Frequency-dependent response and nonlinear polarization in weakly pinned CDW

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Frequency-dependent response and nonlinear

polarization in weakly pinned CDW

M. V. Feigelman, V.M. Vinokur

To cite this version:

Frequency-dependent

response

and

nonlinear

polarization

in

weakly

pinned

CDW

M. V.

_Feigelman

₍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.

(Reçu

le 20 avril 1988, révisé le 22

_septembre

1989,

accepté

le 25

_septembre

₁₉₈₉₎

Résumé. 2014 Nous étudions la

dépendance

dans la

_fréquence

de la conductivité d’ondes de densité de

_charge

faiblement accrochées par des

impuretés,

à une dimension. Les contributions des oscillations dans les vallées et celles des sauts activés entre les vallées sont calculées dans la limite de sur-amortissement. La contribution non linéaire à la

_réponse

diélectrique

statique

est étudiée

et s’avère être fortement non

_analytique.

Abstract. 2014 We

study

the

_{frequency-dependent conductivity}

of 1D

_{charge density}

waves

_weakly

pinned

by

impurities.

The contributions from

intravalley

oscillations as well as from

intervalley

activated transitions are calculated in the

_overdamped

limit. The nonlinear contribution to the static dielectric response is considered and found to be

_{strongly nonanalytic.}

Classification

Physics

Abstracts 72.15N

1. The ac _{response of}

_{charge-density-waves}

_(CDW)

in

_{pinned regime}

has been

_investigated

extensively

over

_past

_{years and it is}

_{generally agreed}

that the anomalous

_{low-frequency}

behavior of

_CDW-systems

should be associated with the existence of a

_large

number of

metastable states of CDW

_pinned

_{by impurities.}

The unusual

_{power-law frequency}

dependence

of the dielectric constant,

e(to),

that was found in recent

_experiments

_[1-3]

departs considerably

from that

_expected

from the

_single

relaxation time model

_[4].

Stokes et al.

_suggested

_[3]

that this feature of

_{low-frequency}

CDW

_dynamics

can be related to the broad distribution of relaxation times associated with the

_thermally

activated

_jumps

between metastable states.

_Moreover,

Stokes et al. observed

_strong

_nonlinearity

in ac

_conductivity

for

ac

_amplitudes

well below the dc threshold field. This

_ac-amplitude

_dependence

was assumed

to arise from a distribution of local

_pinning

fields.

We shall consider the

_{low-frequency}

_{overdamped dynamics}

of the 1D CDW on the basis of

the

_{general approach developed}

in our earlier papers

_[5-8]

(1).

We are aware that one should

take care when

_comparing

our results with

_experimental

data because of the

_essentially

3D

nature of CDW

_pinning

in the

_compounds

involved

_(such

as

_o-TaS3

and blue bronze

Ko.3MO03).

Nevertheless,

the one-dimensional model has the

_advantage

_{of being}

solvable and besides we

_expect

the most

_important

features of one-dimensional

_glassy

CDW

_dynamics

to

(’)

An

_{analogous approach}

was also

successfully

used in a somewhat similar situation

[14].

410

be true also for 3D

systems.

Note that the relaxation of the CDW

_pinned

_by

weak

_impurities

has

_recently

been studied within the mean-field

_theory

_by

Littlewood and Rammal

_[9],

who have

_pointed

out, in

particular,

the

_important

difference between relaxation within a

_single

local minimum of the free energy and processes of transitions between different metastable

states. Some of their results are in conflict with our

_{approach ;}

we discuss this

_point

below at

the end _{of the paper.}

The paper is

organized

as follows. First we consider the ac _{response due} to relaxation within

a

_single

metastable well.

Second,

we calculate the contribution of the

_intervalley

transitions and show that at

_sufficiently

low

_frequencies

the ac _{response is} dominated

_by

these

_thermally

activated processes.

