Diffusion of a particle in the Gaussian random energy landscape: Einstein relation and analytical properties of average velocity and diffusivity as functions of driving force
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Diffusion of a particle in the Gaussian random energy landscape:Einstein relation and analytical properties of average velocity anddiffusivity as functions of driving force
S.V. Novikov
1, 2, ∗ A.N. Frumkin Institute of Physical Chemistry and Electrochemistry,Leninsky prosp. 31, 119071 Moscow, Russia National Research University Higher School of Economics,Myasnitskaya Ulitsa 20, Moscow 101000, Russia
Abstract
We demonstrate that the Einstein relation for the diffusion of a particle in the random energylandscape with the Gaussian density of states is an exclusive 1D property and does not hold inhigher dimensions. We also consider the analytical properties of the particle velocity and diffu-sivity for the limit of weak driving force and establish connection between these properties anddimensionality and spatial correlation of the random energy landscape.
PACS numbers: 05.40.Jc,05.60.Cd,72.80.Le,72.80.Ng ∗ [email protected] . INTRODUCTION Fundamental feature of a simple diffusion process is the validity of the Einstein relation(ER) between the diffusivity D and drift mobility µ . A particular but very important exam-ple of the diffusive transport is the motion of charge carriers in amorphous semiconductorsunder the action of the applied electric field E , where the ER takes form D = kTe µ. (1)Relation (1), apart from the clear fundamental importance, serves as a very useful tool forthe estimation of D in many materials demonstrating hopping charge transport. Indeed,the mobility could be rather easily measured in experiments, e.g. by the time-of-flighttechnique, while the direct measurement of D is much more difficult. At the same time, inmany materials the mobility depends on E and the simple Einstein relation (1) is not valid.It was found that for the case of the 1D transport in disordered materials with the Gaussiandensity of states (DOS) the properly modified Einstein relation is valid [1, 2] D = kTe ∂V∂E , (2)where V is the average carrier velocity. This very relation may be rewritten in a morebeautiful form. Indeed, if we let the magnitude of disorder goes to zero while keeping allother relevant parameters the same, then for the resulting system the simple Einstein relation D = kT µ /e is certainly valid (here the corresponding diffusivity and average velocity are D and v = µ E ), so Eq. (2) is equivalent to DD = ∂V∂v . (3)In this form the modified Einstein relation (mER) contains no parameters such as e , T , etc.In future we will use this very form of the mER. A natural question is whether Eq. (3) couldbe extended to the multidimensional case. In this paper we are going to demonstrate thatthe mER is strictly the 1D relation which could not be extended to higher dimensions.We consider the continuous model of the carrier diffusion in the random energy landscape U ( ~x ). It provides a proper description of the long time behavior of the hopping charge carriertransport. In fact, in the strict sense nether ER nor mER is valid for the lattice model of thehopping transport. Indeed, let us consider the simplest model of the hopping to the nearest2eighbors only for the 1D chain without disorder and for the Miller-Abraham hopping rate[3]. In this case a simple calculation gives for the velocity v and diffusivity Dv = ν a sign( E ) (cid:0) − e −| λ | (cid:1) , (4) D = 12 a ν (cid:0) e −| λ | (cid:1) . (5)here ν is the scale of the hopping rate, a is the lattice scale, λ = eEa/kT . Both Eqs.(1) and (2) are invalid for this model, apart from the limit λ ≪
1. This phenomenon isnot a specific property of the Miller-Abraham hopping rate, because the use of an arbitraryhopping rate leads to the substitution ν → ν f ( | E | ), with some function f ( | E | ) going to aconstant at E →
0. Again, Eq. (1) or (2) hold only in the limit λ ≪
