dashed curve. (a) Calculated real part of the permit- tivity for silver. Here horizontal lines represent the surface plasmon frequencies. (b) Same plot as (a), but including the imaginary part of the permittivity for silver. . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 (color) Distribution of local electric fields shown in yz plane, which is normal to the interface with y (horizontal coordinate) being the direction of surface plasmon polariton propagation, and z (the vertical coordinate) being normal to the surface. The magnitude is coded by colors as shown in a bar at the bottom. For panels (a), (b), (c) and (d) the frequency is 2.01 eV, for (c) and (d) it is 3.0 eV, as indicated. 55

xii 3.5 Vector diagrams of local electric fields in the yz plane, which is same coordinate system as Fig. 3.4. The spatial scale and the corresponding frequencies are indicated in the figure. . . . . . . . . . . . . . . . . . 56 3.6 Spatial quality factor Q as a function of frequency for two different dielectric media, i.e., εd = 1 and 10 as indicated in the figure. The metal is silver, here the experimental data from [17] have been used for the dielectric constant of silver. . . . . . . . . . . . . . . . . . . . 58 3.7 Im[k] as a function of ω for two different dielectric media, i.e., εd = 1 and 10 as indicated in the figure. The metal is silver, here the experimental data from [17] have been used for the dielectric constant of silver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8 Propagation length Lsp for (a) silver-vacuum interface, where εd = 1. and (b) silver-semiconductor interface, where εd = 10. . . . . . . . . 60 3.9 Skin depth, δm (dotted line) and decay length δd (dashed line) for a dielectric, as a function of energy at (a) silver-vacuum interface, where εd = 1. and (b) silver-semiconductor interface, where εd = 10. . . . . 61 3.10 Lifetime of surface plasmons τ (ω), computed from the experimental data of Ref.[17]. Panel (a) shows the data for silver and panel (b) for gold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1 Schematic illustration of the metal grating on the surface of a dielec- tric medium. Here, grating period is d and the grating height is h. The grating consists of periodic strips of metal with dielectric constant εm (ω) and air with dielectric constant εI = 1. The width of a metallic strips is a. The region III is filled by a material with dielectric con- stant εd . Here E is the electric field vector and H is the magnetic field vector. The angle θ is the incident angle. . . . . . . . . . . . . . . . . 68 4.2 Calculated average intensity, Iav , for s-polarized light at distance 10 nm below the grating (i.e., in region III) in units of I0 for different a/d values as indicated in the panel. For all the panels the grating period d = 2µm, the incident angle θ = 450 , and the height of the diffraction grating h = 50nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 The same plot as Fig. 4.2 but for p-polarized light (a) for a = 0.4d and (b) for a = 0.7d. For all the panels the grating period d = 2µm, the incident angle θ = 450 , and the height of the diffraction grating h = 50nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xiii 4.4 The peak frequency, ωpeak , at the maximum value of average intensity is shown as a function of a/d for both s- and p-polarization. The grating period d = 2µm, the incident angle θ = 450 , and the height of the diffraction grating h = 50nm. . . . . . . . . . . . . . . . . . . . . 81 4.5 The peak wavelength, λpeak , is shown as a function of grating period d, for both p- and s-polarization. The ratio a/d = 0.5 and the incident angle θ = 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 The average intensity, Iav , at distance z = 50 nm below the grating (i.e., in region III) in units of I0 for different incident angle θ as indi- cated in the panels. The panels (a)-(c) corresponds to p-polarization and (d)-(f) corresponds to s-polarization. For all the panels the grating period d = 2 µm and the ratio a/d is 0.5. . . . . . . . . . . . . . . . . 84 4.7 The maximum average intensity (peak value), Iav(max) , for the p- and s-polarized light at distance z = 10nm below the grating (i.e., in region III) is shown as a function of incident angle, θ. The intensity is in the units of I0 . The grating period d = 2µm, the ratio a/d = 0.5, and the height of the diffraction grating h = 50nm. . . . . . . . . . . . . . . . 86 4.8 The maximum average intensity (peak value), Iav(max) , for the p-polarized light (scaled by a factor of 5) and s-polarized light at distance z = 10 nm below the grating (i.e., in region III) is shown as a function of incident angle θ. The intensity is in units of the intensity of the inci- dent light The grating period d = 2µm, the ratio a/d = 0.5, and the height of the diffraction grating h = 50 nm. . . . . . . . . . . . . . . . 87 4.9 (a) The average intensity, Iav , for p-polarized light at distance z = 50nm below the grating (i.e., in region III) is shown in units of I0 as a function of h for a/d = 0.5 (a) at ω = 68 THz for θ = 450 and (b) at ω = 67.8 THz for θ = 50 . For all the panels the grating period d = 2µm. 88 4.10 The average intensity, Iav , for s-polarized light at distance z = 50 nm below the grating (i.e., in region III) is shown in units of I0 as a function of h (a) at ω = 75 THz and a/d = 0.5 and (b) at ω = 69 THz and for a/d = 0.3. For all the panels the grating period d = 2µm and the incident angle θ = 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xiv 4.11 (color) The distribution of electric field intensity for p-polarized light is shown in region III. (a) Intensity I(x, z) at peak frequency, ω = 67.95 THz, for θ = 450 and a = 0.3d. (b) The same plot for a = 0.5d. (c) Similar plot for θ = 50 and a = 0.5d. (d) Electric field intensity in the vicinity of the first peak in panel (c) is shown in the x − z plane. The scale of the intensity is indicated by the color bar on the top. For all the panels the grating period d = 2µm and the height of the diffraction grating h = 50 nm. . . . . . . . . . . . . . . . . . . . . . . 91 4.12 (color) The distribution of electric field intensity for s-polarized light is shown in region III. (a) θ = 450 and a = 0.4d; (b) θ = 450 and a = 0.5d; (c) θ = 50 and a = 0.5d. (d) Electric field intensity in the vicinity of the first peak in panel (c) is shown in the x − z plane. The scale of the intensity is indicated by the color bar on the top. For all the panels the grating period d = 2µm and the height of the diffraction grating h = 50 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.13 (color) Distribution of I(x, y) is shown for the grating region (region II) and below the grating region (region III) for (a) p-polarization and (b) s-polarization. For all the panels: d = 2µm, a/d = 0.5, θ = 100 , and h = 50 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1 Local fields (absolute value relative to that of the excitation field) in the equatorial plane of symmetry for linear self-similar chain of three silver nanospheres. The ratio of the consecutive radii is = Ri+1 /Ri = 1/3; the distance between the surfaces of the consecutive nanospheres di,i+1 = 0.6Ri+1 . Inset: the geometry of the system in the cross section trough the equatorial plane of symmetry. . . . . . . . . . . . . . . . . 103 5.2 Local fields (absolute value relative to that of the excitation field) in linear symmetric self-similar chain of six silver nanospheres. The ratio of the consecutive radii is 3; the distance between the surfaces of the consecutive nanospheres 0.6 of the smaller sphere’s radius. Inset: the geometry of the system (in the cross section through the equatorial plane). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xv 5.3 (color) Geometry of the nanoplasmonic waveguide. The propagation direction of the surface plasmon polaritons is indicated by the arrow. Intensity of the local fields relative to the excitation field is shown by color. The scale of the intensities is indicated by the color bar in the center. (b) Local electric field intensity is shown in the longitudinal cross section of the system. The coordinates are indicated in the units of the reduced radiation wavelength in vacuum, =100 nm. The radius of the waveguide gradually decreases from 50 to 2 nm. . . . . . . . . 106 5.4 (color) Snapshot of instantaneous fields (at some arbitrary moment t=0): Normal component Ex (a) and longitudinal component Ez (b) of the local optical electric field are shown in the longitudinal cross section (x − z) plane of the system. The fields are in the units of the far-zone (excitation) field. . . . . . . . . . . . . . . . . . . . . . . . . 108 6.1 The geometry of the nanosystems in the cross section through the xz plane of symmetry: V-shape (a) and random planar composite (b). The units in x and z axes are are nm. The thickness of both the systems in the y direction is set to be 4 nm. . . . . . . . . . . . . . . 117 6.2 Temporal dependencies of local electric optical field (the y component, in units of excitation field E0 ) at the apex of V-shape for values of absolute phase ϕ indicated in panels. The black squares denote the temporal points contributing to the current in Eq. (6.2) where the Θ function arguments are positive; the gray triangles denote the points that do not contribute to the photocurrent. The corresponding exci- tation pulses are shown in the insets at the upper right corners of the corresponding panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3 Distributions of photoelectron current density j(r) over surface of V- shape computed from Eq. (6.5) shown by gray-level density plot for the values of absolute phase ϕ indicated. The highest current is shown by the black and corresponds to the relative value shown at the top denoted “max”. The geometry of the system is shown on the plots by light gray shadows superimposed on the current distribution. The panel (a) corresponds to the maximum and (b) to minimum current depending on ϕ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.4 Total current J as a function of absolute phase ϕ for: (a) V-shape at ω0 = 1.55 eV, and (b) RPC at ω0 = 1.25 eV. The current J(ϕ) is plotted in relative units where its maximum is ascribed a value of 1. . 121 6.5 Same as in Fig. 6.3 but for RPC. . . . . . . . . . . . . . . . . . . . . 122

xvi 7.1 (a) Example of electric field of excitation pulse (τ = 5 fs, T = 8 fs). (b) Geometry of nanosystem in the cross section through the xz plane of symmetry. The units in x and z axes are are nm; the thickness of the system in the y direction is 4 nm. . . . . . . . . . . . . . . . . . . 128 7.2 (color) Density of the integrated photoemission current in the plane of the nanosystem shown in Fig. 7.1, made of silver, for ω = 3 eV. Scales are in nm, the maximum current (in relative but consistent units) and the delay are shown above the corresponding plots. Color scale of the current density is shown to the right of the panels. . . . . . . . . . . . 131 7.3 (color) (a)-(d): Same as in Fig. 7.2 but for the V-shapes of the geometry shown in the plots where the angle of a V-shape is shown on the plots. (e): Integrated current density at the hot spots(relative units) as a function of delay τ at the openings of the two V-shapes, identified by color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.1 The lattice structure of graphene. The unit cell (as shown in figure) contains two carbon atoms from different sublattices denoted by A and B. Two primitive translation vectors are denoted by a1 and a2 , where τ1 , τ2 and τ3 are nearest neighbor vectors. . . . . . . . . . . . . . . . 136 8.2 Energy diagram for the first two s- and p-orbitals of Carbon. . . . . . 138 8.3 Orbital configurations of bonding in Carbon. From [105] . . . . . . . 139 8.4 (color) Electronic band structure of graphene. The conductance band touches the valence band at the K and K points. . . . . . . . . . . . 141 8.5 (color) Band structure of graphene in the vicinity of the Fermi level. . 143 8.6 Reciprocal lattice and first Brillouin zone of graphene (shaded). The vectors b1 and b2 are the basis vectors of the lattice, where K and K are Dirac points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.7 (color) Dispersion relations for (a) linear and (b) parabolic energy spec- tra. The relevant Pauli matrices are indicated. . . . . . . . . . . . . . 147

xvii 9.1 (color) Tunnelling through a potential barrier in graphene. (a) Schematic diagrams of the spectrum of quasiparticles in single-layer graphene. The pseudospin denoted by σ is parallel (antiparallel) to the direction of motion of electrons (holes), which also means that σ keeps a fixed direction along the red and green branches of the electronic spectrum. (b) Potential barrier of height V0 and width D. For the both panels E is the Fermi energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.2 The geometry of the graphene quantum dot. The radius of the dot is R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.3 The energy spectra of an electron in the graphene quantum dot is shown for different angular momenta, m (as indicated) in the complex energy plane. For all the panels ν0 = 20. . . . . . . . . . . . . . . . . 162 9.4 The density of states, g(ε) as a function of ε (for panel (b) in Fig.1), where ν0 = 20 and , m = 3/2. . . . . . . . . . . . . . . . . . . . . . . 168 9.5 The geometry of the graphene ring with the outer radius R and the inner radius a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.6 The imaginary part of the energy, Im[ε], of the quasilocalized state as a function of β for a graphene quantum ring for different values of angular momentum, m, and different strengths of confinement potential, ν0 (as indicated). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.7 The imaginary part of the energy, Im[ε], as a function of β for different values of ν0 (as indicated). The angular momentum is m = 1/2 for panel (a) and m = 3/2 for panel (b). (c) The Bessel function, J1 , and the Neumann function, Y1 , of the first order. . . . . . . . . . . . . . . 175 10.1 The geometry of the coupled graphene quantum dots. The quantum dots have the same radius R. Here d is the distance between the centers of the quantum dots (inter-dot distance). To characterize the position of point P, introduce the polar coordinates r1 , θ1 and r2 , θ2 for each quantum dot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10.2 (a)-(b) the imaginary parts of the energies, Im[ε], of the quasilocalized states are shown as functions of d/R for double quantum dot system in graphene for different values of angular momentum, m, (as indicated). (c)-(d) same diagram for the real part of the energy, Re[ε]. For all the panels ν1 = ν2 = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

xviii 10.3 The imaginary parts of the energies, Im[ε], of the quasilocalized states are shown as functions of d/R for double quantum dot system for angular momentum, m = 3/2, and different strength of confinement potentials, ν1 and ν2 (as indicated). . . . . . . . . . . . . . . . . . . . 189 10.4 The density of states, g(ε) as a function of ε for different values of d/R (as indicated). Here ν1 = ν2 = 20 and m = 3/2. . . . . . . . . . . . . 190 10.5 The geometry of the quantum dots array system. Inter-dot spacing is d and the radius of the dot is R. . . . . . . . . . . . . . . . . . . . . . 191 10.6 The imaginary part of the energy, Im[ε], is shown as a function of the wave vector, k for different values of d/R (as indicated). Here ν0 = 20 and m = 3/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

1 Chapter 1 Fundamentals of nanoplasmonics 1.1 Introduction The study of optical phenomena related to the electromagnetic response of met- als has been recently termed plasmonic or nanoplasmonic. Nanoplasmonics forms a major part of the fascinating field of nanooptics, which is a modern branch of op- tical science that explores how optical frequency radiation can be confined on the nanoscale, i.e., 1 − 100 nm (much smaller than the optical wavelength). Such nanolo- calized fields are due to the interaction processes (oscillation of polarization charges) between electromagnetic radiation and conduction electrons at metallic interfaces or in small metallic nanostructures, leading to an enhanced optical near field of sub- wavelength dimension. Such oscillations on the nanoscale are called surface plas- mons. I will introduce a qualitative description of surface plasmons and local fields

2 associated with them in next two Chapters. History has shown that there are two main components of plasmonics: surface plasmon polaritons and localized surface plasmons . However, in the earlier literature, separation was made rarely between the surface plasmon polaritons [see Chapter 3] which are electromagnetic waves and localized surface plasmons which are purely electric, nanolocalized mechanical oscillations, which will be discussed in Chapter 2. However, now nanoplasmonics is much more developed field in both theoretical and experimental studies. Therefore, there is a necessity to distinguish between them. Throughout this work I will consistently call a mode surface plasmon polariton when it propagates along a system at distances longer than its wavelength. In contrast, a mode is called a localized surface plasmon when its coherent propagation at distances comparable to wavelength does not occur. In that case, the magnetic component of the field, though generally not zero, does not define its properties and can be neglected in the theory [see Sec. 1.2 ]. In addition to that, throughout this work I will use the term surface plasmons to represent both surface plasmon polaritons and localized surface plasmons at the same time. The theoretical description of these surface waves was established around the turn of the 20th century. In particular, the form of radio waves propagating along the surface of a conductor was described by A. Sommerfeld in 1899 [1], and J. Zenneck in 1907 [2]. Also the observation of sudden intensity drop in spectra was observed by R. Wood back in 1902 [3]. However, that observation was not explained until mid-

3 century [4]. A couple of years later, loss phenomena associated with metallic surfaces were also recorded via the diffraction of electron beams at thin metallic foils [5]. Then it was linked with the original work on diffraction gratings in the optical domain [6]. By that time, the excitation of Sommerfeld’s surface waves with visible light using prism coupling had been achieved [7]. More importantly, a unified description of all these phenomena in the form of surface plasmon polaritons was established. The beginning of the modern nanooptics was pioneered by H. Bethe and C. J. Bouwkamp [8, 11]. In H. Bethe’s pioneering work, he showed that if there is a small hole (radius R), much smaller than radiation wavelength λ, i.e., R λ , in an ideal (with infinite conductivity) metal, there will be some tunneling of the electromagnetic radiation through this hole. The total amount of energy, Et , pass through such a small hole is depend on R as, Et ∝ R6 . This result demonstrates two characteristic features of nanooptics: • the possibility to localize optical fields on the nanoscale • the efficiency of the energy concentration is extremely small for R λ. The inefficiency of such a system can be explained by quantum mechanics. From the quantum mechanical point of view, the electromagnetic radiation passing through a small hole is due to tunneling of photons through a classically forbidden region. For the small hole the probability of such a tunneling is very small. Therefore, we need to find ways to improve the efficiency of the optical field concentration on the nanoscale.

