Plasmons, Polarons, Polaritons

Home , Plasmon, Polariton, Polaron, Surface 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

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(,) =4 (,) rt πρext ik⋅= D(, kω ) 4πρ ω ext (, k )

()ρ = ρρext+ ind

Take the Fourier “shuttle” between 2 spaces: dk3 dω E(,)rt= ππ Ek (, ) eikr()⋅−ω t , same for D ∫ (2 )3 2 ω 3 dk d ikr()⋅− t ρ(,)rt= ππρωω (, k ρ ) e ,ω same for ∫ (2 )3 2 ext Dk ( ,ωεω )= ( k , ) Ek ω ( , ) (by definition) ρωεωρω or ext (,kkk )= (, )(, ) (easier to calculate) φωεωφω ω φω or ext (,kkkEkikk )==− (, )(, ) ∵ (, ) (, )... etc

Q: What is the relation between D(r,t) and E(r,t)? EM wave propagation in metal

Maxwell equations ωπ 1 ∂B iω ∇×ED= − ; ∇⋅ = 4πρ ik×+ E= B ε ; ε k ⋅= E 4πρ ct∂ π ext c ion ext 14∂D i 4 ∇×BJB=; + + ∇⋅ =0ik×− B=; E + J ik ⋅= B 0 ct∂ c ext ccion ext JE= σ ωπω2 4 i ( ) →×kkE() ×= −22ion E − E || ccεσ 2 kkE()⋅− kE • Transverse wave 2 2 ωπσ⎛⎞4 i k =+2 ⎜⎟ion υ c ⎝⎠ε c ∵ ==ω , refractiveω index n = ε εωεp kn 4πσi (in the following, ∴ (,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 −it ω vve= 0 emτ / →=−vEe ()t 1− i ωτ →=−jnev =() E σω AC conductivity σω ne2τ 0 σ ()= , σ 0 ≡ 1− i me ωτ εω4 i ()=1+ πσ ω εω AC dielectric function (uniform EM wave, k=0) πσ ωωτ 4 i ⎧1+ 4πσi ω ωτ / for << 1 (0, )=1+ 0 = 0 ⎨ ωω22 ωτ (1− i ) ⎩1− p / for >>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 ⎪⎧200 / for /<< 1 where nI =−⎜⎟ + 1 +() 40 / =⎨ 22 ⎝⎠⎩⎪ 2/00 for />> 1 • Attenuation of plane wave due to n πσ ωI exp(ik⋅= r ) exp( inω / ckˆ ⋅ r ) πσ σ ω ω ˆ ˆ = exp(inR ω / ck ⋅r )exp(−nkI /cr⋅ πσ ) σω ω exponential decay ω ? p < ω = /) ω ω ()

~ ε √ε 01 < p

~ ω >→

ω EM wave propagates with phase velocity C/ What happens near <

ω () ε p ion p ion p 22 22 22 0 if positive ion charges can be distorted → p 11 =− → − = − ~ ω < () / / (

εω ωω ε ωω εω ωω EM wave is damped >>1 τ ω Shuttle blackout • High frequency n only be transverse. Energy dispersion

(a) Gauss’ (a) (b) law, ρ ω )=0. i ω ( = ) Ohm’s law, show that 0, the EM wave ca 0, the EM wave ε ω ≠

ρ ) 0 ω EM wave!

ω ( ()() () = ε

⋅≠ π

4 πρ ω kE 40 () 0 () 0 ε longitudinal ⋅= ≠ p ik E ω =→ Homework: thatAssume equation of continuity, and (c , then from

ω can have Also show that this leads to • On the contrary, if • When , there • , When 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 oscillatesωπ at a frequency = 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 2/3 • Fermi energy: επ()rnr≈ () 32 () F 2m 2 • Electrochemical 2/3 με=−= φ() πrer ()() 3 2 n potential: F 2m 0

3 ne2 ρ ()renrn=− (() − ) ≈− 0 ()r ind 0 2 3 ne2 or ()k =− 0 ()k ind 2 φ ρ μ

μ φ k () () φ ρ 0 2 μ n 0 ne k 3 2 3 2 2 2 me = 0 a k () k () , ρμ 2 ρ kkk 2 π (=1+ eD D 2 πμμ / F 2 == ⎛⎞ ⎜⎟ ⎝⎠ k () k () s ext ind kr ps 22 22 ρ 0 ρ −

μ ) = mk O(1) π F ==−=+ r 6 ε ( ～ 011 ≡ ) s D k 2 s 0 =− ⋅ (,) a ε ik r + 4(),() F Q 22 kk 01 0 (, ) / ; ,) / )(1/k 4 3

where εω ωωε π = 3 22 2 s π kne k dk Q (2 ) k () () =⇔ = =(4/ k () ∫ 2 φπρ () ) ext 2

rQr k Qk ε F =+ φ kk k )1 /k rkee == s ext ext kk () () ( φφπ (k

ε φφ k () 4 but Dielectric function ∴ → • free electron gas, For φπ For ( ) / ( ) 4 / ⇒= = Comparison: • of a point charge Screening 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 τ=−∑ fV||ee i ( E i E f ) if, Scattering 22δ The 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 397.. 352 ω ()k → 0 2 L mm* /..050 043 m* e m* ≈ for smallα mm* /..125 093 pol 16− / pol e α α 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 ω / M

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 ≡ εF 12/ Longitudinal ωυ==kmM,(/) υ υ 3 F 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 singleωω mode of uniform EM wave −−itωω it Et()== Ee00 , then u ue e →−22uu+ = 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 ==+εω22ED,4 Eπ P ωωT − π ε(ω) 4/neM2 Therefore, ( )= 1+ωω 22− T ε(0) If charge cloud distortion ε 4/π neM2 ε(∞) → + of ions is considered, ∞ 22 ωT −ω 4π ne2 • Static dielectric const. ω ω ε 0 = εε∞ +>2 ∞ T L ω M T ω ωεεω 22 22 ωεT ωε0 − ∞ ⎛⎞L − ∴ ()=≡∞ ⎜ ⎟ 22−− 22 ωωT ωω⎝⎠T ωω 220 where L ωω≡ ε T , such that ()L = 0 ∞ ε εω NaCl ω 2 • LST relation L ε 0 ωε2 = (1)> T ∞

Experimental results

NaCl KBr GaAs ε< 0

ωL/ωT 1.44 1.39 1.07 ωT ωL [ε(0)/ε(∞)]1/2 1.45 1.38 1.08

ω=C/ε 1/2k Dispersion relation ω ∞

22 2 2 2 2 Ck() Ck − T ω == 22 () ∞ − L LO mode εω ε ωL ωω TO mode ωω>> ω → ε = (/Ck ) LT, ωω∞ ωT

ωω<polariton. Peierls instability of 1-dim metal chain

metal insulator

• An example of 1-d metal: poly-acetylene (1958) 聚乙炔

•sp2 bonding • Dimerization opens an energy 二聚[作用] • π- electrons delocalized along gap 2-3 eV (Peierls instability) and the chain becomes a semiconductor.