1. We may concludethat the ER or mER are not valid for the lattice hopping models even in the ideal case ofabsolutely ordered 1D lattice. For this reason we limit our consideration to the continuousdiffusion model.To avoid a possible confusion we mention here another generalization of the Einsteinrelation, typically called the generalized ER (gER), which is specifically tailored for thecharge transport in the case of not very low charge density [4–8]. The gER for the GaussianDOS has the form D = g ( n, T ) kTe µ, (6)where the enhancement factor g depends on the carrier density n and T . In contrast to Eq.(6), the relation (2) is valid for n → E , while the relation (6) is valid onlyin the case of field-independent µ which typically implies E → D and V on v for v → E → et al. a striking difference wasfound for the dependence of the hopping carrier velocity and diffusivity for the well knownGaussian Disorder Model (GDM) on v for different dimensionality of space [9, 10]. In thefirst paper the exact solution of the lattice 1D version of GDM has been extensively studiedand it was found that V and D are non-analytical functions of vV = A v ( T ) v + B v ( T ) | v | v + ..., (7) D = A D ( T ) + B D ( T ) | v | + ... V = A v ( T ) v + B v ( T ) v + ..., (8) D = A D ( T ) + B D ( T ) v + ... At the same time, for the 2D and 3D cases the careful numerical simulations and approximateanalytic consideration suggest that Eq. (8) provides the proper description of the dependenceof V and D on v [10]. The reason for the exceptional behavior in 1D case is not clear. Weare going to clarify the situation and try to answer the question whether the 1D case isindeed exceptional. In addition, we are going to study how the analytical properties of V ( v )and D ( v ) depends on the correlation properties of the random energy landscape. There isa natural reason to expect such connection because in 1D case it is well known that thefunctional dependence of V and D on v is directly governed by the correlation function C ( x ) = h U ( x ) U (0) i /σ [11] and computer simulation supports that connection in 3D case,too [12, 13] (here σ = h U i is the variance of the disorder and we define the correlationfunction in such a way that C (0) = 1). From the general point of view the GDM is just oneparticular case of the correlated disorder where for site energies U i the binary correlationfunction is zero for different sites: h U i U j i ∝ δ ij .A major limitation of our approach is the use of perturbation theory (PT). Yet, we willsee that the PT approach for the 1D case gives the proper functional dependence of V and D on v , and the corresponding perturbative coefficients A ptv,D and B ptv,D could be obtained by theexpansion of the exact coefficients in series in the disorder strength parameter g = ( σ/kT ) .At the same time, the result of Ref. 9 provides a reliable anchor point for the comparisonof our results with the exact solution of the particular model. Indeed, the general structureof the functional dependence V ( v ) and D ( v ) for the 1D GDM in the limit case v → g . We will see that this is a generalphenomenon for 1D hopping transport for any type of the correlation function. Thereis a general agreement that the effect of disorder on the charge carrier transport is themost prominent in the 1D case because in this case the path is predetermined and carrierinevitably has to move across all fluctuations of the random energy landscape. As a result,for all transport parameters ( V , D , etc.) the effect of the strength of disorder becomes weakerwhen the dimensionality of space d becomes higher. For example, the renormalization group4nalysis gives the leading asymptotics for µ and D for v → µ, D ≃ − d (cid:16) σkT (cid:17) , (9)which agrees well with the exact solution of the 1D case [2, 11] and computer simulation for3D case [13, 15, 16]. For this reason we may expect that if the functional form of V ( v ) and D ( v ) for v → V ( v ) and D ( v ) for v →
0, then the same istrue for any d . II. EINSTEIN RELATION