4 The spatially-periodic structure of holes in a metal film has been considered as one of the solution, where coherent addition of amplitudes of fields from different holes causes giant enhancement of transmission [9, 10]. A negative side of such systems as concentrators of energy on nanoscale is that the total size of the system is still macro- or microscopic, while the ultimate goal of nanooptics is to operate exclusively on the nanoscale. Another effect that can dramatically improve the efficiency of the nanolocalization of energy is resonant enhancement. 1.2 Nanolocalization of energy It is worth to discuss the possibility of concentrating optical energy on the nanoscale although it is widely known that it is impossible to focus electromagnetic wave en- ergy in regions whose size is significantly less than a half wavelength. A Fabry-Perot resonator is a classical example which has a λ/2 length and ideally reflecting mirrors. Here, the electric field of the wave is zero at the mirrors and maximum in the center of the cavity. Meanwhile, the magnetic field is maximum at the mirrors and zero in the center. If one attempts to squeeze this wave by making the cavity shorter, this elec- tromagnetic mode will disappear. The physical reason for this is that the exchange of energy between the electric and magnetic fields takes a quarter of wavelength in space. This is a relatively slow, long-range process (compared to a nanoscale), which is due to the large term, speed of light, c in the Maxwell equations [see Eq. (1.2)]. Furthermore, this is the coupling term of the electric and magnetic components.

5 Therefore, now the question is, how it is possible to concentrate the electromag- netic energy on the nanoscale in nanooptics? The answer is very simple, it is impossi- ble. The optical energy concentrated on the nanoscale by the means of nanoplasmonics is not electromagnetic but purely electric-oscillation energy. Magnetic fields may be present but in contrast to the electric fields they do not play a significant role in the energy concentration. The reason is the following: light is in a sense one-handed when interacting with atoms of conventional materials. Out of the two field components of light (electric and magnetic) only the electric hand efficiently probes the atoms of a material. Meanwhile, the magnetic component remains relatively unused, due to the fact that, the interaction of atoms with the magnetic field component of light is nor- mally weak. In nanoplasmonics, the scale of energy concentrations is not limited by the λ which is few orders of magnitude larger than the nanoscale. The scale of energy concentrations is only determined by the characteristic size of the metal plasmonic nanosystem. However, there are principal limitations related to the spatial dispersion and Landau damping [see Sec. 1.5]. It is time to get a quick snap shot of the very basics of nanooptics to see what optics at the scale of a few nanometers makes perfect sense and is not forbidden by nature. The propagation of light in free space is described by the dispersion relation ω = c k where, k = 2 2 2 kx + ky + kz is three dimensional wavevector, ω is the angular frequency of a photon and c is speed of propagation in free space (speed of light).

6 According to the Heisenberg’s uncertainty relation: ∆x∆px ≥ /2 (1.1) where ∆x is the uncertainty in the spatial position of a microscopic particle in the x direction, ∆px is the uncertainty in the component of its momentum in the same direction and is the reduced plank constant. However, for photons, the momentum ∆p can be written as ∆p = ∆k. Now, for photons the Heisenberg’s uncertainty relation (for the x-direction) can be rewritten as ∆x ≥ (2∆kx )−1 . Therefore, spatial confinement is inversely proportional to the magnitude of wavevector components in the respective spatial direction, here x. The maximum possible kx is the total length of the k (= 2π/λ). Therefore, ∆x ≥ λ/4π which is very similar to the well-known expression for the Rayleigh diffraction limit. This result immediately shows that the spatial confinement that can be achieved is only limited by the spread of k in a given direction. The following mathematical trick (property) can be used to increase the spread of wavevector components. Starting with two arbitrary perpendicular directions in space, e.g. x and z, one can increase one wavevector component to a value beyond the total wavevector, at the same time requiring the wavevector in the perpendicular direction to become purely imaginary. However, we can still fulfill the requirement for the total length of the wavevector k to be 2π/λ. Let’s assume that we choose to increase the wavevector in the x-direction, at the same time the possible range of wavevectors in this direction is also increased. Therefore, the confinement of light is

7 no longer limited by ∆x ≥ λ/4π. However, to achieve this we have to sacrifice some- thing, which in this case is confinement in the z-direction, resulting from the purely imaginary wavevector component in this direction that is necessary to compensate for the large wavevector component in the x-direction. With a purely imaginary wavevec- tor component in the expression for a plane wave, we have exp(iκz) or mathematically equal expression exp(−|κ|z). Therefore, in one direction (positive z-direction) we have an exponentially decaying field, an evanescent wave, while in the opposite direction (negative z-direction) the field is exponentially increasing. However, exponentially increasing fields have no physical meaning, this immediately shows that this result does not make any sense for infinite free space! Instead of taking one medium, one can divide infinite free space into at least two half-spaces with different refractive indices. Now the exponentially decaying field in one half-space can exist without needing the exponentially increasing counterpart in the other half-space. Note that to fulfill the boundary conditions for the fields at the interface different solutions can be considered in each half-space. The conclusion of the above simple argument is the following: in the presence of an inhomogeneity in space, the Rayleigh limit for the confinement of light is no longer strictly valid, but in principle infinite confinement of light becomes, at least theoretically possible. This property is the basis of nanooptics.

8 1.3 Maxwell’s equations and electromagnetics of metal The interaction of metals with electromagnetic radiation is mainly dominated by the free conduction electrons in the metal. According to the simple Drude model, the electrons oscillate 1800 out of phase relative to the excitation electric field. Therefore, metals possess a negative dielectric constant at optical frequencies. As we know from everyday experience, for frequencies up to the visible part of the spectrum metals are highly reflective, i.e., they do not allow electromagnetic radiation to penetrate. Therefore, metals are traditionally used as cladding layers for the construction of resonators and waveguides for electromagnetic radiation at far-infrared frequencies. At low frequencies, due to the lack of penetration into the metal, one can assume the perfect or good conductor approximation. However, at higher frequencies towards the near-infrared and visible part of the spectrum, field penetration increases significantly. Finally, at ultraviolet frequencies, metals acquire dielectric character and allow the propagation of electromagnetic waves, albeit with varying degrees of attenuation, depending on the details of the electronic band structure. Alkali metals such as sodium have an almost free-electron-like response and thus exhibit an ultraviolet transparency. For noble metals such as gold or silver on the other hand, transitions between electronic bands lead to strong absorption in this energy range. The interaction between metals and electromagnetic fields can be firmly under-

9 stood from classical framework based on Maxwell’s equations. Even the metallic nanostructures, which have sizes on the order of a few nanometres, can be described without going into quantum mechanics, because the high density of free carriers re- sults in minute spacings of the electron energy levels compared to thermal excitations of energy kB T at room temperature. However, there are some important quantum effects in nanoplasmonics for which quantum electrodynamics is essential. Among them, notable are spontaneous photon emission by nanosystems and surface plasmon amplification by stimulated emission of radiation [13]. Let’s start with Maxwell’s equations which are two equations that describe the mutual interaction of electric E and magnetic H fields, 1 ∂B ∇×E=− , (1.2) c ∂t 1 ∂D 4π ∇×E = + jext , (1.3) c ∂t c where c is the speed of light in vacuum, and jext is the current of external (with respect to the system) charges. As one can see from the form of this equation, the Gaussian system of units have been used. This system is predominantly used in the fundamental literature. The SI system that is mostly used in engineering literature because it contains an extra unit (Ampere) that leads to the appearance in the equations of unphysical constants: permittivity and permeability of the vacuum. The two vectors, B and D are determined by the material composition of the system, field strengths, etc. These are the electric displacement D and magnetic

10 induction B. The physical field that determines the force, F that acts on a point charge is given by the Lorentz law e F = eE + v × B , (1.4) c where e is the charge and v is its velocity. From this it is obvious that the physical magnetic field is actually B, not H. For the plasmonics, this distinction between B and H is actually not important because in the plasmonics of metal nanosystems in optical spectral region, the magnetic field and induction coincide for any practical purposes, in other words the magnetic permeability µ = 1. However, the magnetic response at optical frequencies is the main subject of a newly developed area in nanooptics of so called left-handed systems or negative-index materials [14]. I will leave left-handed systems outside of the scope of this work. For more details about left-handed systems and applications see Chapter 9 in Ref.[14] and references therein. The other pair of Maxwell equations are the divergence equations, ∇.D = 4πρext , (1.5) ∇.B = 0 , (1.6) where ρext is the density of external (i.e., not belonging to the system) charges. Using Eq. (1.5) and Eq. (1.6) one can immediately obtain the continuity equation for the external current, jext ∂t ρext + ∇.jext = 0 (1.7)

11 In general, the external charges and currents are not present in optics, therefore, I will drop them from the Maxwell equations for the purpose of this work. However, any charges and currents induced in bulk or at the interfaces by the fields are automatically taken into account by the Maxwell equations through the dielectric polarization and magnetic induction. Assuming that fields are weak enough that D and B can be obtained by per- turbation theory, one can obtain integral linear-response relations in terms of the corresponding fields (i.e., relations between D and E, and B and H) ∞ D(r, t) = dt ε(r − r , t − t )E(r , t)dr (1.8) −∞ V ∞ B(r, t) = dt µ(r − r , t − t )H(r , t)dr (1.9) −∞ V where ε(r − r , t−t ) (dielectric function) and µ(r − r , t−t ) (magnetic permeability) denote the response functions in space and time. The displacement D at time t depends on the electric field at time t previous to t (temporal dispersion or frequency dispersion). In addition to that, the displacement at point r also depends on the values of the electric field at neighboring point r (spatial dispersion). A spatially dispersive medium is also called a non-local medium. This effect can be observed at interface between different media or in metallic objects with sizes comparable with the mean-free path of electrons. In most cases of interest, the effect of the spatial dispersion is very weak, therefore, we can assume that the materials of the system are isotropic. Otherwise, both ε and µ would have been tensors, which would make no difficulty in principal but would make the formulas somewhat more complicated.

12 However temporal dispersion is a widely encountered phenomenon and it is important to take it into account accurately. As we discussed above the relations (1.8) and (1.9) are non-local both in space and time. However, one can use a mathematically equivalent description in the Fourier domain which is local, D(k, ω) = ε(k, ω)E(k, ω), B(k, ω) = µ(k, ω)H(k, ω). (1.10) Here we have introduced the corresponding arguments in the Fourier domain which are wave vector k and frequency ω. Therefore, the Fourier transform of the electric field is defined as E(k, ω) = E(k, ω) exp(ik.r − iωt)drdt (1.11) and of course one can get similar expressions for other quantities. 1.4 Characteristic lengths When the characteristic size of the particle becomes comparable to the character- istic scale of the system, the spatial dispersion becomes much more important. One example of such a scale is the lattice constant in metals which is on the order of the electron wavelength at the Fermi surface λF ∼ 1˚. Another important scale in A nanooptics is the Debye screening radius (or Thomas-Fermi screening radius) εh EF rD = , (1.12) 6πne2

13 where εh is the background dielectric constant of the metal that is due to the core (valence) electrons and ion motion (phonons), n is the concentration of electrons in c.g.s units and EF is the electron Fermi energy. 2 2 kF EF = , (1.13) 2m∗ √ 3π 2 n is the electron wavevector at the Fermi surface and m∗ is electron 3 where kF = effective mass in the metal. Using the Bohr radius, aB = εh 2 /m∗ e2 and kF , one can rewrite Eq. (1.12) for most metals, including noble metals, as rD 2.9 × 105 (n−1/6 ) 1˚. A (1.14) The correlation length, lc : that an electron at the Fermi surface travels during a period of the optical radiation, is another important scale for optical interactions in a metal system, which is on the order of ≈ vF /ω, where vF is the electron speed at the Fermi surface; for metals vF ∼ 108 cm/s, and ω ∼ 1015 s−1 is the optical frequency. This yields an estimate lc ∼ 1 nm which is the largest spatial scale compared with the other two scales. Therefore, when a characteristic size a of the nanosystem becomes small, it may become comparable to lc which will make important the nonlocality in the optical responses of the electron system [15].

14 1.5 Nonlocal dielectric constant and Landau damping The well known Lindhard formula [16] is one of the closed solutions in the theory of the Fermi system that explicitly gives the nonlocal dielectric response ε(k, ω). 2 3εh ωp ω ω + kvF ε(k, ω) = εh + 1− ln , (1.15) 2 k 2 vF 2kvF ω − kvF where ωp is the plasma frequency which is defined as 4πne2 ωp = . (1.16) εh m∗ The complex function in Eq. (1.15), ln ω+kvF ω−kvF is defined as ln ω+kvF ω−kvF =ln| ω+kvF |−iπ ω−kvF ω+kvF for ω−kvF < 0. Note that ε(k, ω) has a non-zero imaginary part, which describes optical losses, only when ω < kvF . These optical losses can be connected to the excitation by a field of incoherent electron-hole pairs. This phenomenon is called Landau damping, which is described by Eq. (1.15). The Landau damping actually is dephasing, where coherent field oscillations are transformed into incoherent electron hole pairs, but the total energy of the system is not changed. Landau damping is fulfilled when the size of the system is comparable to or less than the correlation length, lc , then the condition ω < kvF is satisfied. Then Eq. (1.15) has an imaginary part, 2 3πεh ωp ω Im[ε(k, ω)] = (1.17) 2(kvF )3

15 which has the same order of magnitude as Re[ε(k, ω)]. This effect (Landau damping) is not a small effect even at its onset. As explicitly demonstrated in Eq. (1.17) it is obvious that the relaxation and losses in the electron system come only with a strong spatial dispersion (dependence on k). The Landau damping and the spatial dispersion are highly important for the nanoplasmonics, because they are the most pronounced for low-frequencies. In particular, it is in the small-size range of parameters where the nanoplasmonic effects are the strongest and most interesting. Even though these phenomena are very important for nanooptics, it is very difficult to take them into account because the expression for the dielectric response is relatively complex in the k space [see Eq. (1.15)]. Note that the boundary conditions at the surfaces of the nanostructure are to be imposed in the real r space. 1.6 Local dielectric constant The total dielectric constant of a metal is given by the sum of a contribution from the free electrons of the metal and a contribution from the interband transitions. The contribution from the free electrons is given by the Drude free electron model [see Eq. (1.21)].

