11D Mott variable-range hopping with externalfield
Alessandra Faggionato
University La Sapienza, P.le Aldo Moro 2, Rome, Italy, [email protected]
Abstract.
Mott variable-range hopping is a fundamental mechanism forelectron transport in disordered solids in the regime of strong Andersonlocalization. We give a brief description of this mechanism, recall someresults concerning the behavior of the conductivity at low temperatureand describe in more detail recent results (obtained in collaboration withN. Gantert and M. Salvi) concerning the one-dimensional Mott variable-range hopping under an external field . Keywords:
Random walk in random environment, Mott variable-rangehopping, linear response, Einstein relation
Mott variable range hopping is a mechanism of phonon-assisted electron trans-port taking place in amorphous solids (as doped semiconductors) in the regimeof strong Anderson localization. It has been introduced by N.F. Mott in or-der to explain the anomalous non-Arrhenius decay of the conductivity at lowtemperature [20,21,22,23,25].Let us consider a doped semiconductor, which is given by a semiconductorwith randomly located foreign atoms (called impurities ). We write ξ := { x i } for the set of impurity sites. For simplicity we treat spinless electrons. Then,due to Anderson localization, a generic conduction electron is described by aquantum vawefunction localized around some impurity site x i , whose energy isdenoted by E i . This allows, at a first approximation, to think of the conductionelectrons as classical particles which can lie only on the impurity sites, subjectto the constraint of site exclusion (due to Pauli’s exclusion principle). As aconsequence, a microscopic configuration is described by an element η ∈ { , } ξ ,where η x i = 1 if and only if an electron is localized around the impurity site x i .The dynamics is then described by an exclusion process, where the probabilityrate for a jump from x i to x j is given by (cf. [1]) ( η x i = 1 , η x j = 0) exp {− ζ | x i − x j | − β { E j − E i } + } . (1) If you want to cite this proceeding and in particular some result contained in it,please cite also the article where the result appeared, so that the contribution of mycoworkers can be recognised. This can be important for bibliometric reasons. a r X i v : . [ m a t h . P R ] J un Alessandra Faggionato
Above β = 1 /kT ( k being the Boltzmann’s constant and T the absolute tem-perature), while 1 /ζ is the localization length. A physical analysis suggests thatthe energy marks { E i } can be modeled by i.i.d. random variables. In inorganicdoped semiconductors, when the Fermi energy is set equal to zero, the commondistribution ν of E i is of the form c | E | α dE on some interval [ − A, A ], where c isthe normalization constant and α is a nonnegative exponent.At low temperature (i.e. large β ) the form of the jump rates (1) suggeststhat long jumps can be facilitated when the energetic cost { E j − E i } + is small.This facilitation leads to an anomalous conductivity behavior for d ≥
2. Indeed,according to Mott’s law, in an isotropic medium the conductivity matrix σ ( β )can be approximated (at logarithmic scale) by exp (cid:0) − c β α +1 α +1+ d (cid:1) , where de-notes the identity matrix and c is a suitable positive constant c with negligibletemperature-dependence.The mathematical analysis of the above exclusion process presents severaltechnical challenges and has been performed only when ξ ≡ Z d (absence of geo-metric disorder) and with jumps restricted to nearest-neighbors (cf. [7,24]). Thislast assumption does not fit with the low temperature regime, where anomalousconductivity takes place.To investigate Mott variable range hopping at low temperature, in the regimeof low impurity density some effective models have been proposed. One is givenby the random Miller-Abrahams resistor network [1,19]. Another effective modelis the following (cf. [9]): one approximates the localized electrons by classicalnon–interacting (independent) particles moving according to random walks withjump probability rate given by the transition rate (1) multiplied by a suitablefactor which keeps trace of the exclusion principle. To be more precise, given γ ∈ R , we call µ γ the product probability measure on { , } ξ with µ γ ( η x i ) = e − β ( Ei − γ ) e − β ( Ei − γ ) . Then it is simple to check that µ γ is a reversible distribution for theexclusion process. In the independent particles approximation, the probabilityrate for a jump from x i to x j is given by µ γ ( η x i = 1 , η x j = 0) exp {− ζ | x i − x j | − β { E j − E i } + } . (2)It is simple to check that at low temperature, i.e. large β , (2) is well approximatedby (cf. [1]) exp {− ζ | x i − x j | − β | E i − γ | + | E j − γ | + | E i − E j | ) } . (3)In what follows, without loss of generality, we shift the energy so that the Fermienergy γ equals zero, we take 2 ζ = 1 and we replace β/ β . Since the aboverandom walks are independent, we can restrict to the analysis of a single randomwalk, which we call Mott random walk . We give the formal definition of Mott random walk as random walk in a randomenvironment. The environment is given by a marked simple point process ω = D Mott variable-range hopping 3 { ( x i , E i ) } where { x i } ⊂ R d and the (energy) marks are i.i.d. random variableswith common distribution ν having support on some finite interval [ − A, A ].Given a realization of the environment ω , Mott random walk is the continuous-time random walk X ωt with state space { x i } and probability rate for a jumpfrom x i to x j (cid:54) = x i given by r x i ,x j ( ω ) = exp {−| x i − x j | − β ( | E i | + | E j | + | E i − E j | ) } . (4)As already mentioned, one expects for d ≥ β → ∞ . In [3] a quenched invariance principlefor Mott random walk has been proved. Calling D ( β ) the diffusion matrix ofthe limiting Brownian motion, in [8,9] bounds in agreement with Mott’s lawhave been obtained for the diffusion matrix. More precisely, under very generalconditions on the isotropic environment and taking ν of the form c | E | α dE forsome α ≥ − A, A ], it has been proved that for suitable β -independent positive constant c , c , κ , κ it holds c exp (cid:110) − κ β α +1 α +1+ d (cid:111) ≤ D ( β ) ≤ c exp (cid:110) − κ β α +1 α +1+ d (cid:111) . (5)We point out that for a genuinely nearest–neighbor random walk D ( β ) wouldhave an Arrenhius decay, i.e. D ( β ) ≈ e − cβ , thus implying that (5) is determinedby long jumps.In dimension d = 1 long jumps do not dominate. Indeed, the following hasbeen derived for d = 1 in [2] when ω is a renewal marked simple point process.We label the points in increasing order, i.e. x i < x i +1 , take x = 0 and assumethat E [ x ] < ∞ . Then the following holds [2]:(i) If E [ e x ] < ∞ , then a quenched invariance principle holds and the diffusioncoefficient satisfies c exp {− κ β } ≤ D ( β ) ≤ c exp {− κ β } , (6)for β –independent positive constants c , c , κ , κ ;(ii) If E [ e x ] = ∞ , then the random walk is subdiffusive. More precisely, anannealed invariance principle holds with zero diffusion coefficient.We point out that the above 1d results hold also for a larger class of jumprates and energy distributions ν [2]. Moreover, we stress that the above bounds(5) and (6) refer to the diffusion matrix D ( β ) and not to the conductivity ma-trix σ ( β ). On the other hand, believing in the Einstein relation (which statesthat σ ( β ) = βD ( β )), the above bounds would extend to σ ( β ), hence one wouldrecover lower and upper bounds on the conductivity matrix in agreement withthe physical Mott law for d ≥ d = 1.The rigorous derivation of the Einstein relation for Markov processes in randomenvironment is in general a difficult task and has been the object of much inves-tigation also in the last years (cf. e.g. [10,11,13,14,15,16,17,18]). In what follows,we will concentrate on the effect of perturbing 1d Mott random walk by an ex-ternal field and on the validity of the Einstein relation. Hopefully, progresses onthe Einstein relation for Mott random walk in higher dimension will be obtainedin the future. Alessandra Faggionato
We take d = 1 and label points { x i } in increasing order with the conventionthat x := 0 (in particular, we assume the origin to be an impurity site). It isconvenient to define Z i as the interpoint distance Z i := x i +1 − x i . −2 E E E −2 0 1 2 x −2 x x E x −1 E −1 1 x=0 Z Z −1 Z Z Fig. 1.