Let us consider diffusion of a particle in d -dimensional space with the random energylandscape U ( ~x ) having the spatially correlated Gaussian DOS. For the particular realizationof U ( ~x ) the particle Green function G U ( ~x, t ) obeys the equation ∂G U ∂t = D ∇ · [ ∇ G U + βG U ∇ U ] − ~v · ∇ G U , G U ( ~x,
0) = δ ( ~x ) , β = 1 kT . (10)We are going to consider the perturbation theory expansion for the Green function G ( ~k, s ) = D G U ( ~k, s ) E averaged over static disorder ( G ( ~k, s ) is the Fourier transform of the Green func-tion on ~x and Laplace transform on t ), the corresponding approach and diagrammatic ex-pansion are briefly described in the Appendix A. We limit our consideration to the stationarystate s = 0 and do not write the argument s anymore. Averaged Green function at s = 0is perfectly suitable for the description of the dynamics of the particle in a well establishedtransport regime where the initial relaxation is over and experimentally measured particlevelocity and diffusivity do not depend on time anymore. Introducing the self-energy Σ( ~k )and taking into account the usual representation of GG − ( ~k ) = D k + i~v · ~k − Σ( ~k ) , (11)we may calculate the corrections to the effective diffusivity D = D + δD and average velocity V = v + δV as δ ~V = i ∂ Σ ∂~k (cid:12)(cid:12)(cid:12)(cid:12) ~k =0 , δD ab = − ∂ Σ ∂k a ∂k b (cid:12)(cid:12)(cid:12)(cid:12) ~k =0 . (12)Using the first order correction to self-energy Eq. (A3), we obtain δ ~V (1) = i gD (2 π ) d Z d~pC ( ~p ) G ( − ~p ) p ~p, g = ( σβ ) . (13)5 D (1) ab = − gD (2 π ) d Z d~pC ( ~p ) G ( − ~p ) (cid:20)(cid:0) D p + i~v · ~p (cid:1) p a p b − i p ( v a p b + v b p a ) (cid:21) . (14)Diffusion tensor for d > ~v is diagonal D = diag( D || , D ⊥ , ..., D ⊥ ), where D || and D ⊥ are lateral and transversal diffusion coefficients,correspondingly. Hence, X a δD aa = δD || + ( d − δD ⊥ , (15) X a,b δD ab v a v b = δD || v , and δD (1) || = − gD (2 π ) d Z d~pC ( ~p ) G ( − ~p ) "(cid:0) D p + i~v · ~p (cid:1) ( ~v · ~p ) v − ip ( ~v · ~p ) . (16) δD (1) ⊥ = − gD (2 π ) d ( d − Z d~pC ( ~p ) G ( − ~p ) (cid:0) D p + i~v · ~p (cid:1) (cid:18) p − ( ~v · ~p ) v (cid:19) . (17)Let us try to extend the mER to the multidimensional case. It is easy to check that theproper extension for the mER is 1 D X a D aa = X a ∂V a ∂v a , (18)or, in the proper coordinate system with one axis parallel to ~v D (cid:0) D || + ( d − D ⊥ (cid:1) = ∂V∂v . (19)This relation is indeed valid for the first order PT, demonstrates a reasonable tensorstructure and is the only proper valid extension of the mER which is linear in D and V anddoes not explicitly depend on the effective charge g . Unfortunately, this relation does nothold for the second order PT (see Appendix B) X a (cid:18) D aa D − ∂V a ∂v a (cid:19) = O ( g ) . (20)If the mER is invalid even in the second order PT, then we may safely conclude that themER is a strict 1D relation having no reasonable extension to the multidimensional case.Why the 1D mER is valid and what is the difference in the multidimensional case?Diagrammatic approach provides a very clear explanation of this phenomenon. For example,6or the 1D case the relation for δ Σ simplifies δ Σ ( ~k ) = k g D (2 π ) ∞ R −∞ dp dp C ( p ) C ( p ) ( p p ) e G ( k − p ) e G ( k − p − p ) × (21) h e G ( k − p ) + e G ( k − p ) i , here e G ( k ) = ( D k + iv ) − . Transformation of kG ( k ) to e G ( k ) for every diagram of the PTis the specific feature of the 1D case. Important property of e G ( k ) is ∂ e G ∂k = − D e G , ∂ e G ∂v = − i e G , (22)i.e. these derivatives are proportional to each other and the proportionality coefficient doesnot contain k . Another important property of every diagram is that k (apart from beingthe common multiplier) is contained here in the arguments of e G functions, and not in thefactors such as k − p , k − p in the nominator. In the 1D case a general structure of thecontribution A ( k ) of any particular diagram of the n th order to Σ( k ) is A ( k ) ∝ k Z n Y j =1 dp j p j C ( p j ) n − Y m =1 e G k − X l m p l m ! , (23)where every set of l m is a subset of (1 , .., n ) and depends on the structure of the diagram.Calculating the corresponding derivatives in Eq. (12) and taking into account Eq. (22), itis easy to see that the mER is valid, in fact, for any individual diagram.At the same time, for the 1D case there is an exact expression for the average stationaryvelocity V of the particle V = D ∞ R dx exp {− γx + g [1 − C ( x )] } , γ = v/D , (24)which is equivalent to the full summation of the PT series for V and demonstrates nosingularities for any reasonable real-space correlation function C ( x ) (i.e., when C (0) = 1, | C ( x ) | ≤ x >
