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December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix A Thomas–Fermi Approximation and Basics of the Density Functional Theory As stated at the beginning of section 2.7 the total energy is a key function describing the basic physical and chemical properties of materials: the ground state. It consists of both kinetic (describing motion) and potential energy parts. To make a theoretical model realistic it is very important to incorporate all important contributions to both the parts of the total energy. In view of the big number of the particles involved into the model, this is very challenging for the first-principles theory. Different approximations are applied in order to achieve a trade-off between complexity and accuracy. Very successful in realistic modeling of the ground state is the density functional theory, DFT. In this chapter we present the basic ideas of the DFT and demonstrate both advantages and problems for optics within this method. Initially Thomas and Fermi (TF) in the 1920s suggested describing atoms as uniformly distributed electrons (negatively charged clouds) around nuclei in a sixdimensional phase space (momentum and coordinates). This is enormous simplification of the actual many-body problem. It is instructive to consider the basic ideas of the TF approximation before starting with a more accurate theory: the DFT. The basic ideas and results of the TF model in application for atoms are provided here. Following the TF approach the total energy of the system could be presented as a function (functional) of electron density [McQuarrie (1976); Parr and Yang (1989)]. Each h3 of the momentum space volume (h is the Planck constant) is occupied by two electrons and the electrons are moving in an effective potential field that is determined by nuclear charge and by assumed uniform distribution of electrons. The density of ΔN electrons in real space within a cube (nanoparticle) with a side l is given by ρ(r) = ΔN ΔN = 3 . v l (A.1) The electron energy levels in this three-dimensional infinite well are given by: E= h2 h2 2 R̃ , (n2x + n2y + n2z ) = 2 8ml 8ml2 279 nx , ny , nz = 1, 2, 3, . . . (A.2) December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix A 280 Radius R = R̃max of the sphere in the space (nx , ny , nz ) covering all occupied states determines the maximum energy of electrons: the Fermi energy F . The number of energy levels within this maximum value at zero temperature is given by 3/2 π 8ml2 F 1 4πR3 = . (A.3) NF = 3 2 3 6 h2 The density of states is defined as g(E)dE = NF (E + dE) − NF (E) = π 4 8ml2 h2 3/2 E 1/2 dE. (A.4) At zero temperature all energy levels below the Fermi energy are occupied: 1E≤F f (E) = . (A.5) 0E>F Consequently the total energy of the electrons in one cell will be given by F F 4π 3/2 3 Ef (E)g(E)dE = 3 (2m) l E 3/2 dE E= h 0 0 3 π 2l (2m)3/2 F 5/2 . = 5 h (A.6) The Fermi energy F can be obtained from the total number of electrons ΔN in a cell: 3 F π 2l ΔN = 2 f (E)g(E)dE = 3 (2m)3/2 F 3/2 . (A.7) h h 0 Combining Eqs. (A.6) and (A.7) the energy of the electrons in one cell is given by l3 5/3 l3 3 (3π 2 )2/3 ρ = CF , 2 10 (2π) (2π)5/3 3 2 2/3 3π CF = = 2.871. 10 E= (A.8) In Eq. (A.8) we reverted to atomic units e = h = m0 = 1. The electron density is a smooth function in a real space. For systems without translational symmetry it is different for different cells. However, for spatially periodic systems only consideration within the unit cell is required since all unit cells are equivalent. Now adding the contributions from all cells with energies within F , we obtain (A.9) TTF [ρ] = CF ρ5/3 (r)d3 r. Equation Eq. (A.9) represents the well-known Thomas–Fermi kinetic energy functional, which is a function of the local electron density. The functional Eq. (A.9) could be applied to electrons in atoms encountering the most important idea of the modern DFT, the local density approximation (LDA) [Martin (2004)]. Adding to December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Appendix A Gavrilenko˙NanoOptics 281 Eq. (A.9) classical electrostatic energies of electron–nucleus attraction and electron– electron repulsion we arrive at the energy functional of the Thomas–Fermi theory of atoms: ρ(r) 3 ETF [ρ(r)] = CF ρ5/3 (r)d3 r − Z d r r 1 ρ(r 1 )ρ(r 2 ) 3 + d r 1 d3 r 2 . (A.10) 2 |r 1 − r 2 | Note that the nucleus charge Z is measured in atomic units. The energy of the ground state and electron density can be found by minimizing the functional (A.10) with the constrain condition ρ(r)d3 r. N= (A.11) The electron density in Eq. (A.10) has to be calculated in conjunction with Eq. (A.11) from the following equation for chemical potential, defined as the variational derivative according to 5 ρ(r 2 ) 3 δETF [ρ] Z 5/3 = CTF ρ (r) − + d r2 . (A.12) μTF = δρ(r) 3 r |r 1 − r 2 | The Thomas–Fermi model provides reasonably good predictions for atoms. It has been used before to study potential fields and charge density in metals and the equation of states of elements [Feynman et al. (1949)]. However, this method is considered rather crude for more complex systems because it does not incorporate the actual orbital structure of electrons. In view of the modern DFT theory the Thomas–Fermi method could be considered as an approximation to the more accurate theory. For systems like molecules and solids, much better predictions are provided by the DFT. Search for the ground state within the DFT follows the rule that the electron density is a basic variable in the electronic problem (the first theorem of Hohenberg and Kohn [Hohenberg and Kohn (1964)]) and another rule that the ground state can be found from the energy variational principle for the density (the second theorem of Hohenberg and Kohn [Kohn (1999)]). According to the DFT the total energy could be written as E[ρ] = T [ρ] + U [ρ] + EXC [ρ], (A.13) where T is the kinetic energy of the system of noninteracting particles and U is the electrostatic energy due to Coulomb interactions. The most important part in the DFT is EXC , the exchange and correlation (XC) energy that includes all many-body contributions to the total energy. The charge density is determined by the wave functions, which for practical computations could be constructed from single orbitals, φj (e.g. antisymmetrized product—the Slater determinant, atomic or Gaussian orbitals, linear combinations of plane waves). The charge density is given by 2 |φj (r)| , (A.14) ρ(r) = j December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix A 282 where the sum is taken over all occupied j orbitals. In the spin-resolved case there will be orbitals occupied with spin-up and spin-down electrons. Their sum gives the total charge density and their difference gives the spin density. In terms of the electron orbitals the energy components are given in atomic units as 1 ∗ T =− φj (r)|∇2 |φj (r)d3 r, (A.15) 2 j U =− N n α j + + 1 2 φ∗j (r) Zα φj (r)d3 r (Rα − r) φ∗i (r 1 )φ∗j (r 2 ) i,j N N α β<α 1 φi (r 1 )φj (r 2 )d3 r 1 d3 r 2 (r 1 − r 2 ) Zα − Zβ . |Rα − Rβ | (A.16) The first term in potential energy (A.16) stands for the electron–nucleus attraction, the second term describes for electron–electron repulsion, and the third term represents nucleus–nucleus repulsion. In Eq. (A.16) Zα refers to the charge on nucleus α of N −atom system. The third term in Eq. (A.13) describes the exchange and correlation energy. Rather simple for computations but surprisingly good approximation is the local density approximation, LDA, which assumes that the charge density varies slowly on the atomic scale, i.e., the effect of other electrons on the given (local) electron density is described as a uniform electron gas. The XC energy can be obtained by integrating with the uniform gas model (see e.g. [Ceperley and Adler (1980)]): ∼ (A.17) EXC = ρ(r)ẼXC [ρ(r)]d3 (r), where ẼXC [ρ(r)] is XC energy per particle in a uniform electron gas. For many systems a good approximation provides analytic expression for ẼXC [ρ(r)] suggested by Perdew and Wang (1992). In practical calculations through minimization of the total energy Eq. (A.13) one determines self-consistently the electron density and the actual XC part. A variational minimization procedure leads to a set of coupled equations proposed by Kohn and Sham [Kohn and Sham (1965)]: 1 2 (A.18) − ∇ − VN + Ve + μXC (ρ) φj = Ej φj , 2 with μXC = ∂ (ρEXC ) , ∂ρ (A.19) The solution of the Kohn–Sham equation provides the equilibrium geometry and the ground-state energy of the system. However, eigen functions and eigen energies of the Kohn–Sham equation cannot be interpreted as the quasiparticle December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Appendix A Gavrilenko˙NanoOptics 283 quantities needed for optics. The term quasiparticle refers to a particle-like entity arising in certain systems of interacting particles. If a single particle moves through the system, surrounded by a cloud of other interactiong particles, the entire entity moves along somewhat like a free particle (but slightly different). The quasiparticle concept is one of the most important in materials science, because it is one of the few known ways of simplifying the quantum mechanical many-body problem describing excitation state and is applicable to an extremely wide range of many-body systems. Calculation of the ground state from the Kohn–Sham equation does not result automatically in correct prediction of excitation energies required for optics. For example, in nonmetallic systems the predicted value of the energy difference (energy gap) between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) in most cases is underestimated (gap problem). Special corrections (quasiparticle, QP corrections) are required to get more accurate excitation energies [Onida et al. (2002)]. Without corrections in semiconductors and insulators the local density approximation, LDA, substantially underestimates forbidden gap values. In this chapter we present LDA results for optics with different QP corrections avoiding, however, detailed analysis of