Hopping transport in disordered solids

Hopping transport in disordered solids Dominique Spehner Institut Fourier, Grenoble, France Workshop on Quantum Transport, Lexington, March 17, 2006 p. 1

Outline of the talk Hopping transport Models for electron-phonon interactions Link with classical Markov processes Random walk in a random environment Joint work with: J. Bellissard, A. Faggionato, H. Schulz-Baldes, R. Rebolledo Workshop on Quantum Transport, Lexington, March 17, 2006 p. 2

Outline of the talk Hopping transport Workshop on Quantum Transport, Lexington, March 17, 2006 p. 3

Anderson localization H = + V ω = electron Hamiltonian in a disordered solid. The low energy part of spect(h) is a.s. pure point, H ψ i = E i ψ i, E i [, ] The eigenfunctions ψ i are exponentially localized around random points x i R d. At low temperature k B T, the relevant states for transport are close to the Fermi level E F = 0 Energy Valence band (empty) E F kt H e = P [, ] HP [, ] Conduction band (full) spect(h e ) pure point (a.s.) Workshop on Quantum Transport, Lexington, March 17, 2006 p. 4

Coupling with phonons Apply to the solid a small constant electric field E H e H e,e = H e + EX, X= position operator. spect(h e,e ) a.c. with resonances E i (E) close to R-axis. electons in perturbed states ψ i (E) remain localized no transport! At temperature T > 0: coupling with phonons allows for electronic jumps between the localized states ψ i (E) Energie Ε F Ε Mott Ε i Ε j distance hopping transport r Mott Workshop on Quantum Transport, Lexington, March 17, 2006 p. 5

Conductivity σ β Let E j E i k B T, x i x j 1 = localization length. Jump rate from ψ i to ψ j : γ i j e x i x j e β max{e j E i,0} Effective jump rate (Mean Field): c i j = γ i j f i (1 f j ) f i = ( e β(e i µ) + 1 ) 1 = mean of electrons in ψi for E = 0 Ε F Energie Ε Mott Ε i r Mott c i j e x i x j e β( E i E j + E i + E j ) Ε j distance Optimal jumps for x i x j r Mott and E i E j ɛ Mott, r Mott = const. β 1/(d+1) ɛ Mott = const. β d/(d+1) (const. depends on DOS at E F ). } Mott law: σ β σ 0 exp { const.β 1 d+1, β 1 [Mott 68] Workshop on Quantum Transport, Lexington, March 17, 2006 p. 6

Outline of the talk Hopping transport Models for electron-phonon interactions Workshop on Quantum Transport, Lexington, March 17, 2006 p. 7

Electron dynamics (exact) Electron Hamiltonian Ĥe = i E i a i a i (2 nd quantization) Electronic observables A e belong to a CAR algebra A e = C -algebra generated by the creation op. a i = a (ψ i ). Electron-phonon interaction Hamiltonian (for N atoms): Ĥ e ph = λ ( ) νq ê iqx b q + b q N b q q = creation op. of a longitudinal acoustic phonon with momentum q and frequency ν q = c s q λ = coupling constant. Assume the phonons are in state ω ph at time t = 0 and are uncorrelated with the electrons ( ) A e (t) = ω ph e it(ĥ e +Ĥ ph +Ĥ e ph ) A e 1 ph e it(ĥ e +Ĥ ph +Ĥ e ph ). Workshop on Quantum Transport, Lexington, March 17, 2006 p. 8

Van Hove limit (weak coupling) Phonons = bath at equilibrium with inv. temperature β initial state ω ph = β-kms state for free bosons. Van Hove limit: λ 0, t = λ 2 τ [Davies 74] A e (λ 2 τ) e τ(l e+d) (A e ) 0. Liouvillian : L e (A e ) = i [ Ĥ e, A e ] D(A e ) = i j = Lindblad generator commutes with L e γ i j ( a i a ja e a j a i 1 2 { }) a i a i a j a j, A e a i, a i = creation & annihilation op. of an electron in state ψ i γ i j = jump rate from ψ i to ψ j as given by Fermi golden rule. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 9

When is weak coupling OK? Let f, g L 2 (R d, d 3 q), (H ph f)(q) = ω q f(q) b(f) = annihilation op. of a phonon in state f. Phonon correlation time τ c : If f, g are analytic in a strip around the R-axis, the phonon correlation function ( ) ω ph b (f)b(e iτh ph g) = f e iτh ph(e βh ph 1) 1 g decreases exponentially to 0 as τ with a rate 1/τ c. The electron dynamics is well-approximated by the semigroup of completely positive maps (e t(le+d) ) t 0 if (1) τ c t λ 2 τ 1 c (Markov limit + perturbation theory) (2) E λ 2 τ c with E = smallest energy difference E i E j for relevant jumps (adiabatic limit). Here E ɛ Mott, τ c = β λ β (2d+1)/(2d+2). Workshop on Quantum Transport, Lexington, March 17, 2006 p. 10

