Interacting Electrons in Disordered Quantum Wires: Localization and Dephasing. Alexander D. Mirlin

Transcription

1 Interacting Electrons in Disordered Quantum Wires: Localization and Dephasing Alexander D. Mirlin Forschungszentrum Karlsruhe & Universität Karlsruhe, Germany I.V. Gornyi, D.G. Polyakov (Forschungszentrum Karlsruhe) cond-mat/ , to appear in PRL + cond-mat/

2 Outline : High T : Weak localization and dephasing in a Luttinger liquid Low T : What happens with conductivity σ(t ) in the strongly localized regime? (applicable also to quasi-1d, 2D)

3 Disorder, interference + not too much interaction strong interaction, non-fermi liquids Mesoscopics Strongly Correlated Electron Systems Mesoscopics of Strongly Correlated Electron Systems Disorder and interaction effects especially strong in 1D (quantum wires)

4 Mesoscopics: Quantum interference in disordered conductors weak-localization correction to resistivity G i A i 2 = A i 2 + i j A i A j generically A i A j 0 but: time-reversed paths: A i = A i mesoscopic conductance fluctuations (δg) 2 ( i j A i A j) 2 i j A i 2 A j 2 1 Key objects: diffusons & cooperons Π D (q, ω) Dq 2 iω Weak localization: G/G d d q/dq 2 = Π D (r, r) return probability Conductance fluctuations: (δg) 2 /G 2 d d q/(dq 2 ) 2 = drdr Π 2 D (r, r ) e-e interaction dephasing rate τ 1 φ, cuts off quantum interference

5 Strongly correlated system: single-channel quantum wire Interaction, no disorder: Luttinger-liquid Common wisdom: Non-Fermi liquid, electronic excitations do not exist, proper language: bosons (density field) Disorder, no interaction: Localization (ξ l), no diffusion Luttinger liquid + disorder σ(t ) =? Are notions of mesoscopic physics (weak localization, dephasing) applicable to a Luttinger liquid? Merging mesoscopics with strong correlations Gornyi, ADM, Polyakov, cond-mat/ , to appear in PRL

6 Quantum wires: Carbon nanotubes Dekker group, Delft

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

8 Semiconductor quantum wires: Cleaved edge overgrowth Auslaender et al (Weizmann Inst./Bell Labs), Science 2002

9 Quantum wires: Quantum Hall edges longest 1D quantum wires ever measured (several mm) Courtesy of M. Grayson (Walter Schottky Inst.)

10 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 V (0)/2πv F 1 No e-e backscattering White-noise weak ( E F τ 0 1 ) disorder, U(x)U(x ) = δ(x x )/2πν 0 τ 0 σ(t ) =? (Weak) localization? Dephasing?...

11 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 {U ψ +,σψ,σ + U ψ,σψ +,σ + H f

12 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

13 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 zero-t localization length ξ(t = 0) τ 1 2α 0 BUT! T τ 1 onset of localization Localization: L ϕ ξ 1D: τ /τ ϕ 1

14 Disordered Luttinger liquid is Fermi-liquid? 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 1: Luttinger liquid physics renormalization of disorder ( Virtual transitions with energy transfer T < ɛ < ɛ F ) interaction still important inelastic scattering, dephasing ( Real transitions with energy transfer ɛ < T ) Step 2: Dephasing: Back to fermions!

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

16 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 ImV µν(ω, q) e {iq [ x i (t 1 ) x j (t 2 ) ] iω(t 1 t 2 )} 0

17 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

18 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

19 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

20 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! quasi-1d: Ludwig, ADM 04

21 Part 2. Low temperatures: Strong localization Applicable not only to 1D but also to quasi-1d, 2D,... (with some modifications) Gornyi, ADM, Polyakov, cond-mat/ v1, cond-mat/ Basko, Aleiner, Altshuler, cond-mat/ (granular model)

22 From Metal to Insulator Philip W. Anderson Disorder-induced localization Anderson insulator D 2 all states are localized D 3 metal-insulator transition Sir Nevill F. Mott interaction-induced gap Mott insulator The Nobel Prize in Physics 1977 How does the interaction affect an Anderson insulator at finite T?

23 Anderson localization + electron-electron interaction + vanishing coupling to the external world (phonons, etc.) finite T 0 : σ(t ) =? No interaction = σ(t ) 0 for any T σ(t ) is (possibly) nonzero due to e-e interaction only! Variable-Range Hopping? Fleishman, Licciardello, Anderson 78,... No! Activation? No!

