Plasmonic and excitonic effects in restricted geometries

Many electron systems are well described by Landau's Fermi liquid theory. However, such a decent theory breaks down in one dimension due to the correspondingly \unusual" shape of a Fermi surface. A one-dimensional electron system is thus called a Luttinger liquid, which was first proposed by Tomonaga in 1950 and later developed by Luttinger, Mattis, Haldane and others. The most notable difference in 1D is that low energy excitations are massless fermions with linear dispersion. Correspondingly, electron-electron interaction significantly alters transport properties in 1D. Graphene is a two-dimensional layer material with a honeycomb structure, and its low energy excitations are massless fermions. Accordingly, carbon nanotube, as the roll-up sheet of graphene, has been regarded as an ideal platform for testing Luttinger liquid effects. In Chapter 2, we studied the excitonic eect in metallic nanotubes. Exciton is formed when a particle and a hole is bounded by their attractive Coulomb interaction. In bulk metals, exciton hardly exists due to the strong screening effects. However, the situation changes in 1D where Coulomb interaction remains largely unscreened. The relatively ineffective screening yields a sufficiently large radius of excitons that is about 10 times larger than the radius of nanotube Rex  10R. Therefore, excitons in metallic nanotube become a well-defined 1D problem. The problem has been studied in an analytical manner. In Chapter 3, we discussed many-body effects in the depolarization effect. For a cylinder exposed in an external electric field perpendicular to its axis, the electric field inside the cylinder is found to be \suppressed." The suppression is due to the depolarization effect. Same phenomena occur in nanotube, which originates from a dipolar Coulomb interaction. The prediction is within the theory electrostatics or Random Phase Approximation(RPA) in quantum theory. However, what makes the problem more complex is Luttinger liquid effects, which enter the RPA series as vertex correction. Post-RPA effects, or many-body effects, are often hard to treat exactly. One often resorts to numerical approach or renormalization group in the qualitative fashion. However, by virtue of a hybrid approach, which combines the advantage of bosonization and perturbation theory, we examined many-body modification of the line shape in a depolarization effect. In addition to single particle excitations in the electron system, plasmon excitations as a novel excitation also attract researchers. Under the external field, electrons in metals are oscillating back and force to gain a balance against the external field. Such oscillations exist even without the presence of external fields, which are called plasmon excitations. Therefore, plasmon waves are collective oscillation, namely all electrons are moving as a whole. For the plasmon waves that are localized near the interface of two materials while propagating along the surface, they are called surface plasmon or surface plasmon polariton if retardation eects are considered. The geometry of the boundary mostly determines the property of the surface plasmon. In Chapter 5, a novel surface plasmon in hyperbolic boundary was discussed in many perspectives.

PLASMONIC AND EXCITONIC EFFECTS IN RESTRICTED GEOMETRIES by Lei Shan A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2018 Copyright c Lei Shan 2018 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Lei Shan has been approved by the following supervisory committee members: Eugene Mishchenko , Chair(s) 30 July, 2018 Date Approved Oleg Starykh , Member 30 July, 2018 Date Approved Sarah Li , Member 30 July, 2018 Date Approved Zheng Zheng , Member 30 July, 2018 Date Approved Steven Blair , Member 30 July, 2018 Date Approved by Peter E. Trapa , Chair/Dean of the Department/College/School of Physics and Astronomy and by David B. Kieda , Dean of The Graduate School. ABSTRACT Many electron systems are well described by Landau’s Fermi liquid theory. However, such a decent theory breaks down in one dimension due to the correspondingly “unusual” shape of a Fermi surface. A one-dimensional electron system is thus called a Luttinger liquid, which was first proposed by Tomonaga in 1950 and later developed by Luttinger, Mattis, Haldane and others. The most notable difference in 1D is that low energy excitations are massless fermions with linear dispersion. Correspondingly, electron-electron interaction significantly alters transport properties in 1D. Graphene is a two-dimensional layer material with a honeycomb structure, and its low energy excitations are massless fermions. Accordingly, carbon nanotube, as the roll-up sheet of graphene, has been regarded as an ideal platform for testing Luttinger liquid effects. In Chapter 2, we studied the excitonic effect in metallic nanotubes. Exciton is formed when a particle and a hole is bounded by their attractive Coulomb interaction. In bulk metals, exciton hardly exists due to the strong screening effects. However, the situation changes in 1D where Coulomb interaction remains largely unscreened. The relatively ineffective screening yields a sufficiently large radius of excitons that is about 10 times larger than the radius of nanotube Rex ≈ 10R. Therefore, excitons in metallic nanotube become a well-defined 1D problem. The problem has been studied in an analytical manner. In Chapter 3, we discussed many-body effects in the depolarization effect. For a cylinder exposed in an external electric field perpendicular to its axis, the electric field inside the cylinder is found to be “suppressed.” The suppression is due to the depolarization effect. Same phenomena occur in nanotube, which originates from a dipolar Coulomb interaction. The prediction is within the theory electrostatics or Random Phase Approximation(RPA) in quantum theory. However, what makes the problem more complex is Luttinger liquid effects, which enter the RPA series as vertex correction. Post-RPA effects, or many-body effects, are often hard to treat exactly. One often resorts to numerical approach or renormalization group in the qualitative fashion. However, by virtue of a hybrid approach, which combines the advantage of bosonization and perturbation theory, we examined many-body modification of the line shape in a depolarization effect. In addition to single particle excitations in the electron system, plasmon excitations as a novel excitation also attract researchers. Under the external field, electrons in metals are oscillating back and force to gain a balance against the external field. Such oscillations exist even without the presence of external fields, which are called plasmon excitations. Therefore, plasmon waves are collective oscillation, namely all electrons are moving as a whole. For the plasmon waves that are localized near the interface of two materials while propagating along the surface, they are called surface plasmon or surface plasmon polariton if retardation effects are considered. The geometry of the boundary mostly determines the property of the surface plasmon. In Chapter 5, a novel surface plasmon in hyperbolic boundary was discussed in many perspectives. iv For my wife Yaxin Zhai, my son Mason Shan, and my parents. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii CHAPTERS 1. 2. 3. INTRODUCTION TO CARBON MATERIALS . . . . . . . . . . . . . . . . . . . . 1 1.1 Carbon bonds and hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Polarization function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Polarization function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Optical response of carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 4 5 6 7 7 EXCITONIC STATE IN METALLIC CARBON NANOTUBES . . . . . . 9 2.1 Introduction to exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Frenkel and Wannier exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Direct- and indirect-bandgap in semiconductors . . . . . . . . . . . . . . . . . . . 2.1.3 Photoluminescence and polariton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Excitonic state in metallic nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Singular potential in one dimension and Loudon model . . . . . . . . . . . . . 2.2.2 Failure of effective-mass approximation in semiconducting SWNTs . . . . 2.2.3 Binding energy in metallic SWNTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Relaxation of exciton in metallic SWNTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Finite lifetime of exciton in metallic SWNTs . . . . . . . . . . . . . . . . . . . . . 2.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 11 12 12 15 16 18 20 21 24 LUTTINGER LIQUID AND MANY-BODY EFFECTS IN METALLIC NANOTUBES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Introduction to Fermi liquid and Luttinger liquid . . . . . . . . . . . . . . . . . . . . . 3.2 Luttinger liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Scattering processes in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Renormalization group in Luttinger liquid . . . . . . . . . . . . . . . . . . . . . . . 3.3 Many-body effects in the depolarization effect of metallic nanotubes . . . . . . . 3.3.1 Depolarization effect and random phase approximation . . . . . . . . . . . . . 25 27 27 32 34 37 39 3.3.2 Bosonization in metallic nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Time-ordered Green’s function and the singular interaction . . . . . . . . . . 3.3.4 Second order expansion and self-energy disaster . . . . . . . . . . . . . . . . . . . 3.3.5 Poorman’s summation and beyond RPA . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. INTRODUCTION TO PLASMONICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1 Plasmon excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Drude model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Plasmon oscillation in the formalism of quantum theory . . . . . . . . . . . . 4.2 Surface plasmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Planar surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Other simple geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Surface plasmon in a film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Surface plasmon polariton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 43 46 52 55 58 59 60 60 62 63 64 64 65 67 SURFACE PLASMON AND ENERGY TRANSFER IN HYPERBOLIC GEOMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Plasmon spectrum of a single hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Elliptic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Plasmon dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Field distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Geometry of two hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Splitting of the plasmon spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Comparison of the field distributions for two geometries . . . . . . . . . . . . 5.3.3 Two co-directed hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Interaction of two emitters at the metal-air surface . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 71 71 72 75 77 77 81 83 84 87 APPENDIX: IMPORTANT CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 91 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 vii LIST OF FIGURES 1.1 Graphene has a honeycomb geometry in both real and reciprocal space. . . . . . 2 1.2 Global electronic spectrum of π-electrons in graphene. Dispersion has a linear structure near Dirac points (touching points in the plot). . . . . . . . . . . . . . . . . . 4 1.3 Feynman diagram for the polarization bubble corresponding to Eq. (1.11). . . . 5 1.4 Nanotubes are characterized by two integers (n, m). The red line in the plot indicates the circumferential direction of a nanotube. . . . . . . . . . . . . . . . . . . . . 6 The plot shows selection rule in nanotube: transverse electric fields induce interband transitions and longitudinal electric fields induce intraband transitions. 8 1.5 2.1 Spectrum of exciton states in semiconductor. At n → ∞, energy of exciton coincides with band continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 The line shape with excitonic effect (solid red line) and without excitonic effect (dashed red line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Vertical transition and nonvertical transition. In the latter case, a single photon is not sufficient and the mismatch of momentum can be provided from a phonon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 The exciton state emerges due to the final state interaction of an electron and a hole with energy below the continuum edge. . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Two Coulomb interactions: the left one yields binding energy of exciton while the right one leads to finite lifetime of exciton. . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 Particle hole continuum in 3D (left) and 1D (right). Dashed lines are plasmon dispersions: quadratic in 3D with a gap and linear in 1D being gapless. In one dimension, plasmon dispersion complete sits in the particle-hole continuum which indicates the decay of electron-hole excitation into collective plasmon modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 In one dimension, one can always linearize dispersion near the Fermi surface. Excitations with low energy(particle-hole pair near the Fermi surface) in one dimension are well defined in energy and momentum, because it only depends on q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Four scattering processes with low energy in one dimension. For spinless particles, g1 and g2 are the same. All processes obey conservation of momentum except the left bottom one, which is the umklapp process, since it involves interaction with the lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Renormalization flow of Sine-Gordon Hamiltonian. The right diagonal line is a marginal irrelevant line or phase transition line. Interaction is relevant above it and irrelevant below it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Geometry of a nanotube in a transverse electric field (left) and corresponding absorption transitions induced by such field (right): from filled gapless states (m = 0) into unfilled gapless states (|m| = 1) and from the filled gapped states (|m| = 1) into empty movers (m = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Four scattering processes in V1 -interactions. Wiggle lines carry angular momentum m = 1 due to the conservation of angular momentum at vertices. Double- and single-solid lines represent massive and massless propagators, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.7 † † First order corrections of the most singular interaction ψ̂−1,α ψ̂1,β ψ̂r,β ψ̂r,α in a one-channel system (top) and a multichannel system (bottom). For this type of interaction, it is required that two loops must be in different neutral channels to display as a more singular correction. The interaction V0 (dashed line) is account to all orders (not shown in the top one). . . . . . . . . . . . . . . . . . 51 3.8 Two contributions at second order with neutral phases in an alternating order. The three-loop diagram (top) reproduces RPA in the limit γ → 0, while the self-energy diagram (bottom), being divergent correction to first zeroth order, renormalizes band gap energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.9 Absorption lineshape for different values of V1 = 4πvλ1 /N . For small V1 values (solid blue line) the suppression (depolarization) induced overcomes the enhancement due to V0 being sufficiently close to the threshold. For larger V1 values the depolarization effect dominates everywhere. . . . . . . . . . . . . . . . . 59 4.1 A film exposed in vacuum which resembles a three-layer system. . . . . . . . . . . . 65 4.2 Plasmon dispersion for a film immersed in vacuum. . . . . . . . . . . . . . . . . . . . . . 66 4.3 Plasmon dispersion of SPP for planar surface. . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1 The plasmon spectrum of these structures can be found analytically if metallic surfaces are confocal hyperbolas. Simplest examples of multiconnected plasmonic structures: a) “neck” and b) “gap.” 2a is the distance between the foci. c) Schematic view of a plasmonic array[1]. The field distribution in the array is determined by the points of contact of metallic islands. . . . . . . . . . . . . . . . . 70 5.2 Geometry of a single hyperbola. In elliptic coordinates ξ, η, defined by Eq. (5.3), the metal with dielectric constant, ε(ω), occupies the domain (−ηp , ηp ). Outside the hyperbola there is air with ε = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3 Plasmon spectrum of a single hyperbola for two opening angles 2η0 = π/10 (red) and 2η0 = π/3 (blue) is plotted from Eq. (5.11) for symmetric mode and Eq. (5.12) for antisymmetric mode. At a large wavenumber, m, both branches √ approach the flat surface plasmon frequency, ωp / 2. . . . . . . . . . . . . . . . . . . . . 74 5.4 Distribution of Ex (for symmetric mode) and Ey (for antisymmetric mode). (a) The components of the electric field along the x-axis is plotted from Eqs. (5.18)-(5.20) for the wavenumber m = 2 and the opening angle 2η0 = π/3. The inset shows the large-x oscillating tail of Ex . (b) The same as (a) for two hyperbolas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ix 5.5 Density plot of the field intensity of the plasmon modes for a single hyperbola with the opening angle 2η0 = π/3. The upper row corresponds to a symmetric mode, for which the potential distribution is determined by Eqs. (5.5) and (5.6). The lower row corresponds to an antisymmetric mode with the potential described by Eqs. (5.8) and (5.9). Left, central, and right panels correspond to the wavenumbers, m = 0.1, m = 0.4, and m √ = 0.7, respectively. With increasing m the plasmon frequencies approach ωp / 2, while the field concentrates near the metal-air surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.6 The geometry of two hyperbolas with the opening angles 2η0 and 2η1 . A dipole emitter close to the tip and polarized along the interface excites the dipoles located to the left and to the right from the tip at distances much bigger than the focal distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.7 Density plot of the field intensity of the plasmon modes in the geometry of two symmetric hyperbolas with the same opening angle, 2η0 = π/3, as in Figure 5.6. The upper row corresponds to a symmetric mode for which the potential distribution is determined by Eqs. (5.21) and (5.26). Left, central, and right panels correspond, respectively, to the wavenumbers, m = 0.1, m = 0.4, and m = 0.7, the same as in Figure 5.6. While the fields of individual hyperbolas are disconnected along the x-axis, they overlap along the y-axis. The lower row corresponds to antisymmetric modes with the same m-values. The field of individual hyperbolas overlap, predominantly, along the y-axis. . . 81 5.8 Comparison of the plasmon spectra in the geometry of two hyperbolas for two values of the opening angle. For η0 = π/6 < π/4 (upper panel) the frequencies of both symmetric modes (blue) are smaller than the flat surface plasmon √ frequency ωp / 2, while √ the frequencies of both antisymmetric modes (red) are bigger than ωp / 2. The relative signs of the oscillating charge density along the metal surfaces are schematically illustrated to the left of the graph. For η0 = 2π/5 > π/4 (lower panel) the positions of the upper symmetric mode and lower antisymmetric mode with respect to ωp invert. . . . . . . . . . . . . . . . . . 82 5.9 Illustration of the plasmon spectrum in the geometry of two co-directed hyperbolas. Blue and red curves correspond to symmetric and antisymmetric plasmon modes, respectively. The spectrum is plotted from Eq. (5.33) for ηc = δη = π/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.10 Geometric factor for the energy transfer is plotted as a function of frequency from Eq. (5.42) for the values of the opening angle 2η1 = 0.2π, 0.24π, 0.28π (a), and 2η1 = 0.56π, 0.6π, 0.64π (b). The value of the opening angle, 2η0 , is chosen 2η0 = 4π/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 A.1 Logarithmic function in the complex plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.2 Contour integral of multivalue function is chosen to be closed in the upper half plane. For the lower plane, there is no singularities and thus no absorption for frequency ω < ∆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A.3 “X-ray” absorption in metallic nanotubes: conduction band consists of right/leftmoving states and the lower massive subband with |m| = 1 chosen to be core-hole states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 x LIST OF TABLES 3.1 Comparison of all scattering processes in interaction V1 . . . . . . . . . . . . . . . . . . 53 3.2 Interpolation of odd power polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ACKNOWLEDGEMENTS I would like to give my deepest gratitude to my advisor Dr. Mishchenko for guiding me in one of the most interesting directions in condensed matter physics, and giving me professional training in thinking and solving physical problems. Eugene’s personality and patience inspire me all the time in my PhD life. I would like to thank Dr. Raikh for collaborating our surface plasmon paper. I am deeply grateful to our former department secretary Jackie Hadley, who helped me through the darkest period of my life. I would like to thank Dr. Liu and his family who treated me with kindred love and tasty feasts. I would like to thank all members of my committee for their kind concerns. Finally, I want to thank my parents for their unconditional support and love throughout my life. CHAPTER 1 INTRODUCTION TO CARBON MATERIALS Carbon nanotubes are cylinders rolled by honeycomb lattice. Owing to its special geometry, carbon nanotubes have unusual properties, which attracts many people’s attention in optics, nanotechnology, electronics and so forth. Electronic structure of a single-walled nanotube can be either metallic or semiconducting which depends on chirality. Metallic carbon nanotubes have low energy states with a linear spectrum, which has been regarded as a good candidate in examining Luttinger liquid effects. 1.1 Carbon bonds and hybridization A carbon atom is the 6th element in the periodic table with 6 protons and 6 electrons around the nucleus. Since two electrons stay in the 1s orbital with antiparallel spin directions in the lowest energy state, four outer electrons are valence electrons. In the ground state, electron configuration is 1s2 2s2 2p2 (two electrons in the lowest energy state n = 0, l = 0, two electrons in the n = 1, l = 0 state, and two electrons in n = 1, l = 1 state). In the excited state, a carbon atom can form covalent bonds with other atoms, namely two atoms can share one or several electrons. This is called hybridization. The most energetically favorable state is sp3 hybridization, where all four valence electrons form σ bonds in sp orbital (a hybrid orbital by mixing of one s and three p orbital) with bond angle 109.5 degree. σ bond is responsible for the robustness of lattice structure in all allotropes. Such tetrahedral geometry leads to lowest energy state, and thus being extremely stabilized such as diamond (when all atoms are carbon) and methane (CH4 ). If there are three electrons in the hybridized orbital with angle 120 degree between each other, it is called sp2 hybridization. There is then one valence electron left in the 2p orbital forming a π bond with other “free” electrons and therefore this electron is called a π electron. π-electrons are supposed to endow metallic properties to the material. Besides, there is sp1 hybridization 2 when two valence electrons are in the hybridized orbital and two in the 2p orbital. 