On the stochastic transport in disordered systems

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On the stochastic transport in disordered systems

M.V. Feigel’Man, V.M. Vinokur

To cite this version:

M.V. Feigel’Man, V.M. Vinokur. On the stochastic transport in disordered systems. Journal de

Physique, 1988, 49 (10), pp.1731-1736. �10.1051/jphys:0198800490100173100�. �jpa-00210854�

On the stochastic transport ⁱⁿ disordered systems

M. V. Feigel’man (1) and V. M. Vinokur (2)

(1) Landau Institute for Theoretical Physics, ^Moscow 117334, U.S.S.R.

(2) Institute for Solid State Physics, 142432, Chernogolovka, ^Moscow region, ^U.S.S.R.

(Requ le 30 novembre 1987, révisé le 20 avril 1988, accepté ^{le 1 er} juin 1988)

Résumé.

Nous considérons la marche aléatoire à une dimension dans un champ de force aléatoire (RWRF)

en présence ^d’un petit champ alternatif. Nous calculons la réponse alternative de la particule ^se propageant suivant la loi sublindaire x ~ tkW, 03BA ^1, où tw ^{est le} temps ^{total du mouvement de la} particule. ^{Nous montrons}

que la partie imaginaire ^{de la} susceptibilité alternative ~" ^{obéit à une loi de} puissance ~"(03C9) ~ 03C9K pour

tw ~ ~ et ~" (03C9) ~ 03C9-03BA pour tw ~ ^{03C9-1. Nous} suggérons ^la propriété de loi d’échelles de la susceptibilité pour des temps longs mais finis : ~" ~ 03C903BA f(1/03C9t1-03BAw). Nous discutons l’analogie ^{entre ces résultats et les} phénomènes ^de transport dispersifs dans les milieux amorphes ^et le vieillissement des verres de spin.

Abstract.

The one-dimensional random walk in random-force field (RWRF) in the presence of the small ^ac field is considered. The ac response of the particle propagating according ^{to the} sublinear law x ~ tkw,

03BA 1, where tw is the total time of the particle’s motion, is calculated. The imaginary part ^{of ac} susceptibility,

~", ^{is shown to} obey the power law : X"(03C9) ~ ^{03C9k at} tw ~ ^{~ and} ~"(03C9) ~ ^{03C9 -k} at tw ~ 03C9-1. The scaling

behaviour of susceptibility ^for large but finite times tw ^is suggested : ~" ~ 03C9k f(1/03C9t1w-k). ^The analogy

between these results and phenomena ^of dispersive transport ⁱⁿ amorphous ^{media and} ageing ⁱⁿ spin glasses ^is

discussed.

Classification

Physics ^Abstracts 05.40 - 71.55J

1. We report ^{some new results} concerning ^the properties ^{of the} one-dimensional random walks in the random force field (RWRF). The interest in the

subject ^is based upon the fact that 1D RWRF

appeared ^{to mimic} surprisingly ^well the time evolu- tion phenomena ^{of a wide} variety of disordered systems. ^The dispersive transport ⁱⁿ amorphous

semiconductors [1-2], spin-slip dynamics ^{on fractals} [3] ^or relaxation of magnetization ⁱⁿ spin-glasses [4, 5] ^{can be mentioned as the most} striking examples.

In this paper ^we consider the particle propagating

over 1D random force medium under the action of extra driven force and find the ac response of this system. ^The low-frequency response appears ^to be

nonstationary ^and strongly ^affected by ^{the total} propagation time, tW, ^{of the} particle, provided ^the particle ^{motion is} governed by thermal activation

over large-scale ^barriers (in ^other words, ^{at suffi-} ciently ^low temperatures). We find the ac suscepti- bility, ^X (úJ ), ^to demonstrate the anomalous power- law frequency dependence im X (to ) - w - ’ at ^initial

stages ^{of the evolution of the} system ^and

im X (w ) - w ’ when the total time of particle ^mo- tion tw is very long : tw >>>w - 1.

The observed tw dependence ^{of the} finite-fre- quency response X (w ) is in fact the kind of ageing phenomena, ^{which are} well-known in spin-glass ^and polymer-glass physics [4, 5, 6]. ^It should be noted that the power-law ^behaviour 3m x ( w ) ~ w K itself could be easily interpreted ^{in terms} of the multitude of modes with an exponential distribution of energy

barriers ; however, ^the fact that this behaviour is

accompanied by ageing is less trivial and indicates the highly correlated nature of dynamics ^involved.