Points x i , energy marks E i and interpoint distances Z i . We make the following assumptions:(A1) The random sequence ( Z k , E k ) k ∈ Z is stationary and ergodic w.r.t. shifts;(A2) E [ Z ] is finite;(A3) P ( ω = τ (cid:96) ω ) is zero for all (cid:96) ∈ Z \ { } ;(A4) There exists some constant d > Z ≥ d ) = 1.Note that we do not restrict to the physically relevant energy mark distribu-tions ν which are of the form c | E | α dE on some interval [ − A, A ].Given λ ∈ [0 ,
1) we consider the biased generalized Mott random walk X ω,λt on { x i } with jump probability rates given by r λx i ,x j ( ω ) = exp {−| x i − x j | + λ ( x j − x i ) − u ( E i , E j ) } , x i (cid:54) = x j , (7)and starting at the origin. Above, u is a given bounded and symmetric function.Note that we do not restrict to (4). The special form (4) is relevant when studyingthe regime β → ∞ , on the other hand here we are interested in the system ata fixed temperature (which is included in the function u ) under the effect of anexternal field.In the rest, it is convenient to set r λx,x ( ω ) ≡
0. We also point out that onecan easily prove that the random walk X ω,λt is well defined since λ ∈ [0 , Proposition 1 [5]
For P –a.a. ω the random walk X ω,λt is transient to the right,i.e. lim t →∞ X ω,λt = + ∞ a.s. Theorem 1 [5]
Fix λ ∈ (0 , .(i) If E (cid:2) e (1 − λ ) Z (cid:3) < ∞ and u is continuous, then for P –a.a. ω the followinglimit exists v X ( λ ) := lim t →∞ X ω,λt t a.s. and moreover it is deterministic, finite and strictly positive. D Mott variable-range hopping 5 (ii) If E (cid:2) e − (1+ λ ) Z − +(1 − λ ) Z (cid:3) = ∞ , then for P –a.a. ω it holds v X ( λ ) := lim t →∞ X ξ,λt t = 0 a.s. As discussed in [5], the condition E (cid:2) e (1 − λ ) Z (cid:3) = ∞ does not imply that v X ( λ ) = 0. On the other hand, if ( Z k ) k ∈ Z are i.i.d. (or even if Z k , Z k +1 areindependent for every k ) and u is continuous, then the above two cases (i) and(ii) in Theorem 1 are exhaustive and one concludes that E (cid:2) e (1 − λ ) Z (cid:3) < ∞ if andonly if v X ( λ ) >
0, otherwise v X ( λ ) = 0.The above Theorem 1 extends also to the jump process associated to X ω,λt ,i.e. to the discrete time random walk Y ω,λn with probability p λx i ,x k ( ω ) of a jumpfrom x i to x k (cid:54) = x i given by p λx i ,x k ( ω ) = r λx i ,x j ( ω ) (cid:80) k r λx i ,x k ( ω ) . In particular, if E (cid:2) e (1 − λ ) Z (cid:3) < ∞ and u is continuous then the random walk Y ω,λn is ballistic ( v Y ( λ ) > E (cid:2) e − (1+ λ ) Z − +(1 − λ ) Z (cid:3) = ∞ then the randomwalk Y ω,λn is sub–ballistic (i.e. v Y ( λ ) = 0). We point out that indeed the resulthas been proved in [5] first for the random walk Y ω,λn and then extended to thecontinuous time case by a random time change argument.We write τ x ω for the environment ω translated by x ∈ R , more precisely weset τ x ω := { ( x j − x, E j ) } if ω = { ( x j , E j ) } . We recall that the environment viewedfrom the walker Y ω,λn is given by the discrete time Markov chain ( τ Y ω,λn ω ) n ≥ .This is the crucial object to analyze in the ballistic regime. The following resultconcerning the environment viewed from the walker Y ω,λn is indeed at the basisof the derivation of Theorem 1–(i) as well as the starting point for the analysisof the Einstein relation. Theorem 2 [5,6]
