Quantum transport in disordered systems: Part II. Alexander D. Mirlin

Transcription

1 Quantum transport in disordered systems: Part II Alexander D. Mirlin Research Center Karslruhe & University Karlsruhe & PNPI St. Petersburg mirlin/

2 Renormalization Electron-electron interaction effects: renormalization and dephasing Virtual processes, energy transfer T become stronger when T is lowered; in low d singular in the limit T 0 Altshuler, Aronov 79 (review: 85) ; Zala, Narozhny, Aleiner, Phys. Rev. B 64, (2001) ; see also Gornyi, ADM, Phys. Rev. B 69, (2004) Cf.: Fermi-liquid renormalizations, Kondo, Luttinger-liquid power laws, Fermi-edge singularity... Dephasing Real inelastic scattering processes, energy transfer T become weaker when T is lowered, vanishing as T 0 Altshuler, Aronov, Khmelnitskii 82 ; see also Aleiner, Altshuler, Gershenson, Waves Random Media 9, 201 (1999)

3 Interaction correction to conductivity: Exchange f d h g e b c a

4 Interaction correction to conductivity: Hartree a b c d e g f h

5 Interaction correction to conductivity: Diffusive regime q, ω diffusive regime: T τ 1 typical diagram (exchange): p q,ε ω R p, ε Α R p, ε Α p, ε Ω Ω e 2 σ(ω) 2 Ω2π (dp)v 2 x [GR ɛ (p)]2 [G A ɛ (p)]2 (dq) Ω 0 dɛ 2π ɛ dω iu(q, ω) 2π (Dq 2 iω) 2 τ 2 (dp)v 2 x [GR G A ] 2 = 4πτ 2 νd σ(ω) = σ 0 π 2 Ω short-range interaction σ(ω) σ 0 U(0) U(q, ω) U(0) l 1 l 1 Ω Ω 0 dɛ (dq) (dq) Dq 2 e 2 νu(0) ɛ iu(q, ω) dω (Dq 2 iω) 2 ln Ωτ, d = 2 ( ) Ω d/2 1, d 2 D

6 For T Ω : Interaction correction to conductivity (cont d) Ω T Coulomb interaction, all exchange diagrams: 1 2π ln T τ, d = 2 2 σ(t ) = e 2 ( ) T d/2 1, d 2 D Exchange Hartree: overall factor 1 3 F σ 0 Fermi liquid interaction constant, 1 < F σ 0 < 0 [ 1 ln(1 F 0 σ) ] F0 σ Sign of σ(t ): exchange: negative, usually dominates; Hartree: positive, wins for F σ 0 > 0.45 Ballistic regime T τ 1, 2D : ln T τ (E F T )τ In the ballistic regime, prefactor of the Hartree term depends not only on F0 σ, but also on all F m ρ, F m σ with m 0 and on disorder range

7 Interaction correction to tunneling DOS q, ω Tunneling DOS: ν(ɛ) = 1 A π Im GR (r, r; ɛ) p q, ε ω = 1 π Im (dp)g R (p; ɛ) p q, ε ω p q p" q A R p p" R R p, ε p, ε p, ε p, ε ν(ɛ) ν 0 = 1 π Im (dq) ɛ dω [ ] 1 U(q, ω) 2U(p p, 0) (Dq 2 iω) 2 For short-range interaction ν(ɛ) ν 0 = 1 π [ U(q = 0) 2U(p p )] Dq 2 (dq) (Dq 2 ) 2 ɛ σ(ω) Ω ɛ 2 σ 0 Manifestation: Zero-bias anomaly in tunneling di/dv

8 Interaction correction to TDOS: Coulomb interaction Dynamically screened Coulomb interaction, 2D : U(q, ω) = q κ 2πe 2 Dq2 Dq 2 iω Exchange correction to TDOS:, κ = 4πe 2 ν poor screening for Dq 2 ω ν(ɛ) = 1 2πe 2 ν 0 2π 2Im q dq dω ɛ q(dqκ iω)(dq 2 iω) dq dω In the range Dq 2 ω Dqκ the integral is of log 2 type: Dκqω ν(ɛ) = 1 ] [ln 2 Dκ2 ln 2 Dκ 2 τ ν 0 8π 2 νd ɛ This does not happen for σ! Reason: gauge invariance There is a range of parameters where σ is still small, whereas ν is not! ν(ɛ) ν 0 = exp { 1 8π 2 νd [ln 2 Dκ2 ɛ ]} ln 2 Dκ 2 τ Finkelstein 83