This type ^of dynamics ^is usually ^{assumed to be}

characteristic for glassy systems (see e.g. [7, 8]).

Therefore, ^{we believe our results to be more} general

than the simplest 1D model considered.

2. The 1D RWRF has been extensively ^studied by

numerous authors [9-19] and the unusual behaviour of mean displacement (x) ’" t K, ^K 1 under the action of driven force [11, 12, 14] ^{as well as the} anomalously slow diffusion law (without ^{the driven}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490100173100

1732

force) (X2) -- ^{In’ t} [13] have been found to occur for the discrete version of the model. The continuous version of RWRF was introduced by ^{one of the}

authors [17] for the kink propagation along ^the

dislocation line (note that kinks on the dislocation represent ^the only known direct physical realization of the 1D RWRF). ^The exponential distribution of barriers was suggested ^{and the} sublinear motion law, x -- t K, ^{K 1 was} obtained from the very simple

Arrhenius arguments ([17]).

Continuous RWRF is governed by ^a Langevin- type equation of motion :

where ( f (x ) ) ^{= 0,} ( f (x ) f (x’ ) ) ^{= y} 5(x-x’), ^F

represents the driven force, q is the thermal noise :

TJ (t) TJ (t’» T = 2 T 5 (t - t’ ).

The long-time dynamics ^{reflects the} thermally

activated particle ^{motion over} large fluctuations of the random potential U(x) ^and is determined by ^the probability distribution density, p (E), of the energy

barriers, E, ^{for a} system ^{of fixed} length, ^{L. The} particle energy variations, e, when moving through

random potential ^{relief can} be described with Marko- vian process :

(the quenched ^disorder f (x ) represents ^{« noise » in} the random walks over the energy axis). Therefore, p (E) ^can be written in the form of path integral

The scales of energy, Eo, ^and length, xo, ^can be found from equation (3) ^{and are} given by [17] :

In a large system ^{there is a finite} probability ^{to find}

the barrier with E > Eo. ^This probability ^{can be}

calculated by saddle-point integration ⁱⁿ equation (3).

The saddle-point trajectory that minimizes the

« action » in equation (3) obeys ^the equation ^of

motion

with the solution ^E (x ) ^{= Fx,} 0 -- X -- fE ⁼ E/F, describing ^the large fluctuational barrier retarding

the particle and solution E ^{= -} Fx representing ^the

mean slop of the random potential v (x ). Performing

the steepest ^descent integration ^{one finds}

The delay time distribution corresponding ^to p(E) ^from equation (5) is

This expression ^{is valid for} ’r > Tj 1=

To exp (1 / K ), were 71 1 is the time for the particle ^to

overcome a typical ^barrier Eo.

Consider first the high-temperature region, ^K > 1.

The mean total time which the particle ^{needs to} get through ^the system ^of length ^{L is}

,

(the ^factor L/xo represents ^{the mean number of} relevant traps ^{in the} system). By using equation (6)

one gets immediately :

Hence, ^{the mean} displacement (x) ^{as a} function of t is

When ^{K -+-} 1, the total time ttot (L ) ^the particle

travels over the system diverges. ^{This means that the}

contribution of the traps ^{ceases to} be additive. At

K 1 the motion of the particle is dominated by

very ^rare, but extremely deep potential wells. The

divergence ⁱⁿ (7) ^{should be cut off at r =} ’rmax(L),

where ?max (L ) ^{is the} largest probable waiting ^time

the particle encounters on the distance L. ’rmax(L)

can be found from a simple ^{estimate :}

The meaning of this relation is that the particle ^at

most once is captured by ^the deepest trap correspon-

ding ^to ’Tma.(L)- ^{It follows} immediately ^from (10)

that

Inverting (11) ^one obtains the ^mean displacement x

in time t for ^K 1 :

The origin of this sublinear drift as well as the

anomalous logarithmical ^diffusion (in the absence of

driven force) ^can be understood as a result of

subsequent captures ^{of the} particle by ^{the more and}

more deep ^{and wide} potential wells the particle

encounters when travelling ^{over 1D random}

medium. Attention is to be called that such a motion

can be alternatively ^{described as} dwelling ^{within the} potential well with walls growing up ^as the total time of particle ^motion, tW, (that ^{is the «} age ^» of the

system) ^increases.