Fix λ ∈ (0 , . Suppose that E (cid:2) e (1 − λ ) Z (cid:3) < ∞ and that u is continuous. Then the environment viewed from the walker Y ω,λn admits aninvariant and ergodic distribution Q λ mutually absolutely continuous w.r.t. P .Moreover, it holds v Y ( λ ) = Q λ (cid:2) ϕ λ (cid:3) and v X ( λ ) = v Y ( λ ) Q λ (cid:104) / ( (cid:80) k r λ ,x k ) (cid:105) , where ϕ λ denotes the local drift, i.e. ϕ λ ( ω ) := (cid:80) i x i p λ ,x i ( ω ) . We point that for λ = 0 the probability distribution Q defined as d Q = (cid:80) k r λ =00 ,x k E [ (cid:80) k r λ =00 ,x k ] d P is indeed reversible for the environment viewed from the walker Y ω,λ =0 n , andthat v Y (0) = v X (0) = 0. In what follows, when λ = 0 we will often drop λ fromthe notation (in particular, we will write simply r x i ,x j ( ω ), p x i ,x k ( ω ), X ωt , Y ωn ). Alessandra Faggionato
The proof of Theorem 2 takes inspiration from the paper [4]. The maintechnical difficulty comes from the presence of arbitrarily long jumps, whichdoes not allow to use standard techniques based on regeneration times. Followingthe method developed in [4] we have considered, for each positive integer ρ , therandom walk obtained from Y ω,λn by suppressing jumps between sites x i , x j with | i − j | > ρ [5]. For this ρ -indexed random walk, under the same hypothesis ofTheorem 2, we have proved that there exists a distribution Q ( ρ ) λ which is invariantand ergodic for the associated environment viewed from the walker, and that Q ( ρ ) λ is mutually absolutely continuous w.r.t. P . In particular, the methods developedin [4] provide a probabilistic representation of the Radon–Nykodim d Q ( ρ ) λ d P , which(together with a suitable analysis based on potential theory) allows to prove that Q ( ρ ) λ weakly converges to Q λ , and that Q λ has indeed the nice properties statedin Theorem 2. Let us assume again (A1), (A2), (A3), (A4) as in the previous section. Theprobabilistic representation of the Radon–Nykodim derivative d Q ( ρ ) λ d P mentionedin the above section is the starting point for the derivation of estimates on theRadon–Nykodim derivative d Q λ d Q : Proposition 2 [6]
Suppose that for some p ≥ it holds E (cid:2) e pZ (cid:3) < + ∞ . Fix λ ∈ (0 , . Then sup λ ∈ [0 ,λ ] (cid:13)(cid:13)(cid:13) d Q λ d Q (cid:13)(cid:13)(cid:13) L p ( Q ) < ∞ . The above proposition allows to prove the continuity of the expected value Q λ ( f )of suitable functions f : Theorem 3 [6]
Suppose that E [e pZ ] < ∞ for some p ≥ and let q be theconjugate exponent of p , i.e. q satisfies p + q = 1 . Then, for any f ∈ L q ( Q ) and λ ∈ [0 , , it holds that f ∈ L ( Q λ ) and the map [0 , (cid:51) λ (cid:55)→ Q λ ( f ) ∈ R (8) is continuous. Without entering into the details of the proof (which can be found in [6]) we givesome comments on the derivation of Theorem 3 from Proposition 2. To this aim,we take for simplicity p = 2. Hence we are supposing that E (e Z ) < ∞ and that f ∈ L ( Q ), and we want to prove that f ∈ L ( Q λ ) and that the function in (8)is continuous. For simplicity, let us restrict to its continuity at λ = 0. The factthat f ∈ L ( Q λ ) follows by writing Q λ ( f ) = Q ( d Q λ d Q f ) and then by applyingSchwarz inequality and Proposition 2. The proof of the continuity at λ = 0 D Mott variable-range hopping 7 