16 1.6.1 Drude-Sommerfeld theory Here we assume that particles are non-interacting and the background dielectric constant is εh . For large systems where system size lc , one can obtain the Drude- Sommerfeld model for the free-electron gas. Therefore, the equation of motion of such a particle (electron) is [23] m∗ ∂t r + m∗ Γ∂t r = eE0 e−iωt 2 (1.18) where e and m∗ are the charge and the effective mass of the free electrons, E0 and ω are the amplitude and the frequency of the applied electric field and Γ is the relaxation constant (damping constant) of the electron motion. Note that the equation of motion contains no restoring force since free electrons are considered. The damping term, Γ, is very small compared to the frequency of the excitation field in plasmonic region, i.e., Γ ω. The presence of an electric field leads to a dipole moment d according to d = er. The cumulative effect of all individual dipole moments of all free electrons results in a macroscopic polarization per unit volume P = nd = ner, where n is the number of electrons per unit volume. Seeking a solution as r(t) = r0 e−iωt , from Eq. (1.18) one can write P as e2 n 1 P=− E . ∗ ω(ω + iΓ) 0 (1.19) m Using well known expression D = E0 + 4πP = ε(ω)E0, (1.20)

17 we can obtain the classical Drude formula 2 2 2 ε h ωp ε h ωp ε h ωp Γ εDrude (ω) = εh − = εh − 2 +i , (1.21) ω(ω + iΓ) (ω + Γ2 ) ω(ω 2 + Γ2 ) 4πne2 where ωp = m∗ is the volume plasma frequency. Because of its simplicity, this Figure 1.1: Real part (a) and imaginary part (b) of dielectric permittivity for silver as function of excitation frequency ω. The dashed curves represent a fit to Drude formula Eq. (1.21) with parameters ωp = 9.24 eV, Γ = 0.02 eV and εh = 1. The solid curves display the experimental data from [17]. classical Drude formula is commonly used in theoretical computations. Therefore, it is very important to understand its limitations in describing the experimental data for metals, especially noble metals. On the other hand silver is known to possess the lowest dielectric losses in the visible and near-infrared (near-ir) spectral area and has the best pronounced plasmonic behavior. Therefore above model can be used to describe the behavior of the dielectric constant of the silver metal. A comparison is

18 shown in Fig. 1.1 of both the experimental data [17] and a theoretical fit to these data. A relatively close relationship between the Drude formula and the experimental data for Re[ε(ω)] can be seen from the Fig. 1.1(a). Here the real part of the dielectric constant is negative. One obvious consequence of this behavior is the fact that light can penetrate a metal only to a very small extent since the negative dielectric constant √ leads to a strong imaginary part of the refractive index n = . In contrast to the Re[ε(ω)], the general agreement with Im[ε(ω)] is much worse. A quantitative agreement takes place only between 1 and 2 eV. The strong disagreement can be seen around ω ≥ 3.5 eV, and this is due to the electron transitions from the d- band to the conduction sp-band. These almost localized d electrons do contribute to the optical absorption in the blue and near-ultraviolet (near-uv) spectral region. However, they do not participate in the plasmonic oscillations. The cause of the strong deviation from the Drude behavior in the red to near-ir spectral region seen in Fig. 1.1 is not quite clear (the frequency still appears too high for the phonons to contribute significantly). A possible cause may be the Landau damping due to electron scattering from grain boundaries in the metal. Note that the condition of the Landau damping a lc = vF /ω is easier to fulfill in the red to near-ir region than for higher frequencies. Because the dielectric losses described by Im[ε(ω)] play a decisive role in the optical enhancement processes by metal plasmonic nanostructures, the Drude formula cannot be used, even for silver, for quantitative theories except, perhaps for a narrow spectral range 1 − 2 eV.

19 1.6.2 Interband transitions In real metals, besides the contribution of the free electrons, one must consider the contribution from interband transitions. For the noble metals such as silver and gold, the influence of interband transitions in the visible range has been considered. For silver and gold, the interband transitions are 4d → 5sp and 5d → 6sp, respectively, and the free electrons are in the 5s and 6s states, respectively [18]. The dielectric con- stant, ε(ω) of a real metal is given by the sum of the contribution for the free electron, εDrude (ω) (Eq. (1.21) and the contribution of the interband transitions, εib (ω), ε(ω) = εDrude (ω) + εib (ω). (1.22) Although the Eq. (1.21) gives quite accurate results for the optical properties of metals in the infrared regime, it has to be corrected in the visible range by the response of bound electrons. Bound electrons in metals exist, in lower-lying shells of the metal atoms. Using the same method for the free electrons, one can get the equation of motion for a bound electron as [18] m∗ ∂t r + m∗ Γ∂t r + βr = eE0 e−iωt . ˜ 2 ˜ ˜ (1.23) Here, m∗ is the effective mass of the bound electrons, which is in general different ˜ from the m∗ , Γ is the damping constant describing mainly radiative damping in the ˜ case of bound electrons, and β is the spring constant of the potential that keeps the electron in place. Using the same steps as before one can get the contribution of

20 bound electrons to the dielectric function ε h ωp 2 ˜ εib (ω) = εh + . (1.24) (ω0 − ω 2 2 ) − iΓω ˜ 4π˜ e2 n Here ωp = ˜ m∗ ˜ and n is the density of the bound electrons. ωp is introduced in ˜ ˜ analogy to the plasma frequency in the Drude-Sommerfeld model, however, obviously here with a different physical meaning and ω0 = β/m∗ . Finally, we can rewrite ˜ Eq. (1.24) to separate the real and imaginary parts εh ωp 2 (ω0 − ω 2 ) ˜ 2 ˜˜ ε h Γω p 2 ω εib (ω) = εh + +i ≡ Re[εib (ω)]+i Im[εib (ω)] (ω0 − ω 2 )2 + Γ2 ω 2 2 ˜ (ω0 − ω 2 )2 + Γ2 ω 2 2 ˜ (1.25) Finally we have the both contributions to the dielectric constant, namely, the effect of the free electrons through Eq. (1.21) and the effect of interband transitions through Eq. (1.25), 2 ε h ωp εh ωp 2 (ω0 − ω 2 ) ˜ 2 2 ε h ωp Γ ˜˜ ε h Γω p 2 ω ε(ω) = 2εh − + 2 +i + 2 . (ω 2 + Γ2 ) (ω0 − ω 2 )2 + Γ2 ω 2 ˜ ω(ω 2 + Γ2 ) (ω0 − ω 2 )2 + Γ2 ω 2 ˜ (1.26) 1.6.3 Size effects The dielectric constant discussed in previous section is related to bulk metals. Now the question is, how can we relate this when the dimension of the metal is decreased, i.e., nanoparticles, where the dielectric constant deviates from that for the bulk metal. When the size of the particle becomes smaller than the mean free path of the free electrons, the electrons collide with the boundary of the particle. There

21 have been many attempts to include this effect by adding the rate of the collisions with the surface of the nanoparticle [19, 20] to obtain the damping constant. If this effect is included, the damping constant is given by [28] vf Γ+A (1.27) a where a is the radius of the particle, and A is a dimensionless constant. The value of A depends on the particle shape and is usually around unity. (a) (b) Re[ε(ω)] Im[ε(ω)] 1 2 3 4 5 0 50 ћω (eV) 40 -50 a=2 nm 30 a=5 nm -100 a=10 nm 20 a=50 nm a=100 nm 10 -150 bulk 0 1 2 3 4 5 -200 ћω (eV) Figure 1.2: (color) Real part (a) and imaginary part (b) of dielectric permittivity for silver as a function of excitation frequency ω for different radii as indicated in colors. Here, the size effect has been considered and εa = 1 and A=1. The real part and imaginary part of the dielectric constant for silver is shown in Fig. 1.2. It is illustrated as a function of excitation frequency ω for different radii including the size effect. The experimental data from [17] has been used for the dielectric constant of the silver. Note that, here the dielectric constant of the ambient is 1 and A=1. As one can see there is no significant change in Re[ε(ω)], however, it is

22 a different story for Im[ε(ω)]. For the frequency, ω < 2 eV Im[ε(ω)] is significantly increasing with the decreasing a. However, as the size of a nanosystem decreases, the effects of the spatial dispersion (dependence on k) become more important (as we already know from the above- discussed Lindhart formula in Sec.1.5). For small nanosystems (a ∼ 1 nm), the enhancement of the losses (temporal dispersion) always comes together with a strong spatial dispersion. Again this can be explicitly illustrated by the phenomenon of Landau damping [see Eq. (1.15) and its discussion].

23 Chapter 2 Plasmon modes in metal nanoparticles: Particle plasmons 2.1 Plasmon resonance of small metal particles Here I introduce the fundamental excitation of localized surface plasmons in nanoparticles: known as particle plasmons. Instead of localized surface plasmons the term particle plasmon has been used for plasmons in nanoparticles. In nanooptics we are interested in establishing field confinement in two or even three dimensions. Therefore, it is useful to analyze theoretically the electromagnetic modes associated with small particles called nanoparticles. We will see in the Chapter 3 that surface plasmon polaritons are propagating, dispersive electromagnetic waves coupled to the electron plasma of a conductor at a dielectric interface and the electromagnetic field

24 Figure 2.1: (color) The first nanotechnologists, Ancient stained-glass makers knew that by varying tiny amount of gold and silver in the glass, they could produce the different colors in stained-glass windows. Middle panel shows the size and shape effect to the different colors. From [21]. is strongly localized in one dimension, i.e., normal to the interface. Localized surface plasmons on the other hand are non-propagating excitations of the conduction elec- trons of metallic nanostructures coupled to the electromagnetic field. We will see that these modes arise naturally from the scattering problem of a small, sub-wavelength metal nanoparticle in an oscillating electromagnetic field. The curved surface of the particle exerts an effective restoring force on the driven electrons, so that a resonance can arise, leading to field amplification both inside and in the near-field zone outside the particle. This resonance is called the localized surface plasmon or in case of small particle it is called particle plasmon. Gold and silver particles with diameters on the nanometer scale show very bright colors both in transmitted and reflected light. These resonantly enhanced absorption and scattering falls into the visible region of the electromagnetic spectrum. These

25 Figure 2.2: (color) Ancient Roman Lycurgus cup (4th century AD, now at the British Museum in Landon). The colors originates from metal nanoparticles embedded in the glass. The Lycurgus cup illuminated by a light source from behind [panel (a)]. Light absorption by the embedded gold or silver nanoparticles leads to a red color of the transmitted light whereas scattering by the particles yields greenish colors [panel (b)]. From [22]. bright colors of noble metal such as gold and silver, have fascinated people for many centuries. However, for long time it was not clear what causes theses colors. Today it is known that they are due to gold nanoparticles embedded in the glass for example [see Figs. 2.1, 2.2]. One example of a historical application is the staining of church windows [21] in the Middle Ages [see Fig. 2.1] or the beautiful Lycurgus cup [see Fig. 2.2] manufactured in Roman times, which is now at the British Museum in London [22].

26 2.2 Particle plasmon oscillations: Simple semi-classical model To understand the mechanism of resonant excitation of a collective oscillation of the conduction band electrons in the nanoparticles, consider the plasmon modes in metal nanoparticles. The metal nanosphere is subjected to an excitation field of an external wave with wave vector k. [see a schematic in Fig. 2.3]. +z Direction of polarization Electrons - k Lattice ions + 20 nm Figure 2.3: Schematic of metal nanosphere subjected to electric field of an external electromagnetic wave. Using the following simple semi-classical model, one can qualitatively understand many properties of particle plasmons. Since the skin depth of electromagnetic waves in metals is on the order of the diameter of the nanoparticle (the skin depth at optical frequencies is 30nm for gold and silver), the excitation light is able to penetrate the particle. The field inside the particle shifts the conduction electron with respect to

27 the fixed positive lattice ions, and electrons build up a negative charge on one side of the surface and the lattice ions build positive charge on the opposite side [see Fig. 2.3]. This shift is greatly exaggerated in the figure, for realistic fields it is much smaller. There is another way of analyzing the uniformly polarized nanosphere, which nicely describes the idea of a bound charge [23, 24]. What we have, actually, is two spheres of charge: a positive sphere and a negative sphere. In the absence of the polarization the charges of electrons and lattice overlap in the bulk of the nanosphere that stays electrically neutral. But when the material is uniformly polarized (as shown in Fig. 2.3), all the positive charges shift slightly toward the −z direction, and all the negative charges move slightly toward the +z direction. Now the two spheres no longer overlap perfectly: at the surface there are crescents of uncompensated negatively charged electron and positively charged lattice charges. The reason is called surface, the charges whose density −∇P are all at the surface [23, 24] while the dielectric polarization P is uniform within the volume of the metal nanosphere. The attraction between these uncompensated surface charges generates restoring force between the electrons and lattice ions along with the mass of the electron forms a mechanical oscillator, which is a particle plasmon or localized surface plasmon. In general this force depends on the separation of the surface charges, i.e., particle size, and the polarizability. More importantly the restoring force is directly related to the resonance frequency of the system. The alternating surface charges form an oscillating dipole, which radiates electromagnetic waves. This simple semi-classical

28 model for particle plasmons can be considered as optical antenna [25]. At the same time that system leads to another effect which is the dipole field in the surrounding medium. This dipole electric field is the local plasmonic field. This plasmonic field is localized in the region on the scale of the radius of the nanosphere which is much smaller than the excitation wavelength. 2.3 Particle plasmons in a metal nanosphere In order to keep the analysis simple and more attractive, the discussion will be limited to the quasi-static approximation which neglects retardation: I assumed that all points of an object respond simultaneously to an excitation field. This is a very good approximation if the characteristic size of the object is much smaller than the excitation wavelength. As we discussed in Chapter 1, fields at the nanoscale are governed by quasistatic equations, therefore, the Earnshow theorem is applicable: fields and any of their components (x, y, orz) can have a local or global extremum (minimum or maximum) only at the surfaces or interfaces. In this approximation the Helmholtz equation reduces to the Laplace equation which is much easier to solve. In general the electric field of an oscillating dipole is given by [26], 1 eikr 1 ik E(r, t) = k 2 (r × d) × r + [3r(r.d) − d] 3 − 2 eikr eiωt (2.1) 4πεa r r r

29 where, d is the dipole moment, in the near field zone (kr 1). Equation (2.1) can be approximated to [26] 1 eiωt E(r, t) = [3r(r.d) − d] 3 . (2.2) 4πεa r In other words, this is exactly the electrostatic field of a point dipole. This has only time oscillating term, eiωt , that is why, it is termed quasi-static. In the quasi-static limit potential, φ, has to satisfy the following equations. E = −∇φ, ∇2 φ = 0. (2.3) Let us consider a nanosphere with radius a centered at the origin. The sphere is illuminated by an z-polarized plane wave. The geometry is shown in Fig. 2.4. First z a m y Figure 2.4: The metal nanosphere with dielectric permittivity εm (ω) is illuminated by z-polarized plane wave. The dielectric constant of embedding medium is εa . express the Laplace equation (2.3) in spherical coordinates (r, θ, ϕ) as [23] 1 ∂ ∂ ∂ ∂ 1 ∂2 sin θ r2 + sin θ + φ(r, θ, ϕ) = 0. (2.4) r 2 sin θ ∂r ∂r ∂θ ∂θ sin θ ∂ϕ2

30 The solutions are of the form [23] φ(r, θ, ϕ) = A ,m φ ,m (r, θ, ϕ). (2.5) ,m Here, the A ,m are constant to be determined from the boundary conditions and the φ ,m are of the form [23] ⎧ ⎫⎧ ⎫⎧ ⎫ ⎪ ⎪ r ⎪⎪ ⎪⎪ P ⎪⎪ ⎪ ⎨ ⎬⎨ m (cos θ) ⎪ ⎪ eımϕ ⎬⎨ ⎪ ⎬ φ ,m = , (2.6) ⎪ − −1 ⎪ r ⎪ ⎪ Q m (cos θ) ⎪ ⎪ e−ımϕ ⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭⎩ ⎭ where the P m (cos θ) are the Legendre polynomials and the Q m (cos θ) are the Leg- endre polynomials of second kind [23]. Using the proper boundary conditions and the continuity of the tangential electric fields and the normal components of the displacement vector at the surface of the sphere imply that ∂φin ∂φout = (2.7) ∂θ r=a ∂θ r=a ∂φin ∂φout εm = εa (2.8) ∂θ r=a ∂θ r=a where φin , is the potential inside the sphere and φout , is the total potential, i.e., potential of the incoming field, φ0 , and the scattered fields, φscatter , (φout = φscatter + φ0 ). For the incoming electric field, it has been assumed that the homogeneous z- polarized plane wave and potential is φ0 = −E0 z = −E0 r P1 (cos θ). Finally the 0 solutions are [23, 24] 3εa φin = −E0 r cos θ (2.9) εm + 2εa εm − εa 3 cos θ φout = −E0 r cos θ + E0 a (2.10) εm + 2εa r2

31 and the electric filed can be calculated from first equation in Eq. (2.3) 3εa 3εa Ein = E0 (cos θ ˆ − sin θ θ) = E0 r ˆ x; for r ≤ a ˆ (2.11) εm + 2εa εm + 2εa a3 εm − εa Eout = E0 (cos θ ˆ − sin θ θ) + E0 3 r ˆ ˆ (2 cos θ ˆ + sin θ θ); for r > a. (2.12) r r εm + 2εa The Eq. (2.11) describes the uniform electric field at point r, where r is inside the nanosphere and outside it is a superposition of the uniform excitation field [first part of Eq. (2.12)] and dipolar local field [second part of Eq. (2.12)]. Following the above simple theory, the magnitude of the local optical electric field, |E|, in the vicinity of a silver nanosphere of radius a = 10 nm is displayed in Fig. 2.5, where εa = 1. Here the local fields are represented by bright areas on a dark background. In panel (a) results are shown for a frequency of 1.51 eV, which is significantly off from the surface plasmon resonance. Note that the surface plasmon resonance frequency of a silver nanosphere in vacuum is 3.5 eV. As shown in panel (a) the maximum field enhancement in this case is obviously not very large, but still results in the local intensity enhancement which is proportional to |E|2 by one order of magnitude. The electric field penetrates the metal very weakly due to its high dielectric permittivity at low frequency [see Fig. 1.1]. This field is concentrated to radii of order a, with the maximum at the surface of the metal. The spatial distribution of the field is clearly elongated in the z-axis, which is the direction of the polarization. The same plot for frequency of 3.37 eV is shown in Fig. 2.5(b), which is close to the surface plasmon resonance frequency. The maximum field enhancement in this case is not large, but still results in a local intensity enhancement by two orders of

32 Figure 2.5: Local electric field distributions around a silver nanosphere with a 10nm radius. The polarization of the excitation radiation is linear in the z direction. Panels (a)-(d): For frequencies indicated, magnitudes |E| of the local electric field, calculated from Eqs. (2.11) and (2.12), are displayed as density plots. The corresponding density scales are shown in the right hand side of the each panel.