To handle the motion of the particle ^{in the} permanently evolving random environment it is convenient to define the « renormalized » probability density 1/1’ R (t) ^{for the} particle ^{still to} remain in the trap ^{with the} decay time t up ^to the moment t :

The point ^{to be} emphasized is that 41 (t ) represents the probability ^of finding the barrier with the delay

time t in the arbitrary point ^{of the} system ^while

1/1’ R (t) determines the probability ^to find the barrier with the delay ^{time -- t at the} point ^the particle ^is

located at time t.

At x > 1 1/1’ R (t) is normalizable. Accordingly, ^the

distribution density of energy barriers takes the form

so the most probable ^value of the energy barrier ^on the distance spanned ^is

and remembering ^{that the mean} slope of the relevant barrier is equal ^to F, ^one obtains for the barrier size :

Note that the time t, ^{for the} stationary ^motion regime ^and stationary distribution (6) ^{to be settled}

should satisfy the condition

At low temperatures, ^K 1, ^the normalization

integral ^for ’/J’ R ( T) should be cut off at 7-ma,,(L)

which coincides with the total time of the particle motion, tw (note ^{that L =} (T max (L»K ⁼ tK ^{is the}

total distance spanned by ^the particle). The normali- zed distribution 1/J’ R ( T) ^is given by

This tW-dependence ^{of the} probability density 1/1’ R ( T) implies ^{that at K 1} the motion becomes

non-stationary. The average depth and size of the well the particle finds itself within are now :

3. Now we turn to the calculation of the frequency susceptibility ^{of the} system ^{at low} temperatures,

K 1. Consider first the infinite, tw ^- oo, limit. At

frequencies w >> tw 1 ^the particle ^can be treated as

captured by ^the symmetric potential ^well, UR (x ) :

(the ^{first term} determines the mean slopes ^{of the} well, while the second describes fluctuations due to

quenched random force f (x)). ^{Then the} probability

for the particle ^{to be near the} point x ^{at time t} obeys

the Fokker-Planck equation :

where f eù ^{e- i eùt is the} infinitesimal harmonic force.

We shall look for ^a solution in the form

where Po (x ) ⁼ exp [- UR (x )/T ] ^{is the} equilibrium

solution of the unperturbed, f (J) ⁼ 0, system. ^Then the susceptibility X(10)= - a (x «(ù ) > / a f (J) ⁼

f dx P 1 (x ) x ^{can be expressed in terms of the}

eigenfunctions, a ) , ^and eigenvalues, sa, of the related Schr6dinger equation

as

where 10) ^- Po (x ) ^{is the ground state eigenfunc-}

tionwithso=0.

To bring (22) ^{to a more} conventional form one can

define the operator (3 (x ) by the relation

Hence

Making

use of the relation dJ3 d7 = 1 (HJ3 - J3 H) ^where

T

H is the Hamiltonian corresponding ^to (21 ) ^{one finds}

I I A I k 1

One can see from (24) ^{that X} (0 ) =1 ^{T (0 IX21 0) as it}

should. Introducing spectral density g

1734

one obtains

The spectral density g ( e ) is calculated by ^the

usual methods developed ^{in the} theory ^{of 1D}

disordered media.

The point ^to be noted is that, because of the nonuniform character of the potential UR (x ), ^the

Beresinskii-Gor’kov approach [20] (where ^{the wave}

functions found in regions ^of uniformity ^{are then}

matched at the singularity point) rather than the Schmidt method [21] should be employed. ^The straightforward calculation gives ^{after some} algebra

the result for the low-lying ^{states :}

The physical ^{reasons for the} proportionality g ( E ) oc p ( E ) ^{one can} perceive in the fact that the localization length, f, ^{of the} eigenfunctions I a ) corresponding ^{to the} low-lying ^states, is determined

by ^{the mean} slope of the barrier and can be shown to be independent of £ : e -- K - 1. ^This implies ^{that the}

matrix elements (0 1 x « ) appear ^to be independent of E,, ^for small Ea ^and the calculation of g ( £ )

reduces in fact (for low-lying states) ^to the calcu- lation of the averaged density ^{of states} p ( £ ) _

s p s ( E - ^{E,, ). This quantity was} calculated in [18]

(for ^{the case of « bare »} potential ^field U(x) _

/*v

and the result

that was obtained can be shown to hold for our case as well.