is more involved. We recall that by Kakutani’s theorem balls are compact forthe L ( Q )–weak topology. Hence, due to Proposition 2, the family of Radon–Nykodim derivatives d Q λ d Q , λ ∈ [0 , λ ], is relatively compact for the L ( Q )–weaktopology. In [6] we then prove that any limit point of this family is given by . As a byproduct of the representation Q λ ( f ) = Q ( d Q λ d Q f ) and of the weakconvergence d Q λ d Q (cid:42) , we get the continuity of (8) at λ = 0.We now move to the study of ∂ λ =0 Q λ ( f ). To this aim we introduce theoperator L : L ( Q ) → L ( Q ) as L f ( ω ) = (cid:88) k p ,x k ( ω )[ f ( τ x k ω ) − f ( ω )] , f ∈ L ( Q ) . We recall that a function f belongs to L ( Q ) ∩ H − if there exists C > |(cid:104) f, g (cid:105)| ≤ C (cid:104) g, − L g (cid:105) / ∀ g ∈ L ( Q ) . Above (cid:104)· , ·(cid:105) is the scalar product in L ( Q ). Due to the theory developed by Kip-nis and Varadhan [12], for any f ∈ L ( Q ) ∩ H − , we have the weak convergence1 √ n (cid:0) n − (cid:88) j =0 f ( ω j ) , n − (cid:88) j =0 ϕ ( ω j ) (cid:1) n →∞ → ( N f , N ϕ )for a suitable 2d gaussian vector ( N f , N ϕ ). Above ϕ denotes the local drift ϕ λ with λ = 0 (cf. Theorem 2) and ω j = τ Y ωj ω , i.e. ( ω n ) n represents the environmentviewed from the walker Y ωn .Finally, we need another ingredient coming from the theory of square inte-grable forms in order to present our next theorem. We consider the space Ω × Z endowed with the measure M defined by M ( v ) = Q (cid:104) (cid:88) k ∈ Z p ,x k v ( · , k ) (cid:105) , ∀ v : Ω × Z → R Borel, bounded . A generic Borel function v : Ω × Z → R will be called a form . L ( M ) is knownas the space of square integrable forms . Given a function g = g ( ω ) we define ∇ g ( ω, k ) := g ( τ x k ω ) − g ( ω ) . (9)If g ∈ L ( Q ), then ∇ g ∈ L ( M ). The closure in M of the subspace {∇ g : g ∈ L ( Q ) } is the set of the so called potential forms (its orthogonal subspace isgiven by the so called solenoidal forms ).Take again f ∈ H − ∩ L ( Q ) and, given ε >
0, define g fε ∈ L ( Q ) as theunique solution of the equation ( ε − L ) g fε = f . (10)As discussed with more details in [6] as ε goes to zero the family of potentialforms ∇ g fε converges in L ( M ) to a potential form h f : h f = lim ε ↓ ∇ g fε in L ( M ) . (11) Alessandra Faggionato
Theorem 4 [6]
Suppose E (e pZ ) < ∞ for some p > . Then, for any f ∈ H − ∩ L ( Q ) , ∂ λ =0 Q λ ( f ) exists. Moreover the following two probabilistic rep-resentations hold with h = h f (see (11) ): ∂ λ =0 Q λ ( f ) = (cid:40) Q (cid:2)(cid:80) k ∈ Z p ,x k ( x k − ϕ ) h ( · , k ) (cid:3) − Cov( N f , N ϕ ) . (12)We point out that a covariance representation of ∂ λ =0 Q λ ( f ) as the second onein (12) appears also in [11] and [18].We give some comments on the derivation of Theorem 4 up to the firstrepresentation in (12). Fix f ∈ H − ∩ L ( Q ), thus implying that Q ( f ) = 0.Given ε > g ε ∈ L ( Q ) as the unique solution of the equation ( ε − L ) g ε = f , i.e. g ε = g fε with g fε as in (10). Then we can write Q λ ( f ) − Q ( f ) λ = Q λ ( f ) λ = ε Q λ ( g ε ) λ − Q λ ( L g ε ) λ . (13)The idea is to take first the limit ε →
0, afterwards the limit λ →