1.2 Graphene The most popular material corresponds to sp2 hybridization is graphene. Graphene is made of carbon atoms arranged on a honeycomb structure. Graphene is a two-dimensional allotrope of carbon with three electrons in the σ band and one electron in the p orbital perpendicular to the plane. This electron binds with neighboring carbon atoms into a π band. Since there is only one electron in each p orbital, the π band is thus half filled. 1.2.1 Electronic structure Wallace is the first one who gave the band structure of gaphene and its electronic properties. The structure of graphene is given in Figure 1.1 where the lattice vectors are chosen as, a1 = a √ (3, 3), 2 a2 = √ a (3, − 3), 2 (1.1) where a ≈ 1.42Å is lattice constant between two nearest carbons. Three nearest neighbor vectors are give by, √ ~δ1 = a (1, 3), 2 √ ~δ2 = a (1, − 3), 2 ~δ3 = a(−1, 0). (1.2) Therefore, Brillouin zone has hexagonal geometry as well, where two topologically different points K and K 0 points are given by ሺܽሻ ‫ܤ‬ ‫ܣ‬ ሺܾሻ ‫ܤ‬ ‫ܣ‬ ‫ܭ‬ᇱ ‫ܣ‬ ‫ܭ‬ ߜଵ ߜଷ ‫ܤ‬ ‫ܤ‬ ‫ܭ‬ᇱ ‫ܭ‬ ߜଶ ‫ܣ‬ ‫ܤ‬ ‫ܤ‬ ‫ܣ‬ ‫ܭ‬ᇱ ‫ܭ‬ ‫ܣ‬ Figure 1.1. Graphene has a honeycomb geometry in both real and reciprocal space. 3 K=( 2π 2π , √ ), 3a 3 3a K0 = ( 2π 2π , − √ ). 3a 3 3a (1.3) In the tight-binding model where only the transitions between nearest neighbours are considered, the Hamiltonian is given by, H = −t X [â†i b̂j + c.c] (1.4) i,j √ P ~ Recalling Fourier transformation that âi = 1/ N k exp(−ik · δi )âk , and relation that P 0 1/N x exp[i(k − k )x] = δk,k0 , Hamiltonian can be written in a more instructive form, X 0 −tf (k) ak † † † H = −t [f (k)âk b̂k + c.c] = ak bk (1.5) ? −tf (k) 0 bk k where f (k) = P j ~ e−ik·δi . The Spectrum of graphene under tight-binding model can be found through diagonalization procedure, s √ X √ 3aky 3akx −ik·δ~j E(k) = ±t| e | = ±t 3 + 2 cos( 3aky ) + 4 cos( ) cos( ). 2 2 (1.6) j Utilizing Eq. (1.3) and Eq. (1.2), the dispersion can be calculated in the vicinity of K point, k = K + q, f (q) = 3a (iqx + qy )eiKx a . 2 (1.7) √ Since eiKx a exhibits a plane wave at point K = (2π/3a, 2π/3 3a), it can be removed safely. Accordingly, Hamiltonian becomes, H = ~vF σ̂q = ~vF 0 qx − iqy qx + iqy 0 (1.8) where σ̂ is Pauli matrix and vF = 106 m/s is Fermi velocity of electrons in graphene (under tight-binding model), which is 300 times smaller than the speed of light c. Diagonalization √ of the Hamiltonian yields the dispersion near K = (2π/3a, 2π/3 3a) as, E(q) = ±~vF |q|. (1.9) The linear dispersion of low energy excitations indicates the existence of massless fermions (or Dirac fermions) which mimics quantum electrodynamics for massless fermions. The Fermi surface of undoped graphene is right at zero energy where conduction (upper cone) and valence bands (lower cone) touch. From Hamiltonian in Eq. (1.5), we see that wave 4 functions of graphene are two component spinnors. It is easy to find that for right/leftmoving π electrons, pseudospin matrix has form, 1 1 ψ̂r (x) = ψ(x) ψ̂l (x) = ψl (x) 1 −1 (1.10) Since electrons are fermion and there are two topologically different valleys K and K 0 , the degeneracy of π-electrons is N = 4. The complete spectrum of graphene is shown in Figure 1.2. Although low energy electrons have “metallic” dispersion, there are only six touching points in the Brillouin zone which limit the conductivity of graphene. In other words, graphene is a hybrid of metal and insulator – semimetal with vanishing density of state at Fermi surface ρ ≈ |E|/v 2 . 1.2.2 Polarization function The polarization function in one loop can be calculated from the diagram shown in Figure 1.3, Π(iω, q) = X N Tr G(0) (i + iω, p + q)G(0) (i , p). βV (1.11) i ,p In graphene, bare Green’s function of π-electrons in the low energy sector consists of contributions from both upper cone (β = 1) and lower cone (β = −1), Figure 1.2. Global electronic spectrum of π-electrons in graphene. Dispersion has a linear structure near Dirac points (touching points in the plot). 5 Figure 1.3. Feynman diagram for the polarization bubble corresponding to Eq. (1.11). G(0) (i , p) = 1 1 X 1 + β σ̂p = , i − vp · σ̂ 2 i − Eβ (p) (1.12) β=±1 where σ̂p = σ̂ · p/p is the projection of the pseudospin Pauli matrix onto the direction of electron momentum. Dispersion of Dirac electron follows Eβ (p) = βE(p) = βv|p|. Matsubara frequency summation yields, Π(iω, q) = n(Eβ,p ) − n(Eβ 0 ,p+q ) N XX Fβ,β 0 (p, q) , V p iω + Eβ (p) − Eβ 0 (p + q) 0 (1.13) β,β where n(E) follows Fermi-Dirac distribution. Function Fβ,β 0 (p, q) describes the overlap of two wave functions for either intraband or interband transitions. 1 Fβ,β 0 (p, q) = Tr[(1 + β 0 σp )(1 + βσp+q )]. 4 (1.14) In undoped graphene and zero temperature limit, transitions are restricted to the ones from the valence band (lower cone)to the conduction band (upper cone), where ββ 0 = −1. Retarded Green’s function of polarization operator can be obtained through analytical continuation iω → ω + iη. However, since virtually transitions avoid all singular poles, small imaginary constant iη can be safely discarded, Π(ω, q) = Z∞ β[cos(θp+q − θp ) − 1] N X d2 p , 2 8π ω + β[E(p) + E(p + q)] = − β −∞ q2 N . 16 v 2 q 2 − ω 2 (1.15) (1.16) The result of above equation corresponds to intraband transitions within metallic subbands. 1.3 Nanotube Nanotube is the roll-up sheet of graphene in the cylinder structure, and thus it is regarded as quasi-one-dimensional material. Single-wall nanotubes have different varieties 6 depending on the circumferential boundary which is characterized by two integers (n, m), see Figure 1.4. Two special cases are armchair(n = m) nanotube with metallic property and zigzag(m = 0) with insulating behavior (a gap in the spectrum). Others are called chiral nanotubes. Quantitatively, NT is metallic if 2n + m = 3J and semiconductive if 2n + m 6= 3J where J is a positive integer. 1.3.1 Electronic structure Electronic band structure of carbon nanotubes follows from the underlying two-dimensional spectrum of graphene, (p) = ±vp. The components of the quasimomentum p are measured from the corresponding Dirac points in the first Brillouin zone. In the case of rolledup graphene sheets, the circumferential momentum py is quantized giving a set of onedimensional subbands. For “metallic” folds, some cuts pass through the Dirac points, so that p py = m/R, and the resulting spectrum consists of the subbands, m (p) = ±v p2 + m2 /R2 , classified by the angular momentum quantum number, m = 0, ±1, ±2, ...; for (n, n) armchair tubes the NT radius R = 3na/2π where a = 1.4 Å is the distance between carbon atoms. The following Hamiltonian describes the noninteracting band electrons of 2D Dirac fermions, Ĥ0 = −ivσx iv ∂ ∂ − σy , ∂x R ∂θ (1.17) ƌŵĐŚĂŝƌ ࢇଵ ࢇଶ ŝŐnjĂŐ ࢉ ൌ ݊ࢇଵ ൅ ݉ࢇଶ Figure 1.4. Nanotubes are characterized by two integers (n, m). The red line in the plot indicates the circumferential direction of a nanotube. 7 where θ = y/R is the circumferential angle. Eigenstates of metallic subbands (m = 0) and first gapped subbands (m = ±1) are expressed as the following spinnor form, ψ0 (t, x) ψ̂0 (t, x) = √ 4π 1 ±1 ψ±1 (t, x) , ψ̂±1 (t, x) = √ 4π 1 ±i , (1.18) where the hat indicates that the wave function is a pseudospinor in the sublattice space. The above arguments are valid as long as zone-folding approximation is applied. Moreover, pseudospin matrix for massless particles is exact, while for massive subbands is exclusive for low energy electrons near the vicinity of the minimum point of parabolic subbands. 1.3.2 Polarization function Recalling Eq. (1.13), polarization function for a nanotube can be readily obtained. Let py = m/R and qy = µ/R which follows the quantized momentum along the circumferential direction of the nanotube and transform density response from 2D to 1D by multiplying factor 2πR, we find, Z∞ i h (px + qx )px + (m + µ)m/R2 N X p dpx 1 − p Πµ (ω, qx ) = 2π m (p2x + m2 /R2 ) (px + qx )2 + (m + µ)2 /R2 −∞ p p v (p2x + m2 /R2 ) + v (px + qx )2 + (m + µ)2 /R2 p p × . (1.19) ω 2 − [v (p2x + m2 /R2 ) + v (px + qx )2 + (m + µ)2 /R2 ]2 Particularly, transitions within the linear subbands (µ = 0) can be evaluated as, Π0 (ω, q) = N q2v2 . π~v ω 2 − q 2 v 2 (1.20) In the static limit, it approaches a constant. For µ = 1, the only nonvanishing transitions are between linear subbands and first parabolic subbands, m = −1, 0. In the case of static and homogeneous limit where ω → 0 and qx → 0, the polarization operator becomes, N Π1 (0, 0) = − π Z∞ 0 p 1 − vpx / v 2 p2x + ∆2 N p dpx =− , 2 2 2 πv vpx + v px + ∆ (1.21) which is the same as the Π0 (0, 0). 1.3.3 Optical response of carbon nanotubes From Eq. (1.13), “overlap” function in the transition Fβ,β 0 (p, q) goes to zero if it is between states correspond to dispersion = vp and = −vp. It simply implies that such 8 transitions are forbidden in graphene. Same conclusion applies to nanotubes, which can be easily seen from the pseudospin matrix given in Eq. (1.18). The multiplication of wave functions of right- and left-moving states is zero. Considering the intrasubband transition among higher subbands that does not change the angular momentum, the overlap function is vanishing again. Transitions between linear subbands are important in calculating the binding energy of exciton, which we will discuss more in Chapter 2. The allowed transitions can be induced through photon or electric field. For a transverse electric field, only the interband transitions are allowed which requires the change of angular momentum by one. For a longitudinal electric field, only interband transitions are permitted. These are selection rules in carbon nanotube see Figure 1.5. | | 1 ‫ܧ‬௘௫ ‫ܧ‬ி | | 1 | | 1 ‫ܧ‬ி ‫ܧ‬௘௫ | | 1 Figure 1.5. The plot shows selection rule in nanotube: transverse electric fields induce interband transitions and longitudinal electric fields induce intraband transitions. CHAPTER 2 EXCITONIC STATE IN METALLIC CARBON NANOTUBES The difficulty of describing excitons in semiconducting SWNTs analytically lies with the fact that excitons can neither be considered strictly 1D nor 2D objects. However, the situation changes in the case of metallic nanotubes where, by virtue of screening from gapless metallic subbands, the radius of the exciton becomes much larger than the radius of the nanotube. In this chapter, the theory of excitons in metallic nanotubes will be discussed. 2.1 Introduction to exciton Exciton is a bound state of an electron and a hole formed by their Coulomb interaction. It is an elementary excitation in condensed matter physics that can propagate energy without changing the total charge. Excitons are fundamental to the understanding of optical transitions in semiconductors and insulators. 2.1.1 Frenkel and Wannier exciton Excitons can be divided into two categories, which depend on the strength of the interaction. If the distance is of the same order of magnitude as lattice spacing, excitons are strongly bounded with large binding energy. Such excitons are localized at a specific atom or molecule which are called Frenkel excitons. In the other limit for large separation between electron and hole, excitons are highly delocalized and can move “freely” around, which are called Wannier excitons. In the thesis, we mainly talk about Wannier exicton which will be called exciton for short. Wannier exciton hence has a larger radius which behaves like quasiparticle. Exciton can transfer energy, however, not electric current, since electron and hole have opposite charge. Wannier exciton is more attractive in physics, since it can be regarded as a hydrogenlike atom with a larger size and smaller binding energy. This is due to the large radius 10 of interaction and hence effective-mass approximation can be applied. The Hamiltonian describing exciton can be expressed as below, H=− ~2 ∇2 e2 − , 2µ εr (2.1) where Eg is the gap energy, µ is effective mass of exciton and ε is dielectric function which accounts for screening effects. For Wannier exciton, it is necessary to include many-body effects, since the electron and hole are widely separated with a lot of other electrons around. The Hamiltonian above resembles the one for a hydrogen atom and the spectrum can be easily obtained, E = Eg − µ e 2 . 2 ε~n n = 1, 2, 3... (2.2) which leads to a series of excitonic levels, see Figure 2.1. The levels become closer and closer for higher energies and eventually merge with band continuum with n = ∞. The appearance of exciton excitation also significantly modifies the absorption spectrum. In 3D, the lineshape of a noninteracting system can be predicted simply from band theory p as ~ω − Eg . In the presence of excitonic effect, new absorption peaks appear with lower energy than the band continuum, which correspond to excitonic excitations, see Figure 2.2. ∞ 2 1 Figure 2.1. Spectrum of exciton states in semiconductor. At n → ∞, energy of exciton coincides with band continuum. 11 ‫ܣ‬ሺ߱ሻ ߱ െ ‫ܧ‬௚ ‫ܧ‬ଵ ‫ܧ‬ଶ ‫ܧ‬ଷ ‫ܧ‬௚ ߱ Figure 2.2. The line shape with excitonic effect (solid red line) and without excitonic effect (dashed red line) 2.1.2 Direct- and indirect-bandgap in semiconductors For light induced excitons, they must be created by vertical transitions. It originates from the fact that band gap of a semiconductor is narrow and hence involved momentum change is negligible p ∼ ∆/c 1/a(where a is lattice constant). However, for indirectbandgap semiconductor, a photon is not sufficient to ensure the change of momentum. Therefore, a “recoil” phonon is required to be either emitted or absorbed to compensate the mismatch of momentum, see Figure 2.3. ′ WŚŽƚŽŶ WŚŽƚŽŶ WŚŽŶŽŶ Figure 2.3. Vertical transition and nonvertical transition. In the latter case, a single photon is not sufficient and the mismatch of momentum can be provided from a phonon. 12 2.1.3 Photoluminescence and polariton An exciton is created by absorbing a photon. The opposite process is a luminescent transition, which accompanies the annihilation of an exciton. Just as the absorption spectrum, emission spectrum would involve a series of lines as well. A photon is released with the recombination of electron and hole in the luminescent process. Such photon can be traveling through the lattice and reabsorbed as exciton again. Those transitions are going back and forth. Although the picture is oversimplified, it reveals the fact that excitons are unstable with a finite lifetime. Therefore, it is instructive to take exciton and photon as a excitation–polariton pair. More precisely, exciton is a compound of polarization and optical wave in the crystal. It is worth of noting that photoluminescence are usually observed in semiconductors or insulators. In metals, there is barely any “light” can be observed. This is because the bound state cannot be formed in metallic subbands where massless particles have fixed velocity and hence cannot to be captured by Coulomb interaction. 2.2 Excitonic state in metallic nanotubes Excitons typically exist in insulators and weakly doped semiconductors. Metallic and strongly doped semiconducting materials disfavor the formation of excitons. This is the result of two factors. First, as the conduction band is populated (e.g. by doping), screening by free charges diminishes strongly the magnitude of the electron-hole Coulomb interaction and decreases its binding energy. Second, the population of the conduction band reduces the number of states available to accommodate the electron after its (virtual) scattering off the hole, further decreasing the binding energy, to the point where no meaningful bound state may be formed anymore. This situation changes in quasi-one-dimensional systems, such as metallic nanotubes [2], where the formation of excitons occurs in subbands different from the subbands that are responsible for metallic screening. Such separation occurs as a result of quantization of the circumferential momentum. This makes exciton a well-defined excitation. Excitons in metallic single-walled carbon nanotubes (SWNTs) were experimentally observed in Ref. [3, 4]. Their binding energy was found to be about 50 meV, an order of magnitude smaller than the typical bandgaps of semiconducting nanotubes. Theoretically, excitons in metallic SWNTs were first studied by Ando under the effective-mass approximation [5]. Later, 13 the binding energy of exciton was addressed by first principles calculations [6–9], whose results are consistent with experimental measurements. A brief review of excitonic effects in metallic SWNTs was given in Ref. [10]. However, to our knowledge, no simple analytical description of excitons in metallic nanotubes has been developed. Let us illustrate the difficulty of describing excitons in semiconducting SWNTs, which stems from the fact that excitons can neither be considered one-dimensional nor twodimensional excitations. Indeed, consider the lowest energy subbands with the spectrum p ε(p) = ± ∆2 + v 2 p2 , and expand it near the bottom of the subband, p v 2 p2 ∆2 + v 2 p 2 ≈ ∆ + . 2∆ (2.3) The bandgap is typically, ∆ ∼ ~v/R, where R is the radius of the nanotube. (In particular, in the zone-folding tight-binding approximation both the (8,0) and (10,0) zigzag nanotubes have ∆ = ~v/3R). Because the electron-hole interaction energy is U (r) = −e2 /max(r, R), the exciton binding energy Eb can be estimated by minimizing, v 2 p2 e2 Eb ≈ min − , 2∆ max(r, R) (2.4) and taking into account the uncertainty relation, r ∼ ~/p. This yields, |Eb | ∼ e2 ~v 2 ∆. (2.5) Since e2 /~v ∼ 1, we obtain that Eb ∼ ∆ and, consequently, exciton radius Rex ∼ ~v/Eb ∼ R. Because to consider excitons to be 1D one would need to have Rex R, and conversely, to view them as 2D one would require Rex R, the problem falls instead between the two limits where a numerical analysis is necessary. However, as we are going to see below, the situation changes in case of metallic nanotubes where by virtue of screening by the metallic subbands the exciton binding energy decreases significantly (as already evidenced by the experimental data). As a result, the radius of the exciton increases dramatically, Rex R, and treatment of the exciton as a quasi-one-dimensional object becomes possible. This is what makes analytic solution viable. In this paper, we determine the binding energy and lifetime of an exciton in metallic SWNTs taking into account the screening effects within the Random Phase Approximation 14 (RPA). A problematic feature of the1D Coulomb problem is the r−1 singularity in the potential energy, which is not integrable (unlike its 3D counterpart). This feature was first addressed by Loudon [11] and later extensively studied further by others.[12–16] Loudon introduced the truncated Coulomb interaction e2 /(|x| + a0 ) with a positive constant a0 to ensure that Coulomb potential is regular at small distances x → 0. In our problem, the nanotube radius R appears naturally and no other cutoff is needed. Using a variational ansatz with a Gaussian trial function, we show below that the binding energy is E0 ≈ 0.077∆, ∆= v , R (2.6) a result which is in good agreement with experimental observations. Furthermore, we explore stability of excitons in metallic SWNTs, the question that is of fundamental interest but which remains unexplored in the existing theoretical works. The mechanism of a finite lifetime of exciton can be illustrated by Figure 2.1. The m-th subband–corresponding to the integer angular quantum number m–has the energy εm (p) = p ±v m2 /R2 + p2 . Exciton bound states are formed between gapped, m 6= 0, subbands, and the lowest m = ±1 exciton is indicated by a dashed line on Figure 2.4. This energy overlaps with the gapless m = 0 subbands. Accordingly, scattering of the electron and the hole from gapped to gapless subbands opens up a decay channel for the exciton. To better understand the role of the Coulomb interaction in the formation and decay of the exciton, the following picture is helpful. The exciton is produced by multiple virtual Figure 2.4. The exciton state emerges due to the final state interaction of an electron and a hole with energy below the continuum edge. 15 transitions of the electron and the hole within the gapped subbands. Such transitions are controlled by the Coulomb coupling V0 , the subscript indicating that no angular momentum change takes place. These intrasubband transitions determine the binding energy of the exciton, see Figure 2.5, left panel. The intersubband transitions–shown on the right panel in Figure 2.5–are described by “dipolar” Coulomb interaction V1 and occur with a change ±1 of the angular momentum of electron (and the opposite change of the angular momentum of the hole). We obtain that the ratio of the binding energy to the half-width of the exciton spectral function is Re E Eb = ≈ 2.5. Γ −2 Im E (2.7) Such a ratio indicates that excitons in metallic SWNTs are well-defined excitations. Our theoretical prediction is consistent with experimental results[17] obtained through ultrafast luminescence. Below we derive our main results, Eqs. (2.6) and (2.7). 2.2.1 Singular potential in one dimension and Loudon model In nanotubes, the energy of a bound state can be obtained by solving Schrödinger equation in 1D. However, unlike solvable 3D Coulomb problem, the unusual feature of 1D Coulomb problem arises due to the r−1 singularity. It is believed that the usual techniques of quantum mechanics alone are not sufficient to deal the 1D Coulomb potential. It was first studied by Loudon [11] and later has been extensively studied by many other papers [12–16] Figure 2.5. Two Coulomb interactions: the left one yields binding energy of exciton while the right one leads to finite lifetime of exciton. 16 while still remaining controversy. Loudon introduced truncated Coulomb interactions with positive constant a0 so that Coulomb potential is regularized at a distance x → 0. V = e2 (|x| + a0 ) (2.8) The energy spectrum is obtained by sending a0 → 0 after solving the regularized Schrödinger equation. The ground state was found to have infinite negative energy with significantly concentrated wave function δ(x). Other energy levels turn out to be with two-fold degeneracy: a continuous band for odd parity wave function and discrete spectrum for even parity solutions. The breakdown of non-degeneracy for bound states in one-dimension again stems from the singular behavior of Coulomb potential in 1D. 2.2.2 Failure of effective-mass approximation in semiconducting SWNTs In semiconducting nanotubes, the effective mass approximation is not suitable to calculate the binding energy. This is because binding energy is of the same order of bandgap. Exciton in semiconducting nanotubes typically lies in the definition of Frenkel exciton, where one-particle picture is not completely valid. What is even worse, error from using effective mass approximation in semiconducting nanotube results in instability problem. Coulomb problem in nanotubes essentially interpolates between 1D (R → 0) and 2D (R → ∞). In other words, bound states in nanotubes have curvature induced feature. The attractive Coulomb interaction on the surface of a nanotube of two arbitrary points (x1 , θ1 ) and (x2 , θ2 ) is given by, e2 , V (x, θ) = − p x2 + 4R2 sin2 (θ/2) (2.9) where x = x1 − x2 , θ = θ1 − θ2 . As R goes to infinity, the interaction between two arbitrary points on the surface of a nanotube becomes the limiting case of a two-dimensional problem. As R goes to zero, the structure approaches one-dimensional hydrogen studied by Loudon [11]. The Coulomb interaction on the cylinder is solved by Mahan [18] and Petersen [19], which interpolates between 1D and 2D. Considering that the ground state wave function of one-dimensional hydrogen is δ(x), we take a trial wave function of the Gaussian form, Ψ(x) = (α/π)1/4 e−αx 2 /2 , where α is the parameter with respect to which energy will be minimized. Accordingly, the average energy can be calculated as Ēb = T̄ + V̄ , 17 ~2 α − e2 Ēb = 2µ r Z∞ α π Z2π dx −∞ 0 2 e−αx dθ p . 