0, and C ( x ) → x → ∞ ) [11]. Obviously, the corresponding deriva-tive ∂V /∂v is not singular as well. Hence, the equality between corresponding contributionsto D and ∂V /∂v for every diagram leads to the validity of the full mER (3) for 1D case.If needed, we may assume the proper regularization for p → ∞ in every PT order, it doesnot affect the equality between corresponding contributions to D/D and ∂V /∂v , and thesubsequent removal of regularization again leads to the desired mER.7e see that the diagrammatic technique gives a new proof of the validity of the mER,in addition to the original derivation [2]. This new proof is valid for any Gaussian randomlandscape and significantly extends the area of validity of the mER. Our derivation clearlyshows that the mER is exclusively 1D phenomenon as it holds because of a very specificsymmetry of the diagrams, where every scalar product of vectors is equivalent to a trivialmultiplication of real numbers. In multidimensional case the only possibility is to to derive aseries of relations between transport coefficients explicitly taking expansion into the powersof the effective charge in a manner close to Ref. 4. III. BEHAVIOR OF δV AND δD FOR v → Now let us consider the behavior of δD and δV for small v . In this section we will restrictour approach to the first order PT, so we drop the corresponding index. We considerhere only the isotropic random medium with spherically symmetric correlation function C ( ~p ) = C ( p ), and the function dependence of δD and δV on v is governed by the correlationfunction C ( p ). For v → C ( r ) and,therefore, behavior of C ( p ) for p →
0. It is easy to show that all variety of reasonablecorrelation functions (we assume that C ( r ) is a monotonously decreasing function of r ) fallsin three different classes. For example, if C ( r ) ∝ /r α for r → ∞ , then for p → C ( p ) ∝ /p d − α , α < d, ln(1 /p ) , α = d, const , α > d. (25)Correlation functions with more faster decay (e.g., exponential or Gaussian) fall in the sameclass as the power law correlations with α > d , i.e. C ( p ) ∝ const for p → C ( p ) demonstrates even stronger divergence for p → C ( r ) ∝ /r demonstrates theslowest possible decay in such materials [11, 19].Let us consider in detail the correction for δV , and then just summarize briefly theanalogous results for δD ⊥ and δD || . Let us start with the 1D case.8 . 1D case δV = i gD π ∞ Z −∞ dpC ( p ) p D p − iv = i gD π ∞ Z −∞ dpC ( p ) (cid:18) p + iγ − γ p − iγ (cid:19) (26)here γ = v/D and we assume that C ( p ) is an even function of p . Finally δV = − gv + gv D ∞ Z dxC ( x ) e − γx . (27)Hence, if the integral ∞ R dxC ( x ) converges, then the leading correction to the first term inEq. (27) is ∝ v . If the integral diverges (for example, this is the case for the dipolar glassmodel with C ( x ) ∝ /x ), the correction is different. If C ( x ) ∝ /x α and α ≤
1, then theintegral in Eq. (27) is effectively cut off at x c = 1 /γ and it is proportional to ln(1 /γ ) for α = 1 and 1 /γ − α for α <
1. Diffusivity δD may be obtained from δV using the mER.At the same time, for the 1D case we may calculate the asymptotics of V at v → x where C ( x ) →
0, so D V ≈ e g ∞ Z dxe − γx [1 − gC ( x )] = ve g gvD ∞ Z dxe − γx C ( x ) . (28)We see that behavior for γ → g . As we already noted in the Introduction,we may expect that this very behavior remains intact in higher dimensions. In addition, thevery structure of the 1D result for v →