theoretical methods. For advanced reading of the DFT one can recommend original papers [Hohenberg and Kohn (1964); Kohn (1999); Ceperley and Adler (1980)] and the monographs [McQuarrie (1976); Parr and Yang (1989); Martin (2004); Michaelides and Scheffler (2008)]. This page intentionally left blank December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix B Evaluation of Optical Functions within the Perturbation Theory In this section we describe evaluation of the light-filled induced charge within perturbation theory using plane wave representation, which is used in this chapter for calculations of optical functions (see section 2.7). Equilibrium electron charge density is defined through the density operator (using definition of Trace, T r, as a sum of the diagonal elements): neq (r) = eT r[ρ0 , δ(r − r 0 )]. (B.1) Without illumination if the system is periodic (at least in one dimension) the density operator could be defined in energy representation on a set of Bloch functions according to [Davydov (1980)] ρ|s = ρ0 |k, l = f (Ek,l )|k, l, where the equilibrium Fermi distribution function is given by F −Es −1 f (Es ) = e kT − 1 , (B.2) (B.3) The Bloch functions |s = |k, l = uk,l (k)eikr , (B.4) are solutions of an undisturbed Schrödinger equation with periodic potential: 1 2 (B.5) H0 |k, l = − ∇ + V0 (r) |k, l = Ek,l |k, l. 2 In an external optical field when light quanta strike electrically neutral atoms the equilibrium is broken through the deformation of electron clouds. Time-dependent changes of the electron charge density can be represented as Taylor expansion. The number of the terms to be included into the Taylor sum for the induced part of the charge depends on the excitation intensity: n(r, t) = neq (r) + nind (r, t) = eT r[ρ, δ] = eT r[ρ0 , δ] + eT r[ρ(1) , δ] + eT r[ρ(2) , δ] + . . . (B.6) The first- and higher-order corrections to the density operator are determined from the standard perturbation theory: dρ = [H, ρ] = (Hρ − ρH), (B.7) i dt 285 December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix B 286 with H = H0 + V (1) + . . . , ρ = ρ0 + ρ(1) + ρ(2) + . . . (B.8) In Eq. (B.7) for simplicity we neglected the effect of energy dissipation, which could be included through the relaxation time. Plugging (B.8) in (B.7) and choosing terms of the same order on both the left and right side of the equation of motion for the density operator, Eq. (B.7) splits into a series of equations for zero, first, second, etc., orders of perturbations, respectively: iρ̇0 = [H0 , ρ0 ], iρ̇(1) = H0 , ρ(1) + V (1) , ρ0 iρ̇(2) = H0 , ρ(2) + V (1) , ρ(1) (B.9) Dynamic optical response is described through the time-dependent density operator. In an external electromagnetic field the perturbation is harmonic, i.e., ρ(t) = ρ(0)eiωt , iρ̇(ω) = −ωρ(ω). (B.10) It is convenient now to switch to the matrix representation in Eq. (B.10) by projecting the relevant quantities on a set of Bloch functions in Eq. (2.68). To this end one should multiply every term in Eq. (B.10) with the function Eq. (2.68); the complex conjugate of Eq. (2.68) is then multiplied on the left and right side of the relevant equation. Through integration over the entire space and by taking into account orthonormality conditions for Bloch functions, this leads to the following expression for the first-order terms: (1) (1) (1) (1) Vst ρ0ts − ρ0sp Vps , (B.11) −ωρss = (Es − Es )ρss + t p The density operator defined as Eq. (B.2) in matrix presentation has the form (0) ρss = f (Es )δss . (B.12) Equation (B.11) is now transformed into (1) (1) −ωρss = (Es − Es )ρss + [f (Es ) − f (Es )] Vss . (B.13) At zero temperature, optical excitations occur between completely filled and empty states with Fermi functions equal to either 1 or zero, respectively. Consequently, f (Es ) − f (Es ) (1) Vss = (Es − Es − ω)−1 Vss |T =0 . (B.14) ρss (ω) = Es − Es − ω For the second-order perturbation one needs to use the first-order solution Eq. (B.14). Plugging it into Eq. (B.10) after some algebra leads to the following expressions at T = 0: (1) (1) (1) (1) (2) (2) Vss ρs s − ρss Vs s , (B.15) −ωρss = (Es − Es )ρss + s s December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Appendix B or (2) ρss (ω) = Gavrilenko˙NanoOptics 287 (1) (1) 1 1 [Vss Vs s ( Es − Es − ω Es − Es − ω s 1 )]. − Es − Es − ω (B.16) Equations (B.14) and (B.16) can be used now to obtain induced charge density from Eq. (B.6) within the first and second order of external perturbation, respectively. The first- and second-order contributions to the induced charge density in Eq. (B.6) follow from (B.14) and (B.16), respectively. In a system with a periodicity the perturbation potential is given by ∞ V (q + G, ω)ei(q+G)r dω, (B.17) V (r, t) = qG −∞ where G is the reciprocal lattice vector. For Fourier transform of the potential ∞ V (r, ω)eiωt , (B.18) V (r, t) = −∞ the expansion of the potential is given by V (r, ω) = V (q + G, ω)ei(q+G)r . (B.19) qG In a periodic system with all equivalent atoms separated by Ri one has for the charge: neq (r 0 + Ri ) = neq (r 0 ), with n(r) = eiqr n(q) = q eiqr q (B.20) δqG n(G). (B.21) G For induced charge density in (B.6) we have nind (r, ω) = nind (q + G, ω)ei(q+G)r . (B.22) qG Where Fourier transform of the induced charge is given by Fourier integral by (B.23) nind (q + G, ω) = nind (r, ω)e−i(q+G)r d3 r, Linear part of the induced charge in (B.23) follows from (B.6) nind (q + G, ω) = e T r ρ(1) (ω), δ(r − r) e−i(q+G)r d3 r = eT r ρ(1) (ω), e−i(q+G)r , (B.24) The trace of the operator product is calculated according to T r ÂB̂ = m|ÂB̂|m = Amn Bnm . (B.25) m m n December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix B 288 Eq. (B.24) projected on the plane wave basis 2.68 can be represented as nind = nind (q + G, ω) −i(q+G)r =e l , k + q|ρ(1) ω |k, ll, k|δ(r − r)|k + q, l e k+q k,l =e l , k + q|ρ(1) ω |k, ll, k| =e ei(r−r )G e−i(q+G)r |k + q, l G k+q k,l −i(q+G)r l , k + q|ρ(1) |k + q, l . ω |k, ll, k|e (B.26) k+q k,l In (B.26) the bracket notation means space integration. We also used the definition of the δ function: ei(r−r )G . (B.27) δ (r − r ) = G Now Eq. (B.14) can be written as f [El (k + q)] − f [El (k)] l , k + q|V (r, ω)|k, l El (k + q) − El (k) − ω − iη f [El (k + q)] − f [El (k)] = El (k + q) − El (k) − ω − iη × V (q + G, ω)l , k + q|ei(q+G)r |k, l. (B.28) l , k + q|ρ(1) ω |k, l = q,G The complex part of the energy in the denominator of Eq. (B.28) is introduced to prevent unphysical divergences at resonance frequencies. Plugging (B.28) into (B.26) we arrive at the following expression for the induced charge: nind (q + G, ω) = e PG,G V (q + G, ω). (B.29) G Using the notation (k = k + q) the polarization function is defined as PG,G (ω) = l , k |ei(q+G)r |k, ll, k|ei(q+G )r |k , l k ,k l ,l × f [El (k + q)] − f [El (k)] . El (k + q) − El (k) − (ω + iη) (B.30) Evaluation of the full polarization function is given in Appendix C. Full potential in materials can be separated into two parts, the external and induced potential: V (q + G, ω) = Vext (q + G, ω) + Vind (q + G, ω). (B.31) The Eq. (B.31) can be understood as a reduction (screening) of external potential through the induced charge in materials. This can be presented in terms of dielectric function: ε−1 (B.32) V (q + G, ω) = G,G Vext (q + G , ω), G December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Appendix B or Vext (q + G , ω) = εG,G V (q + G, ω), Gavrilenko˙NanoOptics 289 (B.33) G Equations (B.32) and (B.33) can be considered as the microscopic definition of the dielectric function. The described computation of ε presents a transformation from the microscopic (atom-related) quantities to the macroscopic values used in classical electrodynamics theory. For advanced reading related to the definition of the optical function within the first-principles theory, one can recommend [Martin (2004); Yu and Cardona (2010)] or original papers (see e.g. [Onida et al. (2002); Gavrilenko and Bechstedt (1997)] and references therein). The induced potential satisfies the Poisson equation: Vind (q + G, ω) = 4π nind (q + G, ω). |q + G|2 From Eqs. (B.29), (B.31), (B.33), and (B.34) we obtain 4π PG,G V (q + G, ω) V (q + G, ω) = εG,G (ω) + |q + G|2 G δG,G V (q + G, ω). = (B.34) (B.35) G The dielectric function can now be expressed in terms of the polarization function: εG,G (ω) = δG,G − 4π PG,G . |q + G|2 (B.36) This page intentionally left blank December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix C Local Field Effect in Optics of Solids from the First Principles Local field (LF) effect plays a key role in the optics of nanostructures. Description of the LF effect within classical electrodynamics is presented in many monographs and research papers (see Chapter 3 and references therein). The classical approach, however, cannot include optical excitations on the atomic scale in the whole picture. Electronic excitations are increasingly important in the optics of nanomaterials with reduction of the dimensions of nanostructures, thus requiring microscopic modeling. This section presents the evaluation of the optical dielectric function, including the LF effect within the perturbation theory (see section B). The formula for the polarization function (see Eq. (B.30)) can be presented as 2 k ,k ∗k,k Bn ,n (q + G)Bn,n PG,G (ω) = (q + G ) Ω k ,k n ,n f [El (k + q)] − f [El (k)] . El (k + q) − El (k) − (ω + iη) The Bloch integrals in Eq. (C.1) are defined as × (C.1) ,k Bnk ,n (q + G) = n , k |ei(q+G)r |k, n 1 (C.2) = ψn∗ ,k (r)ei(q+G)r ψn,k d3 r. Ω In plane wave representation (2.68) neglecting the umklapp processes (the nonconserving crystal momentum electron–electron scattering) [Bechstedt (2003)] and in the limit of q → 0 the Bloch integrals have an extremely simple form, given by k ,k (G) = d∗c,k (G1 )dv,k (G1 − G). (C.3) Bc,v G1 Indexes c and v in Eq. (C.3) denote empty antibonding (conducting) and filled bonding (valence) electron states, respectively, at zero temperature. By derivation of Eq. (C.3) we used the following properties of direct and reciprocal lattice vectors: Gi Rj = 2πδij eiGj Rj = 1 ei(k −k+q)Ri = δk −k+q,Gi Ri Gi 291 (C.4) December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix C 292 Equation (C.1) at zero temperature takes the following form: k ,k ∗k,k 2 Bn ,n (q + G)Bn,n (q + G ) . PG,G (ω) = Ω El (k + q) − El (k) − (ω + iη) (C.5) k ,k n ,n From the orthonormality of the wave functions follow the properties of Bloch integrals: k ,k Bc,v (0) = d∗c,k (G1 )dv,k (G1 ) = 0, G1 N 2 k ,k Bn ,n (q + G) = 1. (C.6) n =1 k For G = 0 and in the limit q → 0 the Bloch integrals have the following properties: 3 k ,k (q) = δk,k lim i lim Bc,v q→0 q→0 = qα c, k|rα |k, v α=1 3 δk,k lim qα c, k|vα |k, v, Ec (k) − Ev (k) q→0 α=1 (C.7) k ,k Bc,v (qα ) 1 lim = c, k|vα |k, v. q→0 |qα | Ec (k) − Ev (k) Here we used the general definition of velocity (or momentum) [Adolph et al. (1996)] given by 1 H, eiqr . (C.8) v = lim q→0 q At the limit the velocity is given by v = i [H, r] . (C.9) After projecting on the full set of eigen functions of the Hamiltonian it follows that nk| [H, rα ] |n k = nk|H|m, lm, l|rα |n k m,l − nk|rα |m , l m , l |H|n k (C.10) m ,l = (Enk − En k )nk|rα |n k . Equation (C.11) represents the relationship between matrix elements of the velocity and of the induced dipole momentum, which could be used to obtain the relationship between optical functions calculated in velocity and length gauges [Gavrilenko and Bechstedt (1997)]. December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix D Optical Field Hamiltonian in Second Quantization Representation If there are resonances of electromagnetic field within a cavity the entire field can be presented as a superposition of single modes in the following form [Jaynes and Cummins (1963)]: √ pj (t)E j (r), (D.1) E(r, t) = − 4π j √ H(r, t) = 4π ωj qj (t)H j (r). (D.2) j The total energy of the field is given by 1 1 2 H= (|E|2 + |H|2 )d3 r = (p + ωj2 qj2 ) 8π 2 j j (D.3) The Hamiltonian equation of motion is given by ∂H = pj , ∂pj ∂H p˙j = − = −ωj2 qj . ∂qj q˙j = (D.4) Mathematically, quantization of the field is represented by commutations rules for the canonically conjugated coordinates and momenta: [qi , qj ] = 0, [pi , pj ] = 0, [qi , pj ] = iδij . (D.5) The Hamiltonian of an optical field is convenient to present in terms of second quantization operators, the Bosonic operators of creation (â†j ) and annihilation (âj ) of photons using the definitions [Davydov (1976)] ωj (D.6) âj + â†j , pj = 2 ωj ωj qj = i (D.7) âj − â†j , 2 with commutation rule [âi , â†j ] = δij . 293 (D.8) December 22, 2010 294 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix D The properties of âi operators are described through their action on state vector φ(n1 , n2 , . . . ni . . .) according to √ âi φ(n1 , . . . ni−1 , ni , ni+1 . . .) = ni φ(n1 , . . . ni−1 , ni−1 , ni+1 . . .), √ â†i φ(n1 , . . . ni−1 , ni , ni+1 . . .) = ni + 1φ(n1 , . . . ni−1 , ni+1 , ni+1 . . .). The Hamiltonian of the quantized optical field is now given by 1 † ωi âi âi + H= . 2 i (D.9) December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix E Surface Plasmons and Surface Plasmon Polaritons Collective electronic excitations (also known as plasma excitations) at metal surfaces and/or metal/dielectric interfaces play a key role in the optics of nanomaterials, ranging from physics and materials science to biology. A unified theoretical description of these phenomena is based on the many-body dynamical electronic response of solids (see section B of the Appendix), which underlines the existence of various collective electronic excitations at metal surfaces, such as the conventional surface plasmon, multipole plasmons, and the acoustic surface plasmon. A detailed description of the surface plasmon polariton (SPP) phenomena and its applications in modern optical spectroscopy is out of the scope of the present book. Several specialized monographs and reviews can be recommended for advanced reading [Tudos and Schasfoort (2008); Pitarke et al. (2007); Liebsch (1997); Ritchie (1973); Venger et al. (1999); Raether (1988)]. Here we present the basic conditions and properties of SPP that follow from classical electrodynamics. Consider a model consisting of two semi-infinite nonmagnetic media with local (frequency-dependent) dielectric functions εd (ω) (dielectric) and εm (ω) (metal) separated by a planar interface at z = 0 [Pitarke et al. (2007)]. The full set of Maxwell’s equations in the absence of external sources can be expressed as follows [Jackson (1975)]: 1 ∂ En, c ∂t 1 ∂ H n, ∇ × En = − c ∂t ∇ · (εn E n ) = 0, ∇ × Hn = ε ∇ · H n = 0, (E.1) (E.2) (E.3) (E.4) where the index n = d in dielectric (at z < 0) and n = m in metal (at z ≥ 0). Within the classical picture, the metal can be treated as a semi-infinite electron gas with an abruptly terminated profile of the electron density function. The surface charge density (see Eq. (B.1)) on the metal–dielectric interface can be presented as Liebsch (1997); Raether (1988) n(r, ω) ≈ ejq r δ(z). 