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Plasmons, Polarons, Polaritons

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Plasmons, Polarons, Polaritons Plasmons, polarons, polaritons • Dielectric function; EM wave in solids • Plasmon oscillation -- plasmons • Electrostatic screening • Electron-electron interaction • Mott metal-insulator transition • Electron-lattice interaction -- polarons • Photon-phonon interaction -- polaritons • Peierls instability of linear metals For mobile positive ions Dept of Phys M.C. Chang Dielectric function (r, t)-space (k,ω)-space ∇⋅Ert(,) =4πρ (,) rt ik⋅= E(, kωπρω ) 4 (, k ) ∇⋅Drt(,) =4πρ (,) rt ext ik⋅= D(, kω ) 4πρext (, k ω ) ()ρ = ρρext+ ind Take the Fourier “shuttle” between 2 spaces: dk3 dω E(,)rt= Ek (,ω ) eikr()⋅−ω t , same for D ∫ (2ππ )3 2 dk3 dω ρ(,)rt= ρω (, k ) eikr()⋅−ω t , same for ρ ∫ (2ππ )3 2 ext Dk( ,ωεω )= ( k , ) Ek ( , ω ) (by definition) or ρωεωρω(,kkk )= (, )(, ) (easier to calculate) ext or φωεωφωext (,kkkEkikk )==− (, )(, ) ∵ (, ω ) φω (, )... etc Q: What is the relation between D(r,t) and E(r,t)? EM wave propagation in metal Maxwell equations 1 ∂B iω ∇×ED= − ; ∇⋅ =4πρ ik×+ E= B ; ε k ⋅= E 4πρ ct∂ ext c ion ext 14∂D π iωπ4 ∇×BJB=; + + ∇⋅ =0ik×− B=;ε E + J ik ⋅= B 0 ct∂ c ext ccion ext JE= σ ωπω2 4 i ( ) →×kkE() ×= −22εσion E − E || cc 2 kkE()⋅− kE • Transverse wave 2 2 ωπσ⎛⎞4 i k =+2 ⎜⎟εion c ⎝⎠ω ω c ∵υ ==, refractive index n = ε p kn 4πσi (in the following, ∴εωε(,k )=ion + ω let εion ～1) • Longitudinal wave εω(,k )=0 Drude model of AC conductivity dv v meEtm=−() − eedt τ Assume −itω Et()= Ee0 then −itω vve= 0 emτ / →=−vEe ()t 1− iωτ →=−jnev =σω() E AC conductivity 2 σ 0 ne τ σω()= , σ 0 ≡ 1− iωτ me 4πσi εω()=1+ ω AC dielectric function (uniform EM wave, k=0) 1+ 4πσi / ω for ωτ<<1 4πσi 0 ⎧ 0 εω(0, )=1+ = ⎨ 22 ωωτ(1− i ) ⎩1−ωωp / for ωτ>>1 2 2 where plasma frequency ωp =(4πne /m) • Low frequency ωτ<<1 τ～10-13-10-14, ∴ω can be as large as 100 GHz nnin==+ε RI 1/2 ⎛⎞11 2 ⎪⎧2πσ00 / ω for σ/ ω << 1 where nI =−⎜⎟ +1 +() 4πσ0 / ω =⎨ 22 ⎝⎠⎩⎪ 2/πσ00 ω for σ/ ω >> 1 • Attenuation of plane wave due to nI exp(ik⋅= r ) exp( inω / ckˆ ⋅ r ) ˆ ˆ = exp(inRω / ck ⋅r )exp(−nkIω /cr⋅ ) exponential decay • High frequency if positive ion charges can be distorted ωτ>>1 22 22~ 22 εω()=−11 ωωp / → ε ion − ωω p / = ε ion ( − ωω p /) ~ ~ ω < ω p → ε()ω < 0ω >→ω p 01<ε()ω < EM wave is damped EM wave propagates with phase velocity C/√ε What happens near ω=ωp? Shuttle blackout ω =→ω p ε()ω = 0 can have longitudinal EM wave! Homework: Assume thatkE⋅≠0 , then from (a) Gauss’ law, (b) equation of continuity, and (c) Ohm’s law, show that 4πσ ()()ω ρ ω = iωρ ()ω Also show that this leads to ε(ω)=0. • On the contrary, if ε(ω) ≠0, the EM wave can only be transverse. Energy dispersion • When ik⋅= E 40πρ ≠, there exists charge oscillations called plasma oscillation. (our discussion in the last 2 pages involves only the uniform, or k=0, case) • A simple picture of (uniform) plasma oscillation mu =− eE =−(4π ne2 ) u ∴ u oscillates at a frequency ωπ= 4/ne2 m • For copper, n=8×1022 /cm3 16 ωp=1.6×10 /s, λp=1200A, which is ultraviolet light. Experimental