Finally,

the nonlinear corrections to _{static dielectric response}

_(arising

from

_polarization

in the domains with

_relatively

weak

_pinning)

are calculated. One can

_expect

that

_nonlinearity

in _so

_might

contribute to the

_{amplitude dependence}

of ac _{response observed}

in

_[3].

2. Let us consider relaxation within a

_single

metastable well

_{by using}

the

_approach

that has

already

been

_employed

to

_investigate

the

_underdamped

CDW _response

_{[7, 8].}

The CDW Hamiltonian is

_{[10] :}

where cp is the CDW

phase,

PF

=

ItvF/2

7T is the reduced Fermi

_velocity,

_Vf(x)

and

Vb (x)

are

_potentials

of forward and backward

_scattering

_impurities,

_{respectively,}

h is the

incommensurability

parameter,

Vois

the

_{commensurability}

_potential,

vb (x)

=

V b

y

6

_{(x -}

_Xi)

_{and xi}

are the

_impurities

coordinates. The disorder assumed to be i

strong

as

_compared

with the

_{commensurability potential}

(2) :

the

_parameters

_af and

a b that characterize the

degree

of disorder are

_{large :}

where (Vf(X)Vf(x’»

=

J.L28(x-x’)

and _c is the linear concentration of

backscattering

impurities.

The basic

_quantity

_describing

the

_pinning

_phenomenon

is the

_pinning

_{energy per correlation}

length,13

(hereafter

we call it

_pinning

_energy).

This

_pinning

_energy

_being

a random

_quantity,

to

_complete

the

_description

of CDW

_system

one needs the distribution

_{probability density,}

W(,B ),

of

_pinning

_energies.

It was shown

[5, 6]

that the

_principal

_physical

characteristics of the

system

(in

particular,

ac

_{conductivity)}

are

_{expressed through}

_{W(/3 )}

and the method of calculation of

_{W(13 )}

in

_{weakly-pinned}

one-dimensional CDW

_system

was

_developed.

Below we

_give

some sketch of the

_{approach proposed}

_referring

for more details to the

_original

papers

[5, 6].

To

_begin

_with,

let us fix the value of CDW

_phase

cp in

arbitrary

intemal

point

x of the

_system,

cp (x) =

p, and suppose the distribution of

phases

at the rest

_points

to

minimize the CDW free _{energy. One} can define now _{the free energy}

_{£x( cp ) depending}

on the

(2)

This _strong disorder does not

_imply

the strong

pinning

limit in the

Fukuyama-Lee

sense

_{[15] ;}

our

fixed value cp . The minima of the 2

_7T-periodic

function

Ex{ cp )

_correspond

to metastable states

of CDW

_system.

Consider now the second derivative of

_{e, (cp ) at}

minima :

In the commensurate

_systems

without disorder

_(i.e.

if

_Vb

=

V f

= h =

0)

the

quantities

/3,,

are

_independent

of the

_position

in the

_system

and

_represent

a _{gap in the}

_spectrum

of small oscillations of the CDW

_phase

near the

_ground

state :

_8.,

=

mVJ/2ul/2

[6].

Introducing

then disorder into the

_system

one can follow the smooth crossover from the small

(af,

ab

1)

degree

of disorder where

jg,,

slightly

fluctuates from the

point

to

point

around its

average value

m V J/2 V¥2

to the

_{large degree}

of disorder case where the gap in the

phason

spectrum,

8,,,

fluctuates

_considerably

_{and the average value of}

_{(3 x}

deviates

_strongly

from

mVJ/2vfl/2.

Since the interval

_8cp,

where

_{2 w-periodic}

_{free energy,}

_Ex(cp),

_changes

significantly,

is of the order of

_unity,

_{8 cp -- 1,}

one can

_expect

that the scale of variations in

(3 x

is the same as the scale of variations in

Ex (cp)

near its minima

_points

i.e. the scale of

pinning

energy. Thus the distribution

density

W(,B )

of the

_quantities

_/3,,

can be used as the

distribution of the

_density

of the

_pinning

barriers as well.