295 (E.5) December 22, 2010 296 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix E The electric field associated with the density Eq. (E.5) is given by the Gauss law, which follows from Eq. (E.3) in the presence of charge: ∇ · E(r, ω) = −4πn(r, ω). (E.6) Neglecting the retardation effects only the longitudinal plasma oscillations are considered here. In terms of the scalar potential φ the electric field is given by E(r, ω) = ∇ · φ(r, ω). (E.7) For the choosen model system the potential φ is given by φ(r, ω) = ejq r φ(z, ω) (E.8) The z components of the field decay evanescently into both media. This follows from the electroneutrality since ∇2 φ = 0 must be valid everywhere except at z = 0 [Liebsch (1997)]. Consequently the potential in Eq. (E.8) must be taken in the form φ(r, ω) = φ0 ejq r e−q|z| , (E.9) where q ≡ qz . In this case the field determined by Eqs. (E.7) and (E.9) varies continuously within the interface, however, the normal component is discontinuous. The components above and below the interface are given by Ez (z + 0) = q φ0 ejq r , Ez (z − 0) = −q φ0 ejq r . (E.10) The field and charge distribution for such a surface mode is represented in Fig. E.1. In the long-wavelength limit the boundary condition can be written as εm (ω)Ez (0− ) = εd Ez (0+ ). (E.11) For the metal–vacuum interface (i.e., by εd = 1) the condition for the existence of the surface plasmons followed from Eqs. (E.10) and (E.11) now reads εm (ω) = −1. (E.12) Fig. E.1 Schematic of charge and field distribution for a surface plasmon described by Eq. (E.9). Adapted from [Hofmann (2008)] December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Appendix E Gavrilenko˙NanoOptics 297 Equation (E.12) defines the frequency of the surface plasmon in the q = 0 limit [Liebsch (1997)]. Dielectric dispersion in metal can be described by the Drude model (see Eq. (2.53) in section 2.5), which can be written in the form ωp2 (E.13) ω(ω + jΓ) Pluging the Eq. (E.12) into Eq. (E.13) results ωp ωs = √ (E.14) 2 The plasma frequency ωp is given by Eq. (2.54) (see section 2.5). The existence of the electronic excitations on the interfaces was predicted by Ritchie (1957). If the overlayer has the dielectric constant εd > 1 the condition Eq. (E.12) is given by εm (ω) = 1 − εm (ω) = −εd (E.15) Consequently, the surface plasmon frequency is redshifted according to ωp ωs = √ . (E.16) εd + 1 This redshift has been observed on different metal–dielectric interfaces (see e.g. [Ritchie (1973); Pitarke et al. (2007); Raether (1988)] and references therein). It is a key point of surface plasmon resonance (SPR)-based sensing spectroscopic tools widely used for different fundamental and applied studies (see section 10.5) Consider now the dispersion of surface plasmons. The solution of the system Eqs. (E.1) to (E.4) will be separated into s- (E vector perpendicular) and p-polarized (E vector parallel to the plane of incidence) EM modes. If there is a wave propagating along the interface, it should contain the electric field E component perpendicular to the interface (the p-polarized mode) and thus the s mode is not relevant. Consequently, the problem is now formulated as the search for the conditions of the propagation of the p-polarized EM wave along the interface. Choosing the wave propagation direction along the x-axis the solution should be taken in the form [Pitarke et al. (2007)] 0 0 , 0, En,z )ej(qn x−ωt) e−kn |z| , E n = (En,x Hn = 0 (0, Hn,y , 0)ej(qn x−ωt) e−kn |z| , (E.17) (E.18) where qn denotes a two-dimensional wave vector q of the wave propagating along the interface. Substituting Eqs. (E.17) and (E.18) into Eqs. (E.1) to (E.4) results in the following set of equations: ω (E.19) kd Hd,y = εd Ed,x , c ω km Hm,y = − εm Em,x , (E.20) c and ω 2 . (E.21) kn = qn2 − εn c December 22, 2010 298 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix E The standard boundary conditions require that the components of both electric and magnetic fields must be continuous [Jackson (1975)]. Consequently, Eqs. (E.19) and (E.20) result in kd km Hd,y + Hm,y = 0, εd εm (E.22) Hd,y = Hm,y (E.23) and The system of Eqs. (E.22) and (E.23) has a solution if the determinant is equal to zero: εm εd + =0 (E.24) kd km Equation (E.24) represents the surface plasmon condition [Pitarke et al. (2007)]. The boundary conditions also require continuity of the two-dimensional wave vector q in Eq. (E.21), i.e., qd = qm = q. Based on this condition and combining Eq. (E.24) and Eq. (E.21) one arrives at ω2 ω2 2 2 2 2 (E.25) εd q − ε m 2 = εm q − ε d 2 . c c Equation (E.25) leads to another widely used form of the surface plasmon condition [Ritchie and Eldridge (1962); Ritchie (1973)]: εd εm ω . (E.26) q(ω) = c εd + εm For a metal–dielectric interface with the dielectric constant εd , the solution ω(q) √ of Eq. (E.26) has a slope equal to c/ εd at the point q = 0 and is a monotonic √ increasing function of q, which is always smaller than cq/ εd and for a large q is asymptotic to the value given by the solution of εd + εm = 0. (E.27) This is the nonretarded surface plasmon condition that follows from Eq. (E.24) at kd = km = q. This is valid as long as the