observation: plasma oscillation in Al e metal Plasmon 15.3 eV bulk plasmon 10.3 eV surface plasmon • The quantum of plasma oscillation is called plasmon. • Dielectric function; EM wave in solids • Plasmon oscillation -- plasmons • Electrostatic screening • Electron-electron interaction • Mott metal-insulator transition • Electron-lattice interaction -- polarons • Photon-phonon interaction -- polaritons • Peierls instability of linear metals Electrostatic screening: Thomas-Fermi theory (1927) Valid if ψ(r) is very smooth within λF 2 2 2/3 • Fermi energy: επF ()rnr≈ () 3 () 2m 2 • Electrochemical 2 2/3 με=−=F ()rer φ ()() 3 π n0 potential: 2m 3 ne2 ρ ()renrn=− (() − ) ≈− 0 φ ()r ind 0 2 μ 3 ne2 or ρ ()k =− 0 φ ()k ind 2 μ ρ ()k ρ ()k 3 ne2 φ() k Dielectric function ε(,)k 011==−=+ext ind 0 ρ()k ρμ()k 2 ρ()k but kk2φπρ()= 4 () k 22 knes 2 6π 0 ⎛⎞2 3 n0 ∴ ε (kk)1=+ 2 where s ≡ ⎜⎟==4(),()πμμeD D k μ ⎝⎠2 μ 2 • For free electron gas, D(ε ) = mk /2π2, a = F F 0 me2 2 → (ks/kF) =(4/π)(1/kFa0) ～ O(1) • Screening of a point charge For φφπ(rQr )=⇔ / ( k ) = 4 Qk / 2 ext ext φ ()k Q φπ()k ==ext 4 ε ()k kk22+ s dk3 Q ⇒=φφ()rkee ()ik⋅ r =−krs ∫ (2π )3 r 22 22 Comparison: εω(,01 )=− ωωεps / ;(=1+kkk ,) 0 / Why e-e interaction can usually be ignored in metals? • 221 e KU≈≈, mr2 r Umer2 ≈=r 2 Typically, 2 < U/K < 5 Ka B • Average e-e separation in a metal is about 2 A Experiments find e mean free path about 10000 A (at 300K) At 1 K, it can move 10 cm without being scattered! Why? • A collision event: k1 k3 k2 k4 • Calculate the e-e scattering rate using Fermi’s golden rule: 12π 2 =−∑ fV||ee iδ ( E i E f ) τ if, Scattering 22The summation is over all possible fV|ee | i= kk34 , | V ee | kk 12 , amplitude initial and final states that obey EEE=+; E =+ EE if12 34 energy and momentum conservation. Pauli principle reduces available states for the following reasons: 2 Assume the scattering amplitude |Vee| is roughly of the same order for all k’s, then −1 2π 2 τ ≈ Vee 1 ∑ ∑ E1+E2=E3+E4; kkkk1234,, k1+k2=k3+k4 • Two e’s inside the FS cannot scatter with each other (energy conservation + Pauli principle). At least one of them must be outside of the FS. Let electron 1 be outside the FS: • One e is “shallow” outside, the other is “deep” inside also cannot scatter with each other, since the “deep” e has nowhere to go. • If |E2| < E1, then E3+E4 > 0 (let EF=0) 1 But since E1+E2 = E3+E4, 3 and 4 cannot be very far from 2 the FS if 1 is close to the FS. Let’s fix E1, and study possible initial and final states. 3 (let the state of electron 1 be fixed) 3 • number of initial states = (volume of E2 shell)/Δ k 3 number of final states = (volume of E3 shell)/Δ k (E4 is uniquely determined) -1 3 3 • τ ～ V(E2)/Δ k x V(E3)/Δ k ← number of states for scatterings 2 VE()422≅−π kF || k kF 2 VE()433≅−π kF || k kF -1 3 2 2 2 ∴τ ～ (4π/Δ k) kF |k2-kF|×kF |k3-kF| 3 3 2 Total number of states for particle 2 and 3 = [(4/3)πkF / Δ k] • The fraction of states that “can” participate in the scatterings 2 = (9/kF ) |k2-kF|× |k3-kF| 2 ～ (E1/EF) (1951, V. Wessikopf) Finite temperature: 2 -4 ～ (kT/EF) ～ 10 at room temperature → e-e scattering