To find

_{W(/3 )}

one should notice that on

_{fixing phase}

at

_{point x}

one

_immediately

breaks the

systems

into two

_independent

_parts

_(this

_{is the direct consequence}

_{of one-dimensionality}

of the

_system)

and those left- and

_right-hand

_parts

of the

_system

know

_nothing

about each other. Then as the next

_step

one should define

_quantities

_{à6 left/,igh,}

(x)

for the end

_points

of the new

semi-infinite

system

and find the

_{corresponding}

distribution densities

_{Wleft (/3 )}

and

Wright

(,B ).

This last

_problem

is solved

_{by constructing}

recursion relations

_connecting,

_say

13 left

(xi)

and 8

Ieft

(xi , )’

where xi

and xi , 1

are the

positions

of

neighbouring impurities.

Then,

since the weakness of

_{pinning provides}

the smallness of

_change

in

_{(31eft (x),}

when

_shifting

over

the

_{impurity spacing,}

one can derive the Fokker-Planck

_type

_equation

for a distribution

density

W;cft(/3)

in a standard manner

[5, 6] (the

variable « x » should be omitted

_now).

Making

use of the

_{relation Px..:--ct,81,,ft(x)+,B,ight(x)}

one

_immediately

_gets

W(/3)

from

distributions for the end

_points

_{[5, 6].}

We would like to stress now the distinction between the

weak

_pinning

condition

_Vb

_CVF,

and the condition of

_large

disorder,

expressed by

inequalities

(2).

As we shall see

_below,

the main contribution to the ac

_conductivity

comes from the metastable states,

separated by

barriers

/3

«,Bo

(where

(3o

is the average value of

pinning

energy)

i.e. from the domains of the

system

where the characteristic

_pinning

_{energy is much} less _{than its average value. The distribution}

_density

for

_W({3)

in the

_region

of low

/3

{3o

pinning energies

was found to be

_{[6, 7]}

where

_{h, =}

_{(4/ ’TT ) (V 0}

_UF)’/2

would be the critical value of the

_{incommensurability}

_parameter

in the absence of disorder. The characteristic

_length

of the domain with small

Q

«

_{a 0 pinning}

energy can be estimated

as e

(/3 )

=

VF/ {3.

Note that this relation also

gives

the size of the

412

In the case of a

purely

incommensurate

_system

with weak

back-scattering impurities

,8()

reduces to the well-known

_Fukuyama-Lee

value

The

_point

to be noticed here is that the scale of

_pinning

energy,

/3 0, is

the

_only

_{energy scale} that characterizes weak

_{impurity pinning}

of CDW

_system

_{[5, 6].}

To obtain the

_{low-frequency conductivity}

let us consider the

_equation

of small oscillations of CDW

_phase

_{0 (x, t)}

near the

_ground

state Cf)

_{(x) :}

where u is the

_{phason velocity,}

_{’U (X) = Vb (X) COS (2 kF X + (p (X» + M2 Vo COS M o (X)@}

a is the kinetic coefficient. The

_overdamped

limit

_{implies m}

« À and the

_{corresponding}

Green function is

Here

_{t/J n (X)}

and _{CJ) n} are the

_{eigenfunctions}

and

_eigen-values

of

_{Schrôdinger-like equation}

(7).

Following

[5, 7]

we find the

_expression

for

_conductivity

As has been shown in

_{[5, 7]}

this

_expression

can be rewritten in terms of the

_density,

of fluctuational

_phason

states,

_{p (co ) =}

_{8 «(ù - (JJ n»’}

which in turn was found to _be

pro-portional

to

_{probability density}

of

_{pinning energies,}

_{W(,S) :}

Making

use of the

_expression

for

_{W(f3) (5)}

we come to the

_following

result :

Here C =

1,

S_L

is the _cross section per

single

chain,

f2 p

=

W p/

is the characteristic

pinning

frequency

in the

_overdamped

_limit,

and

_tùp

=

(u//F)

130...