0, i.e. the possibility to use expansion in gC ( x ) hintsfor the importance of the regime of effectively small g for the formation of the functionaltype of the dependence V ( v ) and D ( v ) for low v and, thus, for the possibility to use the PTfor the evaluation of this dependence. 9 . 2D and 3D cases Isolating the maximal power of p in integral (13), it is easy to see that δV could bewritten as ~v · δ ~V = − gv Ω d (2 π ) d ∞ Z dpp d − C ( p ) (cid:20) d + M v ( p/γ ) (cid:21) = − gv (cid:18) d + ∆ (cid:19) , (29)∆ = Ω d (2 π ) d ∞ Z dpp d − C ( p ) M v ( p/γ ) . Here we performed the integration in Eq. (13) over angles for the isotropic correlationfunction, Ω d = 2 π d/ / Γ( d/
2) is the area of the d-dimensional sphere with unit radius andkernel M v ( x ) → − C d /x for x → ∞ , while M v (0) = − /d . Separation of the term 1 /d in Eq. (30) is motivated by vanishing of the kernel M v ( x ) for x → ∞ . For δD || and δD ⊥ the results which may be easily obtained by the corresponding integration of Eq. (16) andEq. (17) have the same structure apart from the trivial replacement v ⇒ D and, ofcourse, constants C d > δV , δD || , and δD ⊥ . We see that ∆( v ) provides theestimation for the mobility field dependence because δµ ∝ ~v · δ ~V /v .For 2D case M v ( x ) = −
12 + x (cid:18) − x √ x + 1 (cid:19) , C v = 38 , (30)and for the 3D case M v ( x ) = −
13 + x (cid:18) − x arcsin 1 √ x + 1 (cid:19) , C v = 15 . (31)We may obtain very rough estimation of ∆ subdividing the integral over p in two regions:from 0 to γ and from γ to p c ( p c ≃ /l is the effective cut-off for some microscopic lengthscale l , e.g. intermolecular distance). In the first region we set M ( x ) ≈ M (0), and in thesecond one M ( x ) ≈ − C d /x . In both cases we may use for the correlation function C ( p )the asymptotics of small p from Eq. (25). Then we get∆ = ∆ + ∆ ≃ − Ω d (2 π ) d d γ Z dpp d − C ( p ) + C vd γ p c Z γ dpp d − C ( p ) . (32)Hence, for the short range correlations with C ( p ) ≈ C (0) we have (keeping only the leading10erms for γ →
0) ∆ ≃ − Ω d (2 π ) d d C (0) γ d , (33)∆ ≃ − Ω d (2 π ) d C (0) C vd γ ln( p c /γ ) , d = 2 ,γ p c , d = 3 , (34)for the marginal case C ( p ) ≈ A ln( p c /p ) in Eq. (25)∆ ≃ − Ω d (2 π ) d d Aγ d ln( p c /γ ) , (35)∆ ≃ − Ω d (2 π ) d AC vd γ [ln( p c /γ )] , d = 2 ,p c , d = 3 , (36)and for the long range correlation C ( p ) ≈ A/p d − α ∆ ≃ − Ω d (2 π ) d dα Aγ α , (37)∆ ≃ − Ω d (2 π ) d AC vd γ − α ( p c /γ ) − α , α < ,p c ln( p c /γ ) , α = 2 , α − , α > . (38)Analogous results for δD ⊥ and δD || are M ⊥ D ( x ) = − x √ x + 1 (cid:0) x (cid:1) + 3 x − , C ⊥ = 5 / , (39) M || D ( x ) = x √ x + 1 (cid:18) x + x x + 1 (cid:19) − x − , C || = 7 / , (40) M || D ( x ) = − x + x x + 1 + x (5 x + 1) arcsin 1 √ x + 1 − , C || = 1 / , (41) M ⊥ D ( x ) = 52 x − x x + 1) arcsin 1 √ x + 1 − , C ⊥ = 1 / . (42)Hence, the corrections for the field dependences of δD ⊥ and δD || could be obtained fromthe corresponding corrections for ~v · ~V by the trivial replacement of the constant C and v ⇒ D .We see that the behavior for the 3D GDM agrees well with the result of the computersimulation [10], but the 2D case does differ and contains an additional logarithmic factor. Itis rather difficult to catch such slowly varying factor in addition to the major contribution ∝ γ while analyzing the simulation data, especially taking into account the limited accuracy11 -6 -5 -4 -3 -3 -2 -1 D l / D eEa/ σ FIG. 1. Fit of data from Ref. 10 (filled squares) for the dependence
D/D = A + B ( eEa/σ ) ln( E/E ) (solid line), eE a/σ ≈ A ≈ . × − , B ≈ − . × − . Broken linedemonstrates the best fit of the data for the dependence D/D = A + B ( eEa/σ ) of the simulation data. For this reason the logarithmic factor has not been found in Ref.10. To support this statements we provide the fit of the data for 2D longitudinal diffusivityborrowed from Ref. 10 using Eq. (34) (see Fig. 1). We do not pretend to provide a properdescription of the data from Ref. 10 with our formula, this is clearly impossible due tothe limitation ( σ/kT ) ≪ ∝ const + E ln E and ∝ const + E for E →
0. Indeed,a significant difference between both depenedences arises only for fields where parameter eaE/σ becomes comparable to 1.