phase velocity is much smaller than the speed of light, i.e., ω/q c. It is instructive now to analyze the dispersion of the SPP propagation on the interface between metal and dielectric. The q0 = ω/c equation represents the magnitude of the light wave vector. Assume that for the dielectric εd = 1. In this case Eq. (E.26) yields ω 2 − ωp2 ω q(ω) = . (E.28) c 2ω 2 − ωp2 The dispersion relation described by Eq. (E.28) is represented in Fig. E.2 [Hofmann (2008)]. December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Appendix E Gavrilenko˙NanoOptics 299 Fig. E.2 Bold solid lines represent the dispersion of light in the retarded (upper line) and the nonretarded surface plasmon polariton regions (lower line). By the thin line the dispersion of light striking the interface at different angles is shown. The √ thin horizontal lines indicate the values of bulk ωp and surface plasmon frequencies ωs = ωp / 2. (Adapted from [Hofmann (2008)]). The upper solid line in Fig. E.2 represents the dispersion of light in solid. The lower solid line is the surface plasmon polariton dispersion curve, which is given by ω 2 (q) = ωs2 + c2 q 2 − ωs4 + c4 q 4 (E.29) √ where ωs = ωp / 2 represents the classical nondispersive surface plasmon frequency. In the retarded region (q < ωs /c), the surface plasmon polariton dispersion curve approaches the light line (ω = cq , see the thin line in Fig. E.2). At short wavelengths where q ωs /c the surface plasmon polariton √ approaches asymptotically the nonretarded surface plasmon frequency ωs = ωp / 2 (see the horizontal dashed line in Fig. E.2). Important conclusions can be made regarding the excitation of the surface plasmon polaritons corresponding to the lower branch in Fig. E.2. The wave vector of the SPPs has the value of the two-dimensional vector within the interface plane, q . Depending on the angle of incidence it varies from q = 0 (normal incidence) to |q | = q (for grazing incidence, qz = 0). The light dispersion will change from the vertical line to that given by ω = cq (see Fig. E.2). For any other angle the light dispersion is given by ω = c q2 + qz2 . Consequently, the light dispersion line and the surface plasmon polariton dispersion curve never cross, and hence there cannot be any excitation of SPP on an ideal interface considered above. There are two basic approaches to generate SPP. It can be generated on a grating. Additional periodic profile on the surface causes modifications of the wave December 22, 2010 300 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Appendix E vector selection rules (like additional Bragg reflection in superlattices). According to the superperiodicity the dispersion curve will get folded, crossing the light dispersion line and thus allowing excitation of the SPPs. This has been observed experimentally by Wood at the beginning of the last century [Wood (1902)], and he described it as an “anomalous diffraction gratings” effect [Wood (1935)]. The same effect can be achieved by a rough surface that can be viewed as a superposition of many gratings with different periodicities [Ritchie (1973); Venger et al. (1999); Raether (1988)]. The excitation of the SPPs via surface roughness is thought to play a role in surface-enhanced Raman scattering (see Chapter 6). The other way to achieve the coupling is to use an optical system where the value of the photon wave vector will increase, thus reducing the slope of the curve. Optical systems with a total light reflection inside a prism mounted in a short distance over the surface are widely used. In this case, an evanescent electric field penetrates the gap between prism and surface. The field decays exponentially because the wave vector contains an imaginary q value in the z direction (see the dashed line in Fig. E.2). The complex value of the light wave vector causes slope decrease of the light dispersion curve in Fig. E.2 that results in the situation when the light dispersion line and the surface plasmon polariton dispersion curve cross, thus allowing the excitation of the SPPs. Examples of the prism systems generating an evanescent light field are shown in Figs. 3.20 and 10.3a. This design is widely used in SPRbased optical spectroscopic tools for materials characterization, biosensing, etc. (see section 10.5). December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Bibliography Achermann, M., Hollingsworth, J. A., and Klimov, V. I. (2003). Multiexcitons confined within a subexcitonic volume: spectroscopic and dynamical signatures of neutral and charged biexcitons in ultrasmall semiconductor nanocrystals, Phys. Rev. 