The relation (26) ^{holds until £} e- 1,,"K , ^at energies corresponding ^{to the} long-time ^limit t > T 1. ^From

(26) ^and (25) ^we get

The susceptibility (27) corresponds ^to slow relax- ation at large ^times

The results obtained can be referred to as the

stationary dynamic response. For the large ^{but finite} tW > t ^the ground state I 0) ^{and low} lying ^{states with}

E e - 1 /’ should be treated as the quasilocalized

levels rather than strictly ^{localized states. The} decay

rates T a ^{are of the order of} T -1= exp (- E/ T ).

Then, ^{the mean} decay ^{rate r} = r a) R ^{is deter-}

mined by

It seems reasonable to assume that the finite-time

generalization ^of (26) takes the form

where f (x ) ^{is the scaling function} satisfying ^con-

ditions f (0) = 1, f ( oo ) = 0 (the ^latter equality

follows from the fact that for extremely ^small energies, 8 Eo = tW 1 tW-1=- T ^the eigenstates

are absent). Note that being quite plausible ^this assumption ^{is not} proved ^{at the moment.} Combining (27), (29) ^and (30) ^{we obtain} finite-tw response :

and the corresponding relaxation law

At waiting times tw - t ^the susceptibility ^{can be}

obtained by direct Fourier-transformation of the law of the « initial ^» evolution of the system

and takes the form

It is interesting ^{to note} that this result describes the response due ^to transitions between neighbour- ing potential wells. In fact, ^{one can} consider the

absorption of energy by ^{the set} of two-level systems constituted by potential wells with the distribution function of relaxation times 1p’R (T). By ^{means of the}

usual Debye-absorption theory approach ^it is easy ^to

recover the result of (34).

Another point ^to be mentioned is that there exists

an intermediate waiting-time, tw, ^{in the} region :

t tW tl/(1- K ) where the crossover of the fre- quency behaviour from equation (34) ^to equation (32) ^{should take} place.

4. To conclude with, ^{some comments are to be}

made. (i) ^The exponential distribution of relevant barriers (5) ^and resulting algebraic form of distri- bution of delay ^times (6) constitute the ground ^of

our analysis. Analogous distributions have been found in the directed percolation ^model [22]. ^This gives ^{us the} possibility ^{to make an} assumption ^that

our results could be applied ^{to the} systems ^or phenomena allowing description ^{in terms of} percola-

tive motion. The transport phenomena ⁱⁿ amorphous

semiconductors including dispersive transport [1]

and photoinduced absorption [2] should be con-

sidered as probable candidates. The time depen- dence ^{of transient} photocurrent, I (t ), corresponding

to the carrier packet propagation ^{from one side of}

the amorphous ^{film to another one} has been demon- strated to have the form [1, 23]

where the transit time tT ^is thought ^to be the time of the first achievement the absorbing side of the

sample by ^the leading edge of the carrier packet, tT - d’Il", ^{where d} is the thickness of the film. To

explain ^{the data} observed, Montroll and Scher [23]

introduced the algebraic waiting-time distribution function 1Jf (t) ’" t- (1 + a). ^{The data} (35) ^{can be}

related to our formulae if we consider the transient current as a consequence of the hoppings ^between potential wells which form some percolative ^clusters.

Then the two regimes ^observed correspond ^{to the} initial, (x t K, evolution and residual relaxation,

(x (t) - (x (00 ) > ’" ^{t- 1(, of the system.} It is interest-

ing then that the time dependence ^of population ^of trapped ^carriers N ’" t - a and logarithmic ^time-de- pendence ^of depth ^{of the most effective} traps,

Etr - Tin t, and temperature dependence ^{of ex-}

ponent « = T/To ^{which has} been observed by photoinduced absorption measurements can be di-

rectly ^described by ^{our formulae} (17) ^and (18) ^with

x = a.