2π x2 + 4R2 sin2 (θ/2) (2.10) Here the mass is the reduced mass of electron-hole pair and is half of that of the electron’s mass µ = ~/2vR on the parabolic subbands. To calculate the binding energy in terms of the band gap of a nanotube, we divide Eq. (2.10) by ∆ ∼ ~v/R and introduce a constant a = 2N e2 /πv~ ≈ 7. The x-integral can be easily carried out first, we get, Ēb t2 at = − √ ∆ 2 4 πN Z2π 2 dθ e2t sin2 (θ/2) θ K0 (2t2 sin2 ), 2 (2.11) 0 where we introduced dimensionless variable t = √ αR and K0 (x) is the modified Bessel function of second kind. Minimizing above equation numerically with respect to parameter t gives the value of t = 0.95 and the energy value evaluated at this value gives, Ē0 /∆ ≈ −2.15. To calculate the analytical expression of the above integral, we approximate both exponential and Macdonald functions to its zeroth order expansion for small values of t2 = αR2 1, ex ≈ 1 and K0 (x) ≈ ln(1.123/x), Ēb t2 at = − √ ∆ 2 4 πN Z2π ! 1.123 . dθ ln 2t2 sin2 (θ/2) (2.12) 0 The above integral is straightforward to calculate, R2π dθ ln | sin(θ/2)| = −2π ln 2 and we get, 0 ! √ Ēb t2 πat 2.246 = − ln . ∆ 2 2N t2 Again, minimizing the above equation numerically, we find t = (2.13) √ αR ≈ 0.47. Accordingly, we obtain the ratio of ground state energy Ē0 and band gap ∆ = ~v/R, Ēb /∆ ≈ −1.58. (2.14) The binding energy of exciton in semiconducting nanotubes is found to be of the same order of magnitude as band gap ∆. On the other hand, the binding energy is higher than the bandgap ∆, which raises the problem of instability of exciton. The negative energy of exciton indicates that excited state is a totally different one and a gap will be opened. Such process is very similar to cooper-pair in BSC theory. The newly ordered excitation is called excitonic insulator, which has been studied extensively 30 years ago. 18 2.2.3 Binding energy in metallic SWNTs In metallic SWNTs, there is a stronger screening effect from the transitions between gapless subbands. The word stronger emphasizes its comparison with the transitions within the first parabolic subbands. The screening effects influence exciton states much more significant in metallic nanotubes than in semiconducting nanotubes. Although the screening effect is usually highly effective in metals, which prevents the observation of exciton, Coulomb potential remains largely unscreened from free electrons in 1D. In contrast to power-law divergence in 3D electron gas, the screening from metallic subbands in a nanotube diverges logarithmically. Screening effect has been treated within Random Phase Approximation in our calculation. As the polarization functions are representing more conveniently in Fourier space, Schrödinger equation can be easily established with the Coulomb interaction Vm (q) [5, 20] below, Z∞ Vm (q) = dx e −iqx Zπ −π −∞ −e2 dθ −imθ p , e 2π x2 + 4R2 sin2 (θ/2) 2 = −2e Km (qR)Im (qR). (2.15) The attractive Coulomb interaction between an electron and a hole forms an exciton and consequently constructs an additional subband which stays lower than the minimum energy of first parabolic subband, see Figure 2.1. It turns out that interaction V0 yields the corresponding attractive force between electron and hole. Accordingly, the dynamically screened interaction within Random Phase Approximation (RPA) is modified by the free carrier in metallic subbands, V0 (q) → U0 (ω, q) = V0 (q) , 1 − V0 (q)Π0 (ω, q) (2.16) where Π0 (ω, q) = N q2v2 , π~v ω 2 − q 2 v 2 (2.17) accounts for both spin directions and the presence of two Dirac points within the Brillouin zone (armchair NT assumed, N = 4). It appears that the large radius of the exciton makes 2 ) it sufficient to consider ω qv limit. Indeed, frequencies involved ω ∼ Eb = 1/(µRex 19 whereas involved momenta are q ∼ 1/Rex , from where the above inequality follows, because of R Rex . The screened Coulomb interaction in the static limit has form, U0 (q) = 2e2 K0 (qR)I0 (qR) , 1 + aK0 (qR)I0 (qR) a= 2N e2 ≈ 7. πv~ (2.18) Although screened interaction has explicit form, it is still hard to solve Schrödinger equation exactly. In the limit of large radius of exciton Rex R, it is sufficient to use Gaussian trial function in the formalism of variational approach: Ψ(p) = ( 4π 1/4 p2 ) exp[− ] α 2α (2.19) The average energy can be readily found with following form, α − Ēb = 2µ Z∞ dp p2 U0 (p) exp[− ]. 2π 4α (2.20) −∞ Plank constant ~ has been sent to one from now on, since eventually what matters is the comparison of exciton energy and band gap. Before we approach to evaluate the average energy exactly, it is instructive to look at the accuracy of variational approach under a shallow-well profile, U (p) ≈ U (0), r α α µU 2 (0) Ēb = − U (0) =− . 2µ π 2π (2.21) The result is obtained with parameter α = µ2 U 2 /π under minimization procedure. Comparing with exact average energy for shallow-well Ēb = −µU 2 /4, there is a minor error from the prefactor. To evaluate the screened Coulomb interaction introduced in Eq. (2.18) more precisely, the average potential energy becomes, 2e2 Ū0 = − πR Z∞ dz exp[− z2 K0 (z)I0 (z) ] , 4t2 1 + aK0 (z)I0 (z) (2.22) 0 where dimensionless variable z = pR and again t = √ αR. The binding energy can be readily determined numerically. However, in order to achieve an analytical approximation to numerical results, we find that for small t 1, the integral converges at z ∼ t 1, it 20 is legitimate to take the expansion of both K0 (z) ≈ − ln z and I0 (z) ≈ 1 for small z 1. The integral becomes elementary, Ēb = t2 2e2 + 2µR2 πR Z∞ dz e−z 2 /4t2 ln z , 1 − a ln z 0 = t2 2µR2 2e2 +√ t ln c1 t . πR 1 − a ln c2 t (2.23) In the last line, we replaced the variable z by c1 t and c2 t in the fraction, since logarithmic function varies much slower than exponential. Two coefficients are determined as c1 = 0.68 √ and c2 = 0.66 in accord with numerical plot and t = αR ≈ 0.36. The binding energy is then found to be, Ēb = 0.077 ∆. (2.24) The theoretical prediction fits experimental measurements perfectly well. Recalling the binding energy without screening effect in Eq. (2.14), the screening effects from metallic subbands increase the radius of exciton to be ten times larger than the radius of a nanotube. The result in turns ensures the validity of Gaussian trial function. The angle dependence is rather unimportant in the scale of Rex R. In addition, due to its fast convergence, Coulomb interaction remains largely unscreened in a considerably large distance in real space which is consistent with comparably weak screening effects in 1D. 2.3 Relaxation of exciton in metallic SWNTs From the last section, exciton subband appears when V0 interaction between upper (electron) and lower (hole) parabolic subbands is considered. Exciton sits below the parabolic continuum with an infinite lifetime. However, when the upper parabolic subband gets populated due to the many transitions from lower parabolic subbands, the energy of exciton might be higher than ∆ and thus intersects with parabolic subbands, see Figure 2.5. In the former case, exciton has chance hoping to its nearby parabolic subband with the same energy. Furthermore, from the perspective of Coulomb interaction, dipolar interaction V1 could break the angular symmetry and consequently cause electron-hole pair jumping from parabolic subbands to metallic subbands where it disappears with fixed velocity. The later yields a finite lifetime of excitons in metallic nanotubes exclusively. In this section, we are going to evaluate the lifetime of exciton in metallic nanotubes, more precisely the ratio of 21 lifetime and binding energy. Unlike most of the metals, exciton is truly existing in metallic nanotubes with sufficient long lifetime. 2.3.1 Finite lifetime of exciton in metallic SWNTs The screening effect to V0 makes binding energy smaller than the case of bare Coulomb interaction. Similarly, V1 also gets screened from intersubband transitions with the change of angular momentum by one, which could make the lifetime smaller than what it is supposed to be. The competition between lifetime and binding energy is essentially the one between V1 and V0 . For the sake of fairness or the fact that exciton oscillates several periods before it attenuates, it is necessary to sort out the modification of V1 from intersubband transitions. This can be simply found from the polarization function in graphene. We leave this part in the appendix to avoid breaking the continuity of the paper. It turns out that Π1 (0, 0) = Π0 (0, 0) in the static limit. Therefore, V0 and V1 are both screened by the same polarization function in the formalism of RPA, πv V1 ≈ . 1 − V1 Π1 (0, 0) N (2.25) In the limit of small wavelength qR 1, U (0) ≈ U1 . The role of V1 is crucial in U1 = understanding the lifetime of excitons in metallic nanotubes. Indeed, V1 interaction leads to intersubband transition which could destroy the electron-hole pair created on the parabolic subbands. Screening effect to V1 means a longer lifetime for the exciton. In order to quantitative evaluate the lifetime of excitons, we start with the following Hamiltonian, Ĥ = i 1X X h p2 ( + ∆)(â†p âp − b̂†p b̂p ) + vp(R̂p† R̂p − L̂†p L̂p ) + U0 (p − k)â†k b†p b̂k âp 2m L p p,k U1 X † † + (R̂k bp L̂k âp + a†p L̂†k b̂p R̂k + L̂†k b†p R̂k âp + a†p R̂k† b̂p L̂k ) 2L (2.26) p,k Here âp and b̂p are the operators for the particles residing on the upper and lower subbands with m = 1 (or m = −1); the operators R̂p and L̂p correspond to right and left moving particles of massless (m = 0) subbands; the U0 interaction describes scattering within the massive subbands whereas U1 coupling describes processes where scattering occurs between massive and massless subbands. Note that of all the Coulomb interaction terms we have retained only those that are responsible for the formation of the exciton with zero total 22 momentum: for example, the U0 -term describes scattering of the electron with momentum p (âp ) and the hole with momentum −p (b̂†p ) into a pair of new states with momenta k and −k. The intersubband terms in the Hamiltonian (2.26) have the extra prefactor 1/2 compared with the intrasubband transitions. The origin of this difference lies in the pseudospin nature of the underlying Hamiltonian of the two-dimensional graphene crystal that forms the nanotubes. (The pseudospin arises from the existence of two atomic sublattices in the graphene honeycomb arrangement of carbon atoms.) As a result, two states of the same energy and opposite momenta are orthogonal to each other. More generally, a transition between the states having the same sign of the energy and momenta p and p+q is suppressed by the factor cos[(θp+q − θp )/2], where θp is the angle that the momentum makes with the x-axis. When graphene is rolled into a nanotube, the circumferential momenta are quantized, py = m/R. The gapless states m = 0 are those that move along the x-axis: p = (p, 0) with θp = 0 or π. In contrast, in the gapped subbands, p = (p, m/R), where near the bottom of the subbands p 1/R. Accordingly, the relevant states are those that have θp ≈ ±π/2. Correspondingly, each particle transitioning between a gapless state √ and a gapped state (close to the bottom of the subband) introduces a factor 1/ 2 into the amplitude of the scattering. For the two-particle Coulomb interaction, the overall coefficient in the U1 -term is thus 1/2. Besides, in order to be consistent with Hamiltonian Eq. (2.26) which consists of electron and hole operators, one needs to flip the sign of Coulomb interaction introduced in Eq. (2.15). The system can be correspondingly described by the wave function below, |ψi = X q fq â†q b̂q |0i + X q>0 gq R̂q† L̂q |0i + X gq R̂q† L̂q |0i. (2.27) q<0 where vacuum state |0i represents initial state of the system: upper parabolic subband is empty while lower counterpart is completely populated. Accordingly, the first term in Eq. (2.27) describes a electron-hole pair on the upper and lower parabolic subbands and the second state is not stable because massless particle has invariant velocity and thus cannot be captured by Coulomb potential. Schrödinger equation can be solved by applying two different bras h0|b̂†q0 âq0 and h0|L̂†q0 R̂q0 on the LHS which give rise to the following two 23 equations, 1X U1 X U1 X E − 2∆ − q 2 /µ fq = − U0 (q − p)fp − gp − gp , L p 2L 2L p>0 (E − 2v|q|) gq = − (2.28) p<0 U1 X fp . 2L p (2.29) The first equation gets simplified by solving gq in Eq. (2.29) and substituting in Eq. (2.28). R P The summations are changed to the integrals as p → L dp/2π. Z∞ q2 (E − 2∆ − )fq = µ dp h U2 fp −U0 (p − q) + 1 2π 2 −∞ Z∞ i 1 dp0 . 2π E − 2p0 v + iη (2.30) 0 Infinitesimal imaginary constant iη is added to make the integrand well-defined. The second term in the above equation contains double integrals over p and p0 . The integral of p0 can be carried using the following identity, 1/(E − 2p0 v + iη) = P (1/(E − 2p0 v)) − iπδ(E − 2p0 v). The imaginary part of the integral can thus be easily evaluated while the principal part is calculated using finite upper cutoff as 1/R. This gives the second order contribution to the binding energy coming from the virtual transitions to other states and goes as U12 /8πv as compared to the main contribution coming from U0 . The correction to the binding energy can thus be neglected and we get, q2 fq = − E − 2∆ − µ Z∞ dp h U2 i U0 (p − q) + i 1 fp . 2π 8v (2.31) −∞ It is obvious to see that the dipolar interaction V1 enters into Schrödinger equation as an imaginary potential iU12 /(8v), which yields the finite bandwidth of exciton. If we assume that U0 (q) changes slowly , namely for a shallow-well potential U0 (q) = U (0), the energy of exciton can be approximated as, E = 2∆ − µh µh U 2 i2 U0 U12 i U0 + i 1 ≈ 2∆ + −U02 − i 4 8v 4 4v = 2∆ − 0.15∆ − i0.03∆ . (2.32) The imaginary part of energy describes decay rate of exciton Γ = −2Im E ≈ −0.06∆. The ratio of binding energy and bandwidth is about 2.5. To evaluate the energy more 24 precisely, we resort to variational approach by using the Gaussian trial function introduced in Eq. (2.19) while taking account of imaginary Coulomb potential in Eq. (2.31), we have, Ē = 2e2 t ln c1 t t U12 t2 √ √ + − i . 2µR2 πR 1 − a ln c2 t πR 8v (2.33) Assuming that the imaginary part in Eq. (2.33) is small comparing with its real part, we √ can still use the value of parameter that c1 = 0.68, C2 = 0.66 and t = αR = 0.36. The ratio of binding energy and bandwidth is found as Eq. (2.7). Shallow-well approximation matches surprisingly well with variational approach. 2.4 Summary and conclusions Because of the presence of the gapless subbands, excitons in metallic carbon nanotubes acquire unique features that distinguish them from excitons in other solid state systems. First, the quasi-one-dimensional nature of nanotubes makes screening by conduction electrons less effective than in conventional metals. As a result, the electron-hole interaction remains significant enough to ensure the formation of a bound pair. Second, the separation (in the momentum space) of gapless states (m = 0) from the subbands where the exciton is formed (|m| = 1) and the fact that the latter subbands are fully gapped allow the electron and the hole to explore fully the gapped subbands, unlike what happens in a conventional doped semiconductor where filling of the conduction band quickly depletes the number of available electron states. Third, the screening by the gapless states is nonetheless significant enough so that the radius of the exciton is greater than the nanotube radius with the binding energy of the order of 0.1∆. This allows to treat excitons as quasi-one-dimensional objects, unlike excitons in semiconducting nanotubes, which are neither one-dimensional nor two-dimensional objects. Fourth, the presence of the gapless subbands opens up a channel for exciton attenuation where electron and hole can scatter off each other into the gapless states. The presence of this channel leads to a considerable broadening of the exciton but not so significant as to smear it away completely. CHAPTER 3 LUTTINGER LIQUID AND MANY-BODY EFFECTS IN METALLIC NANOTUBES In quantum wires, such as metallic nanotubes, the optical absorption of the transverse polarization is controlled by the depolarization effect, which stems from the redistribution of conduction electrons around the circumference of the system. The traditional electrostatics treatment of the depolarization effect relies on approximating the system by a cylinder with some effective dielectric permittivity. We demonstrate that this simple intuitive picture does not adequately describe optical absorption near its threshold, as the depolarization effect becomes dominated by many-body correlations which strongly modify the spectral dependence of absorption. 3.1 Introduction to Fermi liquid and Luttinger liquid Quantum theory has been established for almost 100 years. It is believed that single particle behaviors have already been well understood. However, it is hard to apply it to many-particle system even with the help of the most advanced computer on the earth. Interactions among particles endow so much vitality to the quantum world. There is one misunderstanding of many-particle physics: the complicated many-body problem can be interpolated from properties of elementary particles and thus there is no requirement for new concepts and laws. In fact, there are always new physics appearing at each level of complexity. In condensed matter physics, mean field theory (Hartree-Fock approximation) was introduced as a basic method in dealing interacting electron system. However, it is only a good approximation in high density limit for long range interaction (e.g. Coulomb interaction). In fact, most of the metals do not fall in that region (1.8 < rs < 6 where rs is the radius of a sphere that encloses a unit electron), therefore, it is necessary to introduce a more sophisticated effective theory to target electron system. Random Phase Approximation(RPA) came into being after, whose corresponding static limit is equivalent 26 to Thomas-Fermi(TF) theory which suffices to remove the divergence encountered in the second order perturbation of Hartree-Fork approximation. Thomas-Fermi theory only applies to long wavelength limit where the screened interaction has Yukawa form with exponential decay exp(−qTF r)/r. A more complete description from RPA gives rise to Friedel oscillation cos(2kF r)/r3 which is the manifestation of impurities in metals. The oscillating feature originates from the sharpness of Fermi surface at q = 2kF . Of course, TF theory is too simple to give such correlation effect for being a static and long wavelength expansion of RPA. Another important feature in electron gas is the collective excitations where dielectric function is chosen to be zero. It is called zero sound for short range interaction (e.g. helium) and plasmon excitation for long range interaction (e.g. Coulomb interaction). Electron system in high dimensions has been well treated by Landau’s Fermi liquid theory. By turning on the Coulomb interaction, free electrons are dressed with clouds (density fluctuation) around them which are called quasiparticles, namely equivalent excitations in “noninteracting” system. In one dimension, however, any individual excitation has collective behavior which shows a sharp contrast against “free” quasiparticle excitation in high dimensions. This can be easily inferred from the fact that any propagation has to push its neighbors in one dimension. Accordingly, a single fermionic excitation will split into a sound wave (charge) and spin wave (spin) propagating with different velocities. Spin-charge separation is a unique signature in one-dimensional electron system. A very important quantity is susceptibility χ(ω = 0, Q = 2kF ) that measures the response with respect to charge density, which is found to be divergent. This is known to give rise to Friedel oscillation in one dimension. Recalling the same quantity in high dimensions where the divergence occurs at higher order of derivatives, which implies that our beloved Fermi liquid theory is in the shaky ground. This is due to the shape of Fermi surface in one dimension(degenerating from sphere or circle in high dimension to two points). Nesting property (q) = − (q + 2kF ) is always satisfied in one dimension, so singular behavior of polarization function is thus more significant. Another significant change is the particle-hole continuum, see Figure 3.1. This is again due to the particular shape of Fermi surface in one dimension. The Fermi surface reduces to two points where only forward (q = 0) and backward scattering (q = 2kF ) are allowed. All these peculiar phenomena suggest that the ground state is totally different under the Coulomb 27 ௞ ௞ 0 0 2 ி 2 ி Figure 3.1. Particle hole continuum in 3D (left) and 1D (right). Dashed lines are plasmon dispersions: quadratic in 3D with a gap and linear in 1D being gapless. In one dimension, plasmon dispersion complete sits in the particle-hole continuum which indicates the decay of electron-hole excitation into collective plasmon modes. interaction no matter how weak it is. The mystic mask of one-dimensional electron gas attracted so many people’s attention and a new theory need to be built up to resolve all the issues. 3.2 Luttinger liquid theory Luttinger liquid is an effective theory for one-dimensional interacting electrons, which is a necessity due to the break down of Fermi liquid theory in one dimension. The theoretical model was first introduced by Tomonaga in 1950. Later Mattis and Lieb [21] proposed a correct solution to it. The basic idea is to map the interacting electrons to noninteraction bosons near the Fermi surface, so Luttinger liquid theory is only valid at low temperature where dispersion is linear, see Figure 3.2. 3.2.1 Scattering processes in one dimension In order to reach a good feeling of Luttinger liquid theory, it is instructive to take a look at the Hamiltonian first. The noninteracting part can be cast into the usual quadratic form in the bosonization representation(we will see how convenient it is later), 1 H0 = 2π Z h i dx vF (πΠ(x))2 + (∇φ(x))2 , (3.1) where ∇φ(x) measures density fluctuation at point x and ∇θ(x) measures the current. The definitions are given below, 28 ‫ܧ‬௞ ሺ‫ݍ‬ሻ ‫ܧ‬ሺ‫ݍ‬ሻ ൌ ‫ݒ‬ி ‫ݍ‬ ‫ܧ‬ሺ‫ݍ‬ሻ ൌ െ‫ݒ‬ி ‫ݍ‬ െ݇ி ݇ி ‫ݍ‬ Figure 3.2. In one dimension, one can always linearize dispersion near the Fermi surface. Excitations with low energy(particle-hole pair near the Fermi surface) in one dimension are well defined in energy and momentum, because it only depends on q. ∇φ(x) = −π[ρr (x) + ρl (x)], ∇θ(x) = π[ρr (x) − ρl (x)], Π(x) = 1 ∇θ(x), π (3.2) where ρr and ρl are right and left density operators. Now we add electron-electron interactions to the system, Z Ĥint = = dx dx0 V (x − x0 )ρ(x)ρ(x0 ), 1 X V (q)c†k+q c†k0 −q ck0 ck . 