9 Anderson transitions: Renormalization by e-e interaction Renormalization Group: Finkelstein 83 σ-model with interaction Anderson transitions: depending on the symmetry class and the character (short- vs. long-range) of the interaction, different situations: interaction RG-irrelevant; fixed points and critical indices unchanged Example: broken spin-rotation invariance short-range interaction interaction RG-relevant; new fixed point and critical indices; phase diagram qualitatively unchanged Example: broken spin-rotation invariance 1/r Coulomb interaction interaction not only RG-relevant but also leads to new phases Example: preserved spin-rotation invariance short/long-range interaction Reviews: Finkelstein 90 ; Belitz, Kirkpatrick 94

10 RG for σ-model with interaction Coulomb interaction, N degenerate valleys, preserved T and S RG equations: { [ ]} dt 1 d ln L = t2 N 1 (2N 2 γ2 1) ln(1 γ 2 ) 1 γ 2 dγ 2 ln L = t (1 γ 2) 2 2 t = 1/2πg dimensionless resistivity γ 2 = F σ 0 /(1 F σ 0 ) interaction Finkelstein 83; Punnoose, Finkelstein 02

11 MIT in a 2D gas with strong interaction Kravchenko et al 94,... Punnoose, Finkelstein, Science 05 (number of valleys N 1; in practice, N = 2 sufficient)

12 Dephasing by e-e interaction inelastic e-e scattering processes electron decays, τ 1 φ interacting electrons Nyquist noise FDT: ϕϕ q,ω = ImU(q, ω) coth ω Noise can be considered as classical only for ω T 2T T p U(q, ω) = 1 0 (q) Π(q, ω) Π 1 (q, ω) = Dq2 iω ImU ω νdq 2 νdq 2 U 1 τ 1 φ l 1 T? T (dq) Equivalently: τ 1 νdq 2 φ 1 νd d/2t T? dω ω d/2 2 2D, quasi-1d : IR (low-q) divergence! weak localization: self-consistent cutoff: q min = l 1 φ, ω min = τ 1 φ τ 1 φ ( T νd 1/2 ) 2/3, quasi-1d τ 1 φ T g ln g, 2D Altshuler, Aronov, Khmelnitskii 82

13 Dephasing by e-e interaction (cont d) IR divergence: cutoff depends on geometry of the phenomenon. Aharonov-Bohm oscillations: harmonics decay as exp{ 2πnR/l AB φ } trajectories should encircle the ring! cutoff: q min R 1 1/τ AB φ T R νd, lab φ = ( Dτ AB φ ) 1/2 ν 1/2 D T 1/2 R 1/2 Ludwig, ADM 04 Near Anderson transition: dynamical scaling finite frequency ω length scale l ω ω 1/z smearing of the transition, width ω ζ ζ = 1/νz e-e interaction dephasing length scale l φ T 1/z T smearing of the transition, width T κ κ = 1/νz T

14 Elelectron transport in disordered Luttinger liquid Gornyi, ADM, Polyakov, PRL 05, PRB 07

15 Single-wall carbon nanotubes: Mesoscopics Man, Morpurgo (Delft), PRL 2005 conductance fluctuations magnetoconductance

16 Single-wall carbon nanotubes: Diffusive transport Li, Yu, Rutherglen, Burke,, Nano Letters 04 Electrical properties of 0.4 cm long single-walled carbon nanotubes

17 Semiconductor quantum wires: Auslaender et al (Weizmann Inst./Bell Labs), Science 2002

18 Quantum wires: Quantum Hall edges quantum wires with length of several mm Courtesy of M. Grayson (Walter Schottky Inst.)

19 Model: Disordered Luttinger liquid Single-channel wire: right(left) movers ψ µ, µ = ± Spinless (spin-polarized, σ = ) or spinful (σ = ±) electrons Linear dispersion, ɛ k = kv F Short-range weak e-e interaction, α U(0)/2πv F 1 No e-e backscattering White-noise weak ( E F τ 0 1 ) disorder, V (x)v (x ) = δ(x x )/2πν 0 τ 0 σ(t ) =? (Weak) localization? Dephasing?...