IV. EXPERIMENTAL EVIDENCE FOR THE VALIDITY OR INVALIDITY OFTHE EINSTEIN RELATION IN AMORPHOUS ORGANIC SEMICONDUCTORS
The Gaussian DOS is considered as the most appropriate model for amorphous organicmaterials [20]. Validity of the ER in amorphous organic semiconductors demonstratinghopping charge transport is still a controversial question. There are reliable theoreticalresults showing that the ER cannot hold for the materials having the Gaussian DOS anddemonstrating the non-linear average velocity dependence on E or having a non-negligible12oncentration of charge carriers [2, 5]. Invalidity of the Einstein relation in amorphousmaterials is supported also by computer simulation [21].For the experimental test of the validity of the ER the most suitable is the so-called quasi-equilibrium transport regime where all initial carrier relaxation is over and carrier velocity(averaged over short time intervals) becomes constant. One of the widely used techniquefor a direct measurement of the charge carrier velocity is the time-of-flight experiment [20].In this experiment the quasi-equilibrium regime manifests itself by the development of theplateau of the current transient indicating the constant average carrier velocity.Recent paper by Wetzelaer et al. (Ref. 8) states that in quasi-equilibrium regime thesimple ER perfectly holds if we remove the influence of deep traps. They made the conclusionusing rather indirect experimental evidence on the luminance of the organic light-emittingdiodes. Very probably, the approximate validity of the simple ER is due to the low appliedelectric field, where the ER indeed holds (see Fig. 3 in Ref. 8, where E < × V/cm whichis rather weak field). We should note also that for some materials studied in Refs. 8 and 22(for example, for poly(9,9-dioctylfluorene)) the reported mobility differs by approximatelytwo orders of magnitude with the previously reported values [23, 24]. This difference hintsto the rather unusual structure of the thin transport layers used in light emitting diodes(may be, the structure of the layer is not spatially uniform), which provides an additionalcomplicating factor.We believe that the papers of the Nishizawa group provide much more clear direct evi-dence on the validity of the ER [25–29]. They extracted µ and D by fitting the experimentaltime-of-flight transients in various molecularly doped polymers with the solution of classicdiffusion-drift equation. Typically, the quality of fits is rather good (see Refs. 25, 26, and28). Moreover, obtained transport parameters µ and D show no dependence on the thick-ness of transport layers, thus indicating a well-established quasi-equilibrium transport regime[26]. At the same time, the difference between fitted D and calculated using the simple ormodified ER is about two orders of magnitude (see Fig. 2). Such huge difference stronglysupport the idea of the invalidity of any variant of Einstein relation for 3D charge transportin amorphous materials with the Gaussian DOS.Unfortunately, the direct comparison of our results for the behavior of V and D in thelimit of weak driving force with the experimental data on charge carrier transport cannotbe done due to the total lack of the reliable data for very weak field region.13 -10 -9 -8 -7 -6 -5
350 400 450 500 550 600 D , c m / s E , (V/cm) { { { FIG. 2. Field dependence of the diffusivity in molecularly doped polymer. Points are borrowedfrom Ref. 25, the temperature is indicated at the left. Dotted lines are shown as a guide for aneye. Solid lines show the diffusivity calculated from the experimental mobility values assumingln µ = A + B √ E and using the mER Eq. (2). Broken lines show the diffusivity calculated usingthe simple ER Eq. (1). V. CONCLUSION
We considered the diffusive motion of a particle in the random spatially correlated energylandscape having the Gaussian DOS. For such system the average particle velocity in thequasi-equilibrium regime is a nonlinear function of the driving force and the simple Einsteinrelation is certainly not valid. Using the perturbation theory we found that the modifiedEinstein relation [2] is an exclusively 1D property and does not hold for higher dimensions d >
1. For this reason a usual estimation of the diffusivity from the mobility could beapproximately valid only for a low force region because the simple Einstein relation whichis certainly valid at zero driving force serves as a kind of anchor point here.We provide also a new proof of the mER for 1D case which is completely different fromthe previous one [1, 2]. This new proof extends the validity of the mER to arbitrary Gaus-sian random landscape and does not depend on the assumption of the particular type ofcorrelation function, thus covering a more wider variety of possible random landscapes.We obtained also the leading corrections for the average velocity and diffusivity in the14imit of weak driving force and demonstrated how such corrections depend on the dimen-sionality and correlated properties of the random landscape. For the short range correlationwe obtain results which agree well with the corresponding dependences for the lattice model[9, 10]. At the same time, the results show that the case d = 1 is not exceptional one andthe functional form of the corrections vary in some regular way with the variation of d . ACKNOWLEDGEMENTS
Financial support from the Program of Basic Research of the National Research Univer-sity Higher School of Economics is gratefully acknowledged.