2L 0 (3.3) k,k ,q The first line and second lines are interactions in real space and Fourier space respectively. It is worth of noting that interesting physics only happens around Fermi surface. Because valence band is completely filled, it is not possible to remove an electron deep inside the Fermi sea with arbitrary small energy. Accordingly, there are basically four types of scattering, see Figure 3.3, which are labeled in the g-ology as g1 , g2 , g3 and g4 . However, only g1 , g2 and g4 conserve momentum. Process g4 couples fermions on the same side of the Fermi surface (q ∼ 0). Process g2 couples fermions from two sides of the Fermi surface (q ∼ 0). So both g2 and g4 correspond to forward scattering. Process g1 couples fermions 29 ସ ~0 ଶ ~0 ଷ ~4 ி ଵ ~2 ி Figure 3.3. Four scattering processes with low energy in one dimension. For spinless particles, g1 and g2 are the same. All processes obey conservation of momentum except the left bottom one, which is the umklapp process, since it involves interaction with the lattice. that exchange sides (backscattering q ∼ 2kF ). For spinless fermions, g1 and g2 are identical but with a minus sign. In the bosonization representation, g2 and g4 has the following form, Z i h g2 2 2 , (3.4) H2 = dx (∇φ(x)) − (∇θ(x)) (2π)2 Z h i g4 2 2 H4 = dx (∇φ(x)) + (∇θ(x)) . (3.5) (2π)2 Now we see that, g2 and g4 remain quadratic. The total Hamiltonian can be rewritten into following form, H = H0 + Hint 1 = 2π Z h i u dx vF uK(πΠ(x))2 + (∇φ(x))2 , K (3.6) where u is renormalized electron velocity and K is a dimensionless parameter, u = vF [(1 + y4 /2)2 − (y2 /2)2 ], s 1 + y4 /2 − y2 /2 , K = 1 + y4 /2 + y2 /2 (3.7) (3.8) 30 where y = g/(πvF ) is a dimensionless coupling constant. For people who are familiar with Luttinger parameter g, it is easy to see that g = 1/[(1 + y4 /2)2 − (y2 /2)2 ]. So g > 1 or K > 1 correspond to attractive interaction, and g < 1 or K < 1 corresponds to repulsive interaction. g = K = 1 corresponds to marginal point where interaction switches off. We will stick with parameter K in the first half part of this chapter. In the spinless case, interactions can be trivially diagonalized with the help of bosonization. Now we turn to the system with spin. For the convenience, we introduce following unitary transformation, 1 1 √ [φ↑ + φ↓ ], φσ = √ [φ↑ − φ↓ ], 2 2 1 1 √ [θ↑ + θ↓ ], θσ = √ [θ↑ − θ↓ ]. 2 2 φρ = θρ = (3.9) (3.10) (3.11) Two fields corresponding to ρ and σ are commuting with each other. Formulas above describe spinful electrons (with two-fold degeneracy). We will see a more generalized form later (e.g. four-channel system, such as nanotubes). In the presence of spin channels, kinetic energy splits into two parts, H0 = Hρ + Hσ . Meanwhile, g2 and g4 interaction still remain quadratic, 1 H2 = (4π)2 Z dx (g2,k + g2,⊥ ) h (∇φρ (x))2 − (∇θρ (x))2 i i (∇φσ (x))2 − (∇θσ (x))2 Z h i 1 2 2 H4 = dx (g + g ) (∇φ (x)) + (∇θ (x)) ρ ρ 4,⊥ 4,k (4π)2 h i + (g4,k − g4,⊥ ) (∇φσ (x))2 + (∇θσ (x))2 . + (g2,k − g2,⊥ ) h (3.12) (3.13) For g1 process, as we mentioned before, g1,k is identical to g2,k up to a change of sign, which is due to the permutation of two operators. However, g1,⊥ is new, we can write it in the bosonization form which yields, H1 = − g1,k (4π)2 Z h i Z √ 2g1,⊥ dx (∇φρ (x))2 − (∇θρ (x))2 + dx cos(2 2φσ (x)). 2 (2πα) (3.14) The first term is the same to g2,k but with minus sign, however, the second term is nonquadratic. This is our first time facing Sine-Gordon Hamiltonian, which is a headache 31 however not very rare in one dimension. Now we can put all quadratic terms together into the same form as Eq. (3.15), the total Hamiltonian becomes, √ 2g1,⊥ 2φσ (x)), H = Hρ + Hσ + dx cos(2 (2πα)2 Z i h 1 uν Hν = (∇φν (x))2 . dx vF uν Kν (πΠν (x))2 + 2π Kν Z (3.15) (3.16) where ν = ρ for charge part and ν = σ for spin part with parameter in the following form, u = vF [(1 + y4ν /2)2 − (yν /2)2 ] s 1 + y /2 + y /2 4ν ν K = 1 + y4ν /2 − yν /2 gν = g1,k − g2,k ∓ g2,⊥ g4ν = g4,k ± g4,⊥ yν = gν /(πvF ), (3.17) where again upper and lower sign corresponds to charge sector ρ and spin sector σ. It is easy to see that charge density wave (CDW) and spin density wave (SDW) are propagating with different velocities. More specifically, when all coupling constants are the same, the velocity of CDW/SDW is faster/slower than the velocity of bare electrons, and vice versa. In general, velocities depends on relative values of yν . What complex the problem is g1,⊥ . This interaction is unimportant even though it is painful to deal with. The issue can be resolved from two perspectives. Firstly, considering long-range (or at least nonlocal) interaction (varies logarithmically in q-space), the coupling constant V (0) V (2kF ) is usually satisfied and thus backscattering can be ignored. Secondly, since the electrons in the lowest subbands with low energy are spread out around the circumference of an armchair nanotube, the probability to have two electrons near each other is of order 1/n(where (n, n) indicates the chirality of armchair nanotubes). In contrast, forward scattering processes involve small momenta exchange or long-range part of the interaction and thus no 1/n suppression [22]. Keeping only g2 and g4 indicates that chirality cannot be changed by the interaction which suggests that only diagrams with fermion bubbles with two interaction lines contribute: RPA is exact in one dimension. However, when g1 becomes important, one needs to know how to deal with it. Nonquadratic terms are usual in one dimension, and there are in fact many ways to tackle the problem. We are going to analyze it later by using 32 renormalization group. Finally, this is one interaction left that does not obey conservation of momentum, g3 process. g3 corresponds to umklapp process with momentum increasing or decreasing in both processes, so the total change of momentum is 4kF . Accordingly, this process does not conserve momentum among electrons, because it involves the interaction with lattice. Umklapp process is known to give rise to Metal-insulator transition (Mott insulator) [23]. When the repulsive interaction g3 among electrons is much larger than the kinetic energy, the uniform density wave will be replaced by an insulating state where particles are localized, one electron per site. In the bosonization language, g3 process has Sine-Gordon form, 2g3 H3 = (2πα)2 Z √ dx cos( 8φρ (x)) (3.18) this is not the case for a nanotube under zone-folding approximation where the reciprocal √ constant is G = 8π/3 3a other than the required one G = π/a (where a is lattice spacing constant). Hubbard model is a good platform to practice with umklapp process. Even though umklapp process leads to very rich physics, it is out of the scope of the thesis. 3.2.2 Bosonization Different with high dimensional electron system, the dispersion of Luttinger liquid can always be linearized close to the Fermi surface. The most important feature of LL is that interacting electron system can be mapped onto noninteracting bosons. The corresponding unitary transformation is being called bosonization. Mathematically, density fluctuations ρ̂(q), as superpositions of particle-hole excitations, can be related to boson operators âq and b̂q in the following manner, âq ∼ ρr (q) = X b̂q ∼ ρl (−q) = X ĉ†k ck+q , â†q ∼ ρ†r (−q), ĉ†k ck−q , b̂†q ∼ ρ†l (q), k>0 (3.19) k<0 where upper/lower sign corresponds to right/left-moving particles. With the help of bosonization, Hamiltonian becomes purely quadratic which makes the system exactly solvable, Ĥ = v X q(â†q âq + b̂†q b̂q ) + q>0 = X 1 u|q|(γ̂k† γ̂k + ), 2 1 X qV1 (â†q âq + b̂†q b̂q + âq b̂q + â†q b̂†q ) 2π q>0 (3.20) 33 where γ is an operator which can be expressed in terms of boson operators aq and bq through Bogoliubov transformation for bosons. However, u = v/g is not the velocity of free fermion v but the renormalized one of noninteracting bosons due to Coulomb interaction, where g < 1 is Luttinger parameter. Recalling scattering processes in the last section, only g2 and g4 are counted and g2 = g4 in bosonization procedure. All the effort made so far is to show that plasmon excitation is exact in one dimension. However, it is not sufficient to explore transport and tunneling property where single particle operators are needed. Single particle operators in LL is given by, Ur,l ±ikx e exp ψ̂r/l (x) = √ 2πα r 2π X e−αk/2 ik(x−vt) √ [e âk − e−ik(x−vt) â†k ] . L k k>0 (3.21) where again upper/lower sign is for the right/left-moving electrons respectively. Ur/l is fermionic counting operator to ensure the proper anti-commutation relation for electrons. It is worth of noting that one need to take α → 0 at the end of the calculation. α introduced by Haldane is just an artificial parameter to make integral well-defined, but the final result is independent of it. One simple but very practical exercise is to prove that bosonization ansatz is equivalent to Green’s function for free particles. GR (t, x) = −ihψ̂r† (t, x)ψ̂r (0, 0)i = −ieikx , 2πi(x − vt − iα) (3.22) where GR (t, x) is retarded Green’s function. It is then straightforward to obtain single particle operator in the presence of Coulomb interaction through simple modifications, r r 2π X Ur,l ±ikx g ik(x−ut) −αk/2 (3.23) e exp e [e âk − e−ik(x−ut) â†k ] . ψ̂r/l (x) = √ L k 2πα k>0 The transformation is made by substituting v → u and âk → √ gâk , which is in accord with renormalization from Bogoliubov transformation Eq. (3.20) in the total Hamiltonian. Let us find out the influence of Coulomb interaction in the one-dimensional electron system. A quantity that can be observed in the experiments is the density of states, which is constant in the noninteracting electron system. However, now it becomes energy dependent via g = v/u, X (1−g)2 dI 2g ∼ ν(ν, ω) = A(ν, ω) ≈ A(0, ω) ∼ G> (ω) ∼ ω r dV ν (3.24) The above is for the particular case where the spatial variation of tunneling tip is small compared with charge fluctuation of electrons x = x0 = 0. It is easy to see that by 34 switching off Coulomb interaction, constant density of state in the noninteracting system is reproduced. In the presence of interacting, density of states vanishes at Fermi energy (ω → 0) in the form of power law. This is just another manifestation that single particle excitations are converted into collective excitations under the influence of Coulomb interaction. 3.2.3 Renormalization group in Luttinger liquid From previous sections, we see that g2 , g4 and g1,k interaction remain quadratic and thus can be trivially diagonalized without spending much effort. However, g1,⊥ and g3 (umklapp process) interaction are not amenable for bosonization. In this section, we are going to learn a very decent technique to deal with nonquadratic terms in the Hamiltonian. It is called renormalization group(RG). The basic idea of RG is by changing the cutoff while keeping the system invariant. Apparently, coupling constant needs to be changed as well, which will thus generate one or several RG equations or β function. For example, dg = (2 − 4U )g dl (3.25) where g is coupling constant of interaction, and U (just like K or g in LL) is a dimensionless parameter. When U < 1/2, g is increasing with l. Apparently, g flows to infinity as l → ∞. In other words, U < 1/2 refers to strong coupling limit and accordingly interaction is relevant in this regime. Relevant flow means the divergence will be amplified along the flow, it usually refers to a phase transition (e.g. from gapless state to insulating state). For U > 1/2, g is decreasing to a fixed point and thus interaction is irrelevant. Irrelevant flow indicates that the system does not take any qualitative change from the interaction. Accordingly, the physics will be the same as the fixed point under trivial renormalization (recalling g2 and g4 processes). When U = 1/2, the interaction is marginal. Marginal irrelevant interaction does not cause a gap and vice versa [24]. From the simple example, we learned some RG terminology such as relevant, irrelevant and so forth. However, the concepts are still too abstract. Now we turn to the application of RG onto Sine-Gordon Hamiltonian in one dimension. The calculation will not be detailed, however, it will give us a deeper understanding of RG as well as one-dimensional physics. Let us recall backscattering process g1,⊥ , Z H1 ⊥ = dx √ 2g1,⊥ cos(2 2φσ (x)), 2 (2πα) (3.26) 35 The starting point is the following correlation function. If there is no interaction, correlation function yields a power-law result, √ heia √ 2φ(r1 ) −ia 2φ(r2 ) e 2 KF (r −r ) 1 1 2 i = e−a (3.27) where K is dimensionless parameter introduced in Eq. (3.17) and F1 (x) = Log(α/x). At the time being, we focus on g1 process where Kρ = Kσ , since charge sector and spin sector are differed by g2,⊥ and g4,⊥ . For convenience, denote Kρ and Kσ by K and g1,⊥ by g. Accordingly, Kρ = Kσ > 1 refers repulsive interaction(g1,⊥ > 0) and Kρ = Kσ < 1(g1,⊥ < 0) indicates attractive interaction. In the presence of g1,⊥ process, the correlation function cannot be carried out exactly. However, one can perform a perturbative expansion for small g1,⊥ where the first nonvanishing term comes from second order. After laborious calculation, the correlation function becomes, √ √ ia 2φ(r1 ) −ia 2φ(r2 ) he e −a2 KF1 (r1 −r2 ) i=e Z∞ i h g 2 K 2 a2 F1 (r1 − r2 ) 2 2 −4KF1 (r) (3.28) 1+ d r e 4π 3 u2 α4 α The second order expansion can be regarded as first order expansion of an exponential function with effective parameter Keff , Keff y2K 2 =K− 2 Z∞ dr r 3−4K ( ) a a (3.29) α If the integral cutoff varies from α to α0 = α + dα, we have, Keff y 2 K 2 dα y 2 K 2 =K− − 2 α 2 Z∞ dr r 3−4K ( ) a a (3.30) α0 To keep Keff invariant, one needs to make two modifications, y 2 K 2 dα , 2 α (3.31) α0 4−4K(α) ) . α (3.32) K(α0 ) = K(α) − y 2 (α0 ) = y 2 (α)( Rescaling parameter α = α0 el , two renormalization equations are obtained, dK(l) dl dy(l) dl = − y 2 (l)K 2 (l) , 2 = (2 − 2K(l))y(l). (3.33) (3.34) 36 Now we use these equations to analyze g1,⊥ backscattering interaction. Recalling K in Eq. (3.17) and expanding it for small g⊥ , we have K ≈ 1 + y⊥ /2. Accordingly Eq. (3.33) becomes, dyk (l) dl dy(l) dl = −y 2 (l), (3.35) = −yk (l)y(l). (3.36) These equations are expansions of Eq. (3.33) for small coupling constant. However, the insights they provide are very fruitful. Firstly, it is easy to see that, yk2 − y 2 = A, (3.37) which means that for positive A2 , yk and y construct hyperbolas on the left and right of the plane, see Figure 3.4. For negative A2 , they form another hyperbola on the top of the plane (we do not care about negative y due to the symmetry). Let us examine it in several cases. (I). For A = 0, the flow is a decreasing straight line. More specifically, it means yk = y and hence it is a separatrix(or phase transition line) between irrelevant and relevant regime. On the separatrix, we have, ଶ 0 ଶ 0 ଶ 0 1 1 1 Figure 3.4. Renormalization flow of Sine-Gordon Hamiltonian. The right diagonal line is a marginal irrelevant line or phase transition line. Interaction is relevant above it and irrelevant below it. 37 yk (l) = y(l) = y0 . 1 + y0l (3.38) So the flow will be decreasing to a fixed point at yk? = y ? = 0 and K ? = 1 (index ? refers to coordinate of the end of the flow, namely fixed point). The line yk = y is called marginally irrelevant, since y has been renormalized out to a fixed point with “noninteracting” Hamiltonian(K ? = 1). (II). For the right half hyperbola in red color in Figure 3.4, where A > 0 and yk > 0, the flow decreases to the fixed point y ? = 0 and yk? = A. In this case, the fixed point does not correspond to a truely noninteracting state with parameter K ? = 1 but a noninteracting state with renormalized parameter K ? = 1 + A/2. In other words, g1,⊥ can be “absorbed” into noninteracting Hamiltonian in the same manner as g2 and g4 processes. Accordingly, the correlation function can be directly obtained, √ heia √ 2φ(r1 ) −ia 2φ(r2 ) e 2 K ? F (r −r ) 1 1 2 i = e−a (3.39) Comparing it with Eq. (3.27), only the exponent has been changed to renormalized parameter K ? . However, g1,⊥ is still irrelevant in this regime. (III). The upper and left hyperbolas(green lines in Figure 3.4) are flows to strong √ coupling regime, since as K → 0, y → ∞. In order to keep interaction invariant, cos( 8φσ ) requires to approach zero as y → ∞. In other words, φσ is locked into one of the minima of the cosine. Note that kinetic energy is quadratic, meanwhile g1,⊥ process is fluctuating with Sine-Gordon form. In the strong coupling regime, flow is trying to order the field φσ to make the fluctuation more difficult by decreasing parameter K. In the strong coupling regime where interaction becomes relevant, the system undergoes a phase transition to massive phase. The behavior of gap can also be analyzed with the help of RG, see [25]. We are not going to discuss it here. It is easy to see that RG gives us a complete picture of how does interaction influence the system in a qualitative manner. In fact, the sine-Gordon problem can be solved exactly, see [26]. 3.3 Many-body effects in the depolarization effect of metallic nanotubes Metallic carbon nanotubes have low energy states with linear spectrum. Such states are known to be amenable to the Luttinger Liquid theory [25, 27–30]. The latter builds 38 on the divergencies encountered in an attempt to treat electron-electron interactions perturbatively, and predicts strong many-body renormalization of virtually all low-energy observables. The exact solvability of Luttinger liquid model originates from the fact that longitudinal polarizability of interacting electrons with linear spectrum is simply equal to that of a system of noninteracting particles. As a result, plasmon excitations, that occur in any dimensionality, take on further significance as they become exact eigenstates in one dimension. Electrons are coherent combinations of plasmon modes. Plasmons propagate with velocity u which is several times greater than the band electron velocity v = 8 × 105 m/s. Furthermore, the tunneling density of states is strongly suppressed as ν0 ∝ α , since the electron “assimilates” via excitation of an infinite number of bosonic modes. The exponent α = (1 − g)2 /2N g depends on the total number of channels N and effective coupling constant g = v/u as the ratio of bare Fermi velocity and velocity of a collective charged mode plasmon. Observations of metallic nanotubes’ low-energy properties [31–33] are consistent with the picture of Luttinger liquid. In addition to the linear states, nanotubes have subbands with nonzero angular momentum m 6= 0. Tunneling into “massive” subbands is suppressed [34], ν1 ∝ ( − ∆)−1/2+β , with suppression exponent β = (1 − g 2 )2 /2N g. However, the suppression is not due to the absence of well-defined particle-hole excitation but orthogonality catastrophe which occurs in any dimension. The dipole transition selection rules in nanotubes allow transitions with zero change of the angular momentum, m → m, only for the longitudinal polarization of the electric field. However, transitions between linear subbands, 0 → 0 are forbidden (unless magnetic flux is present). In addition, transverse electric field induces transitions with the change of the angular momentum, m → m ± 1. Recalling the suppression of tunneling density of states for massless and massive subbands, one would naively expect a stronger suppression under the transverse electric field. However, this is known to be erroneous due to the work [35]. In sharp contrast, Coulomb interaction yields singular enhancement for perpendicular polarization (while canceling out for longitudinal polarization). Moreover, there exists purely geometric suppression of the local electric field Ei at the surface of a nanotube compared with its value at a distance E0 , which stems from the redistribution of electrons around the tube’s circumference. This phenomenon is known as depolarization effect which has been studied in [36–39]. For an ideal metal cylinder the suppression would be 39 complete, but for a nanotube it results in a finite depolarization factor Ei = 2E0 /(1 + ε⊥ ) where ε⊥ is effective dielectric permittivity. In terms of effective transverse susceptibility α⊥ = (ε⊥ − 1)/4π, the electric field inside the nanotube can be restated as, Ei = E0 = E0 [1 − 2πα⊥ + (2πα⊥ )2 − ...] 1 + 2πα⊥ (3.40) Microscopically, the suppression predicted from electrostatic is equivalent to random phase approximation (RPA) built in the frame of quantum field theory. The existing theory [36–39] of depolarization effect, while taking electron-electron interactions into account to some extent (within Hartree-Fock approximation) disregards all Luttinger Liquid effects. The difficult originates from the insufficiency of single particle Green’s function to account for the many-body effects exactly. In this section, we are going to show the breakdown of classical electrostatics in the depolarization effect. 3.3.1 Depolarization effect and random phase approximation In the presence of transverse external electric field E0 to the axis of the nanotube, it is allowed to have transitions between the subbands with different angular momentum, see Figure 3.5. Microscopically, such picture amounts to the redistribution of free carriers around the circumference of nanotubes. The field-driven redistribution of electrons reduces the value of the electric field inside the wire, which is known as depolarization effect. However, the suppression is not complete due to the limited number of conducting channels N in nanotubes. In the minimal electrostatic model, one can treat the wire as a solid cylinder with effective permittivity ε⊥ , the electric field inside the wire is uniform, Ei = 2E0 /(1+ε⊥ ). As for an ac field with low frequency ω c/R for the retardation to be disregarded, the electric field E0 (θ, t) = E0 (ω) cos θe−iωt induces surface charge with 2D density σ(θ, t) = σ(ω) cos θe−iωt where θ is the circumferential angle. The induced density produces additional electric field which is homogeneous inside the nanotubes, Ed (r < R) = −2πσ(ω)e−iωt . The average of the induced surface charge density σ(ω) can be phenomenologically related to the total electric field via effective transverse susceptibility α⊥ (ω), hσ(ω)i = α⊥ (ω)Ei (ω). (3.41) 40 Figure 3.5. Geometry of a nanotube in a transverse electric field (left) and corresponding absorption transitions induced by such field (right): from filled gapless states (m = 0) into unfilled gapless states (|m| = 1) and from the filled gapped states (|m| = 1) into empty movers (m = 0). The total electric field inside the nanotube is thus given by the sum of applied field and induced field, Ei (ω) = E0 (ω) = E0 [1 − 2πα⊥ + (2πα⊥ )2 − ...] . 1 + 2πα⊥ (ω) (3.42) The formula Eq. (3.42) reproduces the suppression in the static limit with identity α⊥ = (ε⊥ − 1)/4π. The consecutive geometric series originates from endless competition between the external field and induced field. This seemingly trivial model, however, is equivalent to Random Phase Approximation which is built in the formalism of quantum field theory. For a weak perpendicular field to the axis of a nanotube with corresponding electric potential ϕ(θ, t) = −ϕ(ω) cos θe−iωt , the external perturbation is represented as the following term to Hamiltonian, 0 Z2π Ĥ (t) = 0 Z∞ dθ 2π dx eρ̂(x, θ, t)ϕ(x, θ, t), −∞ Z∞ = eϕ0 (t) dx i 1h ρ̂1 (x, t) + ρ̂−1 (x, t) , 2 (3.43) −∞ where ρ(x, θ, t) = 2πRσ(x, θ, t)/e is the effective 1D particle density, which is a more convenient quantity given a quasi-one-dimensional nature of the problem. Electron density 41 operator ρ̂1 (x, t) is expressed as a sum over the channels with different azimuthal angular P † momenta: ρ̂1 (x, t) = m ψ̂m (x, t)ψ̂m+1 (x, t). Similar to the relation Eq. (3.41), the system responds to Fourier Harmonics ϕ(ω)eimθ with the variation of the particle density ρm (ω) via density response χm (ω), 1 hρ̂m (ω) + ρ−m (ω)i = eϕm (ω)χm (ω). 