The factor

that had

_appeared

in

₍₁₁₎

is the localization

_{length corresponding}

to

_low-lying

state with

frequency to

and

_represents

the

_spatial

size of metastable state with

_pinning

energy

R

=

(UF/U)Cù.

where _co==

_{c(0) ==}

_{e2 UF/4}

w

_Sl

Q)

is the static

_{permeability.}

The real

part

of 8

(w )

for the

_{low-frequency}

_region

can be obtained

_{by using}

Kramers-Kronig

relation :

Making

use of the definition of Im s

(co )

from

(13)

and first line of

_{equations (11)}

and

(12)

one

_gets

Then on

_changing

of the

_subsequence

of

_integration

one can be convinced that

notwithstanding

that both

_quantities

have been calculated with the order of

_magnitude

accuracy.

3. Now we turn to the contribution from the transitions between different metastable states.

We consider the interval of intermediate

_frequencies

where the

_only

essential transitions are

those between

_neighbouring

metastable states

_(the

discussion

_concerning

more

_long-time

processes can be found in Ref.

_[9]).

This

_implies

that we assume our

_system

to behave as a set

of two-level

_systems

_(TLS).

_Any

TLS is characterized

_by

_{the activation barrier energy,}

U,

level

_splitting,

_2l,

and

_spatial

size

_{ÎU n=}

_{vF/ U.}

The scale of both U and A is of the order of the local value of

_{{3 x. {3 x’}

as has been mentioned

_above,

characterizes the scale of CDW free

energy and takes different values at different

_spatial

domains of the

_length

_{e {3}

_UF/,B

of the

system.

The contribution of TLSs to the

_imaginary

_part

_{of dielectric response} can be estimated

(see

[11]

and

_[12])

as

where TLS are

_supposed

to be

_classical,

_{r (U) .:2:t f2 p ’ exp (Ul T)}

is the time needed for

thermally

activated transition over barrier U and

R (U, à)

is the distribution

density

of

U and d. The factor

_eu

in the

_integrand

stems from the

_squared

matrix element of

transition,

et,

multiplied by

the TLS linear concentration

_Llfu,

where L is the total

_length

of CDW chain

_{and tu}

is

_just

the

_length

associated with a

_given

TLS.

414

where

_W(U)

coincides with the

_pinning

_energy

_{probability density}

_W(J3)

_{given by}

equation (4).

As the

_dependence

of

_{T (U)}

on U _{is very}

_sharp,

the main contribution to the

integral

(15)

comes from TLSs with w T

(U) =

1.

_Finally

one

_gets

This result is valid in the

_frequency

interval

_{f2 min « w ’ f2 p,}

where

flmin

=

(np

T /(30)

e - fi oiT.

In this interval TLS with small

_f3

dominate

_(relevant

_f3

are much smaller

_{than f3 0). lm 8 «(ù)}

grows as (o decreases and is of the order of

_{(TIB 0) 60 at}

W = f2

min’ that is the transition rate over average barrier.

Following

from

(12)

conductivity

is

At lower

_frequencies

the effective

_density

of the TLS becomes

_{exponentially}

small due to

the

_{exponentially}

fast

_decay

of the

_{probability density}

of

_high

_(/3

>

_f3o)

barriers and transitions in

_many-level

_systems

should be taken into account. In this

_region

of ultra-low

frequencies conductivity

can be

_roughly

estimated as

where ao m

u 2/,k UF

is the static

_conductivity

in the absence of

_{pinning potential}

_{CO (x).}

By

comparing

the results of

₍₁₁₎

and

₍₁₈₎

one can estimate the crossover

_frequency

f2

_{* -- d2 p (TI.8 0)’/In}

_(f3o/T).

In the

_frequency

_range

_{np (ù}

>-- f2 * the main contribution to

the ac _response comes from the

_intravalley

CDW

_relaxation,

while at * $: f2 _min the

intervalley

transitions dominate.

4. To obtain the nonlinear contribution to static

_polarization

let us consider domains with

small,

8

«

_{f3 0’}

_pinning

_{energy and of the}

_length

fp

=

UF/,S.