33 magnitude. The spatial distribution of the field has the same elongated behavior in the direction of polarization. The panel (c) shows the same plot for a frequency of 3.5 eV which is the surface plasmon resonance frequency for a silver nanosphere in vacuum [17]. In this case the local intensity is enhanced by a factor of ≈ 400, which is due to the resonance of the excitation field with the surface plasmon oscillations of the conduction electrons. This high enhancement can be related to the high value −Re[ε] of the resonance quality factor Q for silver , which is defined as: Q = Im[ε] . One can clearly see that the field penetrates metal significantly which is an indication of breakdown of the qualitative behavior of the metal as an ideal conductor at the surface plasmon resonance frequency. At the surface plasmon frequency, the local fields are also localized, with a maximum at the surface; the localization radius is on the order of radius a of the nanosphere. Now consider the case, for a frequency slightly higher than the surface plasmon resonance frequency. This situation is displayed in Fig. 2.5(d), and the picture is significantly different: the field is mostly concentrated in the metal. In contrast to the case of the plasmon resonance frequency, here it is enhanced by relatively small factor, i.e., a factor of ≈ 4, as expected. Using the well known relation, the electric fields can be combined with optical polarizability, α(ω) [24]. The excitation optical fields E0 induce a dipole moment d oscillating with optical frequency on the nanosphere (a λ). Therefore, εm (ω) − εa d = εa α(ω) E0 , α(ω) = a3 . (2.13) εm (ω) + 2εa In applications the dispersion (frequency dependence) of the dielectric medium

34 surrounding the metal can be ignored and one can assume a constant εa . However, the dielectric function of the metal is highly dispersive. Therefore the solution for the fields is characterized by the denominator εm + 2εa . The α(ω) has maximum value when the real part of its denominator vanishes, which defines frequency ωsp of the surface plasmon resonance through an equation Re εm (ωsp ) = −2εa . (2.14) 2.4 Scattering by a small metal sphere The scattering problem for an arbitrary sphere has been solved exactly by Mie [27]. However, the quasi-static approximation is more appropriate if one wishes to obtain a clear physical insight. The scattering cross-section of the sphere is obtained by driving the total radiated power of the sphere’s dipole by the intensity of the excitation plane wave. The time average of the energy flow can be expressed by the real part of the ˜ Poynting vector, S [23] ˜ 1 S = Re Eθ × H∗ r − Er × H∗ θ . ˆ φ ˆ φ (2.15) 2 Here H is magnetic field and H∗ is the complex conjugate of H. Starting with Eq. (2.2) and using Eq. (2.13), the total radiation power, P can be written as [23] 2π π ωk 3 |d2 | ωk 3 εa P = |S|r 2 sin θdθdφ = ˜ = |α(ω)|2E0 2 . (2.16) 0 0 12πεa 12π

35 -6 σsca/a6 (10 nm-4 ) a=5 nm a=10 nm 60 a=20 nm a=50 nm a=100 nm 40 20 0 1 2 3 4 ω (eV) Figure 2.6: (color) Scattering cross section, σsca , of a silver nanosphere in vacuum for different radii (as indicated). Here the σsca is normalized by a6 . ˜ From the Eq. (2.15), the energy density of the excitation field S0 can be written as [18] ωεa 2 S0 = E0 . (2.17) 2k Therefore, the scattering cross-section σsca is given by [18] P k4 σsca = = |α(ω)|2, (2.18) ˜0 S 6π The power loss from the excitation field due to the presence of a particle is not only due to the scattering but also due to the absorption. Therefore, we need to take into account, the power that is dissipated inside the particle. The power dissipated by a point dipole can be expressed as (ω/2)Im[d.E∗ ] by the Poynting’s theorem [23]. 0 Assuming εa is real, then the absorption cross-section has the form

36 σabs = kIm[α(ω)]. (2.19) The spectra of the scattering cross section of silver spheres of various radii are shown -1 σabs/ V (nm ) 0.5 a=2 nm a=5 nm a=10 nm 0.4 a=20 nm a=50 nm 0.3 0.2 0.1 0 1 2 3 4 ω (eV) Figure 2.7: (color) Absorption cross section, σabs , of a silver nanosphere in vacuum for different radii (as indicated). Here the σabs is normalized to volume, V. in Fig. 2.6. The scattering cross section σsca has been normalized by a6 , because σsca is proportional to a6 . The maximum peak appearing around 3.5 eV is caused by the localized surface plasmon resonances. At this frequency the real part of the denominator of the polarizability is zero [see Eq. (2.14)]. Due to the high imaginary part of the dielectric constant of small spheres, low peak height can be seen for the small spheres. As expected, large particles have a low and broadened peak due to the retardation effect. The absorption cross section of silver spheres of various radii are shown in Fig. 2.7. The absorption cross section σabs has been normalized by the

37 volume of the sphere V for the same reason as for the scattering. The figure is some what similar to the spectra of the scattering, but here the absorption is due to the interband transition, which we discussed in Chapter 1. 2.5 Beyond the quasi-static approximation In this section I will briefly discuss the plasmon resonance of particles beyond the quasi-static regime. Here, the proprieties of polarizability will be discussed qualita- tively. The plasmon resonance of particles beyond the quasi-static regime is described by two competing processes [28] [see Fig. 2.8]: a radiative decay process into photons shown in Fig. 2.8(a) (this process dominates for large particles), where the quasi-static approximation-breaks down due to retardation effects. Radiative decay is caused by a direct radiative route of the coherent electron oscillation into photons [31], which is the main reason for the weakening in the strength of the dipole plasmon resonance as the particle volume increases [32]. The second process: non-radiative process is shown in Fig. 2.8(c). The non-radiative decay is a result of the creation of electron-hole pairs via either intraband excitations within the conduction band or interband transitions from lower-lying d-bands to the sp-conduction band (for noble metal nanoparticles), which is due to the Pauli exclusion principle: the electrons can only be excited into empty states in the conduction band. In other words, non-radiative decay is due to a dephasing of the oscillation of individual electrons. The first TM modes of Mie theory for the polarizability αm of a sphere of volume

38 Figure 2.8: Schematic of radiative (a) and non-radiative (c) decay of localized surface plasmons in noble metal nanoparticles. The non-radiative decay occurs via excitation of electron-hole pairs either within the conduction band (intraband excitation) or between the d band and the sp conduction band (interband excitation). V , can be expressed as [29], (εm (ω)+εa )x2 1− 10 + O(x4 ) αm (ω) = 2 ε3/2 V V, (2.20) (εm (ω)+εa )x2 εm (ω) 3(εm (ω)−εa ) − 10 − i 4π 3 a λ3 + O(x4 ) where x = πa/λ is the so called size parameter, which combines the radius a with free-space wavelength λ. In contrast to the simple quasi-static solution Eq. (2.13), this expression has couple of additional terms, where each term has a distinct physical meaning to it. The x2 term in the numerator describes the effect of retardation of the exciting field over the volume of the sphere, leading to a shift in the plasmon res- onance. A similar term in the denominator includes an energy shift of the resonance, due to the retardation of the depolarization field [30] inside the particle.

39 Chapter 3 Surface plasmon polaritons in nanostructured systems 3.1 Introduction The aim in this Chapter is to present a discussion of the propagating part of the surface plasmon, which are called surface plasmon polaritons. Surface plasmon polari- tons are electromagnetic waves propagating at and bound to surfaces and interfaces between two different media [12]. The electromagnetic fields of surface plasmon po- laritons are typically localized to within a few wavelengths of a surface, in the sense that their amplitude is a maximum at the surface and decays exponentially away from it. Therefore, these waves are evanescent waves. These electromagnetic surface waves arise via coupling of the electromagnetic fields to oscillations of the conduc-

40 tor’s electron plasma. The specific properties of the surface polaritons depend on the characteristics of the material, normally as described by their dielectric function. Robert Wood back in 1902 reported the observation of sudden drop of the light intensity, scattered from a metallic diffraction grating, from maximum to minimum which occurred within a very narrow range of wavelength [3]. Wood was unable to explain his own results and therefore named them singular anomalies. However, now one of these anomalies is known to correspond to the excitation of surface plasmon polaritons. The coupling of light to these oscillations results in guided polariton modes that are confined and propagate along the interface [12]. The generation of surface plas- mon polaritons by incident light is forbidden for translationally invariant interfaces due to a mismatch between the momentum of the incident light and that of the sur- face plasmon polaritons at the same frequency, since the surface plasmon polaritons have greater momentum than a free-space photon. The general approach to provide additional momentum and satisfy the momentum conservation law for the coupling of the incident light and the surface plasmon polaritons is to introduce some inho- mogeneous structure at the interface. Such structures can be subwavelength defects (holes), or periodic corrugation, i.e., grating, on the metallic surface. Placing a grat- ing on top of the plasmon waveguide can facilitate an additional wave vector which is equal to a multiple of the grating vector [12]. The generation of surface plasmon polaritons and coupling with light has been discussed in detail in Ref. [12]

41 An important application of surface plasmon polaritons is related to the enhance- ment of an electro-magnetic field near a metal-dielectric interface due to the gener- ation of surface plasmon polaritons. Such enhancement opens up the possibility of manipulating the interaction strength between light and matter. One of the applica- tions of surface plasmon polaritons is strong signal enhancement in surface-enhanced Raman spectroscopy [36, 37], where a molecule is placed near the metallic nanostruc- ture. The enhancement of optical effects by generation of plasmon polaritons at the surface of small metallic objects was the topic of broad research not only in physics, but also in biology, chemistry, and material science. Recently, there has been a great deal of interest in studying the optical properties of semiconductor layered systems using grating couplers [38, 39, 40] with surface plasmon effects [55, 56]. Grating couplers are a widely used and promising tool in the semiconductor nano-systems to design optoelectronic devices [55, 56]. Taking the wave equation as a starting point, this chapter describes the fun- damentals of surface plasmon polaritons both at a single, flat interface and in a metal/dielectric interface. 3.2 Mathematical description In order to understand the physical properties of surface plasmon polaritons we can apply Maxwell’s equations, Eqs. (1.2)-(1.3) and (1.5)-(1.6) to the flat interface between a conductor and a dielectric. To present this discussion more clearly, it is

42 advantageous to cast the equations first in a general form applicable to the guiding of electromagnetic waves, the wave equation. Without the external charge and current densities [see Sec. 1.3], the curl equations Eq. (1.2) can be combined to yield [24] 1 ∂2D ∇×∇×E=− (3.1) c2 ∂t2 Using the identities ∇ × ∇ × E ≡ ∇(∇.E) − ∇2 E, ∇.[ε(r)E] ≡ E.∇ε(r) + ε(r)∇.E and taking into account the absence of external charge and current densities, ∇.D = 0 one can rewrite the Eq. (3.1) as 1 1 ∂2E ∇ − E.∇ε(r) − ∇2 E = −ε(r) 2 2 . (3.2) ε(r) c ∂t It has been shown in Chapter 1 that the variation of the dielectric profile ε = ε(r) over distances on the order of one optical wavelength is negligible. Therefore the Eq. (3.2) can be simplified to the following equation, ε ∂2E ∇2 E − = 0. (3.3) c2 ∂t2 In the homogeneous environment one can solve the Eq. (3.3) separately and the obtained solutions have to be matched using appropriate boundary conditions. To obtain the description of confined propagating waves, we assume in all generality a harmonic time dependence E(r, t) = E(r)e−iωt of the electric field. Then we have Helmholtz equation [24], ∇2 E + k0 εE = 0, 2 (3.4) where, k0 = ω/c is the wave vector of the propagating wave in vacuum.

43 z x y direction of propagation Figure 3.1: The geometry, used to specify the propagation of surface polaritons at an interface. Now it is time to define the propagation geometry. Consider the geometry shown in Fig. 3.1, where it has been assumed that there is a single interface at z = 0 separating two isotropic media. A one-dimensional problem has been considered for simplicity, i.e., ε depends only on one spatial coordinate. Also the direction of the wave propagation is taken along the y-axis, and shows no spatial variation in the perpendicular, in-plane x-direction [see Fig. 3.1]; therefore ε = ε(z). Then, the propagating waves, which travel along the interface, z = 0, can be described as [24] E(x, y, z) = E(z)eiky y . (3.5) The complex parameter ky , is the propagation constant of the traveling waves. The following form of the wave equation can be obtained by combining the Eq. (3.4) and Eq. (3.5). ∂ 2 E(z) + (k0 ε − ky )E = 0. 2 2 (3.6) ∂z 2 One can get similar equation for the magnetic field H. Using the following proper- ties: (1) harmonic time dependence; i.e., ∂ ∂t = −iω , (2) for propagation along the

44 ∂ ∂ y−direction; i.e., ∂y = iky and homogeneity in the x-direction, i.e., ∂x = 0, it can easily be shown that Eq. (3.6), allows two sets of self-consistent solutions with differ- ent polarization properties of the propagating waves. The first kind is the transverse magnetic (TM) electromagnetic mode solution, i.e., a mode where the magnetic field is perpendicular to the plane of propagation of the radiation field (also known as p-polarization, here the p stands for parallel), where only the field components Ey , Ez and Hx are nonzero. The second kind is the transverse electric (TE) electromagnetic mode solution, i.e., a mode where the electric field is perpendicular to the plane of propagation of the radiation field (also known as s-polarization, here the s stands for German word “senkrecht” meaning perpendicular), with only Hy , Hz and Ex being nonzero. Finally, for the TM modes the system of governing equations can be written as ∂ 2 Hx (z) 2 + (k0 ε − ky )Hx = 0, 2 2 (3.7) ∂z ky Ez = − Hx , (3.8) εω 1 ∂Hx Ey = − . (3.9) εω ∂z Similarly, for TE modes ∂ 2 Ex (z) 2 + (k0 ε − ky )Ex = 0, 2 2 (3.10) ∂z ky Hz = Ex , (3.11) ω 1 ∂Ex Hy = i . (3.12) ω ∂z

45 Now we have enough mathematical background to start the discussion of the proper- ties of surface plasmon polaritons at a metal-dielectric interface. 3.3 Surface plasmon polaritons at a metal-dielectric interface The necessary conditions for the existence of surface plasmon polaritons at the plane interface of two (m-metal and d-dielectric) semi-infinite isotropic media are that one of the two media separated by such an interface should have a negative dielectric permittivity, Re[ε] < 0 [see Sec.3.3.1], and the other should have a positive dielectric permittivity, . As we discussed in Chapter 1, in a wide spectral region from near- ultraviolet to mid-infrared [17] , the real part of the dielectric permittivity of metals including noble metals such as silver, gold,etc., is negative, Re[ε] < 0. Therefore, materials with Re[ε] < 0 can be treated as metals. 3.3.1 Equations and solutions In this Chapter, it has been assumed that the magnetic permeability µ = 1 and that the dielectric function is independent of the wave vector, k, i.e., the absence of spatial dispersion effects. This is true for many materials. The directions of co- ordinate axes are chosen as in Fig. 3.2, the z-axis is perpendicular to the media interface and z = 0, corresponds to the interface; the y-axis is directed along surface