rate ∝ T2 • need very low T (a few K) and very pure sample to eliminate thermal and impurity scatterings before the effect of e-e scattering can be observed. insulator conductor 3 types of insulator: 1. Band insulator (1931) [due to e-lattice interaction] e-e interaction is not always unimportant! Si:P alloy 2. Mott insulator (1937) A sharp (quantum) [due to e-e interaction] phase transition insulator metal 1977 3. Anderson insulator (1958) [due to disorder] • Localized states near band edges 1977 localized states disorder Insulators, boring as they are (to the industry), have many faces. Band insulator (Wilson, Bloch) Mott Anderson Quantum Hall Topological insulator insulator insulator insulator Peierls Hubbard Scaling theory transition model of localization 1930 1940 1950 1960 1970 1980 1990 2000 2010 2D TI is also called QSHI • Dielectric function; EM wave in solids • Plasmon oscillation -- plasmons • Electrostatic screening • Electron-electron interaction • Mott metal-insulator transition • Electron-lattice interaction -- polarons • Photon-phonon interaction -- polaritons • Peierls instability of linear metals Electron-lattice interaction - polarons 極化子 • rigid ions: band effective mass m* • movable ions: drag and slow down electrons • larger effect in polar crystal such as NaCl, smaller effect in covalent crystal such as GaAs • The composite object of “an electron + deformed lattice (phonon cloud)” is called a polaron. • # of phonons surrounding the electron KCl KBr deformation energy 1 ≡ α α 397.. 352 ω ()k → 0 2 L mm* /..050 043 m* e m* ≈ for small α mm* /..125 093 pol 16−α / pol e Phonons in metals • Regard the metal as a gas of electrons (mass m) and ions (mass M) The ion vibrating frequency appears static to the swift electrons 22 ∴=+use ε el(,)kkk01 s / However, to the ions, 22 εωion (,014 )=− πne / M ω total ε(k,)ω = ε + ε − 1 For a derivation, el ion see A+M, p.515. k 2 4πne2 = 1+ s − k 2 Mω 2 • The longitudinal charge oscillation (ion+electron) atε=0 is interpreted as LA phonons in the Fermi sea. 2 2 6π ne • For long wave length (both ω and k are small), we have ks ≡ ε F 12/ Longitudinal ωυ==kmM,(/) υ3 υF sound velocity For K, v=1.8×105 cm/s (theory), vs 2.2×105 cm/s (exp’t) Resonance between photons and TO phonons: polaritons 電磁極化子 (LO phonons do not couple with transverse EM wave. Why?) EM wave (ω=ck) ω • dispersion of T(k) ignored in the active region ω • ignore the charge cloud distortion of ions Optical mode • assume there are 2 ions/unit cell resonance + - + - Acoustic mode Dipole moment of a unit cell k peuu= ()+−−≡ eu then M ukuu +−=( ) eEt ( ) ++ + − M −−ukuu−−=−( + − ) eEt ( ) →+= Mu ku eE, −−−111 where MMM=++− is the reduced mass Consider a single mode of uniform EM wave −−itωω it Et()== Ee00 , then u ue e →−ωω22uu+ = E, where ω 2 ≡ kM/ T M T • Total polarization P=Np/V, N is the number of unit cells 2 ne/ M Damping region ∴ P ==+22ED,4 Eπ P ωωT − ε(ω) 4/π neM2 Therefore, εω( )= 1+ ωω22− T ε(0) If charge cloud distortion 4/π neM2 ε(∞) → ε + of ions is considered, ∞ 22 ωT −ω 4π ne2 • Static dielectric const.
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