Edge Transport in Quantum Hall Systems

Transcription

1 Lectures on Mesoscopic Physics and Quantum Transport, June 15, 018 Edge Transport in Quantum Hall Systems Xin Wan Zhejiang University

2 Outline Theory of edge states in IQHE Edge excitations Hall conductance Bosonization Chiral Luttinger liquids in FQHE Haldane pseudo potential Chiral Luttinger liquid theory Edge tunneling field-theoretical calculation Weak vs strong tunneling Interference Noise current

3 Quantum Motion of Electron in a B-Field D electrons in a perpendicular B: the quantization of the cyclotron motion a discrete spectrum with macroscopically large degeneracy chiral edge current Magnetic length: lb = ℏc B 1/ eb Cyclotron frequency: insulating bulk ωc = eb B mc Landau level degeneracy: Nφ = A = Φ B Φ0 π lb

4 Skipping Orbits and Landau Levels The edge of a sample can be represented by a confining potential; the details of the electron motion depend on the form of the potential. The motion of an electron in a sample with a rigid wall confinement can be understood in classical mechanics as a skipping orbit along the edge, which arises from the reflection of a cyclotron orbit at the edge. E(FR) E(FL) = ev

5 Hall Conductance The electron velocity along the current direction v nk 1 de n = ℏ dk The electric current flowing between the two contacts can be summed by I = e n dk v nk f nk π At T = 0, den dk den e I = e v nk = π π de h occupied occupied n The integration over energy extends from L to R I = e ( R) ( L) E F EF ] [ h occupied nocc e = V = σh V h ( )

6 IQH Edge Excitations EF EF Ground state DM = 0 EF DM = (single Slater det.) DM = E 5 EF 3 DM = DM = Degeneracy lifted when nonlinearity is considered DM

7 Bosonization EF Ground state Excited state Linearizing the low-energy excitations, H = v EF dk k ψ + (k) ψ(k) π Introducing edge density operator ρ( x) = : ψ + ( x) ψ( x): = H = v dx ψ + (x)i x ψ( x) 1 ϕ (x) π x After normal ordering, the Hamiltonian becomes H = π v dx ρ ( x) Kane & Fisher in Perspective in Quantum Hall Effects, ed. Das Sarma & Pinczuk.

8 Chiral Fermi Liquid L R This is consistent with defining electron operator Ψ ( x) = ei ϕ (x ) [ϕ ( x), ρ( x ')] = iδ (x x ') + + [ρ( x), Ψ ( x ')] = δ (x x ') Ψ (x ') The Lagrangian density is L(ϕ ) = s.t. 1 x ϕ ( t v x )ϕ 4π One can show that the current for a two-terminal measurement is e I = (V R V L) h Kane & Fisher in Perspective in Quantum Hall Effects, ed. Das Sarma & Pinczuk.

9 Model Hamiltonian for the Laughlin State Laughlin wavefunction is the ground state of N q 1 H hardcore= i i< j δ ( z i z j ) Its LLL projection has a simple pseudopotential form Two-particle wavefunction ( z 1 +z ) M ( z 1 z )m Ψ L = i< j ( z i z j ) e When N = particles approach the same point, the wavefunction vanishes as q = 3 powers. Interaction can be written, in general, as Hi = 3 i z i / 4 m V m P m (1,) One produces the 1/3 Laughlin factor by V1 > 0 only In general, the Laughlin state is the zero-energy ground state of q 1 H = V m P m (i, j) m=0 i< j

10 Energy Spectrum for the Model Hamiltonian Laughlin ground state edge states Electron confinement or interaction will lift the degeneracy generically.

11 Hydrodynamical Approach Due to non-zero Hall conductance j = σ z^ E, xy Electrons near the edge drift with velocity E c B v = σ xy = ν e /h The propagation of the edge wave obeys t ρ v x ρ = 0 x ϕ ρ( x) = nh( x) = π h(x): displacement of edge The Hamiltonian of the edge wave is H = 1 dx (eρ)( E h) = dx π vν ρ where we set ℏ = 1 so h = ρ/n = ρ πc eb Wen, Quantum Field Theory of Many-Body Systems, Oxford (004).