Current density j Apply to the solid a small constant electric field E L e L e,e = i[ĥe + E ˆX, ] D D E depends on E. ˆX = i ψ i X ψ j a i a j = 2 nd quantized position op. Solid with finite volume Ω connected to two reservoirs with chemical potentials µ µ. Dynamics in van Hove limit: semigroup (Φ t,e ) t 0 with generator L e,e + D E + electron exchanges with reservoirs Current density: j = 1 Ω lim ϕ ( eq Φt,E (L e,e + D E )( ˆX) ). t }{{} µ i E x = 0 j velocity Workshop on Quantum Transport, Lexington, March 17, 2006 p. 11 µ

Outline of the talk Hopping transport Models for electron-phonon interactions Link with classical Markov processes Workshop on Quantum Transport, Lexington, March 17, 2006 p. 12

Invariant commutative algebra Electron Hamiltonian Ĥ e = i I E i a i a i Eigenvectors in Fock space η = i I ( ) a ηi i 0, η {0, 1} I The dissipative part D of the generator commutes with L e = i[ĥe, ] (adiabatic approximation) If (i) Ĥ e has simple eigenvalues (ii) e t(le+d) has a unique stationary state ϕ. Then 1. ker L e = {Ĥe} = commutative invariant algrebra for the semigroup (e t(le+d) ) t 0. ( 2. ϕ Le (A) ) = 0 for any A A e. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 13

Decoherence Thm : If (i) Ĥ e has simple eigenvalues (ii) e t(le+d) has a unique stationary state ϕ 1. e t(l e+d) { Ĥ e defines a Markov semigroup with generator } ( Lcl f ) ( (η) = η D η f(η ) η η ) η = d dt E η( f(ηt ) ), f C({0, 1} I ) 2. The corresponding Markov process (η t ) t 0 is an exclusion process on {x i } with the jump rates γ i j µ = invariant measure. 3. For any state ϕ and any observable A A e, lim ϕ( e t(le+d) A ) = η A η dµ (η). t Workshop on Quantum Transport, Lexington, March 17, 2006 p. 14

Classical current Current density: j = 1 Ω lim ϕ ( eq Φt,E (L e,e + D E )( ˆX) ) t = 1 η L cl,e ( Ω ˆX diag,e ) η dµ,e (η) ˆX diag,e = ψ i (E) X ψ i (E) a i (E)a i(e) i E ψ i ψ j j (x i ) <0<(x j ) ( γ i j µ ( ηi (1 η j ) ) ( i j )). see [Miller & Abrahams 60] Workshop on Quantum Transport, Lexington, March 17, 2006 p. 15

Outline of the talk Hopping transport Models for electron-phonon interactions Link with classical Markov processes Random walk in a random environment Workshop on Quantum Transport, Lexington, March 17, 2006 p. 16

Random environment Let x i be random distinct points in R d with a stationary and mixing distribution ˆP. N B = points in B R d (B bounded Borel set) E ˆP (N B) = ρ B, ρ = mean density, ρ <. EX : (stationary) Poisson process (i) ˆP(N B = n) = (ρ B ) n e ρ B /n! (ii) the N B are independent for disjoint B s. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 17

Random environment (2) To each point x i is associated a random energy E i [ 1, 1]. The E i are independent and have all the distribution ν. Pick up (at random) a point among {x i } and choose it as the origin new distribution = Palm distribution ˆP 0 EX : ˆP = stat. Poisson process ˆP 0 is obtained by adding 1 (deterministic) point at x = 0. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 18

Random walk Configuration of the environment ξ = {x i, E i }. A particle located at X t at time t, starting from X 0 = 0, walks randomly on {x i } : X t Jumps are possible between any pair of points (x i, x j ), with the rate c xi x j = e x i x j e β( E i E j + E i + E j ) β = inverse temperature. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 19

Random walk (2) X t For a given configuration ξ = {x i, E i } of the environment, let P ξ be the distribution of the Markov process (X t ) t 0. x i x j, t, t 0 0, P ξ (X t0 +t = x j X t0 = x i ) = t c ξ x i x j + O(t 2 ). No explosion if E ˆ P0 (N 2 B ) <. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 20

Main results Of interest here: diffusion constant Thm 1 : in dimension d 2, 1 D β = lim t t E P 0 (E P ξ(xt 2 )) (i) D β exists and D β > 0 (normal diffusion) (ii) the process Y t = εx tε 2 converges weakly in probability as ε 0 to a Brownian motion W D. Thm 2 : Let d 2, energy distribution s.t. g 0 > 0, α 0, Then ν( E i E) g 0 E 1+α. { ( β exp D β c β (α+1)(d 2) α+1+d (c, β 0 = constants independent of β). β 0 ) α+1 α+1+d } Workshop on Quantum Transport, Lexington, March 17, 2006 p. 21

Low temperature limit Energy distribution for E 0 : ( β ln D β β 0 ν( E i E) g 0 E 1+α ) α+1 α+1+d, β [Mott 68] (heuristic). [Ambegoakar, Halperin, Langer 71] For β, the jump rates c xi x j = e x i x j e β( E i E j + E i + E j ) fluctuate widely with x i, x j the particle follows with high probability one of the optimal paths. Workshop on Quantum Transport, Lexington, March 17, 2006 p. 22