Optical Conductivity of 2D Electron Systems

Transcription

1 Otical Conductivity of D Electron Systems Eugene Mishcheno University of Utah E.M., M. Reizer and L. Glazman, PRB 69, (004) A. Farid and E.M., cond-mat/ TAMU, Mar 8 006

2 Outline -- Quasiarticles and collective modes (lasmons) in Fermi Liuid -- Absortion, otical conductivity, Landau daming -- RPA -- Plasmon attenuation: beyond RPA -- D system with sin-orbit couling: interlay of sin-orbit couling and many-body effects

3 Fermi Liuid ξ = F m m i t Γt ψ e ξ F uasiarticles are well defined: Γ ξ + Γ = π dd r r U n(1 n+ )(1 n ) = + RPA Γ = ξ / E, 3D F ( ξ / EF)ln( EF / ξ), D Quinn & Ferrel (1958) Chali (197), Zheng & Das Sarma (1996) Jungwirth & MacDonald (1996)

4 Continuity euation: Euation of motion: Poisson euation: Collective modes (lasmons) Hydrodynamic aroach (euivalent to RPA) r ωδρ = r j r r mω j/ e= eϕ eϕ = V δρ ω = n V m Short-range interaction: V = V ω zero sound Unscreened interaction: 3D V D V 1 1D V ln ω = const ω ω = ln

5 Absortion in Fermi Liuid ω + Single-article channel: Landau Daming r r ω = ξ ξ v < v + F ω ω vf No Landau Daming Landau Daming Plasmons do not decay via the single-article channel F Go beyond RPA!

6 Beyond RPA: two-article channel Landau daming: ω + ω = r v r < v F Two-air channel: Q+ ω + Q Two airs moving in oosite direction can have large energy while having negligible total momentum

7 Beyond RPA: two-article channel = + Π Π = RPA screened RPA interaction Π = D: DuBois & Kivelson, 1969

8 D diagrammatic calculations, Reizer & Vinour: 000 wrong Our method: many-body transitions in the resence of external field r φ( xt, ) = φ e + φ e r r r r iωt+ i x * iωt i x 0 0 dw = absorti o n emission dt dw = dt φ0 σ '( ω, ) ω all real transitions real art of the otical conductivity Q+ + Q

9 Relation between lasmon attenuation and otical conductivity otical conductivity σ '( ω, ) measures absortion at freuency ω and wavevector lasmon attenuation ω ω ω = n V m Γ = V '(, ) e σ ω = + Π Π ( ω, ) = i σ( ω, ) ωe Ward identity

10 Formalism: Golden Rule Two-article wave function 1 Ψ = ( ψ ( x1) ψ ( x) ψ ( x) ψ ( x1) ) Need transition robabilities: Ψ Ψ ' ' Ψ ih = H0 Ψ+ H1 Ψ t H = eφ ( x r ) + eφ ( x r ) + V( x r x r ) H 0 h h = m m 1 ω Second-order time-deendent erturbation theory in H 1 Q+ + Q

11 Transition amlitudes π r r r r r W ' ' = M δξ ( + ξ ξ' ξ ' + hωδ ) ( + ' ' + h) h ' ' M = + ' ' ' ' + ' ' + ' + exchange grahs + V ± V ' ' eφ 0 ξ ξ + + ω How to read grahs: = ' eφ 0 virtual 1 = ( E E ) i V = V

12 Otical conductivity σ '( ω, ) = 1 φ 0 dq dt dw dt abs = W n n (1 n )(1 n ) ' ' ' ' ' ' dq dw dw dt dt dt dw dt dw = Detailed balance dt rincile abs em = + em abs ex( ω / T ) M = + ' ' ' + ' ' + ' + ' ' ' Each matrix element is not small in but they twice interfere airwise almost canceling each other leading to M Otical conductivity vanishes in the homogeneous limit as σω (, ) 0

13 Plasmon attenuation e EF σ ω ω π π ω ω '(, ) = ( T) ln 1 + F V Γ = = + EF σ '( ω, ) ( ) ln ω πt e 4π F E F ω ω Γ ω 5/ Plasmon attenuation is indeendent of the electron charge: Screened interaction (unitary limit)