46 z Ez Delectric ksp z SP Hx εd Ey δd EZ y δm y εm Metal (a) (b) Figure 3.2: (a) Interface between metal and dielectric media with dielectric functions εm and εd . The interface is defined by z = 0 in a Cartesian coordinate system, and (b) shows the penetration into each medium (skin depth). A complete discussion of skin depth will be given at the end of this chapter. plasmon polaritons propagation, and the x-axis lies in the interface plane. Consider an interface between two media with dielectric permittivities εm (for metal) and εd (for dielectric). In each half-space only a single p-polarized wave will be considered, because we are looking for homogeneous solutions that decay exponentially with dis- tance from the interface. Therefore, the solutions of Maxwell’s equations near the surface have the form [24]: Hx (z) = Ad exp(iky y − κd z) , z > 0, (3.13) Hx (z) = Am exp(iky y + κm z) , z < 0, (3.14) where κm and κd are evanescent exponents for the metal and dielectric media, respec- tively, ky is the wave vector component along the direction of propagation. Such a field satisfies Maxwell’s equations together with the boundary conditions for p-polarization

47 (TM modes). The field with s-polarization (TE modes) at the two media interface with µ = 1 cannot satisfy the boundary conditions at any field wave vector (for more details see the end of this section). Therefore, in the surface plasmon polariton field, the magnetic field is perpendicular to the yz-plane H = Hx and the electric field vector lies in the yz plane. From the boundary conditions of the continuity of the tangential components of fields Hx and Ey at the interface, it follows that the wave vector k and amplitudes below and above the interface are correspondingly equal. Taking this into account, setting ε = εm in Eq. (3.9) for the medium z < 0 and ε = εd for the medium z > 0, the following two equations can be obtained. Ad = Am , (3.15) κd κm =− . (3.16) εd εm Substituting these into the wave equations, Eq. (3.7), one can get the following equa- tions relating the wave vector k and evanescent exponent κ for each medium, κ2 = ky − εd k0 , d 2 2 (3.17) κ2 = ky − εm k0 , m 2 2 (3.18) where the sign is chosen in such a way that Re[κd ]> 0 and Re[κm ]> 0 since in the system under consideration, the wave is decaying as it moves away from the surface. Equations (3.16),(3.17) and (3.18) form a system of three equations for the three

48 unknown constituents of the wave vector ky , κd , and κm , εd εm ky = k0 , (3.19) εd + εm ε2 κd = k0 − d , (3.20) εd + εm ε2 κm = k0 − m . (3.21) εd + εm Having derived Eqs. (3.19)-(3.21) we are in a position to discuss the conditions for an interface mode to exist. We are interested in interface waves, which are well localized at the interface, and propagate along the interface. This requires real values for the coefficient, ky . This can be achieved if the sum and the product of the dielectric functions in the Eq. (3.19) are either both positive [see Eq. (3.22)] or both negative [see Eq. (3.23)]. εd + εm > 0, εd εm > 0. (3.22) εd + εm < 0, εd εm < 0. (3.23) Both εd and εm are almost real in absence of damping, i.e., with no losses. In order to obtain a bound solution for well-defined surface plasmon polariton modes, the normal components of the wave vectors have to be considered as purely imaginary in both media given rise to exponentially decaying solutions. By looking at Eqs. (3.20)- (3.21) this can be fulfilled if the summations in the denominator of Eqs. (3.20)-(3.21) are negative, i.e., εd + εm < 0. (3.24)

49 From Eqs. (3.22)-(3.24) we can conclude that the conditions for an interface mode to exist are the two inequalities in the Eq. (3.23), which means that at least one of the dielectric functions must be negative with an absolute value exceeding that of the other. As we discussed in Chapter 1, metals, especially noble metals such as silver and gold, have a large negative real part [see Fig. 1.1(a)] of the dielectric constant along with small imaginary part [see Fig. 1.1(b)]. Therefore, metals, in particular, noble metals are perfect candidates for the good plasmonic behavior. Thus, the well- defined surface plasmon polariton modes exist at the interface between a good metal and a good dielectric. In the case of small dielectric losses in either or both of the two materials, the above conditions can be used without any corrections, but should specify the corresponding real parts. Finally, we can summarize that the conditions for good plasmonic behavior are: Re[εd ] + Re[εm ] < 0, Re[εd ] · Re[εm ] < 0, (3.25) Im[εd ] |Re[εd ]| < 0, Im[εm ] |Re[εm ]| < 0. (3.26) Before discussing the properties of the dispersion relation (3.19) in more details, now I will briefly analyze the possibility of TE (or s-polarization) modes. For the TE modes, the corresponding field components are [similar to the Eqs. (3.8) and (3.9) for TM modes] Ex (z) = Ad exp(iky y − κd z) , z > 0, (3.27) Ex (z) = Am exp(iky y + κm z) , z < 0. (3.28)

50 From Eq. (3.12) we can get similar equations for Hy . Therefore, continuity of Ex and Hy leads to the condition Ad (κd + κm ) = 0. (3.29) Since confinement to the surface requires Re[κd ] > 0 and Re[κm ] > 0, this condition is only fulfilled if Ad = 0, so that also Am = Ad = 0. Thus, no surface modes exist for TE (or s-polarization) modes. Surface plasmon polaritons only exist for TM (p-polarization) modes. 3.4 Dispersion relation From here onward in this Chapter, I will consider the frequency dependent dielec- tric function for the metal as εm (ω) = Re[εm (ω)] + iIm[εm (ω)], (3.30) and permittivity for the dielectric medium is real (negligible losses), i.e., εd ≡ Re[εd ]. To accommodate losses associated with electron scattering the term Im[εm (ω)] has been introduced. Using the real values for ω and εd with the condition |Re[εm (ω)]| Im[εm (ω)], one can rewrite the Eq. (3.19) as complex wave vector: ky (ω) = Re[ky (ω)] + iIm[ky (ω)] (3.31) in particular in more details manor, 1/2 3/2 ω εd Re[εm (ω)] ω εd Re[εm (ω)] Im[εm (ω)] ky (ω) ≈ +i . c εd + Re[εm (ω)] c εd + Re[εm (ω)] 2(Re[εm (ω)])2 (3.32)

51 The Re[ky (ω)] determines the surface plasmon polariton wavelength, while the Im[ky (ω)] contributes the losses associated with the damping of the surface plasmon polariton as it propagates along the interface. As I mentioned earlier, for Re[ky (ω)] one needs Re[εm (ω)] < 0 and |Re[εm (ω)]| > εd , which can be fulfilled in a metal and also in a doped semiconductor. From Eq. (3.19), we can see that under conditions (3.25) and (3.26) the surface plasmon polariton dispersion relation lies outside the light cone for the surrounding dielectric, which means that ω√ ky (ω) > εd . (3.33) c √ The dispersion relation [see Fig. 3.3] approaches the light line εd ω/c at small √ ky , but remains larger than εd ω/c. Therefore the surface plasmon polaritons are dark modes/nonradiative, i.e., they do not couple to light waves in the surrounding dielectric and cannot be either emitted or excited by electromagnetic radiation from the far zone. From Eq. (3.19) it is obvious that surface plasmon polariton wave vector ky be- comes very large compared to that of wave vector in free space when the real part of the denominator becomes small and vanishes at a particular frequency, ωsp , which is the so-called surface plasma frequency for flat surfaces. On the other hand this is the upper limit of the plasmonic region, since at higher frequencies Eq. 3.23 is violated. The ωsp satisfies the following condition, Re[εm (ωsp )] = −εd . (3.34)

52 (eV) (eV) 3 (a) 3 (b) 2 2 1 1 0.2 0.6 1 0.2 0.6 1 6 -1 6 -1 k×10 (cm ) y Re[k y]×10 (cm ) Figure 3.3: The relationship [see Eq. (3.19)] between ω vs. ky for silver (εm ) and a dielectric (εd ) interface. Where εd = 1 for the solid curve and εd = 10 for the dashed curve. (a) Calculated real part of the permittivity for silver. Here horizontal lines represent the surface plasmon frequencies. (b) Same plot as (a), but including the imaginary part of the permittivity for silver. The surface plasmon polariton wavelength, λSP P , can be defined as: 2π Re[εm (ω)] + εd λSP P = ≈λ , (3.35) Re[ky (ω)] Re[εm (ω)] εd where λ is the wavelength of the excitation light in vacuum. Under the condition Re[εm (ωsp )] → −εd , the wavelength λSP P of the surface plasmon polariton modes becomes very small. In other words, under the condition (3.34) the effective refractive index neff = λ/λSP P for surface plasmon polaritons becomes very large. This may reduce the theoretical diffraction limit of resolution with surface plasmon polaritons, which is determined by λSP /2. This value may reach a scale of a few nanometers. Note that, a wave cannot be localized much shorter than half of its wavelength λ/2 in vacuum. However, this has to be corrected as λ/2n for any medium other than vacuum due to the shorter wavelength λ/n of light in the

53 medium with refractive index n. From the practical point of view, the Ohmic losses in a metal limit the wavelength and propagation length of the short wavelength surface plasmon polaritons, therefore, this will severely limit the resolution. The behavior of the surface plasmon polariton dispersion relation given in Eq. (3.19) is illustrated in Fig. 3.3. The panel (a) shows the relations for an ideal metal, i.e., no losses are present. The solid curves represent the results for the silver-vacuum interface, where εd = 1 and for silver, the experimental dielectric values of the bulk silver have been used [17]. However, the imaginary part of the experimental dielec- tric values of the bulk silver have been ignored. The results in Fig. 3.3(a) and the previous discussion of the properties of surface plasmon polariton dispersion relation are in good agreement. Here, the horizontal solid line represents the upper limit of the plasmonic frequency, ω → ωsp , for the silver-vacuum interface ωsp ≈ 3.67 eV. In other words, at this frequency range the wave vector related to the surface plasmon polariton becomes infinity, i.e., k → ∞. As a result, both the phase velocity, vp , and group velocity, vg , of surface plasmon polaritons tend to zero, which corresponds to standing surface plasmons [87]. Note that the phase velocity and group velocity of surface plasmon polaritons are defined as for any other wave by the following relations [24], ω ∂ω vp = Re , vg = Re (3.36) k ∂k The dashed curve in Fig. 3.3(a) is similar to the one we discussed above, i.e., silver- vacuum interface, but for the silver-semiconductor interface, where εd = 10. The

54 panel (b) shows the relations for an actual metal, i.e., when losses are present, where the dispersion relation is significantly different. In this case, the wave vector ky is complex, therefore, Re[ky ] has been shown in Fig. 3.3(b). As shown in panel (b) the dispersion curve ends at some maximum value of k, with a cusp which is limited by the dielectric losses at the surface plasmon frequency. The maximum value of k, which is related to the dielectric losses, defines the minimum possible localization size of the surface plasmon polaritons in a nanosystem. 3.5 Surface plasmon polaritons at a metal-vacuum interface Now consider the properties of the electromagnetic fields at metal-vacuum inter- faces that exist due to surface plasmon polaritons propagation. The relative distribu- tions (relative magnitude) of the local electric field for a silver-vacuum interface are shown in Fig. 3.4. Panels (a), (c) and (e) show the transverse electric field component, Ez , and (b), (d) and (f) display the longitudinal electric field component, Ey . The evanescent nature of the fields in the normal direction to the interface can be seen in all the panels. In all the panels the penetration of the fields into the vacuum is much greater than the surface plasmon polariton wavelength [see Eq. (3.35)], but the pen- etration into the metal (skin depth) is much shorter than the wavelength, which can be clearly seen in panels (c) and (d). Note that this is for the same frequency as for

55 = 2.01 (eV) 1000 1000 500 500 z (nm) 0 z (nm) 0 -500 Ez -500 Ey (a) (b) -1000 -1000 0 400 800 1200 0 400 800 1200 y (nm) y (nm) = 2.01 (eV) 0 0 -20 -20 z (nm) z (nm) -40 -40 -60 -60 Ey Ez (c) (d) -80 -80 0 400 800 1200 0 400 800 1200 y (nm) y (nm) = 3.0 (eV) 1000 1000 500 500 z (nm) 0 z (nm) 0 -500 Ez -500 Ey (e) (f) -1000 -1000 0 400 800 1200 0 400 800 1200 y (nm) y (nm) -1 0 +1 Figure 3.4: (color) Distribution of local electric fields shown in yz plane, which is normal to the interface with y (horizontal coordinate) being the direction of surface plasmon polariton propagation, and z (the vertical coordinate) being normal to the surface. The magnitude is coded by colors as shown in a bar at the bottom. For panels (a), (b), (c) and (d) the frequency is 2.01 eV, for (c) and (d) it is 3.0 eV, as indicated.

56 ħω = 2.01 (eV) 400 nm 550 nm ħω = 3.0 (eV) 300 nm 350 nm Figure 3.5: Vector diagrams of local electric fields in the yz plane, which is same coordinate system as Fig. 3.4. The spatial scale and the corresponding frequencies are indicated in the figure. the panels (a) and (b). It can be seen, from the comparison of the panels (a) and (b) with (e) and (f) that the plasmon polaritons propagation wavelength and evanescent extensions decrease close to the plasmonic frequency region. To get a better look at the local electric fields, the vector behavior of the local electric field E is shown in Fig. 3.5. As one can see, the local electric field is neither longitudinal nor transverse. The vector of E changes in each quarter period between transverse and longitudinal, alternating its direction each half period. As given in Eq. (3.8) the magnitude of the magnetic field Hx is proportional to the transverse component of the electric field Ez .

57 3.6 Propagation length, skin depth and plasmon lifetime Before we discuss the propagation length and skin depth of surface plasmon po- laritons, let’s consider spatial quality factor Q of the surface plasmon polariton waves which is defined as Re[ky (ω)] Q= , (3.37) Im[ky (ω)] where Re[ky (ω)] and Im[ky (ω)] can be found in the Eq. (3.32). Physically, the quality factor determines the number of oscillations that a surface plasmon polariton wave undergoes before it dissipates due to the dielectric losses in real metal. As we can see from the Fig. 3.6, the quality factor dramatically decreases with increasing fre- quencies. When ω → ωsp , the quality factor Q → 1, which means that: closer to the plasmonic frequency surface plasmon polaritons do not propagate. In other words, at this point they do behave like localized surface plasmons, which we discussed in Chapter 2. The effect of different dielectric media are also shown in the figure. With increasing dielectric constant of the dielectric medium both quality factor and stoping frequency of polaritons drop significantly. The propagation length, Lsp , of the surface plasmon polariton along the interface is determined by the Im[ky (ω)], which is given in second part of the Eq. (3.32). Im[ky (ω)] is shown in Fig. 3.7, as a function of ω for two different dielectric media, i.e., εd = 1 and 10 as indicated in the figure. The metal is silver, and here the experimental data

58 Q = Re[ky( )]/Im[ky( )] 4 10 d 1 3 10 d 2 10 1.0 2.0 3.0 (eV) Figure 3.6: Spatial quality factor Q as a function of frequency for two different di- electric media, i.e., εd = 1 and 10 as indicated in the figure. The metal is silver, here the experimental data from [17] have been used for the dielectric constant of silver. from [17] have been used for the metal dielectric εm . As one can see, the Im[ky (ω)] significantly increases around the plasmon resonance frequency, where it becomes on the same order of magnitude as Re[ky (ω)] (not shown). Therefore, according to the Eq. (3.37) we have not the surface plasmon polaritons but the localized surface plasmons. Also note that, due to the dielectric screening Im[ky (ω)] is significantly higher for high εd [see the curve for εd = 10 ]. The electric field of surface plasmon polaritons along a smooth surface decreases as exp(−Im[ky (ω)]y) [18]. Therefore, the propagation length Lsp is defined as: the length after which the electric field decreases to 1/e, which is given by, 1 Lsp = . (3.38) Im[ky (ω)] The propagation length of the surface plasmon polariton for silver is shown in

59 -1 Im[k](cm ) 6 10 5 10 εd =10 4 10 3 10 2 εd =1 10 1 10 1.5 2 2.5 3 3.5 4 ħω (eV) Figure 3.7: Im[k] as a function of ω for two different dielectric media, i.e., εd = 1 and 10 as indicated in the figure. The metal is silver, here the experimental data from [17] have been used for the dielectric constant of silver. Fig. 3.8, where panel (a) shows the results for the silver-vacuum interface while (b) shows the same results as panel (a) but for the silver-dielectric interface, where εd = 10. This shows very good agreement with what we discussed under quality factor. In particular, one can see from the figure that when ω → ωsp , the propagation almost comes to an end. Note that the ωsp = 3.67 eV for the silver-vacuum interface, while ωsp = 2.46 eV for silver-dielectric interface, where εd = 10. More importantly, contrast to the nano-scale the propagation length of the surface plasmon polariton is very large, which is one of the most important properties in nanoplasmonics. There is another important length parameter, which is called decay length (or skin depth, for metal), δ, of the surface plasmon polariton electric fields. In the propagation length we considered the fields along the interface (along the y-axis), while in the decay length we consider the surface plasmon polariton electric fields