Appendix A: Diagrammatic technique for the particle diffusion in random medium
We consider the diffusion of the charged particle in the random environment having thespatially correlated Gaussian DOS. For the particular realization of the random potential U ( ~x ) the particle Green function G U ( ~x, t ) obeys Eq. (10). If we consider the Laplacetransform according to t and Fourier transform for ~x , then the corresponding equationbecomes G U ( ~k, s ) = G ( ~k, s ) (cid:20) − βD (2 π ) d Z d~pG U ( ~k − ~p, s ) U ( ~p ) (cid:16) ~k · ~p (cid:17)(cid:21) , G − ( ~k, s ) = s + D k + i~v · ~k, (A1)here G ( ~k, s ) is the Green function for the zero disorder. In future we are going to considerthe stationary case s = 0 only, and use the simplified notation G U ( ~k,
0) = G U ( ~k ). Consider-ing the PT expansion of Eq. (A1) and making the average over disorder, we may write downthe diagram expansion for the averaged over disorder ‘Green function G ( ~k ) = D G U ( ~k ) E . De-tails of the diagram technique may be found in excellent Bouchaud and Georges review [30].The trivial difference between out and their notations is that they used the random force ~F = −∇ U instead of U .The basic building blocks of a diagram are shown in Fig. 3. For every inner moment ~p there is an integration π ) d R d~p , and dotted line with ~p going into the vertex provide − ~p for the vertex weight because of the momentum conservation D U ( ~k ) U ( ~k ) E = (2 π ) d σ δ ( ~k + ~k ) C ( ~k ) , (A2)15 ( ~k, s ) = a) σ C ( ~p ) = b) βD ( ~k · ~p ) = c)FIG. 3. a) Bare Green function G ( ~k, s ). b) Correlation function of the Gaussian random field. c)Interaction vertex βD ( ~k · ~p ). where C ( ~k ) is the Fourier transform of the spatial correlation function C ( ~x ) = h U ( ~x ) U (0) i /σ .We assume that the integrals converge at p → ∞ , as it is the case for fast decaying C ( p ).Moreover, keeping in mind the possible application of the theory to the diffusion of particlesin amorphous material we should expect an inevitable cut-off at p ≃ /l , where l is sometypical atomic or molecular scale.Here we briefly show only the expansion for the self-energy Σ( ~k ) = G − ( ~k ) − G − ( ~k )where only the strongly connected diagrams should be taking into account (the diagramswhich cannot be disconnected by cutting a G line). The first order contribution to Σ( ~k ) is(see Fig. 4) δ Σ = FIG. 4. First order contribution to Σ( ~k ). Σ = − gD (2 π ) d Z d~pC ( ~p ) G ( ~k − ~p )( ~k · ~p ) h ( ~k − ~p ) · ~p i , g = ( σβ ) , (A3)and the second order one is (see Fig. 5) δ Σ = + FIG. 5. Second order contribution to Σ( ~k ). δ Σ = g D (2 π ) d R d ~p d ~p C ( ~p ) C ( ~p ) G ( ~k − ~p ) G ( ~k − ~p − ~p )( ~k · ~p ) h ( ~k − ~p ) · ~p i × nh ( ~k − ~p − ~p ) · ~p i R ( ~k, ~p ) + h ( ~k − ~p − ~p ) · ~p i R ( ~k, ~p ) o , (A4) R ( ~k, ~p ) = ( ~k − ~p ) · ~pG ( ~k − ~p ) . We mostly use the first order approximation for Σ( ~k ), and the second order one is used onlyfor the test of the validity of the modified Einstein relation. Appendix B: Second order PT approximation