14 1. Discussion of otical conductivity e EF σ ω ω π π ω ω '(, ) = ( T) ln 1 + F Logarithmic enhancement trace of one-dimensional singularities in 1D ' θ ' Almost collinear scattering Exchange rocesses are dominated by direct rocesses by virtue of the logarithm

15 . Discussion of otical conductivity e EF σ ω ω π π ω ω '(, ) = ( T) ln 1 + F Otical conductivity vanishes in the homogeneous limit 0 Galilean invariance Parabolic sectrum ε ( ) = m Electric current deends only on the total momentum and is not modified by electron-electron collisions in the homogeneous limit r j = e r m Many-body effects have no effect on σ ( ω,0)

16 Sin-orbit couling breas Galilean invariance r 1 Particle moving in electric field: E = e U In the reference frame moving with the electron velocity there is a magnetic field r r r v E B = c This magnetic field leads to the Zeeman energy which is momentum-deendent g HSO = U s r r mc r v = r m Electric current is sin-deendent and is not conserved during electron-electron interactions r r j const m = const r r j = e v = e r + H m SO Homogeneous otical conductivity can robe many-body effects

17 D electron system doants GaAs/AlGaAs heterojunction g H SO = U s r r mc H = λ( s r r ) SO z confined electrons = λ( s s) x y y x Electric field Rashba S-O Hamiltonian (Vas o 79, Bychov & Rashba 84)

18 Electron eigenstates ε ( ) Sin degeneracy is lifted by H = λ( s s ) SO x y y x E F Eigenvalues: y x ε ( ) = + aλ chirality a = 1, 1 m Eigenstates: Different Fermi momenta: a = F amλ F 1 e = ae iθ / a ψ iθ /

19 Modification of Landau daming ε ( ) Direct transitions between subbands are ossible: E F 1) Conventional Landau daming ) Combined or chiral resonance Energy constraint ω = ε ( ) ε ( ) = λ 1 Partition constraint σ '( ω) e 16 4mλ mλ < < + mλ F F λ F ω

20 Two-article channel with sin-orbit ω Q+ + Q Direct rocesses Exchange rocesses

21 Our method alied π a b c d r r r r r W ' ' = M δξ ( + ξ ξ' ξ ' + hωδ ) ( + ' ' + h) h M =, a, b +, f ', c ', d = eφ 0 A A V A af fc fd, + +, ' ', ' ξ ξ + ω a f + A ab, ' 1 = = + b a ( ) ( i( θ θ )/ i( θ θ )/ χ ) ' χ e abe ' ' Projection of sin states before and after scattering How to read grahs:, a ', b or, a ab ab A, ' A, ' ', b

22 Otical conductivity with Sin-Orbit couling ' ' M = + ' ' + ' ' + ' + ' ' The four terms interfere only once leading to M For a short-range interaction V = const e λ m V σ '( ω) ω + ( πt ) v ω F This contribution is the result of the interlay of sin-orbit couling and interaction

23 Discussion: I e λ m V σ '( ω) ω + ( πt ) v ω F Absence of logarithmic factor suression of collinear scattering ' θ ' No interlay of interactions and sin-orbit couling in 1D ε ( ) EF Chirality of colliding articles is conserved: No change in the total velocity (thus, current) during a collision in 1D

24 Discussion: II ' θ ' Large-angle scattering: θ 1 Exchange rocesses are eually imortant as the direct rocesses

25 Discussion: III e λ m V σ '( ω) ω + ( πt ) v ω F Two-air contribution is divergent for T ω 0 Three- and higher-article rocesses are imortant

26 Conclusions Sin-orbit σ '( ω) couling results in a single-air absortion 1 which is narrow in freuency σ '( ω) e 16 4mλ λ F ω Combined effects of sin-orbit couling and electron-electron interactions result in a broader contribution from many-article excitations Sin-orbit couling maes otical conductivity a robe for many-body effects