2 (3.44) The average of particle density hρ̂m (ω)i in the interacting system can be transformed into the average under ground states of Ĥ0 through Kubo formula. Utilizing the identity that hρ̂m (ω)i = hρ̂−m (ω)i, χm (x, t) = − iE iΘ(t) Dh , ρ̂m (x, t), ρ̂1 (0, 0) + ρ̂−1 (0, 0) 2 0 (3.45) which is only nonvanishing for |m| = 1. To make the definition of density response more natural, we take off the prefactor 1/2 (which will be tracked back later on). In frequency space, for m = 1 we get, Z∞ χ(ω) = −i dt e iωt Z∞ dx Dh iE ρ̂1 (x, t), ρ̂−1 (0, 0) , 0 (3.46) −∞ 0 which is a correlator of the electron density operators. [To ease notations, we omit the subscript 1 in χ1 (ω)]. Without Coulomb interaction, Eq. (3.46) follows Lindhard formula for a metallic nanotube, χ (0) Z∞ (ω) = −i2N 0 dp 2π Z∞ G( + ω, p)G( , p), 2π (3.47) 0 where N = 4 accounts for the flavor degeneracy of π-electron in nanotubes: two orbital valleys and two spin directions. Prefactor 2 comes from the fact that, at zero temperature, transitions with the change of angular momentum +1 are those between metallic states p = ±vp and first gapped states | | = ∆2 + v 2 p2 : two transitions from the filled gapless movers into the m = 1 gapped empty subbands, and two transitions from the m = −1 filled gapped subbands into empty movers. Four transitions contribute equally with amplitude 1/2. Green’s function G( + ω) and G( , p) correspond to massless and massive subbands respectively, 1 , + ω − vp + iη 1 G( , p) = . 2 + ∆ + p /2m? − iη G( + ω, p) = (3.48) (3.49) 42 The integral can be evaluated easily for large mass m? and small momentum p, (0) χ (ω) = N π Z∞ dp , ω − vp − ∆ − p2 /2m? + iη 0 = i Nh ∆−ω ln |Ω| − iπΘ(−Ω) , Ω = , vπ ∆ (3.50) Consider now the Coulomb interaction of electrons in a nanotube, Z 1X † † V̂ ≈ Vm (q) dx ψµ+m (x)ψν−m (x)ψν (x)ψµ (x), 2 µνm (3.51) where Vm (q) is the Fourier component of Coulomb potential between two arbitrary points on the surface of a nanotube, e2 p , x2 + 4R2 sin2 (θ/2) (3.52) Vm (q) = 2e2 I|m| (qR)K|m| (qR), (3.53) V (x, θ) = where I|m| (x) and K|m| (x) are the modified bessel functions of the first and second kind of the order |m|. In the long wavelength limit qR 1 with a metallic gate located a distance d ∼ 1/q away, Vm =  2  2e ln (d/R), m = 0.   e2 , |m| m 6= 0. (3.54) Suppose that only V1 is retained, which accounts for the transverse interaction among electrons. In the large channel limit of N 1, one can keep only the diagrams with the maximum possible number of the loops, which yields the RPA series for the density-density correlation function Eq. (3.46), χ(ω) ≈ χRPA (ω) = χ(0) . 1 − V1 χ(0) (3.55) RPA naturally reproduces the result of electrostatics Eq. (3.42) with the identity that ε⊥ (ω) = 1 − 2e2 χ(0) (ω). The imaginary part of χ(ω) determines the absorption spectrum. The amount of Joule heat generated per unit length of the tube is given by the time-averaged product of the azimuthal component −E0 sin θ of the total field inside the tube and the 43 electric current of induced charges. Utilizing continuity equation, ∂t σ(t, θ) + R1 ∂θ j(t, θ) = 0, one can find the amount of Joule heat dissipated in the wire, R Q(ω) = − Re 2 Z2π dθ sin θE0? (ω)j(θ, ω), 0 1 = − e2 R2 ω|E0 (ω)|2 χ00 (ω). 4 (3.56) Depolarization effect embodied in Eq. (3.50) and Eq. (3.55) predicts the absorption near the threshold ω = ∆, which is suppressed logarithmically, 00 χRPA (m = 1, ω) = − N h v Θ(−Ω) i2 1 − πv ln |Ω| + N e2 . (3.57) N 2 e4 v2 The result above includes four scattering processes with the change of angular momentum +1. However, ignoring V0 is problematic. Luttinger liquid effect is very important which will be investigated in the following sections. 3.3.2 Bosonization in metallic nanotubes Ignoring V0 is inadequate to describe one-dimensional systems. In nanotubes, the density of states of gapped and gapless subbands are both being suppressed due to the Luttinger liquid effects. Longitudinal Coulomb interaction V0 corresponding to the LL effects along the axis of nanotube reads, V0 V̂0 = 2 Z∞ 2 † dx ψ̂r† ψ̂r + ψ̂l† ψ̂l + ψ̂±1 ψ̂±1 , (3.58) −∞ where ψ̂r/l and ψ̂±1 describe the one-dimensional propagation of massless and massive particles. Neglecting backscattering and Umklapp scattering ensures the validity of Luttinger liquid model (the reason is described at the beginning of this section). Accordingly, the interaction is screened by the electron-hole excitation in the metallic subbands, V0 → V0 ω2 − q2v2 . (ω + iη) − q 2 u2 (3.59) The poles of Eq. (3.59) correspond to collective eigenmodes of plasmon which propagates p with velocity u = v 1 + N V0 /πv = v/g rather than v the velocity of band electrons. Within the approximation of LL, excitations of electrons are naturally bosons. It is conve- 44 nient to write electronic operators of right/left moving electrons in the gapless subbands in the exponential form, ψ̂r/l (t, x) = √ 1 Ûr/l ek̂r/l (t,x) , 2πR (3.60) where Ûr/l are fermionic counting operators. The phase operators are given by, k̂r/l (t, x) = i √ X 1 ± g sgn q h −i|q|ut+iqx p π âq e − c.c 2gN |q|L q −1 X i √ NX Θ(±q) h p + 2π b̂iq e−i|q|vt+iqx − c.c , N |q|L i=1 q where the upper/lower sign is for the right/left-moving electrons respectively. (3.61) In the bosonization scheme plasmons âq represent total charge mode of the system, while the remaining N −1 modes b̂iq are charge neutral (they account for spin and/or band degeneracy) and propagate with the Fermi velocity. The Hamiltonian of the interacting light electrons P −1 P P † is thus simply, Ĥ = u q |q|â†q âq + N q |q|b̂iq b̂iq . i=1 v For massive particles, plasmons are accompanied by electric field, or corresponding scalar potential in quantized form, eϕ̂0 (t, x) = X ϕq √ âq eiqx−i|q|ut + c.c , L q (3.62) where the amplitude ϕq will be determined below. The scalar potential describing the coupling of longitudinal particle density to V0 as eϕ̂0 (t, x) = V0 ρ̂(t, x). Utilizing continuity equation ∂t ρ̂(t, x) + ∂x ĵ(t, x) = 0, the total Hamiltonian of charged massless particle can be written as the following quantized form, π Ĥ = 2vN = Z∞ h i dx u2 ρ̂2 (t, x) + ĵ 2 (t, x) , −∞ 2 2πu ϕ2q X † âq âq vV02 N q (3.63) 1 X 1 + = u|q| â†q âq + . 2 2 q p Recalling the Luttinger parameter that g = 1/ 1 + N V0 /vπ = v/u, we find that ϕq = p V0 g|q|N/(2π). Then the quantized scalar field Eq. (3.62) becomes, s eϕ̂0 (t, x) = u(1 − g 2 ) X q π|q| iqx−i|q|ut âq e + c.c . 2gN L (3.64) 45 Schrödinger equation, describing interaction of the massive particle close to the bottom of the subband with the fluctuating electric field (3.64), has the form, ∂ 1 ∂2 i −∆+ ψ̂±1 (t, x) = eϕ̂0 (t, x)ψ̂±1 (t, x), ∂t 2µ ∂x2 (3.65) where the effective mass is µ = 1/vR. Take the trial solution in the form, (0) ψ̂±1 (t, x) = ψ̂±1 (t, x)eK̂(t,x) . (3.66) (0) Here ψ̂±1 is the solution of Eq. (3.65) for eϕ̂0 = 0. For heavy particles, neglecting spatial derivatives of the phase K̂, we obtain the following solution, K̂(t, x) = (1 − g 2 ) Xr q π âq e−i|q|ut+iqx − c.c . 2N g|q|L (3.67) In the plasmon representation above, Green’s function of dressed particles in gapped states D E (0) can be represented as the average of bosonized phases, G> (t, x) = G> eK̂(t,x) e−K̂(0,0) , 0 (0) where bare Green’s function G> (without Coulomb interaction V0 ) is, (0) G> r = iµx2 µ e−i∆t+ 2t . 2πit (3.68) Utilizing Feynman theorem for exponential form operators that exp(Â+B̂) = exp(Â) exp(B̂) exp(−[Â, B̂]/2) and letting β = (1 − g 2 )2 /2N g, G> (t, x) = R β r 2 µ e−i∆t+iµx /2t . 2πit (u2 t2 − x2 ) β2 (3.69) The density of states of gapped states near the threshold ω ≈ ∆ is suppressed as ν1 (ω) ∼ ( − ∆)−1/2+β comparing with van hove singularity ( − ∆)−1/2 . With the same fashion, the density of states of gapless states is also found to be suppressed, ν0 (ω) ∼ ( −∆)α , other than a step function Θ( −∆) in the noninteracting system, where α = (1−g)2 /2N g. Suppression of density of states indicates that single particle excitation decays into an infinite number of low energy plasmon modes. More importantly, optical absorption between different subbands is found to be conversely enhanced. Consider the imaginary part of density response Eq. (3.46) in the presence 46 of V0 interaction (with four transitions corresponding to the angular momentum change ∆m = +1), Z∞ Z∞ D E i 1 i(ω−∆)t (0) −k̂r (x,t) K̂(x,t) −K̂(0,0) k̂r (0,0) , dt dx e G (x, t) e e e e Im χ(0) (ω) = − > V 2πR 0 h 1 = − 2πR −∞ Z∞ −∞ dt e i(ω−∆)t R 1−γ , ivt + R −∞ (3.70) where γ = (2 − g − g 3 )/2N . Since the spatial integral converges on distances x ∼ p t/µ and t ∼ 1/(ω − ∆), we observe that x vt, ut. It is therefore sufficient to set x = 0 in the correlation Eq. (3.70). The branch cut is chosen along the positive half of the imaginary t axis where the branch point iR/v resides. Performing the time integral by deforming the contour to follow the sides of the branch cut and utilizing Kramers-Kronig relation, Lindhard formula Eq. (3.50) under the influence of interaction V0 is modified as, γ N Γ(γ) ∆ (0) χV (ω) = − . vπ ω−∆ (3.71) All these nonperturbative renormalizations stem from the decomposition of single electron state into infinite low-lying plasmon modes. It is worth of noting that the gapped state created in the interband transition acts similarly to a core-hole in the conventional Xray edge singularity problem, studied by Mahan and Nozières and De Dominicis. The notable difference is that in additional to core-hole interaction, electrons are interacting with themselves. The total exponent of MND problem can be found naturally through bosonization technique. We leave the calculation in the appendix. 3.3.3 Time-ordered Green’s function and the singular interaction Naively, if one uses RPA mean field approach embodied in Eq. (3.55) and Eq. (3.71), the absorption rate would be suppressed as power-law with the same exponent Q ∝ Ωγ |E0 |2 . In fact, the suppression is found to be much stronger, Q ∝ Ω3γ |E0 |2 . Because independentloop approximation is not adequate in describing interloop vertex corrections from V0 . In order to sort out the modification of the electrostatic solution by V0 , we adopt a hybrid approach, namely treating dipolar interaction V1 perturbatively in the usual diagrammatic technique while using bosonized electron operators Eq. (3.60) and Eq. (3.66). The advantage 47 of such approach is that the correlation function can be calculated exactly to all orders. Although bosonization ansatz is equivalent to Green’s function method, its application in perturbation theory is distinct and has not been used in any existing literature yet. Therefore, caution is necessary: random phase approximation must be reproduced when V0 is switched off. This is equivalent to say that Wick’s theorem cannot be violated in “noninteracting” system. Accordingly, it is natural to introduce the following time-ordered Green’s function by treating V1 perturbatively, Z∞ Z∞ Z∞ Z∞ ∞ 1X n−1 dx1 ...eiωt dt dx dt1 (−i) χt (ω) = 2 n=0 −∞ −∞ −∞ −∞ D E × Tt ρ̂1 (x, t)ρ̂−1 (0, 0)V̂1 (x1 , t1 )V̂1 (x2 , t2 )... , 0 (3.72) with the subscript in χt indicating that it is time-ordered Green’s function. The correlation function only accounts for topologically different connected diagrams. However, it turns out that only the term obeying t > t1 > t2 > ... > 0 corresponds to absorption process, which is unique in a system with gapped states. Indeed, the spatial average is mostly determined by Gaussian functions in Eq. (3.68) and goes to e−i∆(t−t1 ) e−i∆(t1 −t2 ) ... = e−i∆t , whose corresponding Fourier component iΘ(ω − ∆) means that the transition describes absorption process with requirement that ω > ∆. Such particular time sequence analytically transforms time-ordered Green’s function into required retarded Green’s function, ∞ 1X χV (ω) = (−i)n−1 2 n=0 Z∞ dt 0 Zt1 Zt dt1 0 D E dt2 ... ei(ω−∆)t ρ̂1 (t)ρ̂−1 (0)V̂1 (t1 )V̂1 (t2 )V̂1 (t3 )... . 0 0 (3.73) The dipolar interaction V̂1 (x, t) has two different types of terms (α and β label pseudospin of the electrons), V̂1 (x, t) = i V1 h † † † † ψ̂0,β (x, t)ψ̂0,α (x, t)ψ̂−1,α (x, t)ψ̂1,β (x, t)+ψ̂0,α (x, t)ψ̂1,β (x, t)ψ̂0,β (x, t)ψ̂1,α (x, t) . 2 (3.74) As gapless particles are chiral, both interactions in Eq. (3.74) can be either forwardscattering or backscattering, see Figure 3.6, which contribute equally in the absence of interaction V0 . Under the influence of LL effects, it is a formidable task to calculate the 48 Figure 3.6. Four scattering processes in V1 -interactions. Wiggle lines carry angular momentum m = 1 due to the conservation of angular momentum at vertices. Doubleand single-solid lines represent massive and massless propagators, respectively. correlation function exactly to all orders. The complexity comes from the nonquadratic terms in V1 -interaction which is the pain in bosonization. But such cases are not rare in one-dimension. For V0 interaction, as long as backscattering is discarded, the system can be trivially diagonalized with the help of Bogoliubov transformation. This is ensured by the large radius of nanotubes (N 1) where the zone-folding approximation is applied. However, V1 brings new processes in V1 that cannot be treated in the same way as V0 . For a two-channel system in the bosonization language, one of the scattering processes in V1 reads, X † † ψ̂−1,α (t, x)ψ̂1,β (t, x)ψ̂r,β (t, x)ψ̂r,α (t, x) + c.c = α,β h i h i 2 cos −2θ̂1 (t, x) + 2θ̂r (t, x) + 2 cos −2θ̂1 (t, x) + 2θ̂r (t, x) + 2φ̂s (t, x) , (3.75) where θ̂r (t, x) and θ̂1 (t, x) are related to bosonized phase of charge part for right-moving electrons and massive particles (|m| = 1) respectively, and φ̂s (t, x) is connected to spin phase in Eq. (3.61). ik̂r/l (t, x) = θ̂r/l (t, x) + φ̂s (t, x), iK̂(t, x) = θ̂1 (t, x). (3.76) (3.77) 49 The first term on the RHS of Eq. (3.75) corresponds to the “spinless” case (α = β) while the second term is in the “spinful” case (α 6= β). It is easy to see that spin-charge separation does not work for the first term. This is not surprising, because spin-charge separation is distinct in one dimension while V1 has nonone-dimensional character as it involves transitions to parabolic subbands. Now we are facing a new situation that does not occur in the Luttinger liquid model. For the second term, it only involves the difference of charge phases of light and heavy particles which has no counterpart in LL model. The situation is a little relieved for the other forward-scattering term in V1 , X † † ψ̂r,β (t, x)ψ̂1,α (t, x)ψ̂r,α (t, x)ψ̂1,β (t, x) + c.c = 2ψ̂r† (t, x)ψ̂r (t, x)ψ̂1† (t, x)ψ̂1 (t, x) α,β +2 cos 2φ̂s (t, x) . (3.78) The first two terms, corresponding to the spinless case (α = β), are reminiscent of core-hole interaction in the X-ray problem. They give rise to trivial renormalization the same way as V0 interaction. We leave a detailed discussion of this term in the appendix. The third term has typical Sine-Gordon form, which only depends on spin phase. This interaction preserves spin-charge separation. The usual treatment of the third term is by using renormalization group. RG shows that Sine-Gordon term is irrelevant for repulsive interaction and relevant for attractive interaction. In the latter case, the interaction opens a gap under phase transition. Therefore, for electron-electron interaction, there is no need to worry about the third term. In fact, the spinful term (third term) does not reproduce first order expansion in RPA. It means that something goes wrong if Eq. (3.55) is not reproduced in the static limit (V0 → 0). We will give a more detail discussion about this point later. Overall, this forward-scattering term is easy to treat. For the other two backscattering terms in the bosonized form, X † † ψ̂−1,α (t, x)ψ̂1,β (t, x)ψ̂r,β (t, x)ψ̂l,α (t, x) + c.c = α,β h i h i 2 cos −2θ̂1 (t, x) + θ̂r (t, x) + θ̂l (t, x) + 2 cos −2θ̂1 (t, x) + θ̂r (t, x) + θ̂l (t, x) + 2φ̂s (t, x) , (3.79) 50 X h i † † ψ̂r,β (t, x)ψ̂1,α (t, x)ψ̂l,α (t, x)ψ̂1,β (t, x) + c.c = 2 cos θ̂r (t, x) − θ̂l (t, x) α,β h i +2 cos θ̂r (t, x) − θ̂l (t, x) + 2φ̂s (t, x) . (3.80) It is easy to notice that all backscattering terms do not respect spin-charge separation. Eq. (3.78) is the only interaction that we can deal with by taking the advantage of onedimensional physics. Then it raises a serious question: how do we treat other terms in V1 -interaction? In this section, we resorted to the spirit of conventional RPA to sort out the most singular interaction in V1 and only the correlation function of this specific interaction at all orders will be calculated. At first order, we compared the density response for all four scattering processes. It can be verified that as γ → 0, all exponents vanish and first order expansion of random phase approximation ln Ω is reproduced. As we pointed out earlier, RPA must be reproduced when V0 is switched off. This is because we are looking into the case of large channel limit where RPA is exact. † † ψ̂−1,α ψ̂1,β which yields the largest ψ̂r,α The most singular interaction is found to be ψ̂r,β exponent 3γ. This very distinct term constructs two kinds of diagrams in one-channel (α = β) and multichannel system (α 6= β) respectively, see Figure 3.7. However, within the RPA phase approximation, one should only preserve the diagrams with the most loops to each order. Indeed, the two-loop diagram has N (N − 1) degree of freedom which is N − 1 times larger than the one-loop diagram. The corresponding correlation function at first order is calculated from the following equation. (1) χV (ω) V1 N (N − 1) =− 4π 2 R2 Z∞ Zt dt 0 D E 0 0 dt0 ei(ω−∆)t e−kr (t) eK(t) e2kr (t ) e−2K(t ) e−kr (0) eK(0) . 0 (3.81) Prefactor N (N − 1) accounts for the fact that the second loop must have different pseudospin index than the first one. Correlation function in Eq. (3.81) can be evaluated from the total charge and charge neutral part respectively. For a multichannel system, neutral phase represented in Eq. (3.61) is not adequate. In order to account for N = 4 fold band degeneracy, we introduce the following four modified bosonized phases for particles residing on the gapless subband, 51 † † Figure 3.7. First order corrections of the most singular interaction ψ̂−1,α ψ̂1,β ψ̂r,β ψ̂r,α in a one-channel system (top) and a multichannel system (bottom). For this type of interaction, it is required that two loops must be in different neutral channels to display as a more singular correction. The interaction V0 (dashed line) is account to all orders (not shown in the top one). ik̂r/l,k↑ (x, t) = θ̂r/l (x, t) + φ̂(x, t) + φ̂s (x, t) + φ̂a (x, t), ik̂r/l,k↓ (x, t) = θ̂r/l (x, t) − φ̂(x, t) + φ̂s (x, t) − φ̂a (x, t), ik̂r/l,k0 ↑ (x, t) = θ̂r/l (x, t) + φ̂(x, t) − φ̂s (x, t) − φ̂a (x, t), ik̂r/l,k0 ↓ (x, t) = θ̂r/l (x, t) − φ̂(x, t) − φ̂s (x, t) + φ̂a (x, t), (3.82) where k and k 0 denoting two orbital valleys while ↑ and ↓ showing two spin directions for the electrons. Four electron operators belong to four different band degeneracy. θ̂(x, t) is the total charge phase built in Eq. (3.61), i √ X 1 ± g sgn q h −i|q|ut+iqx p âq e − c.c . θ̂r/l (t, x) = i π 2gN |q|L q (3.83) Again, the upper/lower sign is for the right/left- moving electrons respectively. Meanwhile, there are three independent neutral modes described by φ̂(x, t), φ̂s (x, t) and φ̂a (x, t) with the same structure as the one in Eq. (3.61), 52 i √ X Θ(±q) h p φ̂(t, x) = i 2π b̂iq e−i|q|vt+iqx − c.c , N |q|L q i X √ Θ(±q) h p ĉiq e−i|q|vt+iqx − c.c , φ̂s (t, x) = i 2π N |q|L q i √ X Θ(±q) h p φ̂a (t, x) = i 2π dˆiq e−i|q|vt+iqx − c.c , N |q|L q (3.84) where operators b̂iq , ĉiq and dˆiq are independent and commuting with each other. The mathematical recipe here can be understood as a generalization of spin 1/2 system to N = 4 channel system. It is easy to verify that the first equation in Eq. (3.82) reproduces Eq. (3.61) by taking all neutral phases the same – converting from multichannel to one-channel. At first order, two-fold degeneracy would suffice, such as k ↑ and k ↓, since there are only two loops. Accordingly, we find, (1) χV Z ∞ Z t V1 N 2 (ω) = − 2 2 dt dt0 ei(ω−∆)t 4π R 0 0 D E −k̂r,k↑ (t) K̂(t) k̂r,k↑ (t0 ) k̂r,k↓ (t0 ) −2K̂(t0 ) −K̂r,k↓ (0) K̂(0) × e e e , e e e e V1 N 2 ∆3γ = i 2 2 1−3γ 4v π (i) Z∞ Zt dt 0 dt0 ei(ω−∆)t h i−γ t 1 . × 0 2 02 (t − t ) t (t − t0 )t0 (3.85) 0 In the last step, we used the definition of band gap that ∆ = v/R. The interaction V0 affects the polarization function through exponent γ = (2 − g − g 3 )/2. The vertex correction going between loops leads to post RPA behavior, see Figure 3.7. By power (1) counting, χV (ω) ≈ Ω−3γ yields the highest negative exponent, see Table 3.1. In the limit of γ → 0, Z∞ Zt dt 0 0 ei(ω−∆)t dt0 ≈ (t − t0 )t0 1/(∆−ω) Z t−1/∆ Z dt0 dt 1/∆ 1 = ln2 Ω−1 (t − t0 )t0 (3.86) 1/∆ The first order correction of RPA is being recovered nicely. (Calculation above involves the crossover between interacting and noninteracting systems which will be discussed later.) 3.3.4 Second order expansion and self-energy disaster It seems like subsequent orders should be obtained in the same way. However, an unexpected divergence occurs right at second order. Consider the second order expansion 53 Table 3.1. Comparison of all scattering processes in interaction V1 . (1) ∼ v 2 χV /V1 Interactions in V1 † † ψ̂−1,α ψ̂1,β ψ̂r,β ψ̂r,α Ω−3γ † † ψ̂r,β ψ̂1,α ψ̂r,α ψ̂1,β Ω−γ ln Ω−1 † † ψ̂r,α ψ̂1,β ψ̂l,β ψ̂1,α Ω−γ−(1−g)/N † † ψ̂−1,α ψ̂1,β ψ̂r,β ψ̂l,α Ω−3γ+(1−g)/N Forward-scattering Backscattering below, V12 N 3 χ(2) (ω) = i V 16π 3 R3 Z∞ Zt dt 0 dt0 0 Zt0 E D 0 00 dt00 ei(ω−∆)t e−K̂(t) e2K̂(t ) e−2K̂(t ) eK̂(0) , 0 (3.87) 0 where we used short notation K̂(x, t) = k̂r (x, t) − K̂(x, t). If one follows what is being used in the first order expansion with the charge neutral channel in an alternating fashion, the corresponding correlation function yields, D 0 0 00 00 e−K̂k↑ (t) eK̂k↑ (t ) eK̂k↓ (t ) e−K̂k↓ (t ) e−K̂k↑ (t ) eK̂k↑ (0) h i−γ h (ivt − ivt00 )2 (ivt0 )2 ivt(ivt − ivt0 )2 (ivt0 − ivt00 )4 (ivt00 )2 E 0 = i 1 1 + . (ivt − ivt0 )(ivt0 − ivt00 )ivt00 ivt(ivt0 − ivt00 )2 (3.88) The first part describes the effect of interaction V0 with weak exponent γ. The second part corresponds to noninteracting Green’s function in real space. The Fourier component of first fraction 1/(t − t0 )(t0 − t00 )t00 generates ln−2 (ω − ∆) which is exact second order expansion of RPA, see Figure 3.8. However, the second fraction 1/t(t0 − t00 )2 is divergent P P at t0 = t00 . Its Fourier component is found to be Σ(ω) ∝ k,q iε,iΩ G0 (iΩ, k)G0 (iω −iε, p− q) × G1 (iΩ − iε, k − q), as a self-energy to second order. The divergence of self-energy is out of the scope of low energy physics, which should be understood as a renormalization of the bandgap ∆. 54 Figure 3.8. Two contributions at second order with neutral phases in an alternating order. The three-loop diagram (top) reproduces RPA in the limit γ → 0, while the self-energy diagram (bottom), being divergent correction to first zeroth order, renormalizes band gap energy. In addition, the divergence can also be avoided by noting that the self-energy diagram originates from double pairings of gapless particles between interacting times t0 and t00 . The lack of adequate channels lead to two possible pairings, D D 0 0 00 00 e−K̂k↑ (t) eK̂k↑ (t ) eK̂k↓ (t ) e−K̂k↓ (t ) e−K̂k↑ (t ) eK̂k↑ (0) 0 0 00 00 e−K̂k↑ (t) eK̂k↑ (t ) eK̂k↓ (t ) e−K̂k↓ (t ) e−K̂k↑ (t ) eK̂k↑ (0) E 0 E 0 , (3.89) . (3.90) Two external operators in Eq. (3.89) are pairing with themselves which essentially form so-called disconnected diagram. Disconnected diagrams are supposed to be canceled out and subtracted at the formalism of time-ordered Green’s function Eq. (3.72). The divergent self-energy originates from the fact that the bosonized average is not able to exclude undesired pairings at higher order expansion which yields something unphysical. Therefore, 55 pseudospin plays a critical role by introducing a new degree of freedom and complements the shortcoming of bosonization approach. Accordingly, Eq. (3.91) can be isolated by letting the third pairing having a pseudospin index different than the former two, such as k 0 ↑, D E 00 0 0 00 e−K̂k↑ (t) eK̂k↑ (t ) eK̂k↓ (t ) e−K̂k↓ (t ) e−K̂k0 ↑ (t ) eK̂k0 ↑ (0) . (3.91) 0 Three-fold degeneracy excludes all other possibilities, and hence the second order expansion can be correctly obtained, (2) χV V12 N 3 ∆5γ (ω) = −i 16v 3 π 3 (i)1−5γ Z∞ dt ei(ω−∆)t Z t dt0 0 0 Z t0 h dt00 0 i−γ (t − t00 )2 t02 t(t − t0 )2 (t0 − t00 )4 t002 × 1 (t − t0 )(t0 − t00 )t00 . (3.92) The integrands in Eq. (3.85) and Eq. (3.92) have similar structure: the noninteracting Green’s function 1/(t − t1 )(t1 − t2 )...tn reproduces depolarization effect in the electrostatic limit, Z∞ Zt dt 0 0 dt0 Zt0 dt00 ei(ω−∆)t (t − t0 )(t0 − t00 )t00 0 1/(∆−ω) Z ≈ t−1/∆ Z t0 −1/∆ Z 1/∆ 1/∆ dt0 dt 1/∆ 3 = ln Ω −1 dt00 1 (t − t0 )(t0 − t00 )t00 , (3.93) and the V0 -correction with exponent −γ that describes many-body effect. 