Optical Conductivity and Plasmons in Graphene

Transcription

1 Otical Conductivity and Plasmons in Grahene Eugene Mishchenko University of Utah E.M., PRL 98, (007); E.M., arxiv: S. Gandgadharaiah, A-Kh. Farid & E.M., arxiv: Institute for Nuclear Theory, Seattle, Jan 008

2 Allotroes of carbon grahene grahite nanotube buckyball

3 Grahene band structure Tight-binding model ik τ AB ( ) ik τ AB ( ) ε A= γ e B ε B= γ e A ε γ ik AB =± e τ hoing amlitude Dirac sectrum: ε ( ) =± v

4 Grahene Flakes AFM image SEM image From Geim & Novoselov, Nature Materials 6, (007)

5 D electron system doants GaAs/AlGaAs heterojunction g H SO = U s m c H = λ( s ) SO z confined electrons = λ( s s ) x y y x Electric field z Rashba S-O Hamiltonian (Vas ko 79, Bychkov & Rashba 84)

6 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 θ i /

7 Analogy with DEG with sin-orbit couling DEG ε ( ) y E F H = ( sx y sy x) m + λ Grahene H = v σ x Magaril, Chalik & Entin, (001) σ ( ω) e 16 Chiral resonanse ħ λ F 4mλ ω Minimal conductivity σ ( ω ) = e N 16ħ Color degeneracy ( N = 4)

8 Minimal conductivity Relation of olarization oerator to conductivity Π (0) ( ω, ) = + ie ω σ ( ω) = lim 0 Π( ω, ) Π ( ω, ) = 16 N v ω σ ( ω ) = e N 16

9 Interaction corrections Two ways to calculate interaction corrections: = vσ current vertex = 1 density vertex g σ ' a = σ a σ 0 ln / ( Kv ω) g σ ' b = σ b+ σ 0 ln / ( Kv ω) σ σ a ie ω σ ( ω) = lim 0 Π( ω, ) g 8 3π σ = σ g 6 = 0 a π σ = σ 0 1+ g σ ( g) Large logarithms aear that ultimately cancel No logarithmic corrections aear

10 Higher orders 19 6π σ / σ 0 = 1+ g = σ ( g) RG redicts that g is a running constant g e g = ħv g 1+ ln / 4 ( Kv ω) Cg σ / σ 0 = 1+ g 1+ ln / 4 ( Kv ω) Sheehy & Schmalian, 007 Herbut, Juricic & Vafek, 007 Gonzales et al. 1994; Son 007; Herbut et al. 007 Sheehy & Schmalian, 007 C 1 5 = π Herbut, Juricic & Vafek 007 C 0.01 Mishchenko, arxiv:

11 DEG Electron-hole air excitations ω ω ω= ξ ξ v< v + F ReΠ> 0 ReΠ< 0 + F Plasmon excitations Electric field induces charge fluctuations eϕ ω = Vρ ω (, ) (, ) (, ) = e Π( ω, ) ϕ( ω, ) ρ ω Which induces additional electric field Plasmon euation: 1 V Π ( ω, ) = 0 Polarizaion oerator = + Π( ω, ) Random Phase Aroximation π e n ω = m

12 Grahene Electron-hole air excitations ω ω= v + + v> v ReΠ= 0 ω= v e-h excitations exist ReΠ< 0 Polarization oerator: ( T = 0) Π ( ω, ) = 16 N v ω Plasmon euation: 1 V Π ( ω, ) = 0 does not have real solutions Of course usual lasmon excitations can exist at finite temerature or doing ω e max( T, E F ) v

13 No lasmons in intrinsic grahene true or an artefact of RPA? How good is RPA in grahene? Electron self-energy: RPA g g N ( ) g ln ln K Σ = v a) b) c) exchange rainbow g e = ħvκ Smallness of interaction strength g does not imrove RPA! RPA is exact only in the limit N >> 1 In real grahene N =4

14 Mass-shell singularity Imaginary art of the electron self-energy Lifetime k Energy has to decrease: k< Difference is transferred to create e-h air with energy ω= v( k) and momentum = k Possible if ω v k k Mass-shell singularity due to almost collinear scattering