Calculation of the second order PT corrections to average velocity and diffusivity israther straightforward but produces complicated expressions, so we write down here onlythe relevant ingredients for the test of the validity of the mER Eq. (18) P a ∂δV (2) a ∂v a = g D (2 π ) d R d ~p d ~p C ( ~p ) C ( ~p ) × nh ~F ( ~p ) · ~F ( ~p ) i h ~p · ~F ( ~p + ~p ) i ~p (cid:16) ~p · h ~F ( ~p ) + ~F ( ~p + ~p ) + ~F ( ~p ) i(cid:17) + (B1)+ h ~p · ~F ( ~p ) i h ~p · ~F ( ~p ) i h ~p · ~F ( ~p + ~p ) i (cid:16) ~p · h ~F ( ~p ) + ~F ( ~p + ~p ) i(cid:17)o ,~F ( ~p ) = ~pG ( − ~p ) . D P a δD (2) aa = g D (2 π ) d D R d ~p d ~p C ( ~p ) C ( ~p ) × n ~p · h ~K ( ~p ) + ~K ( ~p + ~p ) + ~K ( ~p ) i h ~p · ~F ( ~p ) i h ~p · ~F ( ~p + ~p ) i h ~p · ~F ( ~p ) i ++ ~p · h ~K ( ~p ) + ~K ( ~p + ~p ) i h ~p · ~F ( ~p ) i h ~p · ~F ( ~p + ~p ) i h ~p · ~F ( ~p ) i −− h ~p · ~F ( ~p ) i h ~p · ~F ( ~p + ~p ) i h ( ~p + ~p ) · ~F ( ~p ) i − (B2) − h ~p · ~F ( ~p ) i h ~p · ~F ( ~p ) i [ ~p · ~p G ( − ~p − ~p )] −− h ~p · ~F ( ~p ) i h ~p · ~F ( ~p ) i h ~p · ~F ( ~p + ~p ) i − h ~p · ~F ( ~p ) i ~p G ( − ~p − ~p ) (cid:27) ,~K ( ~p ) = G ( − ~p ) (2 D ~p − i~v ) . Right parts of Eq. (B2) and (B3) do differ for any d > [1] P. E. Parris, M. Ku´s, D. H. Dunlap, and V. M. Kenkre, Phys. Rev. E , 5295 (1997).[2] P. E. Parris, D. H. Dunlap, and V. M. Kenkre, J. Polym. Sci. B , 2803 (1997).[3] A. Miller and E. Abrahams, Phys. Rev. , 745 (1960).[4] S. A. Hope, G. Feat, and P. T. Landsberg, J. Phys. A , 2377 (1981).[5] Y. Roichman and N. Tessler, Appl. Phys. Lett. , 1948 (2002).[6] T. H. Nguyen and S. K. O‘Leary, Appl. Phys. Lett. , 1998 (2003).[7] N. Tessler and Y. Roichman, Org. Electron. , 200 (2005).[8] G. A. H. Wetzelaer, L. J. A. Koster, and P. W. M. Blom, Phys. Rev. Lett. , 066605 (2011).[9] A. V. Nenashev, F. Jansson, S. D. Baranovskii, R. ¨Osterbacka, A. V. Dvurechenskii, and F.Gebhard, Phys. Rev. B , 115203 (2010).[10] A. V. Nenashev, F. Jansson, S. D. Baranovskii, R. ¨Osterbacka, A. V. Dvurechenskii, and F.Gebhard, Phys. Rev. B , 115204 (2010).[11] D. H. Dunlap, P. E. Parris, and V. M. Kenkre, Phys. Rev. Lett. , 542 (1996).[12] S. V. Novikov, J. Polym. Sci. B 41, 2584 (2003).[13] S. Novikov, Annalen der Physik , 954 (2009).[14] M. Deem and D. Chandler, J. Stat. Phys. , 911 (1994).[15] S. V. Novikov, D. H. Dunlap, V. M. Kenkre, P. E. Parris, and A. V. Vannikov, Phys. Rev.Lett. , 4472 (1998).
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