12 Chiral Luttinger Liquid Theory We expect to get fractional conductance e G = ν h The edge action is then modified to L(ϕ ) = 1 x ϕ ( t v x ) ϕ 4πν The commutation relation becomes ρ( x) = nh( x) = x ϕ π h(x): displacement of edge [ϕ ( x), ϕ (x ')] = i π ν sgn(x x ') i ϕ ( x )/ ν Electron operator: ψ( x) = e Quasiparticle operator: Gapless chiral bosonic charge mode ψ( x) ψ (x ') = ψ( x ') ψ( x)ei π sgn( x x ') / ν ψ1 /m ( x) = ei ϕ ( x ) ψ1 /m ( x) ψ1/ m (x ') = ψ1 /m ( x ') ψ1/ m (x)ei π ν

13 Fractional Charge in Shot Noise Ψ 1/ m ( x,t ) = e i ϕ ( x,t ) De-Picciotto et al., Nature 389, 16 (1997) Saminadayar et al., PRL 79, 56 (1997)

14 Young and Double Slit Interference Double-slit experiment (1801) Thomas Young ( ) Double-slit interference demonstrates the wave nature of light and, later, other quantum particles.

15 Quasiparticle Interference Gates controlling the strength of tunneling edge of the FQH droplet path via point contact 1 path via point contact G t 1 U 1 +t U Ψ 1 = t 1 + t + ℜ { t t e iγ } We will revisit the problem in the non-abelian case in the next lecture.

16 Tunneling Setup See, e.g., Chamon et al. PRB (1997) Tunneling Hamiltonian H tun = Γ j e i ω t Ψ1/+ m ( x L, j, t) Ψ1/ m ( x R, j,t) j Ψ 1/ m (x L ) = e L x1 x i ϕ L (x, t ) e V ωj = ℏ xn R + h.c. J

17 Tunneling Setup See, e.g., Chamon et al. PRB (1997) Charge operator Q L = e dx ρ L ( x) = e dx x ϕ L π Current operator 1 [Q (t), H tun (t)] iℏ L i e i ω t + = Γje Ψ 1/ m (x L, j,t) Ψ 1/ m (x R, j, t ) + h.c. ℏ j I (t ) = J ie I (t) = ℏ i ω J t Γje j V (x j, t ) + h.c. V (t, x) = e i ϕ ( x L,t ) e i ϕ ( xr, t )

18 Perturbation Calculation See, e.g., Chamon et al. PRB (1997) Lowest order in the linear response theory t I (t ) = i dt ' 0 [ I (t ), H tun (t ') ] 0 Four out of the eight terms surviving (charge conservation) The sign of time for two terms switched (t -t) Four terms combined into two integrals from distant past to distant future. They contain, respectively, e iωjt i ω J t e

19 Perturbation Calculation See, e.g., Chamon et al. PRB (1997) Lowest-order perturbation results I (t ) = e j,k * e = e/m, g = 1/ m Γ j Γ k +Γ j Γk g (ω J, x j x k ) P g ( ω J, x j x k ) ] P [ + P g (t, x) = 0 V (t, x)v (0, 0) 0 = ~ ~ P g (ω, x ) = P g (ω, x) = i ωt dt e P g (t, x ) g (ω,0) = lim dt ei ω t P δ 0 Hg( y) = π 1 g g [δ+i (t x)] [δ+i (t + x)] 1 π g 1 = θ(ω)ω Γ[g] (δ+it )g Γ( g) J g 1/ ( y ) g 1/ Γ (g )( y ) ~ = P g (ω, 0) H g (ω x )