3.3.5 Poorman’s summation and beyond RPA Although the calculations of expansions of the correlation function are not carried out for all orders yet, it is not hard to make a guess of high orders. N -th order expansion can be directly inferred as, 56 (n) χV N ∆(2n+1)γ −V1 N n 1 1−(2n+1)γ (ω) = −i πv 4πv i Z∞ dt ei(ω−∆)t 0 × dt1 0 tZn−1 Zt1 Zt dt2 ... 0 n h Y (n+1) × t(−1) dtn 0 n Y n−1 Y 1 j=1 (t − t1 )(tj − tj+1 )tn −2(−1)j+n tj (tj − tk )4(−1) j+k j (t − tj )2(−1) i−γ . j=1 k=j+1 (3.94) For the effective dimensionless dipolar coupling constant −V1 N/(4πv), factor 1/4 can be quickly found by noting that only one out of four possible loops (interactions) has the strongest singularity at ω ≈ ∆. By power counting, perturbative series becomes an oddpower polynomial, ∞ X N −V1 N n −(2n+1)γ Ω , πv 4πv n=0 Z Z Z An (γ) = Γ[(2n + 1)γ] dt1 dt2 ... dtn ... χV (ω) = − An (γ) (3.95) where An (γ) involves senior time integral which is a headache in forming the summation of series. Because there is no regular pattern of An (γ) at all orders. However, it is possible to uniquely construct a series that obeys the following two properties: the n-th order term is an odd-power polynomial in Ω−γ , and the coefficients in the polynomial are chosen in such a way that in the limit of γ → 0 the n-th order term reproduces lnn+1 Ω. Let us go one step back by considering zeroth order expansion below, (0) χV N ∆γ (ω) = − πv Z∞ dt t1−γ ei(ω−∆)t . (3.96) 0 One can find the imaginary part of χ(0) (ω) by extending the integral to negative time and V then obtain the whole thing through Kramer-Kronig relation. On the other hand, the integration can be carried out more simply. Note that the integral is mostly concentrated below t ∼ 1/(∆ − ω). In addition, if one imposes a small-time cut-off by 1/∆, RPA can be reproduced naturally, (0) χV N ∆γ (ω) = − πv 1/(∆−ω) Z dt t1−γ 1/∆ =− N −γ Ω −1 . πvγ (3.97) 57 The approximation shows a direct crossover between noninteracting (γ = 0) and interacting system in the expansion that (Ω−γ − 1)/γ ≈ ln Ω−1 + O(γ), χ(0) (ω) = − V N −γ N ln Ω−1 . Ω − 1 → χ(0) (ω) = − πvγ πv (3.98) It seems like there is nothing new at zeroth order. In fact, such intuitive calculate opens a new way to calculate the correlation by avoiding senior time integral. With the same fashion, the result of the first order in Eq. (3.85) can be quickly established as, (1) χV (ω) = V1 N 2 ∆3γ 4π 2 v 2 1/(∆−ω) Z dt tγ 1/∆ = N2 t−1/∆ Z dt0 , [(t − t0 )t0 ]1−2γ 1/∆ i V1 −3γ −γ Ω − 1 − Ω − 1 . 4π 2 v 2 γ 2 3 h1 (3.99) The second order expansion in terms of small γ yields [(Ω−3γ − 1)/3 − (Ω−γ − 1)]/γ 2 ≈ ln2 Ω−1 . The prefactor of leading order Ω−3γ exactly coincides with rigorous integration. Intuitively, one can interpolate higher order expansion as an odd power polynomial, see Table 3.2. At nth order, the prefactor of the leading term Ann can be uniquely determined as the solution to a linear equation set, Ann = 2 n+1 . n+1 (2γ) 2n + 1 (3.100) Table 3.2. Interpolation of odd power polynomials. n-th order Odd power polynomials RPA limit γ → 0 Zeroth A11 (Ω−γ − 1) ln Ω−1 First A22 (Ω−3γ − 1) + A12 (Ω−γ − 1) ln2 Ω−1 ... ... ... n-th Pn m −(2m+1)γ m=1 An [Ω − 1] lnn+1 Ω−1 58 Keeping only the leading term, the total density response is obtained as, χV (ω) = − λ n N X n+1 1 × Ω−(2n+1)γ . × − πvγ 2n + 1 2γ (3.101) n=0 For convenience, we introduced dimensionless constant λ1 = V1 N/4πv. The summation is equivalent to the following form, ∞ X y n + 1 2n+1 = + arctanh y, 1y 1 − y2 n + 2 n=0 where y = i (3.102) p V1 N/(8πvγ)Ω−γ 1. The first term and the second term can be expanded for large y, 1 1 ≈ − − 3, y y 1 1 + y iπ 2 arctanh y = ln = ± + ln 1 + , 2 1−y 2 y−1 iπ 1 1 ≈ ± + + 3. 2 y 3y y 1 − y2 (3.103) The first nonvanishing term starts with 1/y 3 , ∞ X n + 1 2n+1 iπ 2 ≈ − 3, 1y 2 3y n+ 2 n=0 (3.104) Accordingly, we obtain that the imaginary part of the polarization operator close to the threshold is, Im χV (ω) = − 4N γ (−Ω)3γ sin 3πγ. 3πvλ21 (3.105) The obtained result indicates that the optical absorption close to the threshold ω = ∆ becomes suppressed due to the depolarization effect much more significantly than could be predicted based on the electrostatic mean field theory, ∝ (ω − ∆)γ . This can be view as a much stronger Ei = (ω − ∆)2γ E0 suppression of the electric field acting inside the wire. 3.3.6 Discussion To conclude, we predicted a stronger suppression in the depolarization effect of a nanotube under a perpendicular electric field. The suppression is the interplay between symmetric interaction V0 and dipolar interaction V1 , see Figure 3.9. Many-body effect nontrivially modifies the edge behavior of depolarization effect, which goes beyond the prediction of electrostatics. The absorption given by Eqs. (3.56) and (3.105) should be testable in optical 59 Figure 3.9. Absorption lineshape for different values of V1 = 4πvλ1 /N . For small V1 values (solid blue line) the suppression (depolarization) induced overcomes the enhancement due to V0 being sufficiently close to the threshold. For larger V1 values the depolarization effect dominates everywhere. absorption measurements. Even though previous measurements of transverse absorption [40, 41] do not provide sufficient resolution near the threshold, the modern advances in nanotube manufacturing[42] should make such measurement possible. 3.4 Summary Many-body problems are among the most difficult problems of physics. There are very limited models that are exactly solvable. In this section, we solved a problem with a very distinct method that has not been used in any existing literature yet. Bosonization and Green’s function are two parallel methods in solving many-body problems. The former one transforms an interacting system of electrons into noninteracting bosons; meanwhile, the later is built in the frame of perturbation theory. Although bosonization has its advantage in the calculation and physical intuition, there are not many problems that can be fully bosonized. One often resorts to renormalization group to transform the problem into a familiar and solvable one. However, RG is not an easy subject to harness despite that it is a very efficient method for solving problems. Instead, we used a hybrid approach that combined both Green’s function and bosonization. The advantage is that the correlation function is always easy to be calculated no matter how complicated it is. Nonetheless, one needs to be very careful to discard artificial diagrams. Besides, it is also challenging to sum up the perturbation series. Fortunately, we accomplished all missions and a very nontrivial result show up. All efforts in this project are paid off in the end. CHAPTER 4 INTRODUCTION TO PLASMONICS Plasmon was first proposed by David Pines and David Bohm in 1952, which are collective oscillations of electrons. Just as phonons and single particle excitations, plasmon is being regarded as one of the elementary excitations and hence it is very crucial in understanding electron system. Plasmon can be regarded as quasiparticles since it behaves similarly as harmonic oscillators. Accordingly, plasmon must be quantized just as oscillator quanta. In bulk metals, plasmon excitation is obtained by letting ε = 0. Furthermore, plasmon modes also survive near the interface of metals, which are called surface plasmon. 4.1 Plasmon excitations 4.1.1 Drude model Drude model is an empirical model proposed by Paul Drude to explain transport properties of electrons. Later the theory was further developed by Lorentz, Sommerfeld and Bethe. So the model is also called Drude-Sommerfeld model. Although the model is derived through classic picture, the physical content is fruitful. Consider an electron with momentum p~(t) in a metal or conductor with relaxation time τ , its momentum at later time t + dt is found as, p~(t + dt) = dt dt · 0 + (1 − )[~ p(t) + F~ (t)dt], τ τ (4.1) where dt/t is the probability that electrons are scattered off into zero momentum. Ignoring second order correction (dt)2 , the equation of motion for electrons is obtained, d~ p(t) p~(t) ~ =− + F (t). dt τ (4.2) Even though the equation is derived through classic picture, corresponding physical results are fruitful. Let us consider several particular cases, ~ at long times (t τ ), LHS of Eq. (4.2) can be a). For a DC electric field (F~ = eE) ignored, 61 ~ p~ = eEτ, N e~ p N e2 τ ~ J~ = E. = m m (4.3) With the help of Ohm’s law, the dc conductivity is given by, σ0 = N e2 τ . m (4.4) b). Considering an AC electric field E = E0 e−iωt , electrons would oscillate with the same frequency. Accordingly, a frequency dependent conductivity is found as, p~ = σ = ~ eτ E , 1 − iωτ σ0 , 1 − iωτ N e~ p N e2 τ ~ E, J~ = = m m(1 − iωτ ) (4.5) (4.6) which exhibits that the DC conductivity σ0 is the limiting case of AC conductivity σ at low frequency ωτ 1. c). Let us consider a transverse electromagnetic wave. Maxwell’s equations in a metal are given by, ~ = − ∂ B, ~ ~ = 0, ∇×E ∇·D ∂t ~ = ∂D ~ + J, ~ ∇·B ~ = 0. ∇×H ∂t (4.7) (4.8) ~ = εE, ~ J~ = σ E ~ and ∇ × ∇ × E ~ = For a slow varying field, one can utilize relations that D ~ − ∇2 E. ~ For transverse waves (∇ · E = 0), ∇(∇ · E) ~ + ω 2 µ(ε + ∇2 E iσ ~ )E = 0. ω (4.9) In metals, one can make approximations ε ≈ ε0 and µ ≈ µ0 . Introducing the wave vector k such that, k 2 = ω 2 µ0 (ε0 + iσ ). ω (4.10) Let ω = kv and use identity that (c/v)2 = ε, ε=1+ iσ . ωε0 (4.11) This formula applies to a low-frequency field ω → 0. Recalling that for a complex function ε, the real part is even while imaginary part is odd. So the 1/ω comes as the leading term in 62 the expansion of ω in odd powers. In the limit that ωτ 1, one can take the conductivity as σ ≈ σ0 /(−iωτ ), we have, ε=1− N e2 , mε0 ω 2 (4.12) where ωp2 = N e2 /(mε0 ) is called bulk plasmon frequency. Then Drude formula is reproduced, ε=1− ωp2 . ω2 (4.13) It shows that for low-frequency ω < ωp , the electric field decays exponentially inside the metal with a imaginary wave vector. In the opposite limits where ω > ωp , the metal becomes transparent, waves are propagating without damping inside the metal. Drude model describes a wave vector independent dispersion relation. If the collision is considered, one needs to go one step back to Eq. (4.11) and use the conductivity with finite scattering rates Eq. (4.6), and we get, ε=1− ωp2 ω(ω + τi ) . (4.14) There are two things that we can learn from the Drude model. Firstly, when the frequency is “large” ωτ 1, plasmon is damping less (no imaginary part in ε), because the system is not able to response the fast-changing external field. Secondly, the Drude model applies to the high-frequency regime. Constant 1 is the zeroth order, meaning as ω → ∞, dielectric function ε → 1. The second term proportional to 1/ω 2 comes as the first order expansion for high frequency. However, this is the best accuracy of what Drude model can provide from a purely classical perspective. 4.1.2 Plasmon oscillation in the formalism of quantum theory If one starts from quantum theory, Drude formula Eq. (4.13)can be further refined. Consider Lindhard formula in the “noninteracting” 3D electron gas, Z d3 p nF (εp ) − nF (εp+q ) χ0 (~q, ω) = 2 (2π)3 ω + εp − εp+q + iη (4.15) The real part can be extracted from the principal part. In the long wavelength limit, utilizing following two expansions, εp+q − εp ≈ − ∂nF ∂εp pqλ = vp qλ, ν ≈ δ(εp − εF ) = λ = cos θ δ(p − kF ) vF (4.16) (4.17) 63 the real part of the polarization function is found as, Z Z F − ∂n 1 ∂εp (εp+q − εp ) 2 , Reχ0 (~q, ω) = dp p dλ 2π 2 ω + εp − εp+q Z1 kF2 vF qλ = dλ 2π 2 vF ω − vF qλ (4.18) (4.19) −1 In the high frequency limit, let z = qvF λ ω , Reχ0 (~q, ω) = ≈ we have, kF2 ω 2π 2 qvF2 nq 2 mω 2 qv ZF /ω dz z , 1−z (4.20) −qvF /ω 3 qvF 2 [1 + ( ) ] 5 ω (4.21) (4.22) Recalling the relation between dielectric function and permittivity, ne2 3 qvF 2 e2 χ (~ q , ω) = 1 − [1 + ( ) ] 0 2 2 ε0 q mε0 ω 5 ω ωp2 3 qvF 2 = 1 − 2 [1 + ( ) ] ω 5 ω ε = 1− (4.23) (4.24) where the third term originates from purely quantum effects which can not be captured by Drude model. As ε → 0, there are longitudinal excitations oscillating even without the external field, which are called plasmon waves. 4.2 Surface plasmon The collective plasmon in the last section was studied by Pine and Bohm, which are known as bulk plasmon excitations. In addition, there is another novel type of waves that is localized near the interface of two materials. Consider Maxwell’s equation in the electrostatic limit, ~ = 0, ∇×E ∇ · D = 0. (4.25) Two equations can be unified into a Laplace’s equation, ∇2 φ = 0. (4.26) ~ = −∇φ. At the interface of two materials, where the scalar field is introduced as E continuity condition needs to be satisfied, φ1 (z0 ) = φ2 (z0 ), ε1 ∂ φ1 (r0 ) ∂ φ2 (z0 ) = ε2 ∂z ∂z (4.27) 64 4.2.1 Planar surface For an infinite planar surface, localized surface waves propagating along x-direction have form, φ(x, z) = f (z)eikx−iωt (4.28) Plug it into Laplace’s equation Eq. (4.26), d2 f − k 2 f = 0. dz 2 (4.29) It yields f = f0 e−|k|z for z > 0 half space and f = f0 e|k|z for z < 0 half space, so that electric potential is finite at either z → ∞ or z → −∞. Consequently, the boundary condition leads to, ε1 = ε2 . (4.30) If one side of the material is vacuum while the other part is conductor, the Drude model yields surface plasmon frequency, ωp ω=√ . 2 (4.31) The frequency of surface plasmon is smaller than the one of bulk plasmon. Another difference between surface waves and bulk waves is that surface plasmon is not purely longitudinal but a mixture hybrid. Because the electric field not only exists in the propagating direction (parallel to the interface), but also the direction perpendicular to the surface decaying exponentially at the same rate. Due to the non-dispersive nature, surface plasmon in the infinite planar surface does not carry any energy and momentum. 4.2.2 Other simple geometries Surface plasmon in other simple geometries can be obtained directly in the same fashion. For spherical interface (metal sphere in the vacuum), wave functions inside and outside the sphere have form, φ1 = Arl , φ2 = B 1 rl+1 r<R r>R Using continuity condition Eq. (4.27), the plasmon frequency is found as, r l ω = ωp 2l + 1 (4.32) (4.33) (4.34) 65 such waves are called Mie plasmons. For an infinitely long metallic cylinder immersed in the vacuum, wave functions are conveniently expressed in the polar coordinate system, φ1 = Aρν , φ2 = B , ρν ρ<R (4.35) ρ>R (4.36) √ The dispersion is found to be the same as the one of planar surface, ω = ωp / 2. This is because for an infinitely long cylinder, the boundary coincides with a planar surface. 4.2.3 Surface plasmon in a film All surface waves showed above have zero phase velocity ∂ω/∂k = 0. This is because surface plasmon is seeking in the electrostatic limit. However, the spatial dispersion exists even without considering the retardation effect. For a film with thickness d immersed in the vacuum see Figure 4.1, now there are two surfaces supporting surface waves at z = −d/2 and z = d/2. Wave functions in three regions are seeking in the following form, φ1 (z) = Ae|k|z , z> d , 2 φ2 (z) = B cosh or B sinh, φ3 (z) = Ce|k|z , (4.37) − d d <z< , 2 2 (4.38) d z<− . 2 (4.39) ଴ 1 2 ߝ ଴ 1 2 Figure 4.1. A film exposed in vacuum which resembles a three-layer system. 66 Due to the symmetric geometry of the film, solutions inside the film can be either symmetric or antisymmetric under reflection. Continuity conditions yield two dispersion relations, ωp ω± (k) = √ 1 ∓ e−|k|d , 2 (4.40) where the upper/lower sign corresponds to symmetric/antisymmetric modes, see Figure 4.2. For thick film or short wavelength limit |k|d 1, two modes merge into one, which reproduces the dispersion for planar surface. This is not surprising because thick film resembles the planar surface. For thin film or large wavelength limit, the behavior of two modes are totally different. For antisymmetric mode, the frequency approaches bulk plasmon frequency. ω− ≈ ωp (4.41) This is due to the charge distribution: charges at two surfaces have different signs. For long wavelength, the restoring force thus perpendicular to the surface, and hence the behavior is the same as bulk plasmon. For symmetric mode, one can expand exponential for small |k|d 1, one find, ωp ω+ ≈ √ d|k|, 2 (4.42) ߱௣ ௣ 2 0 Figure 4.2. Plasmon dispersion for a film immersed in vacuum. 67 which approaches the plasmon frequency in 2D. For 2D electron gas, the dispersion of √ collective plasmon behaves as ω ∼ k 4.2.4 Surface plasmon polariton In the previous section, the surface wave is considered in the electrostatic limit or infinite speed of light limit. In the presence of retardation effect, surface waves are named as surface plasmon polariton (SPP). Let us take the simplest case (planar surface) to get a feeling of retardation effects. Now we need to consider Maxwell’s equations with finite speed of light, ~ = iω B, ~ ∇×E ~ = − iωε E, ~ ∇×B c2 (4.43) which allows two kinds of solutions: transverse magnetic (TM) modes (Ex , Hy , Ez ) and transverse electric (TE) modes (Hx , Ey , Hz ). It turns out that SPP does not support TE iβx , H(x, iβx , one would arrive at ~ ~ ~ ~ wave. For a TM wave, E(x, y, z) = E(z)e y, z) = H(z)e following equations for magnetic field, ∂ 2 Hy + (k02 ε − β 2 ) = 0 ∂z 2 (4.44) where k0 = ω/c is the wave vector propagating in the vacuum. One can take trial solutions on two sides of the plane. In the vacuum z > 0, Hy (z) = A2 eiβx e−k2 z iA2 k2 iβx −k2 z e e ω A1 β iβx −k2 z Ex (z) = − e e ω Ex (z) = (4.45) (4.46) (4.47) and in the conductor z < 0, Hy (z) = A1 eiβx ek1 z iA1 k1 iβx k1 z e e ωε A1 β iβx k1 z Ex (z) = − e e ωε Ex (z) = − where k1 = (4.48) (4.49) (4.50) p p β 2 − k02 ε is the wave vector in the conductor and k2 = β 2 − k02 is the wave vector in the vacuum. Electric fields on two sides have different decaying rate due to finite speed of light. Continuity conditions of Hy and Dz yield that A1 = A2 and most importantly, 68 1 k2 =− . k1 ε (4.51) Put the value of k1 and k2 into it, we get the dispersion relation of SPP in the planar surface, 1+ε k2 ω2 = 02 = 2 2 . ε β c β (4.52) In the electrostatic limit c → ∞, the dispersion of surface plasmon in the planar surface is reproduced ε = −1. Plug Eq. (4.52) into the Drude model, s 2 ωp2 ω p + c2 β 2 − c4 β 4 + ω2 = 2 4 (4.53) In short-wavelength limit cβ > ω, surface plasmon in the planar surface is reproduced, see Figure 4.3. Electric fields along x and z have the same amplitude in the surface, because of β = k1 = k2 . At large wavelength limit cβ < ω, Ex Ez inside the conductor which is due to the large dielectric function ε 1. Meanwhile, the surface wave is almost transverse in the vacuum, since Ex Ez . ௣ 2 0 Figure 4.3. Plasmon dispersion of SPP for planar surface. CHAPTER 5 SURFACE PLASMON AND ENERGY TRANSFER IN HYPERBOLIC GEOMETRIES The two central issues of the contemporary plasmonics are manipulation and focusing of light on subwavelength scales, and plasmon-mediated energy transfer, see recent reviews[43– 46]. With regard to the first issue, strong confinement of optical fields at small scales is accompanied by their orders-of-magnitude enhancement. This enhancement can be used, e.g. to boost nonlinear effects or to image and detect[47] small objects. At the core of the second issue is the light-matter interaction. The effect of plasmon-supporting interface on a group of emitters located nearby[48–55] is twofold. Firstly, it strongly modifies the radiative lifetimes of individual emitters. Secondly, virtual plasmon exchange facilitates the dipole-dipole interaction between emitters. A metallic strip or a wire of finite thickness can serve as a plasmonic waveguide. The key approach to a plasmon field focusing utilizes tapering, i.e. gradual narrowing of a waveguide towards one end[56–67]. Field confinement at the end of a tapered metal wire waveguide was demonstrated experimentally[46, 68–70]. Current advances in plasmonics are related to engineering of progressively more complex plasmonic structures[71–76]. Then it is natural to extend the theoretical study of plasmonic waveguides to these complex, in particular, multiconnected, geometries, which contain several disconnected metal-air boundaries. It can be expected that these geometries provide additional control over plasmon fields. Studies of plasmons in multiconnected geometries should encompass calculation of the spectra of plasmonic modes as well as investigation of interaction of dipole emitters mediated by such modes. 70 5.1 Introduction Below we consider the simplest example of a multiconnected geometry illustrated in Figure 5.1. Geometries of a metallic “neck,” (a) in Figure 5.1, and of two tapers separated by a narrow air “groove,” (b) in Figure 5.1, are dual to each other. Both geometries possess a characteristic length scale ∼ a, which is the size of the gap or the width of the constriction. This scale, being much smaller than all other scales in plasmonic structure, see (c) in Figure 5.1, will therefore determine the plasmon spectrum and the plasmon field distribution of the entire structure. We take advantage of the fact that the problem of calculation of the plasmon spectrum in geometry shown in Figure 5.1 can be solved exactly if the metallic surfaces are hyperbolic. A natural unit of momentum (wave number) is q ∼ a−1 ; the meaning of q in the absence of translational symmetry will be clarified below. One advantage of the analytical approach over numerical methods applied to concrete sets of parameters [77] is that an analytical solution allows to establish general properties of the plasmon spectrum. We compare the analytical results for two hyperbolas, Figure 5.1, to the geometry of a single hyperbola. For a single hyperbola, the plasmon spectrum, resembles the spectrum Figure 5.1. The plasmon spectrum of these structures can be found analytically if metallic surfaces are confocal hyperbolas. Simplest examples of multiconnected plasmonic structures: a) “neck” and b) “gap.” 2a is the distance between the foci. c) Schematic view of a plasmonic array[1]. The field distribution in the array is determined by the points of contact of metallic islands. 71 of a finite-width metallic strip. It consists of two branches: the low-frequency branch, √ ω < ωp / 2, where ωp is the bulk plasmon frequency, corresponds to a symmetric mode, √ while the high-frequency branch, ω > ωp / 2, corresponds to antisymmetric mode. Adding the second hyperbola, Figure 5.1, leads to the emergence of two additional branches in the plasmon spectrum. The frequency of the additional symmetric mode is lower than that for a single hyperbola, while the frequency of the additional antisymmetric mode is higher a single hyperbola’s mode. Since the plasmons mediate the energy exchange between the emitters close to the surface, the complex plasmon structure in a multiconnected geometry leads to a nontrivial frequency dependence of this exchange, which we study analytically. 