24 Intermediate T : Power-Law Hopping cf. Gogolin, Mel nikov & Rashba 75 (phonons) Onset of localization (T 1 ): dephasing rate single-particle level spacing ξ in the localization volume T 1 : σ wl σ D, 1/τ wl ϕ ξ 1/τ (1D) T 3 < T < T 1 : Strong localization but σ(t ) power-law (not exponential) function of T T 3 : 1/τ ϕ three-particle level spacing (3) ξ in the localization volume Conductivity mechanism: diffusion over localized states elementary step: inelastic scattering shift by ξ in space τ φ : life time of localized states σ(t ) σ ac (Ω) Ω=i/τφ e 2 νξ 2 /τ φ with τ ϕ from Golden Rule spinless 1D: σ(t ) σ D α 2 T τ T 1+4α

25 Low temperatures: T < T 3 PLH, T > T 3 = 1/ατ : creation of e h pairs conductivity is nonzero decay of an electron state into 3-particle states (2 e + h) possible T < T 3 : interaction matrix element V α ξ < (3) ξ 2 ξ T 3-particle level spacing in localization volume τ 1 φ V 2 / (3) ξ < (3) ξ e ε 1 T 3 < T T 1 3 T < T T T < < 3 h e e

26 T < T 3 : Localization in Fock space cf. Altshuler, Gefen, Kamenev, Levitov 97, ADM, Fyodorov 97, Silvestrov (quantum dots) matrix element V < (3) ξ no transition on the Golden Rule level. Higher orders? a) b) α δ γ β α Have to analyze V (n) / (2n+1) γ 2 γ 1 β n β n β 0 V (n) = diagrams γ 1,...γ n 1 V 1 n 1 i=1 V i+1 E i ɛ γi

27 T < T3: Localization in Fock space (cont d) Let e-h pairs occupy the region mξ, 1 m n V (n) (2n+1) [ α ( m n )1/2 T ξ ] n Error in (n!) in earlier version; acknowledgment to D.Basko, I. Aleiner optimal paths: ballistic, m n ξ V (n) (2n+1) ( T T3 )n Metal-insulator transition at T = Tc T3 σ(t < Tc) = 0

28 Anderson-Fock glass: Mapping onto Bethe lattice Interacting problem in Fock space Anderson model on the Bethe lattice Localization on the Bethe lattice: Abou-Chacra et al 73; ADM, Fyodorov 91 (Anderson model) Zirnbauer 86; Efetov (granular system, σ-model) Metal-Insulator Transition at /V = 4 ln K K: Coordination number K ξ (3) ξ T ξ : Level spacing of n = 1 states: = (3) ξ V : hopping matrix element: interaction matrix element V α ξ n=0 n=1 # n=2 transition temperature T c = ξ α ln α 1

29 Conductivity: Critical behavior T above T c : conductivity σ τ 1 φ = Im Σ level width Statistics of wave functions and local Green functions: ADM, Fyodorov 94, 97 σ(t ) exp { c α ( T Tc T c ) 1/2 } c α ln α 1 Glassy transport: high-order transitions between distant states in Fock space

30 Overview σ(t ): from weak to strong localization Conductivity σ(t ) on the log-log scale σ(τ) Drude σ(τ) Drude AFG PLH WL AFG PLH WL σ(τ)=0 σ(τ)=0 phonons phonons e ph VRH e ph PLH ξ Τ c Τ 3 Τ 1 Τ e ph VRH e ph PLH ξ Τ c Τ 3 Τ 1 Τ quasi-1d, 2D 1D

31 Summary Mesoscopics of a strongly correlated system (Luttinger liquid): weak localization and dephasing in a disordered quantum wire Outlook: Aharonov-Bohm oscillations mesoscopic fluctuations magnetoresistance all-in-one theoretical approach (renormalization + dephasing) 2D strongly correlated mesoscopics Low-T : Anderson insulator + e-e interaction σ(τ) e ph VRH σ(τ)=0 phonons AFG e ph PLH PLH Drude ξ Τ c Τ 3 Τ 1 WL Τ Anderson-Fock Glass, No variable-range hopping Metal-insulator transition, ln σ(t ) (T T c ) 1/2 Outlook: nonequilibrium effects: σ(e) 2D and 3D insulators; short-range vs. Coulomb interaction disordered arrays of quantum dots