15 Imaginary art of the electron self-energy Imaginary art of the electron self-energy Lifetime Im Σ ( ε, = 0) Gonzalez et al., 1996,1999 Hwang et al., 006, 007 Off-shell self energy for Coulomb interaction in grahene Im Σ( ε v) Chubukov & Tsvelik, 006 Mass-shell singularity for short-range interaction, nodal oints of d-wave SCs

16 Mass-shell singularity k + k Σ () ( ε, ) = k Ng 4 = Ng v d ( ) Im Σ (, ) = v Θ( v) ε 0 ε dθ θ ε / v ( ) Σ ε () Im (, ) g e = ħv v v ε

17 Exchange contribution Σ () ( ε, ) = k ( ) ex Im Σ ( ε, ) = π g v ln- Θ( ε v) 3 Coefficients match the coefficients in the real art Ng 4 ( loo ) Im Σ ( ε, ) = v Θ( ε v)

18 Full RPA series N >> 1 Small-angle singularity is increased for higher orders ( RPA) Im (, ) Σ ε RPA ε v v Σ ε = π ε v ( ) Im (, ) ln v v ε

19 Polarization oerator: threshold behavior Π (0) ( ω, ) = + Electron Green s function in subband reresentation + Singular threshold behavior comes from small angle rocesses θ Π (0) ( ω, ) 3/ N 16 v v ω

20 RPA for olarization oerator in grahene Polarization oerator has a singularity at the boundary ω= v Π 0( ω, ) = 16 N v ω Π ( ω, ) = Π (0) ( ω, ) + Π (0) ( ω, ) Π (0) ( ω, ) +... = Π (0) ( ω, ) (0) 1 VΠ ( ω, ) Im Π( ω, ) Singularity is increased in higher orders g = 0 RPA rediction: Im Π (, ) 1 RPA ω N g v ω g= 0.3 v / ω g = 0.1

21 Polarization oerator: first-order terms RPA correction: Self-energy correction: Vertex correction: ' + ' + Almost collinear scattering Velocity renormalization: v v+ v ln ( K / ) θ g 4 + ' + ' ' <<

22 RPA versus vertex correction Π (1) RPA (1) Π V N ln( ω v ) Vertex correction dominates close to the threshold (note sign reversal!) The additional logarithm aears from unscreened Coulomb interaction in the almost collinear scattering of electrons roagating along the external momentum θ + ' + '

23 What haens in higher orders? Π () RPA g N ( ω, ) ( v ω) 3 5/ 3/ Ladder diagram: Π () V ( ω, ) g N ln ( v ω) v ω 3/ at least one interaction line has to carry momentum of order of external exchange diagrams involve interaction lines of momentum of order of external screened ladder diagram: two owers k from interaction lines neutralized by from the inside loo k The ladder diagrams are the more singular for ln( ω v ) >> N

24 Summation of the infinite ladder States roagating along the uer solid line belong to uer cone; lower line to the lower cone. Multile scattering events describe excitonic effect for a given article-hole air The ladder diagrams allow for analytical summation yielding:

25 Infinite Ladder imaginary x ω ω= v ( x= ) ω= u 1< x< x<1 Absortion exists as long as ImΠ 0 : a) Old region: b) New region: ω> v v > ω> u g u v = g g 1 ln, for 1 1. The new domain is the conseuence of the excitonic effect lowering of energies of e-h airs due to attractive Coulomb interaction (though there are no true bound states). Real art of olarization oerator becomes ositive for 1< x<

26 Ladder-Loo summation When but as long as v ln( ) ω v N v ln( ) 1, N 1 ω v the ladder diagrams must be considered together with loos, these are the only contributions to consider Π ( ω, ) = = = 1 V = ΠV ( ω, ) 1 V Π ( ω, ) V sum of only ladder terms

27 Plasmon resonance Π ( ω, ) = ΠV ( ω, ) 1 V Π ( ω, ) V g 1/ N x 1 N VΠ V ( ω, ) = N v ω= v(1 e ) ln ( v ω) non-interacting electrons vertex-dressed RPA bare RPA g = 0.3 N = 4

28 Conclusions Threshold behavior of self-energy and olarization oerator are controlled by small-angle scattering rocesses Non-interacting electron-hole airs: No lasmons! N Π ( ω, ) = < 0 16 v ω Electron-hole airs interacting in the final state (excitonic effect) lower their energy v > ω Π ( ω, ) > 0 and extend into domain where Plasmons can aear!