5.2 Plasmon spectrum of a single hyperbola 5.2.1 Elliptic coordinates We start with a single-hyperbola geometry, Figure 5.2. The boundary of a metal is defined by the equation x 2 y 2 − = a2 . cos η0 sin η0 (5.1) The metal is described with a dielectric permittivity, ε(ω) = 1 − ωp2 , ω2 (5.2) Figure 5.2. Geometry of a single hyperbola. In elliptic coordinates ξ, η, defined by Eq. (5.3), the metal with dielectric constant, ε(ω), occupies the domain (−ηp , ηp ). Outside the hyperbola there is air with ε = 1. 72 and occupies the inner part of the hyperbola, which approaches the asymptotes y = ± x tan ηp at large x. Outside the hyperbola there is air with ε = 1. Surface plasmons correspond to the solution of the Laplace equation, ∇ (ε∇Φ) = 0, for the electrostatic potential, Φ(x, y), which propagates along the boundary of the hyperbola and decays away from the boundary. The variables in the 2D Laplace equation can be separated in elliptic coordinates x = a cosh ξ cos η, y = a sinh ξ sin η, (5.3) where ξ is positive and η changes in the interval −π < η < π. The Lamé coefficients are p the same for both coordinates, hξ = hη = cosh2 ξ − cos2 η. In the new coordinates the Laplace equation, 1 ∇ Φ= 2 hξ 2 ∂2Φ ∂2Φ + ∂ξ 2 ∂η 2 = 0, (5.4) has the structure similar to its usual Cartesian form. The solutions of Eq. (5.4) should satisfy the boundary conditions at η = ηp (upper half of the hyperbola) and at η = −ηp (lower half of the hyperbola): the potential Φ and the normal component of the displacement field, Dn = εh−1 ξ ∂Φ/∂η, should be continuous at this boundary. 5.2.2 Plasmon dispersion The geometry in Figure 5.2 is symmetric with respect to the change of sign of the y-axis. Correspondingly, the solutions of Eq. (5.4) can be classified into symmetric (even), Φs (x, y) = Φs (x, −y), and antisymmetric (odd), Φa (x, y) = −Φa (x, −y). In hyperbolic coordinates the reversal of the sign of y amounts to the change η → −η, so that Φs (ξ, η) = Φs (ξ, −η) and Φa (ξ, η) = −Φa (ξ, −η). Consider first the symmetric modes. For these modes, a general solution of Eq. (5.4) inside the hyperbola, |η| < η0 , which propagates along ξ and grows with η from η = 0 towards the boundary, has the form h i Φs (ξ, η) = A exp(imξ) + α exp(−imξ) cosh(mη). (5.5) In order to satisfy the continuity of the potential at η = ±η0 , the corresponding solution outside the hyperbola, |η| > η0 , must have the same ξ-dependence. It should also decay away from the boundary. This specifies the form of Φs (ξ, η) in the air: h i Φs (ξ, η) = B exp(imξ) + α exp(−imξ) cosh m(π − |η|). (5.6) 73 By matching Φs (ξ, η) and ε∂Φs /∂η at the boundary η = ±η0 , we obtain the dispersion equation for the symmetric modes ε(ω) tanh (mη0 ) coth [m(π − η0 )] = −1. (5.7) At first glance it appears that the constant α in Eqs. (5.5)-(5.6) can be arbitrary. However, from the requirement that the function Φs (ξ, η) is finite and continuous it follows that α = 1. Indeed, for any α 6= 1 the ξ-component of electric field, Eξ = −h−1 ξ ∂Φs /∂ξ, diverges when ξ → 0 and η → 0, since the Lamé coefficient hξ vanishes there. We thus conclude that Φs (ξ) ∝ cos (mξ). Consideration for antisymmetric modes proceeds along the same lines. It is not difficult to see that the solution cos (mξ) sinh(mη) must be discarded as discontinuous across the region of the x-axis, where −a < x < a and ξ = 0: the odd character of sinh (mη) with respect to η → −η implies that it can only by multiplied by a function of ξ that vanishes at ξ = 0. This suggests the following form of the potential, Φa (ξ, η), for antisymmetric modes inside the hyperbola, Φa (ξ, η) = C sin(mξ) sinh(mη). (5.8) Correspondingly, the potential outside the hyperbola, which is odd with respect to y and matches the ξ-dependence of Eq. (5.8) has the form Φa (ξ, η) = D sin(mξ) sinh m(π − |η|). (5.9) The resulting dispersion equation for antisymmetric modes is ε(ω) coth (mη0 ) tanh [m(π − η0 )] = −1. From Eqs. (5.10) (5.7) and (5.10) we can express the plasmon frequencies in terms of the dimensionless wavenumber, m, ( ωs (m) = ωp ( ωa (m) = ωp sinh(mη0 ) cosh[m(π − η0 )] sinh(mπ) cosh(mη0 ) sinh[m(π − η0 )] sinh(mπ) )1/2 , (5.11) . (5.12) )1/2 Examples of dispersions Eqs. (5.11), (5.12) are shown in Figure 5.3. Below we list some general properties of these dispersions: 74 (i) A hyperbola reduces to a plane for η0 = π/2. Then Eqs. (5.11), (5.12) reproduce √ the expected result, ωs (m) = ωa (m) = ωp / 2, for the frequency of a dispersionless surface plasmon. (ii) The spectra Eqs. (5.11)-(5.12) of the symmetric and antisymmetric modes satisfy the “sum rule” relation, ωs2 (m) + ωa2 (m) = ωp2 . (5.13) (iii) In the long-wavelength limit, m → 0, the threshold frequencies are ωs (0) = ωp η 1/2 0 π , η0 1/2 ωa (0) = ωp 1 − . π (5.14) The fact that ωs (0) goes to zero for small η0 could be expected, since at small η0 the hyperbola is effectively a metallic layer with zero thickness, for which the dispersion of the longitudinal surface plasmon does not have a threshold[78]. On the other hand, ωa approaching ωp for η0 → 0 can be interpreted by noticing that the oscillations of the electron density, accompanying this plasmon, are normal to the surface; such oscillations must have the bulk-plasmon frequency ωp . (iiii) Another notable property of the dispersions Eqs. (5.11)-(5.12) is their duality, namely, ωs (π − η0 , m) = ωa (η0 , m). (5.15) Figure 5.3. Plasmon spectrum of a single hyperbola for two opening angles 2η0 = π/10 (red) and 2η0 = π/3 (blue) is plotted from Eq. (5.11) for symmetric mode and Eq. (5.12) for antisymmetric mode. At √ a large wavenumber, m, both branches approach the flat surface plasmon frequency, ωp / 2. 75 Qualitative interpretation of this relation can be given for small η0 , when Eq. (5.15) relates the spectrum in of a sharp metal “edge” with the spectrum of a narrow “funnel”: it suggests that the frequency of a soft symmetric plasmon of the “edge” coincides with the frequency of an antisymmetric plasmon in a “funnel.” Indeed, once the plasmon frequency is low, the absolute value of ε(ω) is big. This, in turn, implies that the normal component of the oscillating electric field in the metal is small. In the funnel geometry, this small field amplitude can be realized only when opposite charges accumulate on the two surfaces of the funnel. The reason is that, for small-angle funnel, such antisymmetric arrangement is similar to the charges of a parallel-plate capacitor; this distribution of charges ensures that the field of the parallel-plate capacitor does not extend outwards. On the other hand, as we mentioned above, the low-frequency plasmon in a sharp-edge geometry is longitudinal, which corresponds to symmetric amplitudes of the charge-density fluctuations for the two surfaces. 5.2.3 Field distribution In addition to the spectrum, ω(m), it is instructive to look at the spatial distribution of electric field of the two plasmon modes. For a given frequency, ω, the value of m is found by √ √ equating it to either ωs (m), when ω < ωp / 2, or to ωa (m), when ω > ωp / 2. The found value of m(ω) is then substituted into the potential distribution Φs = cos(mξ) cosh(mη) or Φa = sin(mξ) sinh(mη), from which the electric field is subsequently calculated. To clarify the physical meaning of the parameter m, we rewrite the potential Φs at large distances from the coordinate origin in the Cartesian coordinates, " !# p h 2 x2 + y 2 yi Φs (x, y)|x,y a = C cos m ln cosh m arctan . a x (5.16) The ratio of the amplitudes of Φs at the boundary, y = x tan η0 , and along the x-axis is equal to cosh (mη0 ). The potential distribution Eq. (5.16) applies inside the metal. In the air, this distri bution differs from Eq. (5.16) by the replacement arctan xy by π − arctan xy and C by C cosh(mη0 )/ cosh [m(π − η0 )]. Consider now a symmetric plasmon propagating with a wavevector, q, in a metallic film of a constant thickness, 2d. For this plasmon the ratio of the potentials at the boundary and at the center is equal to cosh (qd). Identifying d with the length of the arc between the 76 x-axis and the boundary, ρη0 , where ρ = p x2 + y 2 , allows one to specify the local value of the wavevector q(ρ) = m . ρ (5.17) With the ρ-dependent wavevector given by Eq. (5.17), one would expect for the plasmon Rρ phase a “semiclassica” value a dρ0 q(ρ0 ) = m ln(ρ/a), which is indeed the case, as follows from Eq. (5.16). For a symmetric mode, the electric field on the x-axis is directed along x. The domain x > a on the x-axis corresponds to η = 0, so that Φs (ξ) = C cos (mξ). In this domain the behavior of electric field with x = a cosh ξ has the form q i h 2 sin m ln xa + xa2 − 1 √ . Ex (x > a) = Cm x2 − a2 (5.18) The field is finite at x = a, and falls off as 1/x for x a. In the domain a cos η0 < x < a we have ξ = 0 and Φs = C cosh (mη). Differentiating with respect to x = a cos η, we obtain sinh m arccos xa √ Ex (a cos η0 < x < a) = Cm . (5.19) a2 − x2 Finally, in the air, in the domain 0 < x < a cos η0 the x-component of the field is given by ! sinh m(π − arccos xa ) cosh(mη0 ) √ Ex (0 < x < a cos η0 ) = −mC . cosh [m(π − η0 )] a2 − x2 In Eqs. (5.18), (5.19) the frequency and the η0 -dependence of electric field is incorporated in m (ω, η0 ), defined by the condition ωs (m) = ω, where ωs (m) is given by Eq. (5.11). For an antisymmetric mode, the electric field on the x-axis is directed along y. Inside the metal, where Φa = C sin(mξ) sinh(mη), the x-dependence of Ey is the same as the xdependence, Eqs. (5.19), (5.20), of Ex for the symmetric mode. In the air, the x-dependence of Ey is different from Eq. (5.20) and has the form sinh(mη0 ) Ey (0 < x < a cos η0 ) = −mC sinh [m(π − η0 )] ! sinh m(π − arccos xa ) √ . a2 − x2 (5.20) The behavior of Ex (x) and Ey (x) is illustrated in (a) of Figure 5.4. It is seen that the oscillating tail emerges only at large distance ∼ 20a from the tip. It is possible to understand the difference in the distribution of the field intensities for symmetric and antisymmetric modes from simple qualitative arguments. As seen from Figure 5.5, at small momenta, the field of the symmetric mode is predominantly concentrated 77 Figure 5.4. Distribution of Ex (for symmetric mode) and Ey (for antisymmetric mode). (a) The components of the electric field along the x-axis is plotted from Eqs. (5.18)-(5.20) for the wavenumber m = 2 and the opening angle 2η0 = π/3. The inset shows the large-x oscillating tail of Ex . (b) The same as (a) for two hyperbolas. outside the metal[79, 80]. In contrast, the field of the antisymmetric mode is predominantly concentrated inside the metal. This difference can be traced to the continuity of the normal (out) component of the displacement vector at the boundary: Eη (in) = ε(ω)Eη . For small m, the frequency of the symmetric mode is low, so that |ε(ω)| 1. Therefore, the field inside the material is much weaker that the field outside. Conversely, the frequency of the antisymmetric mode is close to ωp . As a result, ε(ω) is small for that mode, effectively reversing the relation of the fields inside and outside. 5.3 Geometry of two hyperbolas 5.3.1 Splitting of the plasmon spectrum Consider now the geometry of two hyperbolas, Figure 5.6. The boundary of the first hyperbola is defined by the same Eq. (5.1); in addition, metal occupies the domain π − η1 < η < π. The ξ-dependence of the potential for the symmetric modes remains the same, 78 Figure 5.5. Density plot of the field intensity of the plasmon modes for a single hyperbola with the opening angle 2η0 = π/3. The upper row corresponds to a symmetric mode, for which the potential distribution is determined by Eqs. (5.5) and (5.6). The lower row corresponds to an antisymmetric mode with the potential described by Eqs. (5.8) and (5.9). Left, central, and right panels correspond to the wavenumbers, m = 0.1, m =√0.4, and m = 0.7, respectively. With increasing m the plasmon frequencies approach ωp / 2, while the field concentrates near the metal-air surface. cos mξ. The η-dependence, which decays away from both boundaries, is characterized by four constants,   A1 cosh(mη), 0 < η < η0 , B1 cosh(mη) + B2 cosh m(π − η), η0 < η < π − η1 , Φs (η) =  A2 cosh m(π − η), π − η1 < η < π, (5.21) These constants, A1 , A2 , B1 and B2 are related by continuity of Φs (η) and ε∂Φs /∂η at η = η0 and η = π − η1 . These continuity conditions read, A1 cosh(mη0 ) = B1 cosh(mη0 ) + B2 cosh m(π − η0 ), ε(ω)A1 sinh(mη0 ) = B1 sinh(mη0 ) − B2 sinh m(π − η0 ), (5.22) A2 cosh(mη1 ) = B1 cosh m(π − η1 ) + B2 cosh(mη1 ), −ε(ω)A2 sinh(mη1 ) = B1 sinh m(π − η1 ) − B2 sinh(mη1 ). (5.23) These relations, together with the explicit form of ε(ω), given in Eq. (5.2), lead to the following characteristic equation: 79 Figure 5.6. The geometry of two hyperbolas with the opening angles 2η0 and 2η1 . A dipole emitter close to the tip and polarized along the interface excites the dipoles located to the left and to the right from the tip at distances much bigger than the focal distance. sinh 2mη0 sinh 2mη1 4 " # 1 1 = − sinh mπ + sinh m(π − 2η0 ) ωp2 2 2 " # ω2 1 1 × − sinh mπ + sinh m(π − 2η1 ) . ωp2 2 2 ω2 (5.24) The two brackets in the right-hand side describe the plasmon dispersion Eq. (5.11) for a symmetric mode, while the left-hand side describes the coupling of the two plasmon branches. The two plasmons decouple when η1 is small. Then the dispersion of the upper symmetric branch is simply ωs (η0 , m), Eq. (5.11). In order to find the dispersion of the lower symmetric branch, it is sufficient to set ω = 0 in the first bracket. This yields ωs− (m) η1 1 h i1/2 = ωp mη1 tanh m(π − η0 ) . (5.25) The general expression for the dispersion of the two coupled symmetric branches reads ( h i ω p ωs± (m) = sinh(mη ) cosh m(π − η ) + sinh(mη ) cosh m(π − η ) 0 0 1 1 1/2 2 sinh(mπ) ) h i1/2 1/2 2 2 ± sinh m (η0 − η1 ) cosh m (π − η0 − η1 ) + sinh (2mη0 ) sinh (2mη1 ) . (5.26) It is easy to see from Eq. (5.26) that for large m 1 both frequencies approach the √ surface plasmon frequency ωp / 2, as in the case of a single hyperbola. The reason is that the short-wavelength plasmon is “local,” a metal surface is locally flat, and the presence 80 of the second surface is of no consequence to the spectrum in this limit. The behavior of ωs± (m) in the limit of long wavelengths is remarkable " #1/2 η η (π − η − η ) 0 1 0 1 ≈ mωp , ωs− (m) η0 + η1 m→0 η + η 1/2 0 1 ωs+ (0) = ωp . π (5.27) The fact that ωs+ (0) is determined by the “net” angle (η0 + η1 ) is consistent with the result Eq. (5.14) for a single hyperbola. As this net angle approaches π, the portion of air in this limit becomes small, and ωs+ (0), approaches the bulk plasmon frequency. The acoustic behavior of ωs− (m) is related to the fact that, unlike for a single hyperbola, in the geometry of two hyperbolas a low-frequency plasmon is not reflected from the tip, but goes “through” the gap into the second hyperbola. For two identical hyperbolas, η0 = η1 , Eq. (5.26) simplifies to ) ( 1/2 sinh(mη0 ) cosh [m(π − η0 )] ± cosh(mη0 ) ± . ωs (m) = ωp sinh(mπ) (5.28) Similar derivation for the antisymmetric plasmons yields ( ωa± (m) = ωp ) 1/2 cosh(mη0 ) sinh [m(π − η0 )] ± sinh(mη0 ) . sinh(mπ) (5.29) From Eqs. (5.21), (5.29) one can trace the evolution of the plasmon spectrum with increasing η0 . For η0 1 the frequencies of both symmetric plasmons are low: # " mπ 1/2 − , ωs (m) ≈ ωp mη0 tanh 2 η0 1 " ωs+ (m) η0 1 ≈ ωp mη0 tanh mπ 2 (5.30) #1/2 , (5.31) while the frequencies of antisymmetric plasmons are close to ωp . As η0 increases and achieves √ the value π/4, the branches ωs+ (m) and ωa− (m) collapse into a single frequency ωp / 2 and become flat. As η0 increases further above π/4, the branches invert: ωs+ (m) is pushed above √ ωp / 2 and ωa− (m) drops below it. This evolution is illustrated in Figure 5.7. For the geometry of two hyperbolas there is a duality relation, π − η0 , m = ωa− (η0 , m) , ωs+ 2 similar to Eq. (5.15) for a single hyperbola. (5.32) 81 Figure 5.7. Density plot of the field intensity of the plasmon modes in the geometry of two symmetric hyperbolas with the same opening angle, 2η0 = π/3, as in Figure 5.6. The upper row corresponds to a symmetric mode for which the potential distribution is determined by Eqs. (5.21) and (5.26). Left, central, and right panels correspond, respectively, to the wavenumbers, m = 0.1, m = 0.4, and m = 0.7, the same as in Figure 5.6. While the fields of individual hyperbolas are disconnected along the x-axis, they overlap along the y-axis. The lower row corresponds to antisymmetric modes with the same m-values. The field of individual hyperbolas overlap, predominantly, along the y-axis. 5.3.2 Comparison of the field distributions for two geometries Hybridization of plasmon fields of individual hyperbolas in the geometry of two hyperbolas is illustrated in Figure 5.4, Figure 5.5, and Figure 5.8. The curves in Figure 5.4 suggest that hybridization along the x-axis is rather weak and becomes progressively weaker √ while the wavenumber increases as the frequency approaches ωp / 2. This behavior is natural, since, the closer the frequency is to that of the flat surface plasmon, the more localized is the plasmon field near the metal-air interface. The density plot of the modes of individual hyperbolas is shown in Figure 5.5. Compared with the latter, the doublehyperbola plot of Figure 5.8 demonstrates that hybridization of individual modes has a “ring”-like character for symmetric modes and the “needle”-like character for antisymmetric modes. Such different nature of hybridization can be interpreted with the help of the patterns of oscillating surface charges in Figure 5.7. For the bottom symmetric mode with 82 Figure 5.8. Comparison of the plasmon spectra in the geometry of two hyperbolas for two values of the opening angle. For η0 = π/6 < π/4 (upper panel) the frequencies √ of both symmetric modes (blue) are smaller than the flat surface plasmon frequency ω / 2, while p √ the frequencies of both antisymmetric modes (red) are bigger than ωp / 2. The relative signs of the oscillating charge density along the metal surfaces are schematically illustrated to the left of the graph. For η0 = 2π/5 > π/4 (lower panel) the positions of the upper symmetric mode and lower antisymmetric mode with respect to ωp invert. low frequency, the positive and negative charges are separated by air, whereas the dielectric function of the metal is large. This expels the electric field lines from the inside of the metal and forces them to go through the air. The field is strong along the y-axis, where it is parallel to the x-axis. For the antisymmetric mode, the upper and lower sides of the metal have opposite charges, while the frequency is close to ωp , so that the dielectric constant of the metal is small. The force lines of electric field are localized inside the metal. Thus, the two metal edges can be viewed as the plates of a parallel-plate capacitor. Correspondingly, the concentration of electric field near the x axis is analogous to the fringe field outside a parallel-plate capacitor. In Sect. II we have established that the field of the plasmon modes at large distance, ρ a, from the origin behaves as 1/ρ. Hence, the field intensity behaves as 1/ρ2 , i.e. it strongly diverges down to the distances ∼ a. This, however, does not translate into a strong R enhancement of the net energy, dr E2 (r), which grows only logarithmically, ∝ ln(λ/a); the upper cut-off is provided by the wavelength of light, λ = 2πc/ω, with the same frequency ω. 83 5.3.3 Two co-directed hyperbolas For completeness, in this subsection we will analyze the plasmon spectrum in the geometry of two co-directed hyperbolas. Assume that the metal occupies the region η0 < η < η1 , while the regions 0 < η < η0 and η1 < η < π are occupied by air, see the inset in Figure 5.9. A straightforward generalization of Eq. (5.24) gives " # ω2 1 1 sinh 2mη0 sinh 2m (π − η1 ) = − sinh mπ − sinh m(π − 2η0 ) 4 ωp2 2 2 " # ω2 1 1 × − sinh mπ + sinh m(π − 2η1 ) . (5.33) ωp2 2 2 To analyze the plasmon dispersion, we introduce the average opening angle and the ”thickness” of the tip, ηc = η0 + η1 , 2 δη = η1 − η0 . (5.34) An example of the plasmon spectrum with the two co-directed hyperbolas is shown in Figure 5.9. In this inverted geometry, symmetric and antisymmetric modes interchange, compared with (b) in Figure 5.7. The symmetric branches in the long-wavelength limit are ωs+ (0) = ωp δη 1/2 π , ωs− (0) = ωp . (5.35) For the upper symmetric mode, ωs+ (0), the signs of the oscillating charges on the opposite sides of each “sleeve” are opposite and the electric field lines are confined inside the metal. Figure 5.9. Illustration of the plasmon spectrum in the geometry of two co-directed hyperbolas. Blue and red curves correspond to symmetric and antisymmetric plasmon modes, respectively. The spectrum is plotted from Eq. (5.33) for ηc = δη = π/6. 84 For this, the dielectric constant of the metal must vanish. For the lower symmetric mode, ωs− (0), the signs of charges on the opposite sides of each sleeve are the same. The field lines mostly stay in the air and do not penetrate into the metal. This, on the other hand, implies that ε(ω) is large, and, correspondingly, the frequency of the mode is small, vanishing in the limit of a very thin “coating,” δη → 0. The acoustic mode, generic for multiconnected geometries, is now found in the antisymmetric part of the spectrum, ωa+ (m) → 0 as m → 0. The second mode, δη 1/2 . ωa− (0) = ωp 1 − π (5.36) √ lies above ωp / 2 and its frequency increases with decreasing δη. 5.4 Interaction of two emitters at the metal-air surface Numerous studies have demonstrated that, due to proximity to the metal, the lifetime of the emitter can be dominated by the excitation of the plasmon modes [48–55, ?–55]. For a metallic nanoparticle plasmons significantly shorten the lifetime when the emitter frequency is close to the frequency of the plasmon dipole (l = 1) oscillations. A less trivial finding[55, ?] is that the coupling to plasmons can dominate the lifetime when the emitter is located in the proximity to a nanowire with a subwavelength radius in which the plasmon spectrum is continuous. When the emitter is positioned close to the tip, the plasmon-induced shortening of the lifetime is even more pronounced[?]