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 Dept of Phys For mobile positive ions M.C. Chang

Dielectric function (r, t)-space Ert (, ) = 4πρ( rt, ) Drt (, ) = 4πρ ( rt, ) ext ( ρ = ρ + ρ ) ext ind (k,ω)-space ik E( k, ω) = 4 πρ( k, ω) ik D( k, ω) = 4 πρ ( k, ω) ext Take the Fourier shuttle between spaces: 3 dk dω i( k r ωt) E( rt, ) = Ek (, ω) e, same for D 3 ( π) π 3 dk dω i( k r ωt) ρ( rt, ) = ρ( k, ω) e, same for ρ 3 ext ( π) π Dk (, ω) = ε( k, ω) Ek (, ω) (by definition) or ρext ( k, ω) = ε( k, ω) ρ( k, ω) (easier to calculate) or φ ( k, ω) = ε( k, ω) φ( k, ω) E( k, ω) = ikφ( k, ω)... etc ext Q: What is the relation between D(r,t) and E(r,t)?

EM wave propagation in metal Maxwell equations 1 B E= ; D= 4 πρ c t 1 D 4π B= + + Jext ; B= c t c Transverse wave Longitudinal wave ( ) ( ) ext ω 4πiω = ε ion σ k k E E E c c k k E k E ω 4πiσ k = + εion c ω ω c υ p = =, refractive index n = k n 4πσ i ε( k, ω)= εion + ω ε( k, ω)= iω ik E= + B ; εionk E = 4πρ c iω 4π ik B= εione + Jext ; ik B = c c ε (in the following, let ε ion ~1) ( J = σ E) ext

Drude model of AC conductivity d v me = ee() t m dt Assume Et () = Ee iωt then iωt v = ve eτ / m e v = E() t 1 iωτ j = ne v = σω ( ) E e v τ AC conductivity σ σω ( ) =, σ 1 iωτ 4πσ i εω ( )=1+ ω ne τ m e

AC dielectric function (uniform EM wave, k=) 4πσ i 1+ 4 πσ i / ω for ωτ<< 1 ε(, ω)=1+ = ω(1 iωτ) 1 ωp / ω for ωτ>>1 where plasma frequency ω p =(4πne /m) Low frequency ωτ<<1 τ~1-13 -1-14, ω can be as large as 1 GHz n= ε = n + in R I 1/ 1 1 πσ / ω for σ / ω << 1 where ni = + 1+ ( 4 πσ / ω ) = πσ / ω for σ / ω >> 1 Attenuation of plane wave due to n I exp( ik r) exp( in / ckˆ = ω r) = exp( in ω / ckˆ r ) exp( nω / ckˆ r) R I exponential decay

High frequency if positive ion charges can be distorted ωτ>>1 ε( ω) = 1 ω / ω ε ω / ω = ε ( 1 ~ ω / ω ) p ion p ion p ω < ~ ω p ε( ω) < ω ~ ω p < ε( ω) > < 1 EM wave is damped EM wave propagates with phase velocity C/ ε What happens near ω=ω p? Shuttle blackout

ω = ω p ε( ω) = can have longitudinal EM wave! Homework: k E Assume that, 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)=. On the contrary, if ε(ω), the EM wave can only be transverse. When ik E = 4πρ, there exists charge oscillations called plasma oscillation. (our discussion in the last pages involves only the uniform, or k=, case) Energy dispersion

A simple picture of (uniform) plasma oscillation mu = ee = (4 π ne ) u u oscillates at a frequency ω = πne 4 / m For copper, n=8 1 /cm 3 ω p =1.6 1 16 /s, λ p =1A, which is ultraviolet light.

Experimental observation: plasma oscillation in Al metal e Plasmon 15.3 ev bulk plasmon 1.3 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 (197) Valid if ψ(r) is very smooth within λ F Fermi energy: Electrochemical potential: /3 ε ( r F ) ( 3 π ( )) n r m μ = εf ( r ) e φ( r ) = 3π m n ( ) /3 ρ or ind 3 ne r e n r n μ φ ( ) = ( ( ) ) ( r ) 3 ne ρ μ φ ind ( k ) = ( k )

Dielectric function ρ ext ( k) ρind ( k) ε( k, ) = = 1 = 1+ 3 ρ( k) ρ( k) ne μ φ( k) ρ( k) but k φ( k) = 4πρ( k) ks 6π ne ε ( k) = 1 + where k 4 ( ), ( ) s = πedμ Dμ = k μ For free electron gas, D(ε F ) = mk F / π, (k s /k F ) =(4/π)(1/k F a ) ~ O(1) 3 n μ a = me Screening of a point charge For φext ( r) = Q/ r φext ( k) = 4 πq/ k φext ( k ) Q φ( k ) = = 4π ε ( k ) k + ks 3 dk ik r Q kr s φ( r) = φ( k) e = e 3 ( π ) r Comparison: ε(, ω) = 1 ω p / ω ; ε( k, ) =1+ ks / k

Why e-e interaction can usually be ignored in metals? K U me r K 1, mr U r = a B e r Typically, < U/K < 5 Average e-e separation in a metal is about A Experiments find e mean free path about 1 A (at 3K) At 1 K, it can move 1 cm without being scattered! Why? A collision event: k 1 k 3 k k 4 Calculate the e-e scattering rate using Fermi s golden rule: 1 π τ = Scattering amplitude i, f ee δ i f f V i ( E E ) = 3 4 ee 1 f V i k, k V k, k ee E = E + E ; E = E + E i 1 f 3 4 The summation is over all possible initial and final states that obey energy and momentum conservation.