60 Lsp(mm) Lsp(mm) εd = 1 0.10 εd = 10 3.0 0.08 2.0 0.06 (a) (b) 0.04 1.0 0.02 1.0 2.0 3.0 1.0 1.5 2.0 2.5 ћω (eV) ћω (eV) Figure 3.8: Propagation length Lsp for (a) silver-vacuum interface, where εd = 1. and (b) silver-semiconductor interface, where εd = 10. away from the interface (along the z-axes in both derections). It can be shown from Eqs. (3.20) and (3.21) that κm and κd can be approximated to first order in Im[εm (ω)]/Re[εm (ω)] as 1/2 ω Re[εm (ω)]2 Im[εm (ω)] κm = 1+i (3.39) c Re[εm (ω)] + εd 2Re[εm (ω)] 1/2 ω [εd ]2 Im[εm (ω)] κd = 1−i . (3.40) c Re[εm (ω)] + εd 2(Re[εm (ω)] + εd ) Following the same argument as before, the decay length, δ can be defined as 1 1 δm = and δd = , (3.41) Re[κm ] Re[κd ] where δm indicates metal and δd indicates dielectric decay length. The skin depth, δm in silver and decay length, δd in dielectric medium is shown in Fig. 3.9 as a function of frequency. In the case of silver in vacuum, δm does not change very much over the entire frequency range, δm 20 − 30 nm. At the both dielectric values, as expected the skin depth is much smaller than the decay length, i.e., δm δd . However, due to

61 δm , δd (nm) δm , δd (nm) 500 εd = 1 εd = 10 100 300 (a) 60 (b) 100 20 1.0 2.0 3.0 1.0 2.0 3.0 ћω (eV) ћω (eV) Figure 3.9: Skin depth, δm (dotted line) and decay length δd (dashed line) for a dielectric, as a function of energy at (a) silver-vacuum interface, where εd = 1. and (b) silver-semiconductor interface, where εd = 10. the dielectric screening with the dielectric medium, where εd = 10, the decay length has been significantly suppressed [see Fig. 3.9] while δm remains at the same order. Therefore, there is no significant difference between δm and δd at high frequencies. Before introducing the lifetime of surface plasmons τ (ω), I first consider the related parameter: the relaxation rate γ(ω) [28] which describes the relaxation due to the dielectric losses in metal. In other words, this is the dephasing of surface plasmons and it also defines their population life time or lifetime τ (ω) through the following relation 1 τ (ω) = . (3.42) 2γ(ω) Since, 1/γ(ω) is amplitude of the relaxation or dephasing time of the surface plasmons, the factor of 2 in the denominator has been taken into account. Therefore, the proper term for the lifetime, τ (ω) would be: the decay time of the energy of the surface plasmons.

62 60 10 8 40 6 4 20 2 0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 Figure 3.10: Lifetime of surface plasmons τ (ω), computed from the experimental data of Ref.[17]. Panel (a) shows the data for silver and panel (b) for gold. The lifetime of surface plasmons τ (ω), computed from the experimental data of Ref.[17] is shown in Fig. 3.10. The results for both silver, panel (a) and gold, panel (b) are shown. The surface plasmons lifetime for silver is much longer than for gold, in particular in between 1 and 2 eV, which is near-IR to red region. For silver τ (ω) is ∼ 60 fs while for gold it is ∼ 10 fs. One of the interesting aspects is the cutoff of the plasmonic region around 2.6 eV for gold [see Fig. 3.10(b)], which is due to the strong absorption by the transitions from the d to sp band in gold. For silver this is around 4 eV, which is not shown in the graph. These highly localized electrons in d band are contributing to the absorption and decay of the surface plasmons, but, they do not contribute directly to the plasmonic nature of noble metals.

63 Chapter 4 Enhancement of optical sensitivity of photodetectors 4.1 Introduction In this Chapter I will study theoretically an enhancement of the intensity of mid- infrared light transmitted through a metallic diffraction grating. I show that for s-polarized light the enhancement of the transmitted light is much stronger than for p- polarized light. By tuning the parameters of the diffraction grating, the enhancement of the transmitted light can be increased by a few orders of magnitude. The spatial distribution of the transmitted light is highly nonuniform with very sharp peaks, which have spatial widths of about 10 nm. The main results in this chapter has been published in the following journals, J. Phys. Condens. [52], Physica E [53] and

64 Infrared Physics and Technology [54]. As we discussed in Chapter 2, it is clear now that due to generation of the surface plasmon polaritons at the boundary between metal and dielectric media the local electromagnetic field can be strongly enhanced [12]. For the surface plasmon polari- tons the enhancement is observed only near the metal-dielectric interface and away from the interface the effect of the surface plasmons is exponentially suppressed. The grating coupler method [12], which is widely used to generate the surface plasmons at the metal-dielectric interface, is based on the diffraction grating placed on the top of the plasmon waveguide [12, 55, 56]. In this case the grating vector provides the momentum required by the momentum conservation between the incident light and the surface plasmons. The diffraction grating has been widely used to generate not only the surface plasmons, but also the plasmon excitations in low dimensional semiconductor systems [57, 58, 45]. In the present Chapter, I will address another important problem related to in- teraction of electromagnetic waves with the excitations near metal-dielectric inter- face. Namely, the enhancement of optical sensitivity of infrared photodetectors in the presence of the metallic surfaces has been studied. For a p-polarized incident light (magnetic field in the wave is parallel to the metal-dielectric interface) this enhance- ment is due to generation of the surface plasmons at the metal-dielectric interface. Similar to other applications of the surface plasmons we should expect a strong local enhancement of the electromagnetic field near the metal surface. In the present work

65 the generation of the surface plasmons is realized by introducing a metallic diffrac- tion grating. The metallic diffraction grating opens also a possibility to enhance the intensity of the s-polarized transmitted light (electric field is parallel to the metal- dielectric interface). This enhancement is due to generation of the local modes of the diffraction grating. In this case these modes are not the surface plasmons. The enhancement of electromagnetic field near the metal-dielectric interface will be con- sidered for both s- and p-polarizations. If the active element of the photodetector is placed in the region of enhancement of the wave field then we should expect the increase of the sensitivity of the photodetector. There are two main types of infrared photodetectors, which are different by their active elements. Namely, in one type of photodetectors the active element is a quantum well [57], while in the other type the active element is a quantum dot [58]. We will see below that the distribution of the light, transmitted through the diffraction grating, is highly nonuniform with very bright spots, which have a size of the order of 10 nm. In this case it is more appropriate to use quantum dots as active elements of the photodetector. This is because the quantum dots have small size, so they can be placed completely inside of the regions of high intensity. This cannot be done for quantum wells. Another advantage of quantum dots is that they can be sensitive to both s-polarized and p-polarized light, while the quantum well is only sensitive to the p-polarized light. The quantum dots can be also used to detect the terahertz radiation [45].

66 Therefore, only the quantum dot photodetectors will be discussed. To characterize the effect of the metal grating on the optical properties of the photodetector, the distribution of the electromagnetic field inside of the active region of the photodetector has been calculated. The exact structure and distribution of quantum dots will not be considered here. I assume that the effect of the quantum dots on the distribution of the electromagnetic field is negligibly small, i.e., the active region of a photodetector has been considered as a homogeneous medium with uniform dielectric constant. The distribution of the electromagnetic field determines the optical, i.e., absorption, properties of the quantum dot system. In this description, the structure and the sizes of the quantum dots will not be discussed. Therefore, my main goal is to calculate the distribution of the electromagnetic field inside the metallic and dielectric media to understand finally how strong the enhancement of the absorption coefficient of quantum dots can be in such systems. The frequency range of the incident light, which I used in the present work, cor- responds to the typical interval spacing in small (∼ 10 nm) quantum dots. Namely, I consider the frequency around ω ∼ 70 THz, i.e., the energy is ∼ 250 meV. The effect of the metal grid gate on the absorption efficiency of the terahertz electromagnetic radiation has been studied recently both theoretically and experi- mentally [59, 60]. The metal grating described in Popov et al. and Shaner et al. has been used to introduce the coupling of the incident light to the plasmon excitations within the double quantum well system of the field-effect transistor [59, 60]. The

67 main difference between my study and the results of Popov et al. and Shaner et al. is that in my case the main effect of the diffraction grating is the redistribution of the electromagnetic field intensity within the active region of an infrared photodetector [59, 60]. As a result of this redistribution we expect the local enhancement of the intensity of the electromagnetic field and finally the enhancement of the absorption efficiency of the photodetector. Therefore, in this work the diffraction grating intro- duces the coupling not between the incident light and the excitations of the active region of photodetector, but between the incident light and the local excitations of the diffraction grating. 4.2 Main system of equations The first step in the analysis of the problem is to find the distribution of the electromagnetic field inside the diffraction grating and the active elements of the photodetector. At this stage I assume that the active region of the photodetector is uniform and does not contain active elements of the photodetector. Therefore, the system under consideration consists of three regions: region-I (air) with dielectric constant εI = 1, region-II (metallic diffraction grating with period d and height h), and region-III (dielectric) with dielectric constant εd . The system is shown schemati- cally in Fig. 4.1. It has been assumed that the dielectric constant of the metal, εm (ω), has the Drude dependence on the frequency

68 Figure 4.1: Schematic illustration of the metal grating on the surface of a dielectric medium. Here, grating period is d and the grating height is h. The grating consists of periodic strips of metal with dielectric constant εm (ω) and air with dielectric constant εI = 1. The width of a metallic strips is a. The region III is filled by a material with dielectric constant εd . Here E is the electric field vector and H is the magnetic field vector. The angle θ is the incident angle. 2 ωp εm (ω) = 1 − , (4.1) ω(ω + i/τ ) where ωp is the plasma frequency and τ is the phenomelogical relaxation time. Below it will be assumed that the metal is gold and ωp = 3.39 × 1015 s−1 and τ = 1.075 × 10−14 s [17]. In the next step the quantum dots and quantum wells are added into the dielectric medium, i.e., region III, at some distance z from the diffraction grating, i.e., from the metal-dielectric interface. To describe the distribution of the electromagnetic field, the coordinate system with axis z orthogonal to the metal-dielectric interface, and axis x in the plane of the interface has been introduced [see Fig. 4.1].

69 The direction of the incident light is characterized by an incident angle θ. The incident light can have two polarizations: s- and p-polarizations. For the p-polarized light the magnetic field of the electromagnetic wave is in the plane of the interface, while for the s-polarized light the electric field of the wave is in the interface plane. To find the distribution of the electromagnetic field, the system of Maxwell’s equations has to be solved with the corresponding boundary conditions. Namely, the amplitude of the incident light (at z > 0) is given and there are no waves incident on the diffraction grating outside of the grating region at z < −h, i.e., in this region there are only waves propagating away from the metal-dielectric interface. Maxwell’s equations for s- and p-polarized lights become decoupled and the electric and magnetic fields can be described in terms of a single function: ψ = Ey in the case of the s- polarized light and ψ = µ0 / 0 Hy in the case of the p-polarized light [61]. To solve the corresponding Maxwell’s equations, the well known modal expansion method [62] has been used. In this method the solutions of Maxwell’s equations are expressed in terms of the eigenmodes of electromagnetic field in all three regions. In regions-I and -III these eigenmodes are simple plane waves, which are characterized by wave vectors. The incident light has an x component of the wave vector equal to kx = k sin θ, (4.2) where k is wave vector of the incident light and k = ω/c. The diffraction grating with the period d introduces the coupling of the incident light with the waves, the x

70 components of the wave vectors, kxn , of which are given by the expression kxn = k [sin θ + nλ/d] , (4.3) where n = 0, ±1, ±2, . . . is a diffraction order, and λ is the wavelength of the incident light. Then the general solution of Maxwell’s equations in regions -I and -III can be written as [56] (I) (I) ψ (I) (x, z, ω) = A(I) ei(kx x−kz z) + Bn ei(kxn x+kzn z) , (I) (4.4) n (III) (III) (III) i(kxn x+kzn z) ψ (x, z, ω) = Bn e , (4.5) n (I) where A(I) is the amplitude of the incident wave, Bn is the amplitude of the nth (III) reflected wave (in region I), Bn is the amplitude of the nth transmitted wave (in region-III), and (I) kzn = εI k 2 − kxn , 2 (4.6) (III) kzn = εd k 2 − kxn . 2 (4.7) The general solution of Maxwell’s equations in region-II can be expressed in terms of the eigenmodes of the diffraction grating. The eigenmodes of the wave equation in the grating region are characterized by parameter κz , which satisfies the following nonlinear equation [61] 2 ξj + 1 sin[β1 a] sin[β2 a] − cos[β1 a] cos[β2 a] = − cos(kx d). (4.8) 2ξj

71 Here j = p, s, where p stands for the p-polarization and s stands for the s-polarization. The following notations have been introduced, ξp = [εm (ω)β2/εI β1 ], ξs = [β2 /β1 ], (4.9) β1 = εm k 2 − κ2 , β2 = z εI k 2 − κ2 . z For a given frequency ω and given parameters of the system, e.g. εm , d, a and θ , we obtain from Eq. (4.8) the infinite set of eigenmodes, which are characterized by the value of κz . These eigenvalues, which are complex, have been found numerically by finding the roots of a transcendental Eq. (4.8). The roots have been found by tracing the roots trajectory in the complex plane with variation of dielectric constant of a metal [63]. The solution of Eq. (4.8) determines infinite number of eigenmodes. If we enumer- ate them by index , i.e. κz is the th solution of Eq. (4.8), then the general solution of Maxwell’s equations in the region-II can be presented in the following form [56] ∞ (II) ψ (x, z, ω) = χ (x) ζ+ (z), (4.10) =1 where the functions χ (x) and ζ+ (z) are determined by the following expressions χ (x) = C1 eiβ1 x + D1 e−iβ1 x ; 0<x<a, (4.11) χ (x) = C2 eiβ2 (x−a) + D2 e−iβ2 (x−a) ; a < x < d , (4.12) (II) −iκz z (II) iκz z ζ± (z) = ± A e ±B e , (4.13)

72 and 1 + ξ eiβ1 a − ei(kx d−β2 a) C1 = 1, D1 = , 1 − ξ e−iβ1 a − ei(kx d−β2 a) (ξ + 1) eiβ1 a + (ξ − 1) e−iβ1 a D1 C2 = , 2ξ (ξ − 1) eiβ1 a + (ξ + 1) e−iβ1 a D1 D2 = . 2ξ (II) (II) Here the numbers A and B are unknown coefficients. (I) (II) (II) (III) The complete set of coefficients Bn , A , B , and Bn were found from the boundary conditions at the interface between different regions. These boundary conditions correspond to continuity of the function ψ and its derivative ∂ψ/∂z for the s-polarized light and continuity of the functions ψ and ε−1 ∂ψ/∂z for the p-polarized light. The boundary conditions at the interfaces between regions -I and -II, and between regions -II and -III result in the following system of equations ∞ (I) (I) (II) (II) A δn0 + Bn = Γ1, n A +B , (4.14) =1 ∞ 1 (I) (I) (II) (II) kz A δn0 − kzn Bn = (I) (I) κz Γ2, n A −B , εI =1 ∞ (III) (III) −ikzn h Γ1, n ζ+ (h) = Bn e , (4.15) =1 ∞ (III) Bn (III) Γ2, n κz ζ− (h) = − kzn e−ikzn h , (III) (4.16) =1 εd

73 for the p-polarized light and ∞ (I) (I) (II) (II) A δn0 + Bn = Γ1, n A +B , (4.17) =1 ∞ (II) (II) kz A(I) δn0 (I) − (I) (I) kzn Bn = κz Γ1, n A −B , =1 ∞ (III) (III) −ikzn h Γ1, n ζ+ (h) = Bn e , (4.18) =1 ∞ (III) Γ1, n κz ζ− (h) = −Bn kzn e−ikzn (III) (III) h , (4.19) =1 for the s-polarization. Here, following notations are being introduced d 1 Γ1, n = dx e−ikxn x χ (x), (4.20) d 0 and d a d 1 χ (x) χ (x) χ (x) Γ2, n = dx e−ikxn x = dx e−ikxn x + dx e−ikxn x . (4.21) d 0 ε(x) 0 εm (ω) a εI The solution of the system of linear Eqs. (4.14)-(4.19) determines the coefficients (I) (II) (II) (III) Bn , A ,B , and Bn in units of the amplitude, A(I) , of the incident light. To make the system finite, the maximum value of the diffraction order, nmax has been introduced, so that |n| ≤ nmax . The value of nmax determines also the maximum number of eigenmodes of the diffraction grating. Namely, ≤ (2nmax + 1). Therefore the final size of the system is 2(2nmax + 1). Since the frequency range, considered in the present paper, corresponds to a very large magnitude of the metal dielectric constant (εm ≈ −847 + 1127i), the numerical procedure of finding the eigenmodes of the diffraction grating becomes highly unsta- ble. Usually the direct numerical solution of nonlinear Eq. (4.8) does not provide all

74 the eigenmodes of diffraction grating, i.e., some of the eigenmodes can be overlooked. To resolve this problem the method of tracing the root trajectory was applied [63], i.e., the roots of Eq. (4.8) were been found by tracing the root trajectory in the com- plex plane of dielectric constant [63]. Namely, at first, all necessary roots of Eq. (4.8) at small value of metal dielectric constant was calculated. Then a straight line in the complex plane of the dielectric constant was introduced. This line connects the initial small dielectric constant and the final large dielectric constant. Finally, the root trajectory along the straight line in the complex dielectric plane was found, i.e., the values of the roots when the dielectric constant is changed along the line were traced [63]. With this method we can find all necessary eigenmodes of the diffraction grating including the hidden modes for the p and s -polarized light [64]. The value of nmax is found from the condition of convergence of the reflection, transmission, and absorption coefficients. Within the range of parameters, considered in the present work, I have found that the method converges at nmax ≈ 15 for the p-polarized light and at nmax ≈ 30 for the s-polarized light. Therefore, throughout the paper I adopted nmax = 25 and nmax = 50 for p-polarization and s-polarization, respectively. With these values of nmax the corresponding system of linear equations have been numerically solved.