20 Disordered Luttinger Liquid: Hamiltonian H = k,µ,σ v F (µk k F )ψ µσ (k)ψ µσ(k) H e e H dis H e e = 1 2 dx { } ψ µ,σ ψ µ,σ g 2 ψ µ,σ ψ µ,σ ψ µ,σ ψ µ,σ g 4 ψ µ,σ ψ µ,σ µ,σ,σ H dis = σ } dx {V ψ,σψ,σ V ψ,σψ,σ H f

21 Bosonization and disorder averaging Giamarchi & Schulz 88 Bosonization: given disorder realization, ψ (fermionic) φ (bosonic) Interaction term quadratic in φ; impurities: cos 2φ Disorder averaging. Quenched disorder: Introduce replicas, φ n Bosonized replicated action, S[φ] = 1 2πv F n v F k 2 F π 2 τ 0 n,m u = v F K K = (1 2α) α dx dτ { [ τ φ n (x, τ )] 2 u 2 [ x φ n (x, τ )] 2 } dx dτ dτ cos[2φ n (x, τ ) 2φ m (x, τ )] powerful without impurities: Gaussian action, interaction treated exactly good for a single impurity inconvenient for disordered systems & Anderson localization

22 Renormalization of disorder Integrate out T < ɛ < ɛ F (cf. Giamarchi & Schulz 88) T -dependent static disorder (Mattis 74, Luther & Peschel 74...) renormalized Drude conductivity τ (T ) = τ 0 (T /ɛ F ) 2α σ D (T ) = e2 n e τ (T ) m T 2α Physical meaning: scattering off Friedel oscillations T τ > 1 : independent renormalization of weak impurities T τ 1 : renormalization stops BUT! zero-t localization length ξ(t = 0) τ 1 2α 0 conductivity is given by Drude formula ( weak.loc. corrections) only if dephasing is strong: L ϕ ξ 1D: τ ϕ τ How to calculate weak localization and dephasing?

23 Disordered Luttinger liquid as Fermi liquid Step 1: Luttinger liquid physics renormalization of disorder ( Virtual transitions with energy transfer T < ɛ < ɛ F ) T < ɛ < ɛ F integrated out: τ 0 τ (T ), ɛ F T : all power-law singularities (ɛ F /T ) 2α now in τ (T ) Step 2: interaction still important inelastic scattering, dephasing ( Real transitions with energy transfer ɛ < T ) Back to fermions! Alternative (all-in-one) approach: functional bosonization

24 Weak localization in 1D: τ φ τ = WL (dressed by e-e interactions ) Minimal Loop : Three-impurity Cooperon f i coherent backscattering L φ

25 Weak localization in Luttinger liquid: Spinless case Hubbard-Stratonovich transformation path-integral for Cooperon in a fluctuating field created by other electrons cf. higher dimensions, Altshuler, Aronov & Khmelnitsky 82 σ wl = 2σ D 0 dt 0 dt a P (t, t a ) exp[ S(t, t a )] P (t, t a ) = (8τ 2 ) 1 θ(t 2t a ) exp( t/2τ ) return probability S(t, t a ) dephasing action, S(t, t a ) = S ff S bb S fb S bf, x f (t) forward, x b (t) backward paths dω dq S ij = 2π 2π t 0 t T dt 1 dt 2 ω ImU µν(ω, q) e {iq [ x i (t 1 ) x j (t 2 ) ] iω(t 1 t 2 )} 0

26 Weak localization in Luttinger liquid (cont d) Interaction propagator in S ij expanded in t/τ : S ij = πα 2 v F T t 0 t dt 1 dt 2 F µν [ x i (t 1 ) x j (t 2 ), t 1 t 2 ], 0 F (x, t) δ(x v F t) (1 t /2τ ) F (x, t) θ(v 2 F t2 x 2 )/4v F τ xf,b (t 1 ) F v F t a 0 b t t 1 F f Interaction line contribution to S (N f N b ) 2 N f, N b number of intersections with forward and backward paths

27 Weak localization in Luttinger liquid: Results S(t, t a ) = 2πα 2 T t a (t 2t a ) τ dephasing action S τ 1 dephasing needs disorder in RPA-interaction propagator! Weak Localization correction: σ wl = 1 ( ) τ wl 2 4 σd φ ln τ τ τφ wl 1 α 2 T ln(α2 T ) WL dephasing rate: 1 τ wl φ ( πt = α τ ) 1/2

28 Conductivity at high T : Summary σ(τ) Drude WL strong localization 1/τ Τ 1 =1/α 2 τ Τ 1 /α ε F Τ T 1 = 1/α 2 τ : localization strong for T < T 1 Drude conductivity : σ D T 2α Localization correction : σ wl = 1 4 σd ( ) 2 τφ wl ln τ τ τφ wl 1 α 2 T ln(α2 T ) Temperature dependence of σ(t ) determined by WL for T < T 1 /α

29 Aharonov-Bohm dephasing: Quantum interference in 1D rings Aharonov-Bohm oscillations gosc cos 2πΦ Φ0 e 2πR/lAB φ 1/τ AB φ = πα 2 T τ AB φ τ wl φ Dephasing in 1D rings and wires parametrically different! cf. quasi-1d: Ludwig, ADM 04