0. By theresults of Kipnis and Varadhan [12] we have that ε Q λ ( g ε ) is negligible as ε → ∂ λ =0 Q λ ( f ) = − lim λ → Q λ ( L g ε ) λ if the latter exists. On the otherhand we have the following identity and approximations for ε, λ small: − Q λ [ L g ε ] λ = Q λ (cid:104) ( L λ − L ) g ε λ (cid:105) = Q λ (cid:104) (cid:88) k ∈ Z p λ ,x k − p ,x k λ ( g ε ( τ x k · ) − g ε ) (cid:105) ≈ Q λ (cid:104) (cid:88) k ∈ Z ∂ λ =0 p λ ,x k h ( · , k ) (cid:105) ≈ Q (cid:104) (cid:88) k ∈ Z ∂ λ =0 p λ ,x k h ( · , k ) (cid:105) = Q (cid:2)(cid:88) k ∈ Z p ,x k ( x k − ϕ ) h ( · , k ) (cid:3) , (14)where L λ f ( ω ) = (cid:80) k p λ ,x k ( ω )[ f ( τ x k ω ) − f ( ω )] and h = h f (see (11)). Roughly,the first identity in (14) follows from the stationary of Q λ for the environmentviewed from Y ω,λn , the first approximation in the second line follows from (11),the second approximation in the second line follows from Theorem 3, the identityin the third line follows from the equality ∂ λ =0 p λ ,k = p ,x k ( x k − ϕ ).The above steps are indeed rigorously proved in [6]. Since, as already ob-served, ∂ λ =0 Q λ ( f ) = − lim λ → Q λ ( L g ε ) λ , the content of Theorem 4 up to thefirst representation in (12) follows from (14).We conclude this section with the Einstein relation. To this aim we denote by D X the diffusion coefficient associated to X ωt and by D Y the diffusion coefficientassociated to Y ωn . D X is the variance of the Brownian motion to which X ωt converges under diffusive rescaling, and a similar definition holds for D Y . Theorem 5 [6]
The following holds:(i) If E [e Z ] < ∞ , then v Y ( λ ) and v X ( λ ) are continuous functions of λ ; D Mott variable-range hopping 9 (ii) If E [e pZ ] < ∞ for some p > , then the Einstein relation is fulfilled, i.e. ∂ λ =0 v Y ( λ ) = D Y and ∂ λ =0 v X ( λ ) = D X . (15)If we make explicit the temperature dependence in the jump rates (7) we wouldhave r λx i ,x j ( ω ) = exp {−| x i − x j | + λβ ( x j − x i ) − βu ( E i , E j ) } , where λ is the strength of the external field. Then Einstein relation (15) takesthe more familiar (from a physical viewpoint) form ∂ λ =0 v Y ( λ, β ) = βD Y ( β ) and ∂ λ =0 v X ( λ, β ) = βD X ( β ) . We conclude by giving some ideas behind the proof of the Einstein relationfor v Y ( λ ) in Theorem 5. We do not fix the details, but only the main arguments.By applying Theorem 4 with f := ϕ and using the first representation in ther.h.s. of (12), one can express ∂ λ =0 Q λ [ ϕ ] as a function of h = h ϕ : ∂ λ =0 Q λ [ ϕ ] = Q (cid:104)(cid:88) k ∈ Z p ,x k ( x k − ϕ ) h ( · , k ) (cid:105) . Recall that v Y ( λ ) = Q λ (cid:2) ϕ λ (cid:3) (cf. Theorem 2) and that v Y (0) = 0. In [6] wehave proved the following approximations and identities (for the last identitysee Section 9 in [6]) : v Y ( λ ) − v Y (0) λ = v Y ( λ ) λ = Q λ [ ϕ λ ] λ = Q λ (cid:2) ϕ λ − ϕλ (cid:3) + Q λ [ ϕ ] − Q [ ϕ ] λ ≈ Q (cid:2) ∂ λ =0 ϕ λ ] + ∂ λ =0 Q λ [ ϕ ]= Q (cid:2) ∂ λ =0 ϕ λ ] + Q (cid:104)(cid:88) k ∈ Z p ,x k ( x k − ϕ ) h ( · , k ) (cid:105) = Q (cid:104)(cid:88) k ∈ Z p ,x k ( x k − ϕ )( x k + h ( · , k )) (cid:105) = D Y . Acknowledgements . It is a pleasure to thank the Institut Henri Poincar´e andthe Centre Emile Borel for the kind hospitality and support during the trimester“Stochastic Dynamics Out of Equilibrium”, as well as the organizers of this verystimulating trimester.
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