75 4.3 Intensity distribution within the active region of photodetector Solution of the system of linear Eqs. (4.14)-(4.19) determines the values of the (III) coefficients Bn . With these values, the wave intensity in the active region of photodetector, i.e., in the region III has been found. In a real quantum dot photodetector there is a layer of quantum dots at some distance z from the metal-dielectric interface. The photoresponse of the photodetector is proportional to the absorption intensity, which is proportional to the intensity of the electromagnetic field at the position of the quantum dots. Therefore, the contribution to the photocurrent due to a single quantum dot at the point with the coordinates x and z can be written as in the following form J(x, z) = αI(x, z), (4.22) where α is the coefficient, which includes the absorption coefficient of the quantum dot and other parameters of the photodetector, and I(x, z) is the intensity of the electromagnetic field at the position of the quantum dot. The real dependence of the photocurrent on the intensity of the light, I(x, z), can be nonlinear and more complicated, but in the present work only linear regime of the photodetector has been considered. In the quantum dots there are two different types of optical transitions. The first one is due to the in-plane component (in our case x or y component) of electric field,

76 and the second one is due to perpendicular component (z component) of electric field. The corresponding optical transitions have different frequencies. For the s-polarized light there is only a y component of electric field and the intensity of electromagnetic field in Eq. (4.22), which is responsible for the optical transitions within the quantum dot, has the following expression 2 I(x, z) = ψ (III) (z, x) . (4.23) For the p-polarized light there are both a z component and an in-plane (x) component of electric field. Therefore, in this case there are two types of optical transitions. Below, in the case of p-polarized light, only the transitions due to perpendicular component have been studied, i.e., the z component, of electric field. Then the intensity I(x, z) in Eq. (4.22) becomes 2 I(x, z) = ∂x ψ (III) (z, x) . (4.24) If the quantum dots within a layer at fixed distance z from the metal-dielectric interface have completely random spatial distribution, then the total light absorp- tion and correspondingly the total photocurrent is proportional to the average wave intensity d 1 Iav (z) = I(x, z)dx. (4.25) d 0 Below I do not discuss the coefficient α in Eq. (4.22) and consider only the intensity distribution I(x, z). To characterize the enhancement of the transmitted light due to the presence of the diffraction grating the intensity I(x, z) will be measured in

77 the units of the intensity, I0 , of electromagnetic wave in the region III without the diffraction grating. The intensity I0 is given by the standard Fresnel’s equations. Namely, 2 4kz I0 = (III) 2 (4.26) [kz + kz ] for the s-polarization and 4kz sin2 θ 2 I0 = (III) 2 (4.27) εd [εd kz + kz ] for the p-polarization [24]. Here kz = k cos θ. Therefore, if I(x, z) is greater than 1 (in units of I0 ) then there is enhancement of the optical absorption by the quantum dot layer due to the presence of the diffraction grating. 4.4 Results and discussion 4.4.1 Average intensity As we have already discussed in the Introduction the enhancement of the inten- sity of the light transmitted through the diffraction grating can be expected. This enhancement is due to generation of the local modes of the diffraction grating. Usu- ally, when such type of enhancement is discussed, it is due to generation of the surface plasmons at the metal-dielectric interface. In the present problem the surface plas- mons are coupled only to the p-polarized incident light, while for the s-polarized light the incident light is coupled to the general modes of the diffraction grating. Therefore

78 in general, much stronger enhancement will be expected for the p-polarized light. It happens that in the present problem the enhancement of the incident light is much stronger for the s-polarized light. This is mainly because I am considering a low fre- quency range for the incident light, within which the dielectric constant of the metal is relatively large. For the parameters of the system considered in the present work the dielectric constant of the metal is ≈ −847 + 1127i. At first, the average intensity [see Eq. (4.25)] of the transmitted light will be analyzed. There are few parameters of the diffraction grating, which determine the properties of the average intensity. These parameters are the period of the diffraction grating, d, the metallic coverage of the dielectric media, a/d, the incident angle, θ, and the height, h, of the diffraction grating. As a function of the frequency of the incident light the average intensity has a maximum. This maximum corresponds to the resonant condition for the generation of the modes of the diffraction grating. In Fig. 4.2 the results of calculations are shown for the s-polarized light and in Fig. 4.3 for p-polarized light for different metallic coverage, a/d, of the dielectric media. The difference between the s- and p-polarized light can be clearly seen. For the p-polarization there is very sharp maximum [see Fig. 4.3]. The position of the maximum has weak dependence on the metallic coverage, a/d. This is an indication that the incident light is coupled to the surface plasmons, since for the surface plasmons the energy of the generated plasmon depends on its wave vector, which is determined only by π/d. Therefore, there is no dependence

79 of the wave vector of the plasmon excitation within the diffraction grating on the parameter a/d. At the same time the average intensity, even at the maximum, is less than 1 (in units of I0 ). It means that, on average, there is no enhancement of the intensity of the transmitted light due to diffraction grating. The results in Fig. 4.2 and Fig. 4.3 are shown for the incident angle θ = 450 . A completely different Iav 3.5 a = 0.3 d 2.5 0.35 d 1.5 0.4 d 0.5 0.5 d 0.6 d 67.5 72.5 77.5 ω (THz) Figure 4.2: Calculated average intensity, Iav , for s-polarized light at distance 10 nm below the grating (i.e., in region III) in units of I0 for different a/d values as indicated in the panel. For all the panels the grating period d = 2µm, the incident angle θ = 450 , and the height of the diffraction grating h = 50nm. behavior is observed for the s-polarized light, see Fig. 4.3. The maximum now is broad and there is a strong dependence of the position of the maximum, ωmax , and the maximum intensity on the value of a/d. With decreasing metallic coverage, i.e.,

80 a/d, the line becomes red shifted, and the maximum intensity increases. For example, if a/d decreases from 0.5 to 0.3 then the maximum intensity increases in six times and becomes 3.5. Since at a = 0 the average intensity should be 1 (in units of I0 ), then at small values of a we should expect a decrease of the maximum intensity with decreasing a/d. It means that there is an optimal value of a/d for which the maximum intensity is the largest. Due to the numerical instability of the problem, it is difficult to go below a/d = 0.3 and find the optimal value of a/d. Iav 0.3 a = 0.4 d 0.2 (a) 0.1 67 68 69 70 ω (THz) Iav 0.10 a = 0.7 d (b) 0.06 0.02 67 68 69 70 ω (THz) Figure 4.3: The same plot as Fig. 4.2 but for p-polarized light (a) for a = 0.4d and (b) for a = 0.7d. For all the panels the grating period d = 2µm, the incident angle θ = 450 , and the height of the diffraction grating h = 50nm. The reason why the maxima in Fig. 4.2 is broad and there is a strong dependence

81 on a/d is that in the case of s-polarization the incident light is coupled not to the surface plasmons but to the modes of diffraction grating. Such modes depend on the actual structure of the diffraction grating, i.e., on the value of a/d. The dependence of the position of the peak, ωpeak , of the average intensity on the metallic coverage, a/d, is shown in Fig. 4.4. This figure illustrates again the facts dis- cussed above. Namely, the frequency ωpeak does not depend on a/d for the p-polarized light, while there is a strong dependence of ωpeak on a/d for the s-polarization. There- fore, only for the s-polarization we can use a/d as a tuning parameter, i.e., by changing a/d we can change the position of the maximum of Iav . Figure 4.4: The peak frequency, ωpeak , at the maximum value of average intensity is shown as a function of a/d for both s- and p-polarization. The grating period d = 2µm, the incident angle θ = 450 , and the height of the diffraction grating h = 50nm. The position of the maxima of the average intensity depends on the period, d,

82 of the diffraction grating for both p- and s-polarized light. These dependencies are shown in Fig. 4.5 and in terms of the wavelength of incident light can be approximately described by the linear functions. Then λpeak = 1.8d + 0.31 (4.28) for s-polarization and λpeak = 2.2d − 0.0047 (4.29) for p-polarization. For p-polarization this linear dependence is universal, i.e., it does not depend on the parameter a/d and on the incident angle (as we will see below). For s-polarization the coefficients in Eq. (4.28) depend on the value of a/d. Another difference between s- and p-polarizations is that the peak wavelength of the s-polarized light is less than the peak wavelength of the p-polarized light. This is due to a different nature of the eigenmodes of the diffraction grating for the different polarizations of the light. The slope of the λpeak (d) dependence for the p-polarized light is 2.2 [see Eq. (4.28)]. This value is very close to the value obtained experimentally in the terahertz frequency range [65]. The experimental result for the slope is 2.19. Although the experimental data [65] have been obtained at a terahertz frequency, i.e., ≈ 30 THz, and our results are for ≈ 70 THz, the fact that the slope is almost the same indicates that at these frequencies the position of the maxima of the average intensity does not depend on the parameters of the metal, i.e., on the dielectric constant of metal. This also indicates that the nature of the diffraction grating modes, responsible for the intensity maxima

83 Figure 4.5: The peak wavelength, λpeak , is shown as a function of grating period d, for both p- and s-polarization. The ratio a/d = 0.5 and the incident angle θ = 450 . of the p-polarized light, is the same across the terahertz frequency range. These modes are the surface plasmons. The dependence of the average intensity on the incident angle, θ is shown in Fig. 4.6. It can be seen from the figure that for p- and s-polarizations the position of the maximum has a very weak dependence on the incident angle. This means that the eigenfrequencies of the modes of the diffraction grating have a weak dependence on θ. At the same time the maximum intensity strongly depends on the angle, θ. We can see that with decreasing the incident angle the maximum intensity increases and at small angles it can be much larger than 1. It means that at small incident angles the

84 Figure 4.6: The average intensity, Iav , at distance z = 50 nm below the grating (i.e., in region III) in units of I0 for different incident angle θ as indicated in the panels. The panels (a)-(c) corresponds to p-polarization and (d)-(f) corresponds to s-polarization. For all the panels the grating period d = 2 µm and the ratio a/d is 0.5. diffraction grating provides strong enhancement of the intensity of the transmitted light. For the s-polarization the diffraction grating can enhance the intensity of the transmitted light by almost two orders of magnitude. At all incident angles the enhancement of the transmitted wave is much stronger for the s-polarization than for the p-polarization. Therefore we can expect strong enhancement of the intensity of the transmitted

85 light at small incident angle. The enhancement is the strongest for the s-polarized light. The behavior of the enhancement as a function of the incident angle for the p- polarized light is unexpected. The modes, which are responsible for the enhancements of the p-polarized light are the surface plasmons. At the same time, it has been known that the coupling of the light to the surface plasmons is determined by the component of electric field orthogonal to the metal-dielectric interface. Therefore we should expect that if we increase the orthogonal component of electric field then generation of the surface plasmons, and correspondingly the enhancement of the transmitted light is increased. But for the p-polarized light we can see in Fig. 4.7 the opposite tendency: with decreasing the angle, i.e., decreasing z component of the electric field, the enhancement is increased. This fact shows that for the p-polarized light the modes of the diffraction grating are not purely surface plasmons. These modes contains both the orthogonal and the in-plane components of the electric field. These modes can be identified as the surface plasmon modes since, as we will see below, the light in this modes is mainly localized near the metal-dielectric interface. As we can see from Fig. 4.7 the enhancement, i.e., the average intensity of the transmitted light in the units of I0 , increases with decreasing the incident angle. At the same time, for the p-polarized light, the intensity I0 has strong dependence on the incident angle, θ, itself. Namely, the intensity I0 decreases with decreasing θ, see Eq. (4.27). Therefore there is an optimal value of the incident angle, θ, which is determined by the condition that the intensity (not the enhancement) of the trans-

86 Figure 4.7: The maximum average intensity (peak value), Iav(max) , for the p- and s- polarized light at distance z = 10nm below the grating (i.e., in region III) is shown as a function of incident angle, θ. The intensity is in the units of I0 . The grating period d = 2µm, the ratio a/d = 0.5, and the height of the diffraction grating h = 50nm. mitted light is maximum. To find the actual intensity of the transmitted light we just need to multiply the data shown in Fig. 4.7 by I0 (θ). In Fig. 4.8 the dependence of the intensity of transmitted light on the incident angle for p- and s-polarized light is shown. For the s-polarized light, a monotonic dependence can be seen on the incident angle and the maximum intensity of the transmitted light is achieved for the normal incident light. For the p-polarized light we can clearly see that there is an optimal value of the incident angle. This angle is around θ ≈ 50 . We can also see a general tendency that the intensity of the s-polarized transmitted light is much larger than the intensity of the p-polarized transmitted light.