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December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics Index additional boundary conditions (ABC), 123 Autler–Townes splitting, 194 dielectric function, 52–55, 289 differential scattering cross section, 176 Dirac point, 38 discrete dipole approximation, 93, 175 DNA-based nanotechnology, 29 dressed state, 186 Drude model, 48, 62 band folding, 216 Bethe–Salpeter equation, 122, 148 bioconjugates, 263 biolabels, 19 biological nanomaterials, 29 biosensors, 272, 275 Bloch functions, 285, 286 Bloch integrals, 291, 292 block conjugated polymers, 255 Bosonic operators, 293 effective mass approximation (EMA), 34, 126 effective medium approximation, 67 electric field induced SHG (EFISH), 211, 215 electro-optical spectroscopy, 215 electromagnetic field enhancements, 68 electromagnetic wave equation, 44 electron charge density, 285 electron–phonon coupling, 151 electroreflectance, 215 entanglement, 182–185, 189 exciton, 119 biexcitons, 144 Bohr radius, 123 Frenkel, 119 singlet exciton, 122 triplet exciton, 122 Wannier–Mott, 119, 120 exciton Raman scattering, 155 C-dot, 19, 263 carbon fibers, 9 carbon nanotubes, 5, 7 charge conservation, 44 chemisorption, 3, 172 chromoionophore, 257 Clausius–Mossotti equation, 67 COIN (composite organic-inorganic nanoparticles), 269 colloidal crystals, 26 conjugated polymers, 29, 243 constitutive relations, 43 continuity equation, 44 Coulomb interaction, 62 fabrication, 1 GaN nanowires, 108 Ag nanoparticles, 26 carbon nanoparticles, 7 CdSe nanocrystals, 18 CdSe-ZnS core/shell nanocrystals, 19 CVD technique, 2 Davydov splitting, 249 deformation potential interaction, 151 delta function, 288 density functional theory (DFT), 50, 279 density of states, 38, 280 density operator, 285, 286 327 December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in 328 DAE-E DX tile nanotubes, 30 FePt nanoparticles, 27 GaMnN nanostructures, 21 GaN nanowires, 21 Stranski–Krastanow growth, 2 titania nanoparticles, 15 Fabry–Perot resonator, 185 Fermi energy, 280 fluorescence emitters, 262 fluorophores, 262 Fourier transform, 287 Fröhlich constant, 155 Fröhlich interaction, 151, 153 fullerenes, 5 graphene, 9, 11, 36 Green’s function, 170 Hückel model, 245 highest occupied molecular orbital (HOMO), 35, 283 Holliday junction, 267 hollow nanoparticles, 23, 75 hot spot, 69, 168, 177 hyper-polarizability, 203 hyper-Rayleigh scattering, 203 interchain polymer distance, 253 invisibility cloak, 82 Jaynes–Cummings model (JCM), 183 jellium approximation, 90 Kirkendall diffusion effect, 23 Laplace equation, 63 LC nanoelement circuit, 223 left-handed materials, 78 lithography, 13 local density approximation (LDA), 282 local field, 53 local field effect, 46, 54, 291 localized atomic orbitals (LCAO), 35 Lorentz force, 222 Lorentz-force field, 224 lowest unoccupied molecular orbital (LUMO), 283 matrix representation, 286 Maxwell’s equations, 43 Index Maxwell–Garnett approximation, 67 metallic carbides, 5 metamaterials, 77, 221 Mie resonance, 224 Mie theory, 62, 65, 91 MOCVD, 22 molecular nanocrystals, 234 Mollow triplet, 192 Moore’s law, 261 nanocomposites, 238, 240 nanoporous carbon, 6 near-field optics, 84 negative-index materials, 81 nonlinear optics, 202 normal modes, 150 oligomers, 235 optical field Hamiltonian, 293 optical functions dielectric constant, 43 dielectric permittivity, 43, 81 displacement, 43 extinction coefficient, 44 index of refraction, 44 magnetic permeability, 43 permeability, 81 polarizability, 46 refraction coefficient, 44 susceptibility, 43, 48 optical labeling, 262 optical loss, 47 optical rectification, 45 organic nanocrystals, 234 organic nanofibers, 235 oscillator strength, 42 para-quaterphenylene, 235 perturbation theory, 285 phase velocity, 45 phonon bottleneck, 147 phonon confinement, 161, 162 phonons, 150 photonic crystal, 190 physisorption, 3 plane wave representation, 291 plasma excitations, 59 plasma frequency, 48 plasmon resonance, 62 electrostatic theory, 63 Gavrilenko˙NanoOptics December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Index plasmonic density of states, 73 plasmonics, 59 PMMA, poly(methyl methacrylate), 4 Poisson equation, 289 polarization function, 45, 46, 52, 288, 291 polymer–metallic nanomaterials, 29 polymers, 4, 239 potential well, 34 Purcell effect, 188 Purcell factor, 188 quantization of the field, 293 quantum confinement, 35, 40 quantum dots, 41 quantum electrodynamics (QED), 181 quantum well, 33 quasiparticle, 34, 283 quasistatic approximation, 63 Rabi oscillations, 184 Raman polarization function, 150 Raman spectroscopy, 149, 160 surface effect, 164 Raman tensor, 150, 173 refractive index, 65, 80, 81 Rydberg atoms, 184 Schrödinger equation, 41, 120, 285 second harmonic generation, 205 second quantization, 293 Gavrilenko˙NanoOptics 329 sensors biosensors, 263 polymer-based sensors, 257 silanization, 263 silicon carbide, 216 single-wall carbon nanotubes (SWNTs), 124 spherical harmonics, 40 split-ring resonator, 223 strong coupling, 183, 188 surfac-enhanced infrared absorption (SEIRA), 271 surface-enhanced Raman scattering (SERS), 149, 165, 219 surface plasmon polariton (SPP), 295 surface plasmon resonance (SPR), 61, 273, 297, 300 surface plasmons, 59 Taylor expansion, 285 Taylor series, 46 Thomas–Fermi approximation, 279 trioctylphosphine oxide, 258 two-level atomic system, 183 velocity operator, 292 vibronic states, 160 Wannier–Mott excitons, 120 weak coupling, 188 Wigner symbols, 128