20 Finite Temperatures Conformal transformation (Shankar, 1990) 1 πt g sin [ π T (δ+i (t ± x))] [δ+i (t± x)] ( P g (T, t, x) = P g (T, t, x) = ) g (π T )g g g [i sinh π T (t x)] [isinh π T (t+ x)] g (T, ω, x) = P g (T,ω, x ) = P g (T, ω,0) H g (T, ω, x) P g (T, ω,0) = P (π T )g i ωt dt e g [isinh π T t ] Hm appears in interference πω = ( π T )g 1 B[g+i ω,g i ω ]e ω = ω /( π T ) δ 0 dt e iωt (π T )g g [sin π T (δ+it )] πω ω δ = ( π T )g 1 B[g+i ω, g i ω ]e e

21 N = 1: Tunneling Current For a single tunneling site (i.e., a single quantum point contact) N =1 g (ω J, 0) P g ( ω J, 0) ] = e Γ π sgn(ω J ) ω J g 1 I tun = e Γ [ P Γ [g] I (V ) V e V ωj = ℏ g 1 At finite temperature T, e V ω J= π ℏT N =1 I tun = e Γ ( π T )g 1 B[g+i ω J, g i ω J ]sinh (π ω J) For weak field V << kbt/e di σ (T ) = dv T V =0 Wen, PRB (1991) g V I

22 Tunneling Current Ohmic F[g_, w_] := *Sinh[Pi*w]*Beta[g + I*w, g - I*w];

23 Weak Tunneling vs Weak Backscattering Electron tunneling i m φl H tun t e t e im φr dt = (1 m)t dl Weak tunneling + h.c. Irrelevant! Integrating the flow equation until the cutoff is of order kt (or ev), one obtains t eff tt m 1, G t eff t T m Quasiparticle tunneling i φl t H tun t e e dt 1 = 1 t dl m ( Weak backscattering ) i φr + h.c. Relevant! (1/m)e /h G t T /m

24 Crossover Behavior v Weak tunneling t Weak backscattering C. L. Kane and M. P. A. Fisher, Edge State Transport, in Perspectives in the Quantum Hall Effect edited by S. Das Sarma and A. Pinczuk (Wiley 1997).

25 N = : Interference For N =, we can define an effective coupling such that N = I tun = e Γ eff ( π T )g 1 B [g+i ω J, g i ω J ]sinh(π ω J) where Γ eff = Γ1 + Γ + Γ1 Γ cos(ϕ 0 + θ AB ) H g (T, ω J, x) : Relative phase of the two paths θ AB : The Aharonov-Bohm phase picked up by the quasiparticle when encircling the interferometer. ϕ0

26 Decoherence Length Unequal arm Exponential decay L = vℏ g πk T Hg[Y_, xbar_] := Sqrt[Pi]*((Gamma[*g]*BesselJ[g - 1/, Y])/(Gamma[g]*(*Y)^(g 1/))) / (Cosh[g*xbar]); (See Bas J. Overbosch, PhD Thesis)

27 Noise Current The noise spectrum of the tunneling current is defined through dt e i ω t 0 { jtun (t), jtun (0)} 0 S (ω) = The main difference from the tunneling current is the anti-commutator instead of the commutator. S (ω, ω J ) = ( e ) j,k For very small measurement frequency ( ω ω J ) S (ω J ) = ( e ) Γ j Γ k +Γ j Γk g (ω J +ω, x j x k ) + P g ( ω J ω, x j x k ) P [ g ( ω J +ω, x j x k ) ] + P g (ω J ω, x j x k ) + P j,k Γ j Γ k +Γ j Γk P g (ω J, x j x k) + P g ( ω J, x j x k ) ] [ Similarly, finite-temperature results can be worked out.

28 N = 1: Tunneling Noise Current Chamon et al. PRB (1995) Again, for a single tunneling site (N = 1) Stun (T = 0, ω J = e V / ℏ) = ( e ) Γ π ω J g 1 Γ[g] At finite temperature T, S tun (T = 0, ω J ) = 4 ( e ) Γ ( π T )g 1 B[g+i ω J, g i ω J ]cosh(π ω J) e V ω J= π ℏT Therefore, S tun (V ) = e I tun (V ), for T = 0 e V Stun (T, V ) = e I tun (T, V )coth, for T 0 T even in V odd in V ( ) Valid for any N for weak backscattering.