87 Figure 4.8: The maximum average intensity (peak value), Iav(max) , for the p-polarized light (scaled by a factor of 5) and s-polarized light at distance z = 10 nm below the grating (i.e., in region III) is shown as a function of incident angle θ. The intensity is in units of the intensity of the incident light The grating period d = 2µm, the ratio a/d = 0.5, and the height of the diffraction grating h = 50 nm. Another parameter of the diffraction grating, which can be varied to optimize the performance of the photodetector, is the height of the grating, h. When the height of the grating is large then the intensity of the transmitted wave should be small and Iav ≈ 0. At zero height, i.e., without a diffraction grating, the average intensity is 1 (in units of I0 ). Then, if at the intermediate values of h there is an enhancement of the intensity of the transmitted light, i.e. Iav > 1, then the dependence of Iav on h is non-monotonic and at some value of h we should see the maximum of the average intensity. This is an optimal value of the height of the diffraction grating. The dependence of the average intensity on the height h, of the diffraction grating is shown in Fig. 4.9, for the p-polarized light. The specific feature of this dependence

88 is that it is non-monotonic even for the parameters of the system, for which the average intensity is less than 1 at all values of h. For example [see Fig. 4.9(a)], at the incident angle θ = 450 the average intensity is less than 1 at all h, but there is a local maximum at h ≈ 50nm. At smaller incident angle, e.g. θ = 50 [see Fig. 4.9(b)], there is an enhancement of the transmitted light and we have an absolute maximum at h ≈ 50nm. The position of the absolute maximum determines the optimal value of the height of the diffraction grating. We can also notice the local minima at small h, h ≈ 8nm. Figure 4.9: (a) The average intensity, Iav , for p-polarized light at distance z = 50nm below the grating (i.e., in region III) is shown in units of I0 as a function of h for a/d = 0.5 (a) at ω = 68 THz for θ = 450 and (b) at ω = 67.8 THz for θ = 50 . For all the panels the grating period d = 2µm. A different behavior is observed for the s-polarized light, see Fig. 4.10. In this case if the average intensity is less than 1 then the dependence of Iav on h is monotonic, i.e., there is no local maximum, which is different from the p-polarization [see Fig. 4.9]. The typical dependence is shown in Fig. 4.10(a), where for a/d = 0.5 the average

89 intensity is less than 1. If the average intensity is greater than 1, then there is an absolute maximum at some finite value of h. For example, at a/d = 0.3 the maximum of the average intensity is achieved at h ≈ 55 nm. Finally, the properties of the average intensity of the transmitted light for the s- and p-polarizations can be summarized. From the data shown above we can conclude that the enhancement of the transmitted light is the largest for the s-polarized light. For s-polarization, the frequency position of the intensity maxima can be tuned by varying the period of the diffraction grating, d, and the metallic coverage of the dielectric media, a/d. For the p-polarized light only the period of the diffraction grating affects the frequency position of the maximum. For p- and for s-polarized light there is no dependence of the position of the intensity maxima on the incident angle. Figure 4.10: The average intensity, Iav , for s-polarized light at distance z = 50 nm below the grating (i.e., in region III) is shown in units of I0 as a function of h (a) at ω = 75 THz and a/d = 0.5 and (b) at ω = 69 THz and for a/d = 0.3. For all the panels the grating period d = 2µm and the incident angle θ = 450 .

90 It is possible to find the optimal parameters of the diffraction grating so that the enhancement of the transmitted intensity is the largest. For the s-polarized light the decrease of the metallic coverage, a/d, and the incident angle, θ, results in increase of the average intensity, while for the p-polarized light only the incident angle changes the intensity of the transmitted light. As a function of the height of the diffraction grating the average intensity has a maximum for both p- and s-polarizations. For the parameters of the system considered in the present work the maximum is achieved at h ≈ 50nm. 4.4.2 Intensity distribution and the modes of the diffraction grating The important property of quantum dots is that they are sensitive to the local electromagnetic field. Therefore it is possible to enhance the sensitivity of the quan- tum dot photodetector by placing the quantum dots at the points with high intensity of the transmitted wave. In this relation it is important to understand the distribution of the transmitted electromagnetic wave in the active region of the photodetector, i.e., in the region III. In this distribution the bright spots, i.e., spots with high intensity can be expected. The distribution of the transmitted electromagnetic field in region III is shown in Fig. 4.11 for the p-polarized light. The main tendency, is that the intensity of the transmitted wave is non-uniform with maxima at points x = nd and x = a + nd, i.e.,

91 Figure 4.11: (color) The distribution of electric field intensity for p-polarized light is shown in region III. (a) Intensity I(x, z) at peak frequency, ω = 67.95 THz, for θ = 450 and a = 0.3d. (b) The same plot for a = 0.5d. (c) Similar plot for θ = 50 and a = 0.5d. (d) Electric field intensity in the vicinity of the first peak in panel (c) is shown in the x − z plane. The scale of the intensity is indicated by the color bar on the top. For all the panels the grating period d = 2µm and the height of the diffraction grating h = 50 nm. at the points corresponding to the boundaries between the metallic strips and air. With increasing enhancement of the transmitted light, i.e., with increasing average intensity, the peaks become more sharp and more localized. There is a weak effect of metallic coverage, a/d, on the distribution of the intensity of electric field [see

92 Fig. 4.11(a) and 4.11(b)]. For large incident angles, i.e., for small values of Iav , the peaks are broad, and the contrasts of the peaks are small, i.e., the ratio of the maximum intensity and the minimum intensity is around 2. When we change the parameters of the grating, which results in the enhancement of the average intensity, then the peaks become very sharp [see Fig. 4.11(c)]. The widths of the peaks are around 10 nm. Such a peak can accommodate one quantum dot. The contrasts of the peaks are large. For example, in Fig. 4.11(c) the ratio of the maximum value of the intensity to the minimum value is around 40. The same results for the s-polarized light is shown in Fig. 4.12. In comparison to the p-polarized light we have much stronger local enhancement of the intensity of the transmitted waves. Another difference from the p-polarization is that now the bright spots are located at the points corresponding to only one side of the metal-air interface, i.e., at x = nd, while at points x = a + nd the enhancement is much weaker. The contrast of the bright spots can be up to 200. The intensity at the peak decreases with the distance from the metal-dielectric interface, and the characteristic distance is around 20 nm. The widths of the peaks are the same as for the p-polarized light and are around 10-20 nm. Similar to the p-polarized light the peaks become more localized when the average intensity is increased. Comparing to the average value of the intensity we can say that the intensity at the peak can be 10-100 times larger than the average value. To clarify the differences between the p- and s-polarized light the distribution of

93 Figure 4.12: (color) The distribution of electric field intensity for s-polarized light is shown in region III. (a) θ = 450 and a = 0.4d; (b) θ = 450 and a = 0.5d; (c) θ = 50 and a = 0.5d. (d) Electric field intensity in the vicinity of the first peak in panel (c) is shown in the x − z plane. The scale of the intensity is indicated by the color bar on the top. For all the panels the grating period d = 2µm and the height of the diffraction grating h = 50 nm. the intensity of electric field in the grating and dielectric regions (regions II and III) is shown in Fig. 4.13. We can see that the peaks for the p-polarized light are more localized in the x direction than the corresponding peaks for the s-polarized light. At the same time for the s-polarization the peaks are more extended in the z direction. Therefore, if we need to use the advantage of the bright spots in the transmitted

94 waves then, for the s-polarization there are more restrictions on the positions of the quantum dots in the growth direction (z-direction). The quantum dots should be placed at the distance from the metal-dielectric interface less than 30-40 nm. Another crucial difference between the p- and s-polarized light is that the number of the bright spots for the p-polarization is twice as large as the number of the bright spots for the s-polarization. This follows from the fact (see Fig. 4.13) that the bright spots for the s-polarized light appear only at one side of the metal-air interface. Figure 4.13: (color) Distribution of I(x, y) is shown for the grating region (region II) and below the grating region (region III) for (a) p-polarization and (b) s-polarization. For all the panels: d = 2µm, a/d = 0.5, θ = 100 , and h = 50 nm. The behavior of the waves inside the diffraction region is also different for the p- and s-polarized light. We can see that electromagnetic field is mainly localized

95 inside the metal region for the p-polarized light and inside the air region (between metallic strips) for the s-polarized light. In the z-direction the electromagnetic field is localized in both cases at the interface with region III. Based on this behavior of the field inside the diffraction grating we can say that the modes, which are responsible for the enhancements of the incident light, are the surface plasmon modes for the p-polarized light and the modes trapped in the air region between two metallic strips for the s-polarized light. For the s-polarization the dependence of the energy of the trapped modes on the width of the air region, i.e., on the value of (d − a), explains the dependence of the average intensity of the transmitted light on the ratio a/d [see Fig. 4.2]. We can see from the figure that with decreasing a/d the frequency position of the intensity maximum decreases. This is because with decreasing a the width, (d − a), of the air regions increases. Then the size of the trapped mode also increases and correspondingly the energy of this mode decreases. With increasing the size of the trapped mode the coupling of the incident light with the mode is increased, which results in increase of the maximum intensity, shown in Fig. 4.2. 4.5 Conclusion It has been shown that the enhancement of the electromagnetic field due to the presence of the diffraction grating is much stronger for the s-polarized light than for the p-polarized light. This is opposite to what we should expect if we assume that the enhancement of the light is due to generation of the surface plasmons, since the

96 surface plasmons can be generated only by the p-polarized light, but not by the s- polarized light. The manifestation of the surface plasmons is not so strong in our system since I am working in the low-frequency range, where the coupling of the incident light to the surface plasmons is weak. For the s-polarized light the enhancement of the transmitted light is due to gen- eration of the modes of electromagnetic field, which are trapped between the metallic strips in the diffraction grating region. For the p-polarized light, the modes respon- sible for the light enhancements are the surface plasmon modes. As a result the light in the diffraction grating is localized in the metallic region for the p-polarized light and in the air region for the s-polarized light. The dependence of the enhancement of the transmitted light on the parameters of the diffraction grating can be summarized as follows • Decreasing the metallic coverage, i.e., a/d, of the dielectric media increases the intensity of the s-polarized light, but does not affect the p-polarized light. For the s-polarized light there is an optimal value of a/d for which the enhancement is maximum. • Decreasing the incident angle increases the enhancement of the light for both p- and s-polarizations. At the same time for both polarizations the incident angle does not change the frequency positions of the intensity maxima. For the actual value of the transmitted light there is an optimal incident angle, θ ≈ 50 , for the p-polarized light, at which the intensity of the transmitted light is the

97 maximum. For the s-polarized light the actual intensity of the transmitted light has a monotonic dependence on the incident angle and the maximum of the intensity is realized at zero incident angle. • The intensity of the transmitted light for both p- and s-polarizations has a nonmonotonic dependence on the height, h, of the diffraction grating. There is an optimal value of h, at which the intensity has a maximum. This value is around 50 nm for the parameters of the system considered in the present work. • The intensity of the transmitted wave decreases with the distance from the metal-dielectric interface. The characteristic length is around 50 nm. • The spatial distribution of the intensity of transmitted light is highly nonuniform with the sharp peaks. The spatial size of the peaks is around 10 nm. For the s-polarized light the intensity at the peak is two order of magnitude larger than the average intensity. Therefore, to have the largest enhancement of the transmitted light in the active region of the quantum dot photodetector we need to use the s-polarized light. In addition, the metallic coverage, a/d, and the incident angle should be small. The height of the diffraction grating should have the optimal value, which is around 50 nm. The strongest effect is achieved if the quantum dots are placed at the bright spots of the transmitted wave. For the s-polarized light the bright spots are near only one type of the metal-air interface. The distance between the quantum dot layer and the

98 metal-dielectric interface should not exceed the characteristic size of the enhancement region in the growth direction. This size is around 30-40 nm. Under these conditions the enhancement of the signal could be a few orders of magnitude. For a general system of quantum dots we can estimate the strength of the local enhancement, Imax , of the transmitted wave so that the net effect of the diffraction grating is the enhancement of the absorption coefficient. If the average distance between the quantum dots is dav , then only part of the quantum dots will experience the high electric field in the photodetectors with the diffraction grating. We can estimate this part as dav /d. Then the maximum enhancement should satisfy the following condition Imax > d/dav . For dav ∼ 50nm and d = 2000nm, Imax > 40 has been obtained. This value can be easily achieved for some values of parameters of the diffraction grating. This results can be also applied to a single quantum dot. Namely, by placing the quantum dot at the bright spots of the diffraction grating we can enhance the optical properties of the dot.

99 Chapter 5 Plasmonic enhancing nanoantennas for photodetection 5.1 Introduction In this Chapter I will discuss the use of plasmonic nanostructured systems as nanoantennas for photodetection. Even though semiconductors and their heterostruc- tures have many useful properties and widely used in photodetection, their electron density is very small compared to that of metals and, therefore, they have low absorp- tion cross sections. The idea of using metal nanostructured antennas is to combine the high optical responses of metals with the functional electric properties of semi- conductors. The main results in this chapter has been published in Infrared Physics and Technology [66].

100 Nanostructured metals have a wide spectral range of surface plasmon resonances from near-ultraviolet to infrared. One particular example is a collection of thin metal wires where the electron density n averaged over the volume is low, and so is the √ electron plasma frequency ωp ∝ n. Metal nanostructures, that have sharp edges or tips also possess low surface-plasmon resonance frequencies. Such nanostructures, when excited by a resonant optical radiation, create high local fields at these sharp features that may be up to three orders of magnitude higher in amplitude than the excitation optical field. For lower THz-range frequencies, the dielectric function of metals becomes mostly imaginary with its real part being positive, which precludes the surface plasmon resonances. In these cases, there are two types of plasmonic systems still available. The first type of the systems is highly-doped semiconductors that possess metallic-like optical behavior in the far-infrared region, Second, dielectrics and intrinsic semiconductors between their TO and LO phonon frequencies which exhibit dielectric permittivity with a large negative real part and relatively small imaginary part (reststrahlen band), which is necessary for a pronounced plasmonic behavior. Microstructures and nanostructures made out of such materials exhibit high-Q plasmonic resonances and enhanced fields. 5.2 Nanoantennas Nanoantennas [66] designed to receive optical energy in the ultraviolet, visible, infrared, and beyond are the smallest detectors available . They can be scaled to

101 a fraction of the radiation wavelength. Such optical antennas are specifically engi- neered to focus micrometer scale light into nanoscale volumes with high spatial and spectral control of the greatly enhanced fields in visible and near-infrared regions [67]. The subwavelength antennas whose size is much smaller than wavelength will be considered. In most cases, resonant conventional antennas in radio-frequency or microwave range have size comparable to and scaling with wavelength. In contrast, plasmonic nanoantennas have internal, plasmonic resonances that do not depend on their size but rather on their shape only. The resonance frequencies of known types of plasmonic nanoantennas span the entire optical range from near-ultraviolet to near-infrared. Nanostructured antennas are related to nanoantennas: they consist of elements with nanoscale sizes but their total size may be comparable or larger than the wavelength. In this case too, the resonance frequency is determined by the nanos- tructure shapes and not necessarily by the size of the entire antenna. A particular type is a nanostructured system which generates propagating electromagnetic waves at the interfaces of plasmonic materials known as surface plasmon polaritons [68]. By nature, the nanoantennas are direction, wavelength, and polarization sensitive elements. There are two types of plasmonic nanoantennas discussed in this work. The first type is based on surface plasmons which are standing quasielectrostatic waves [69, 70, 71]. Such nanoantennas are capable of producing very high local fields in small volumes. They may be used to improve signal/noise characteristic of IR photodetectors. The semiconductor detector should have size on the nanoscale and

102 be positioned near the hot spot (nanofocus) of the nanoantenna. One of the most efficient optical antennas is bowtie plasmonic nanoantenna and their arrays. For example, the fractal bowtie plasmonic nanoantenna array shown in Ref. [70] is designed to detect five bands centered at wavelengths of λ, 2λ, 4λ, 8λ and 16λ . Typically the bowtie nanoantenna consists of two triangular pieces of gold, each about 75 nm long, and whose tips face each other in the shape of a bowtie. The bowtie takes energy from near-infrared light and squeezes it into a 20 nm gap that separates the two gold triangles [67]. As shown in Ref. [67], the result is a concentrated spike of light that is several order of magnitude intense than the exciting field. There has recently been great interest in the spherical metal nanoshells as nanoan- tennas consisting of a dielectric core covered by a thin gold layer. By varying the relative dimensions of the core and the shell, the resonance frequency of the nanopar- ticles can be precisely varied from the near-ultraviolet to the mid-infrared [71]. The plasmon resonance frequency of the nanoshell shifts as a function of nanoshell com- position: for thinner metal shells the resonance shifts toward the red and infrared. 5.3 Chain of metal nanospheres As an efficient nanoantenna, self-similar linear chain of several metal nanospheres with progressively decreasing sizes and separations [69] has been considered. To introduce our idea, consider a finite chain of metal nanospheres where for an ith nanosphere the radius is Ri and its surface-to-surface separation from the (i + 1)th

103 600 a hω = 3.37 eV 400 »E» 200 0 150 b »E» hω = 3.25 eV 100 50 0 10 c hω = 0.77 eV »E» 6 2 Figure 5.1: Local fields (absolute value relative to that of the excitation field) in the equatorial plane of symmetry for linear self-similar chain of three silver nanospheres. The ratio of the consecutive radii is = Ri+1 /Ri = 1/3; the distance between the surfaces of the consecutive nanospheres di,i+1 = 0.6Ri+1 . Inset: the geometry of the system in the cross section trough the equatorial plane of symmetry. nanosphere is di,i+1 . Self-similarity will be assumed, i.e., Ri+1 = κdi,i+1 and di+1,i+2 = κdi,i+1 , where κ =constant; see a schematic in inset in Fig. 5.1(a). Consider κ 1 , so the local field of a given nanoparticle is only weakly perturbed by the next one.

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