As

/3

is the scale of free energy

variation in such domains then even the electric fields with

_magnitudes

well below the threshold

value,

9 «

6T =

_O/UF,

will be

_strong

_enough

to reconstruct the

_ground

state of weak

_pinned

domains,

where

_{js /îp}

₍₈

_{= e}

E and E is the

_physical

electric

_field).

’77’

The

_change

in the

_phase

Ap

associated with the

_applied

field 9 is estimated from the relation

Here l.h.s. is _{the loss in elastic energy}

_arising

from CDW deformation and r.h.s. is the

_gain

in CDW _{energy due} to the

_applied

field. Then the

_{corresponding polarization density}

is

The condition of convergence of the

integral

(22)

at

_{8 --+}

0 determines the slowest

_possible

decrease of

_{W(,B )}

with

_/3

which is

_compatible

with finite

_ET.

We see that nonlinear contribution to the

_polarization

is

_{strongly nonanalytic :}

Note here that the fact of

_{nonanalytical}

behaviour of

_{P (E)}

was also obtained

_by

Fisher

[13].

5. We have shown that in the

_frequency

_range

_{f2 p >}

f2

_{,fl P ( T//3 0 )4}

_(ln

_{(f3o/T) )-1}

the main contribution to ac

_conductivity

arises from

_intravalley

viscous relaxation

_{(Eq. (11)).}

At lower

_{frequencies f2}

* > (0 > fi min =

(T / f3 0) e - J3o/T

ac

conductivity

is dominated

by

intervalley

transitions and is

_{given by equation}

_(18).

_f2.i,,

_corresponds

to the transition rate

over _{average barrier}

_/3o.

The results hold as

_long

as the linear _response

_{approximation}

is valid. The values of ac electric field

_{amplitude leading}

to nonlinear effects can be

_roughly

estimated as

ac

=

/3

2(úJ )/UF

(cf.

Sect.

4),

where

13

(w )

is the scale of

pinning

_{energy in the}

sample regions

producing

the main contributions to _{the response} at

_{frequency w.}

_Using

the results of

section 2,

3 we obtain

The value

_{of F, * (co )}

decreases as to increases in the activated

_regime

_{(Eq. (24b)),}

which is in

rough qualitative

agreement

with the

_experimental

results

_[3].

We have also found that the static nonlinear dielectric _{response is}

_{strongly nonanalytic :}

ôPnonlinear/ô6 161ln2 161

1 (cf.

Eq.

(22)).

Finally

we would like to discuss some

_general

features of CDW

_pinning

which can be

relevant for 3D

systems

as well. The first

_point

is that the total scale of the

_pinning

_energy

_(i.e.

the scale of

_{s (cp )}

_{function variation per}

_(0,

2

_{’TT) interval)}

is small in the « soft »

regions

with

/3

«

_j8o.

In other

words,

in the

_regions

with low local

_pinning

_frequency

_Cùx, all the nonlinear

terms

_(in

the _{free energy}

_expansion

over normal

_modes)

are small too. The second

_important

point

is that the

_spatial

scale of these « soft »

regions

is

large

u /£ô ),

and this leads to

the substantial matrix element

_dependences

on w.

Both these features were not taken into account in reference

_[9]

concerned with 3D CDW

pinning

and we believe this

_point

to be the source of the

_{discrepancies}

between our results and

some of the results of

_{[9] (e.g.}

_Eqs.

_(2.13)

an

_(2.14)).

As our results

_{apply directly}

to the

one-dimensional case

_only,

it is unclear if the same features hold in 3D

_problem

as well.

However,

we believe that such a

possibility

should also be taken into account.

Acknowledgments.

The authors are

_grateful

to R. Rammal for

_providing

us with the

_preprint

_{of his very}

interesting

paper

[9]

prior

to

_publication.

One of us

_(W.M.V.)

would like to thank R. Rammal for fruitful and

_stimulating

discussions and

_hospitality

at the

_CRTBT-CNRS,

Grenoble.

416

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