. This is caused by the field enhancement near the tip. In this regard, to calculate the emitter lifetime in a multiconnected plasmonic structure, we can use the same general scheme as developed in the previous studies. Consider for concreteness, a dipole emitter with frequency ω, located at the metal-air interface, and polarized parallel to the interface, as in Figure 5.6. The structure of the plasmon spectrum in this calculation is captured by Green’s function for electric field, G(ξ, η; ξ 0 , η 0 ; ω) = X Eξ (ξ, η; m)Eξ (ξ 0 , η 0 ; m) m ω − ω(m) , (5.37) where Eξ (ξ, η; m) is the normalized tangential component of the electric field of the plasmonic mode and the summation is taken over all four modes; ω(m) is the plasmon spectrum found above. The decay rate is proportional to the imaginary part of the diagonal value G(ξ, ηi ; ξ, ηi ; ω) taken at the position of emitter. As the summation in Eq. (5.37) is performed over acoustic as well as optical modes, a spike-like feature occurs in the ω-dependence 85 of the decay rate. The origin of this feature is the divergence of the density of the plasmon modes, 1 g (ω) = π + Z ω √p 2 dm δ ω − ω + (m) , (5.38) 0 near the threshold frequency ωs+ (0). Consider for simplicity a symmetric geometry, η0 = η1 < π/4. Expanding Eq. (5.21) at small m, we get ωs+ (m) ωs+ (0) m→0 =1+ π m2 π − η0 − η0 , 3 4 2 (5.39) where ωs+ (0) = (2η0 /π)1/2 . Substituting Eq. (5.39) into Eq. (5.38), one finds " #1/2 1 3 + g (ω) = . π π4 − η0 π2 − η0 ωs+ (0) ω − ωs+ (0) (5.40) The inverse square-root singularity in the density of states translates into a minimum in the emitter lifetime. Similar minimum is expected near ω = ωa+ (0) and near the bulk plasmon √ frequency, ωp . Near ω = ωp / 2 the minimum should be even stronger, since the divergence √ of the plasmon modes diverges as |ω − ωp / 2|−1 . The scheme of calculation of the plasmon-mediated energy transfer between the emitters near the metal-air interface is also well established, see e.g. Ref. [52] for two emitters near a nanoparticle and Ref. [81] for donor and acceptor above the graphene. Here we emphasize the specifics of this transfer in the geometry of two hyperbolas. In particular, we study how the transfer rate between the red emitter near the tip in Figure 5.6 and the blue emitter located on the left hyperbola differs from the transfer rate between the red emitter and the blue emitter located on the right hyperbola, when both blue emitters have the same coordinate, ξ. Quantitatively, the relation of the two transfer rates is described by the ratio of the tangential components of electric field at the two interfaces. From Eq. (5.21) we find !2 " #2 Eξ (η0 , ω) B1 cosh(mη0 ) + B2 cosh m(π − η0 ) T = = . (5.41) Eξ (η1 , ω) B1 cosh m(π − η1 ) + B2 cosh(mη1 ) With the help of the continuity conditions Eqs. (5.22), (5.23) this ratio can be cast in the form " #−2 T = ε(ω) tanh(mη1 ) sinh m(η0 − η1 ) + cosh m(η1 − η0 ) , (5.42) where m(ω) is determined by the dispersion equation (5.26). In Figure 5.10 we plot 86 the factor T as function of frequency for a fixed opening angle of the right hyperbola, 2η0 = 4π/5, and several opening angles of the left hyperbola. Overall, we see that at ω = 0 the geometric factor T rapidly increases with η1 . This can be understood from the electrostatics of ideal metals. Indeed, at ω = 0 the field does not penetrate into the metal at all. Consequently, the sharper the left hyperbola is, the stronger the field near the left tip becomes, resulting in smaller values of T (0). This, in turn, means that the energy transfer from the red emitter in Figure 5.6 to the left blue emitter happens faster than the transfer to the right blue emitter. We also see that the behavior of T (ω) is different for small, (a) in Figure 5.10, and large, (b) in Figure 5.10, values of η1 . When geometrical factor T (0) is small, T (ω) falls off with frequency, suggesting that the left and right hyperbolas become effectively “disconnected”. To the contrary, for larger starting values T (0), the finite-frequency T (ω) Figure 5.10. Geometric factor for the energy transfer is plotted as a function of frequency from Eq. (5.42) for the values of the opening angle 2η1 = 0.2π, 0.24π, 0.28π (a), and 2η1 = 0.56π, 0.6π, 0.64π (b). The value of the opening angle, 2η0 , is chosen 2η0 = 4π/5. 87 grows with frequency, suggesting that the energy transfer becomes more symmetric. The crossover from decay to growth takes place at η1 ≈ 0.19π. This value can be related to the peculiar behavior of the velocity of the low-frequency plasmon Eq. (5.27). Namely, this velocity has a minimum as function of η1 . By setting ∂ωs− /∂η1 = 0, we find the position of this minimum at, η̃1 = (πη0 )1/2 − η0 . (5.43) For η0 = 2π/5, Eq. (5.43) yields η̃1 ≈ 0.23π, close to the crossover value of η1 , which corresponds to ∂T (ω)/∂ω|ω→0 = 0. It is instructive to discuss Eq. (42) from the general perspective of a resonant energy transfer. Conventionally, the enhancement of the transfer rate between two emitters due to neighboring plane, nanowire, or nanoparticle is studied as a function of the distance between the emitters and the distance from the emitters to the plasmon-guiding interface. For the geometry of two hyperbolas with emitters at the interface, the dependence of the fields on the distance is measured by the dimensionless coordinate ξ. Moreover, this dependence has a purely oscillatory form, ∝ cos (mξ). Thus, the effectiveness of the resonant energy transfer depends exclusively on the opening angles and the frequency (in the units of ωp ). The most nontrivial outcome of Eqs. (5.41), (5.42) and Figure 5.10 is that, at frequency √ close to the surface plasmon frequency ωp / 2, the ratio of the transfer rates to “left” and to the “right” switches abruptly from 0 to 1 depending on the relation between the opening angles. 5.5 Summary (i) The prime qualitative finding of the present manuscript is that, for a singly-connected geometry, the opening angle of the hyperbola, 2η0 , defines the minimum frequency, given by Eq. (5.14), below which the plasmon bound to the tip of the hyperbola does not exist. By contrast, in a multiconnected geometry a symmetric plasmon mode, concentrated near the region of the closest contact of the surfaces, exist at arbitrary low frequency. (ii) A geometry somewhat similar to the one considered in this paper was studied in Refs. [82–86]. The authors considered two or more plasmonic wires or radius a in parallel and separated by distance d. They traced numerically how plasmons of individual wires hybridize with decreasing d. In case both a and d are much less than the wavelength, the 88 expected distribution of the field is a universal function of the ratio d/a. However, for the parameters of interest the retardation effects were important. In the exact solution found in this paper it is not possible to incorporate the retardation effects. Understandably, this limitation restricts the applicability of our analytical results to the domain of small enough wavelength or large momenta m (see below). (iii) Originally, the enhancement of the electric field of a plasmon-polariton as it approaches a wedge, has been demonstrated analytically in Ref. [56]. Calculation in Ref. [56], was carried out in polar coordinates, i.e. neglecting the rounding of the wedge near the tip. This curving has been emulated by introducing a cutoff length ∼ a. At distances much greater than a the form of the plasmon field in Ref. [56] is consistent with Eq. (5.16) with one important difference that the plasmon field Eq. (5.16) is not traveling, as in Ref. [56], but is a standing wave instead. This structure is enforced by the boundary condition at the tip. The fact that the plasmon mode near the tip is a standing wave can be viewed from the perspective of focusing of light on subwavelength scale, when the plasmon-polariton with TM polarization (magnetic field along the z-axis) is excited at large (compared to the wavelength, λ) distance, ρ, from the tip[59]. In the geometry of a single hyperbola this polariton will be fully reflected. For small enough opening angles, 2η0 , the transformation of the polariton into small-ρ standing plasmon mode can be traced analytically within the semiclassical (WKB) description[55, 59–61, 63]. Within that description, the metal strip at distances ρ a from the origin is replaced with a planar film with a thickness dρ , equal to the arc distance, 2ρη0 = dρ , between the two metal surfaces. Then the semiclassical R expression for the incident and reflected fields has the form qρ−1/2 exp ±i dρ qρ , where qρ (ω) is the dispersion of the symmetric waveguide mode propagating along the film. This dispersion law satisfies the equation ( ε tanh ω2 qρ2 − ε 2 c !1/2 dρ 2 2 ) =− qρ2 − ε ωc2 qρ2 − ω2 c2 !1/2 . (5.44) 1/2 The relevant question is, how close to the threshold frequency ωp η0 /π is the semiclassical description applicable, i.e. when does the condition dqρ /dρ qρ2 apply? At small frequencies we can replace ε(ω) by −ωp2 /ω 2 and simplify Eq. (5.44) using the identification qρ = m/ρ, Eq. (5.17). Then Eq. (5.44) reduces to the following equation for m 89 " # ωp2 ρ2 1/2 ω 2 ρ2 1/2 ω2 tanh η0 m 1 + 2 2 = 2 1 + 2p 2 , c m ωp c m (5.45) and the semiclassical condition is satisfied provided that m 1. As the distance ρ increases, the solution of Eq. (5.45) grows starting from m = ω2 , ωp2 η0 which is consistent with Eq. (5.11). Thus, for semiclassical description to apply at all ρ bigger than a, it is necessary that this 1/2 minimum m exceeds 1, i.e. ω ωp η0 . The latter condition suggests that the frequency of the incident light, being much smaller than ωp , should still not be very close to the threshold frequency. Another approach to the “delivery” of the light energy to the tip was proposed in Ref. [62]. Namely, one can coat a conical tip of a glass fiber by a silver layer, the geometry similar to the one shown in the inset in Figure 5.9. The wavelength of a plasmon in a silver layer increases away from the tip, and at a certain distance matches the wavelength of the waveguided light propagating towards the tip. At this point the light transforms into the plasmon and heads towards the tip. Experimental realization of a coupling based on this idea was reported in a number of papers, see Ref. [77] and references therein. (iv) Naturally, the structure of the plasmon field at small distances from the tip is not captured in polar coordinates. Meanwhile, this structure is quite nontrivial. To illustrate this, consider the ratio Ex (0)/Ex,tip of the fields at the origin and at the tip for a symmetric plasmon. This ratio can be found from Eq. (5.19). The origin, x = 0, corresponds to η = π/2, while the tip corresponds to x = a cos η0 . Then Eq. (5.19) yields sin η0 sinh π2 m(ω) Ex (0) = . (5.46) Ex,tip sinh [(π − η0 )m(ω)] 1/2 As the frequency grows from the threshold value ωs (0) = ωp η0 /π to the surface plasmon √ frequency ωp / 2, the “wavenumber,” m, changes from m = 0 to m = ∞. Then the ratio √ Eq. (5.46) falls off from π sin η 0/[2 (π − η0 )] monotonically. At frequencies close to ωp / 2 the ratio Eq. (5.46) approaches zero, since the plasmon field is strongly localized near the metal surface. Such a strong change of the field is revealed by the exact solution in elliptic coordinates demonstrated in the present paper. (v) In this paper we assumed that the plasmon field does not depend on z. For a general z-dependence, Φ ∝ exp(iκz), the Laplace equation takes the form, ∂2Φ ∂2Φ + + κ2 a2 (cosh2 ξ − cos2 η)Φ = 0. ∂ξ 2 ∂η 2 (5.47) 90 Separation of variables, leads to the following equation for the potential’s η-dependence, ∂2Φ − (m2 + κ2 a2 cos2 η)Φ = 0. ∂η 2 (5.48) Our results apply for the wavenumbers that exceed some minimum value, m mmin . This minimum value can be estimated from the observation that κ can not be less than H −1 , where H is the “height” of the hyperbola, see Figure 5.1. This determines, mmin ∼ κa ∼ a/H 1. (vi) A finite scattering rate of conduction electrons, γ, sets the maximum distance from the origin, ρmax , where our results apply. The solutions obtained and discussed above decay beyond ρmax . This distance can be estimated by noting that a finite γ is taken into account by replacing ω 2 → ω(ω + i/γ) in the dielectric function, Eq. (5.2). As a result, it leads to a finite imaginary part of m: Im m ≈ γ (∂ω/∂m)−1 , where ∂ω/∂m is the slope of the plasmon dispersion. The plasmons attenuate at distances where ξ Im m ∼ 1. Since at large distances ξ depends logarithmically on ρ, see Eq. (5.16), we conclude that the maximum distance is ρmax ∼ a exp [1/Im m]. (vii) Finally, throughout the paper, we have disregarded the spatial dispersion of the dielectric permittivity, ε(ω, k) ≈ ε(ω). Such approximation is valid as long as the characteristic wave vectors k are small: k ω0 /v, where v is the electron Fermi velocity[87, 88]. Using the value of the plasma frequency for gold[89] ω0 = 1.38 × 1016 s−1 , and its Fermi velocity[90] v = 1.39 × 108 cm/s, we obtain the value of the wave vector for which spatial dispersion must be taken into account, k0 ∼ 108 cm−1 . On the other hand, the smallest wave vector in our wedge geometry is determined by the radius of curvature of the hyperbolas near their sharp points, k ∼ 2π/a. We, therefore, conclude that for a & 1 nm, neglecting spatial dispersion should be a good approximation. APPENDIX IMPORTANT CALCULATIONS In this appendix, several important calculations in Chapter 3 will be give. In particular, contour integrals will be mainly focused. A.1 Hydrodynamics in multichannel system In this section, we are deriving the renomalization of band electrons due to the Coulomb interaction V0 in the multichannel system. Density of charge and current are N -times larger than one channel system, p pl , 2π p 2π p r l j(x, t) = N v(ρr − ρl ) = N v − , 2π 2π ρ(x, t) = N (ρr + ρl ) = N r + (A.1) (A.2) where ρr and ρl are density of states of right and left-moving particle on gapless subbands. And density of states of right and left moving particles are related with momentum as 2πρr/l = pr/l . Similarly, kinetic energy of bare electrons is also N -times larger, Zpr hZ0 dp i dp = N dx − vp + vp , 2π 2π pl 0 Z h i π = dx v 2 ρ2 (x, t) + j 2 (x, t) . 2vN Z Ĥ0 (A.3) If V0 -interaction is considered, Hamiltonian can be formally restated as, Z V0 Ĥ = Ĥ0 + 2 Z π = dx 2vN Z π = dx 2vN dx ρ2 (x, t), h i N vV0 2 (v 2 + )ρ (x, t) + j 2 (x, t) , π h i u2 ρ2 (x, t) + j 2 (x, t) . Velocity of electron is renormalized to be u = v reproduced. (A.4) p 1 + N V0 /(πv). First line of Eq. (3.63) is 92 A.2 Contour integration with branch cut There are two integrals frequently appearing in the paper. In the calculation of correlation function, one always encounters following integral, Z 0 ∞ dq −iqut (e − 1). q (A.5) The integral is ill-defined on the positive half of imaginary t-axis, where exp(−iqut) = exp(qu|t|) grows exponentially fast (but it is well defined at t = 0). However, for real t, the integral is well-defined. For q 1/iut, the integrand vanishes as e−iqut − 1 ≈ −iqut, which indicates that the integral is only important for q > 1/iut. At the other end of limit (for large exponent iqut 1), exponential oscillates fast and barely contributes to the integral. Therefore, it is legitimate to replace the upper limit by an artificial cut-off 1/R (R will be sent to zero eventually), Z∞ dq −iqut e −1 ≈− q 0 1/R Z R dq ≈ ln . q iut + R (A.6) 1/iut The logarithm has the correct two limits of iut at small time and as ln(R/iut) at large times. There is a branch point at t = iR. It is legitimate to choose branch cut on the imaginary axis starting from t = iR to infinity. In the complex t-plane where t = |t|eiφ , the integral can be decomposed into two parts, for iut 1, ln R R R ≈ ln = ln − iφ. iut + R iut u|t| (A.7) where we choose φ = 0 starts from negative half of imaginary t-axis,see Figure A.1. The integral can also be taken in a more rigorous manner (first proposed by Haldane) by introducing an artificial factor e−q/R to ensures the convergence of the integral, where 1/R 1 is the momentum cut-off we used in the last intuitive calculation. Then we take the expansion and integrate term by term, Z∞ Z∞ ∞ X dq −q/R −iqut (−iut)n e e −1 = dq q n−1 e−q/R q n! n=1 0 0 ∞ R X (−iut)n = = ln n(1/R)n iut + R n=1 (A.8) 93 Figure A.1. Logarithmic function in the complex plane. Small constant R in the denominator renders a branch pole at t = iR/u. This is due to the multivalue feature of logarithmic function. Furthermore, similar characteristics applies to function with fractional exponent. Consider the Fourier transform of correlation function under the effect of V0 interaction, (0) Im χV N ∆γ (ω) = − 2πv Z∞ dt ei(ω−∆)t . (t − iR)1−γ (A.9) −∞ For non-integer γ, the integral has a branch point at t = iR. It is convenient to choose the branch cut on the positive half of the imaginary t-axis from iR to infinity, see Figure A.2. Deforming the contour to follow the sides of branch cut (one from zero to infinite along π/2 and the other from infinity to zero along −3π/2), we have, (0) Im χV (ω) = iN ∆ iπγ γe − e−iπγ 2πv Z∞ d|t| e−(ω−∆)|t| , |t|1−γ 0 N Θ(ω − ∆) = − [(−Ω)−γ − 1]. vΓ(1 − γ) (A.10) In the last step, we used identity that Γ(x)Γ(1 − x) = π/ sin(πx). Unit step function Θ(ω − ∆) indicates that absorption occurs for energy above the band gap. 94 Figure A.2. Contour integral of multivalue function is chosen to be closed in the upper half plane. For the lower plane, there is no singularities and thus no absorption for frequency ω < ∆. Let us take another example in the electrostatic case γ = 0. First order expansion of correlation function has form, (1) χ (ω) = V1 N 4π 2 v 2 Z∞ dt 0 = V1 N 4π 2 v 2 Zt Z∞ dt0 ei(ω−∆)t , (t − t0 − iR)(t0 − iR) 0 ei(ω−∆)t h dt t − iR t−1/∆ Z 0 = V1 N 4π 2 v 2 Z∞ 1 dt + t − t0 0 0 dt ei(ω−∆)t × Zt 1 dt 0 + iπ t 0 Zt i dt0 [δ(t − t0 ) + δ(t0 )] , 0 1/∆ 2 ln(∆t) + iπ . t − iR (A.11) 0 Small constant 1/∆ is added to cut-off wherever singularity occurs. Extending above integral to negative t-axis by noting that, Z0 dt e i(ω−∆)t ln(∆t) + iπ =− t + iR −∞ Z∞ dt ei(ω−∆)t ln(∆t) . t − iR (A.12) 0 The integral on the entire t-axis is equal to the imaginary part of χ(1) (ω). Take the branch cut the same as previous one along the positive half of imaginary t-axis, the integration on both sides of the branch cut yields, Im χ (1) V1 N (ω) = Θ(ω − ∆) 2 2 4π v Z∞ dt e−(ω−∆)|t| ln(∆eiπ/2 ) − ln(∆e−3iπ/2 ), (A.13) |t| 0 V1 N = 2πΘ(ω − ∆) 2 2 ln(−Ω−1 ). 4π v (A.14) 95 Combining with its real part by using identity ln x = ln(−x) + iπ, first order correction of RPA is reproduced. χ (1) V1 N (ω) = Θ(ω − ∆) 2 2 4π v Z∞ dt 2πei(ω−∆)t , |t| 0 = V1 N ln2 Ω−1 . 4π 2 v 2 (A.15) Both integrals above are carried out through contour integration in the complex plane. Because integrands contain multivalue functions, the plane is being cut on the the positive t-axis where the branch point iR resides. The corresponding physical meaning of branch pole is related to optical absorption process. A.3 Core-hole interaction in metallic nanotubes As we pointed out before, there is one scattering process has same physics as MND problem, see Figure A.3. In this section, we will derive the result of MND by using bosonization. Consider the following Hamiltonian, Figure A.3. “X-ray” absorption in metallic nanotubes: conduction band consists of right/left-moving states and the lower massive subband with |m| = 1 chosen to be core-hole states. 96 Z∞ Ĥ = −iv −∞ Z∞ +U Z∞ h ∂ i † ∂ † ˆ t) dx ψ̂r ψ̂r (x, t) − ψ̂l ψ̂l (x, t) − ∆ dx dˆ† (x, t)d(x, ∂x ∂x −∞ h i ˆ t), dx ψ̂r† ψ̂r (x, t) + ψ̂l† ψ̂l (x, t) dˆ† d(x, −∞ (A.16) where the last term is the screened Coulomb interaction between light electrons in the conduction band and the core-hole, meanwhile the first three terms are kinetic energy of conˆ t) = ducting electrons and core-holes. The equation of motion for core-hole reads, −i∂t d(x, ˆ t)]. Operator of core-hole dˆ(0) (x, t) = P exp(ipx + i∆t) is thus modified as, [Ĥ, d(x, q Zt h i (0) ˆ ˆ d(t, x) = d exp − iU dt0 (ψ̂r† ψ̂r + ψ̂l† ψ̂l ) , −∞ = dˆ(0) exp hX q i U p âq e−i|q|vt+iqx − c.c . v 2π|q|L (A.17) where we used approximation for low energy subbands that, ρ̂ = ψ̂r† ψ̂r + ψ̂l† ψ̂l . Similarly, conducting electrons under the core-hole interaction is obtained as, hX ψ̂r (t, x) = ψ̂r(0) (t, x) exp q>0,k ψ̂r(0) (t, x) = Û √ r exp 2πR i U ˆ† ˆ −iqx (dk+q dk e − c.c) , qvL hX r 2π q>0 qL i âq e−iqvt+iqx − c.c , (A.18) (0) where ψr (t, x) is the electron operator without core-hole interaction and Ûr is Fermionic counting operator for right-moving particle. Equivalently, electron operator Eq. (A.18) can be obtained through the unitary transformation below, X † U ei|q|vt dˆk−q dˆk , âq = b̂q + p v 2π|q|L k X † U † † −i|q|vt âq = b̂q + p e dˆk+q dˆk . v 2π|q|L k Hamiltonian Eq. (A.16) can be diagonalized into appealing quadratic form, Ĥ = P ∆ k dˆ†k dˆk and Eq. (A.18) can be reproduced. (A.19) (A.20) P q v|q|b̂†q b̂q − 97 Accordingly, density response of X-ray absorption can be found through following correlation function, Z∞ χMND (ω) ∼ D E ˆ 0) . dt eiωt dˆ† (x, t)ψ̂r (x, t)ψ̂r† (0, 0)d(0, (A.21) 0 Put the dressed electron operators Eq. (A.17) and Eq. (A.18) into it, the bosonized average yields, Im χMND (ω) ∼ −(−Ω)−α (1) +α(2) . (A.22) where the negative exponent corresponds to Mahan singularity α(1) = 2NF U = U/(πv) while second exponent α(2) = 2[δl (kF )/π]2 = U 2 /(2π 2 v 2 ) suppresses the threshold behavior due to the orthogonal catastrophe. It is worth of noting that there is no contribution from ˆ dˆ† operators which means that the core-hole interaction only affects “localized” electrons. d, If we further incorporate interactions among electrons in the conducting Fermi sea, Z V̂ = (V /2) dx (ψ̂r† ψ̂r + ψ̂l† ψ̂l )2 (A.23) both electron operators Eq. (A.17) and Eq. (A.18) are modified by the LL effects, h√ X 1 + g sgn q i Û √ r exp π p âq e−i|q|ut+iqx − c.c , 2πR 2gN |q|L q hX U r g i ˆ x) = dˆ(0) exp d(t, âq e−i|q|ut+iqx − c.c . u 2π|q|L q ψ̂r (t, x) = (A.24) (A.25) In this case, line shape of X-ray absorption is modified as, Im χMND (ω) ∼ −(−Ω)−β (1) +β (2) +β (3) . (A.26) where β (1) = U/(πu), β (2) = gN U 2 /(2π 2 u2 ) and β (3) = (1 − g)2 /(2gN ). In the case of one-channel system without LL effects (g = 1 and N = 1), exponent of MND in Eq. (A.22) is reproduced. For V = U , exponent become γ = (2 − g − g 3 )/2N where Eq. (3.71) is being reproduced. A.4 † † A forward-scattering term in V1 : ψ̂r,β ψ̂r,α ψ̂1,β ψ̂1,α † † Although forward scattering of ψ̂r,β ψ̂1,α ψ̂r,α ψ̂1,β is not the most singular interaction in V1 , the physics is rich. For α 6= β, the interaction has Sine-Gordon form which is known to be irrelevant. For α = β, the interaction becomes density operator of right-moving particle. 98 In order to reproduce RPA, one ought to follow the approximation of Luttinger liquid by taking ψ̂r† (x, t)ψ̂r (x, t) = ρ̂r (x, t), otherwise phases of massless particles would be cancelled out and yields the same order of magnitude as zeroth order ln−1 Ω−1 other than ln−2 Ω−1 . Accordingly, first order correction is written as following form, (1) χFWSC (ω) V1 N 2 = − 2 2 4π R Z∞ Zt dt 0 E D dt0 ei(ω−∆)t e−k̂r (t) eK̂(t) ρ̂(t0 )e−K̂(0) ek̂r (0) . (A.27) 0 0 Then the bosonic operators are not fully in the exponential form. As the average of correlation function is taken in the noninteracting system, one should move âq0 to the rightmost and â†q0 to the leftmost. Then the integral can be carried out directly which leads to, (1) Im χFWSC (ω) = V1 N 2 (−Ω)−γ ln(−Ω−1 ). 4π 2 v 2 (A.28) On the other hand, with careful observation, interaction ψ̂r† ψ̂r Ψ̂†1 Ψ̂1 is nothing but core-hole interaction in the X-ray problem. It indicates that we can borrow the result of MND without doing much calculation. Replacing U by V0 +V1 and keeping linear term of V1 in Eq. (A.26), we get, (1) Im χFWSC (ω) ∼ (−Ω)−γ−V1 /πu ≈ V1 (−Ω)−γ ln(−Ω−1 ). πu (A.29) † † To sum up, forward-scattering ψ̂r,β ψ̂1,α ψ̂r,α ψ̂1,β essentially yields renormalization of inter- action V0 in both spinless (α = β) and spinful case (α 6= β). A.5 Exact integration of second order perturbation Although, it is not possible to calculate the integration of senior time to all orders, it is interesting to take a look at the second order correction Eq. (3.92) from mathematical vigor. Introducing dimensionless variables x = t0 /t and y = t00 /t0 , we have, Z 1 Z 1 1 (1 − xy)−2γ V 2 N 3 ∆ 5γ (2) Γ(5γ) dx dy . χV (ω) = −i 1 3 3 16v π ∆ − ω (1 − x)1−2γ x1−4γ 0 (1 − y)1−4γ y 1−2γ 0 (A.30) The complication comes from the factor (1 − xy)−2γ that prohibits the separation of variables. However, the y-integral can be written in terms of hypergeometric function, Z 1 (1 − xy)−2γ Γ(4γ)Γ(2γ) dy = (A.31) 2 F1 (2γ, 2γ, 6γ, x). 1−4γ 1−2γ (1 − y) y Γ(6γ) 0 Since the integral is mostly dominated at two corners y ∼ 0 and x ∼ y ∼ 1, performing the following analytical continuation from x to 1 − x, 99 2 F1 (2γ, 2γ, 6γ, x) Γ(6γ)Γ(2γ) 2 F1 (2γ, 2γ, 1 − 2γ, 1 − x) Γ2 (4γ) Γ(6γ)Γ(−2γ) + (1 − x)2γ 2 F1 (4γ, 4γ, 1 + 2γ, 1 − x), Γ(2γ)Γ(2γ) Γ(6γ)Γ(2γ) Γ(6γ)Γ(−2γ) ≈ + (1 − x)2γ . Γ2 (4γ) Γ(2γ)Γ(2γ) = (A.32) In the last step, we used approximation that 2 F1 (a, b, c, t) ≈ 1 for small t 1. Then the x-integral is easy to be finished, in the limit of γ → 0, Eq. (A.31) is found to be 5/(8γ 2 ). 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