(ii) ^The power-law frequency dependence ^{of the} susceptibility 3m x - (J) I( and the corresponding nonexponential relaxation (28) ^are surprisingly ^ana- logous ^{with the} frequency-dependent response X (w ) ^{and remanent} magnetization. M (t ) ^behaviour

that has been found in recent experiments ⁱⁿ spin- glasses [5, 24, 25]. ^{In fact, it was} shown that

analytical expression which best fits the experiment

can be written as

where t, is the time the system ^is kept ^below freezing

temperature before the applied field is switched off and t is the observation time. In the limits of an

infinite age, tw ^{- oo,} the relaxation reduces to a

power law :

and correspondingly X " (w ) - co ^{" .}

Still more interesting is that the ageing processes in frequency susceptibility ^{X " and} magnetization

relaxation are of the form given by equations (31), (32). ^{Indeed, in} spite of the fact that ageing ⁱⁿ spin glass ^{seems to} be described by ^{the two} independent parameters ^{a and 03BC, the recent data indicate the} relation ^{a =} 1 - g [24], ^{which is} just the result of

our model (where a = K, JL = 1 - K). Obviously

this simplest ^model is very far from real spin glasses,

but the above similarity ^{seems to be} intriguing.

Acknowledgements.

One of the authors (V.M.V.) would like to thank P. Monceau and C.R.T.B.T. for hospitality ^at

C.N.R.S. and also express his gratitude ^{to M.} Papou-

lar and R. Rammal as well as to J. Hammann and M. Ocio for helpful ^and stimulating discussions.

References

[1] ^{PFISTER, G. and} SCHER, H., ^Adv. Phys. ²⁷ (1978)

747.

[2] ^{TAMOR, M. A.,} Solid State Commun. 64 (1987) ^141.

[3] RAMMAL, R., ^J. Phys. ^{France 46} (1985) ^1809.

[4] OMARI, R., PRÉJEAN, J. J. and SOULETIE, J., ^J.

Phys. ^{Lett. France 45} (1984) ^1809.

[5] ALBA, M., OCIO, ^{M. and} HAMMANN, J., Europhys.

Lett. 2 (1986) ^42.

[6] ^{STRUIK, L. C.} E., Physical Ageing ⁱⁿ Amorphous Polymers and Other Materials (Elsevier, ^Scien-

tific Publ., Houston, Tex.) ^1978.

[7] PALMER, ^R. G., STEIN, ^D. L., ABRAHAMS, ^{E. and} ANDERSON, P. W., Phys. Rev. Lett. 53 (1984)

958.

[8] ^{PALMER, R. G.,} Heidelberg ^{Collo. on} Glassy Dynamics, Edis J. L. van Hemmen and I. Morgenstern (Springer-Verlag) ^{Lect. Notes in} Physics (1987) p. 275.

[9] ^{PETUKHOV, B. V., Fiz.} Tverd. Tela 13 (1971) ^1445.

[10] TEMKIN, ^D. E., Dokl. Akad. Nauk SSSR 206 (1972)

27.

[11] KESTEN, H., KOZLOV, ^{M. and} SPITZER, F., Compos.

Math. 30 (1975) ^145.

[12] SOLOMON, F., Ann. Probab. 3 (1975) ^1.

[13] ^{SINAY, Ya. G.,} Theor. Probab. Its Appl. ²⁷ (1982)

247.

[14] DERRIDA, ^{B. and} POMEAU, Y., Phys. Rev. Lett. 48

(1982) ^627.

[15] MAYNARD, R., ^J. Phys. ^{France 45} (1984) ^L-87.

[16] BERNASCONI, ^{J. and} SCHNEIDER, ^W. R., ^Helv. Phys.

Acta 58 (1985) ^597.

[17] ^{VINOKUR, V. M., J.} Phys. ^{France 47} (1986) ^1425.

[18] BOUCHAUD, ^J. P., COMTET, A., GEORGES, ^{A. and}

LE DOUSSAL, P., Europhys. ^{Lett. 3} (1987) ^653.

[19] ^{BOUCHAUD, J.} P., GEORGES, A., ^LE DOUSSAL, P., J. Phys. ^{France 48} (1987) ^1855.

[20] BEREZINSKY, ^{V. L. and} GOR’KOV, ^L. P., ZhETF 77

(1979) ^2498.

1736

[21] SCHMIDT, H., Phys. ^{Rev. 105} (1957) ^425.

[22] DHAR, Deepak, ^J. Phys. A ¹⁷ (1984) ^{L 257.}

[23] ^{SCHER, H. and MONTROLL, E. W.,} Phys. ^{Rev. B 12} (1975) ^2455.

[24] ALBA, U., VINCENT, HAMMANN, J. and OCIO, M., ^J.

Appl. Phys. ⁶¹ (1987) ^4092.

[25] REFREGIER, Ph., VINCENT, E., HAMMANN, ^J. and

OCIO, M., ^J. Phys. ^{France 48} (1987) ^1533.