Pauli principle reduces available states for the following reasons: Assume the scattering amplitude V ee is roughly of the same order for all k s, then π 1 τ Vee k, k k, k 1 3 4 E 1 +E =E 3 +E 4 ; k 1 +k =k 3 +k 4 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. 1 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 E < E 1, then E 3 +E 4 > (let E F =) But since E 1 +E = E 3 +E 4, 3 and 4 cannot be very far from the FS if 1 is close to the FS. Let s fix E 1, and study possible initial and final states. 1 3

(let the state of electron 1 be fixed) number of initial states = (volume of E shell)/δ 3 k number of final states = (volume of E 3 shell)/δ 3 k (E 4 is uniquely determined) τ -1 ~ V(E )/Δ 3 k x V(E 3 )/Δ 3 k number of states for scatterings V( E ) 4 π k k k F V( E ) 4 π k k k 3 F 3 F F τ -1 ~ (4π/Δ 3 k) k F k -k F k F k 3 -k F Total number of states for particle and 3 = [(4/3)πk F3 / Δ 3 k] The fraction of states that can participate in the scatterings = (9/k F ) k -k F k 3 -k F ~ (E 1 /E F ) Finite temperature: ~ (kt/e F ) ~ 1-4 at room temperature e-e scattering rate T (1951, V. Wessikopf) 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.

3 types of insulator: 1. Band insulator (1931) [due to e-lattice interaction] insulator conductor e-e interaction is not always unimportant!. Mott insulator (1937) [due to e-e interaction] insulator metal Si:P alloy A sharp (quantum) phase transition 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 insulator Anderson insulator Quantum Hall insulator Topological insulator Peierls transition Hubbard model Scaling theory of localization 193 194 195 196 197 198 199 1 D TI is also called QSHI

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 deformation energy ω ( k ) m * pol L 1 α * m for small α 1 α / 6 m m * pol KCl KBr α 397. 35. * / m 5. 43. e / m 15. 93. 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 use ε el ( k, ) = 1+ k s / k However, to the ions, εion(, ω) = 1 4πne / Mω total ε( k, ω) = ε + ε 1 el = 1+ k k s ion 4πne Mω For a derivation, see A+M, p.515. The longitudinal charge oscillation (ion+electron) atε= is interpreted as LA phonons in the Fermi sea. For long wave length (both ω and k are small), we have 1 ω = υk, υ = ( m/ 3M) / υf Longitudinal sound velocity k 6 s π ne ε F For K, v=1.8 1 5 cm/s (theory), vs. 1 5 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?) dispersion of ω T (k) ignored in the active region ignore the charge cloud distortion of ions assume there are ions/unit cell Dipole moment of a unit cell k p = e( u+ u ) eu then M u + + + k( u+ u ) = ee( t) M u k( u+ u ) = ee( t) Mu + ku = ee, where + - + - M = M + M 1 1 1 + is the reduced mass EM wave (ω=ck) ω Optical mode resonance Acoustic mode

Consider a single mode of uniform EM wave iωt iωt Et () = Ee, then u= ue e ω u+ ωt u = E, where ωt k / M M Total polarization P=Np/V, N is the number of unit cells ne / M, 4π ωt ω P = ED = E + P Therefore, εω ( ) = 1+ If charge cloud distortion of ions is considered, ε + 4 π ne / M ω ω T 4 π ne / ωt ω M ε(ω) Damping region ε() ε( ) Static dielectric const. 4π ne ε = ε + > ε MωT ω T ω L ω ωε T ωε ωl ω εω ( ) = ε ωt ω ωt ω ε ω ω εω where L T, such that ( L ) = ε

LST relation ω ω ε L = ( > 1) T ε NaCl Experimental results NaCl KBr GaAs ε< ω L /ω T 1.44 1.39 1.7 ω T ω L [ε()/ε( )] 1/ 1.45 1.38 1.8 Dispersion relation ω ω=c/ε 1/ k ω Ck ( Ck) ω ωt = = εω ( ) ε ω ω ω >> ω ω = ( C/ ε ) k LT, ω << ω LT, ω = ( C/ ε ) k ω < ω < ω no solution T L L The composite object of photon+to phonon is called polariton. ω L ω T ω=c/ε 1/ k LO mode TO mode k

Peierls instability of 1-dim metal chain metal insulator An example of 1-d metal: poly-acetylene (1958) 聚乙炔 sp bonding π- electrons delocalized along the chain Dimerization opens an energy gap -3 ev (Peierls instability) and becomes a semiconductor. 二聚 [ 作用 ]