On the current conversion problem in charge density wave crystals. I. Solitons

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On the current conversion problem in charge density wave crystals. I. Solitons

S. Brazovskii, S. Matveenko

To cite this version:

S. Brazovskii, S. Matveenko. On the current conversion problem in charge density wave crystals.

I. Solitons. Journal de Physique I, EDP Sciences, 1991, 1 (2), pp.269-280. �10.1051/jp1:1991130�.

�jpa-00246320�

Classification

Physics

Abslracts

71.45L 71.45J 61.70G

On the current conversion problem in charge density

_wave

crystals. _{I. Solitons}

S. Brazovskii and S. Matveenko

L-D- Landau Institute for Theoretical

Physics, Kosygjna,

2, GSP-I, Moscow, 117940, U-S-S-R-

(Received

21Seplember

1990,

accepted

ii October

1990)

Abstract. The

theory

of solitons in

crystallographically

ordered

charge density

_waves

(CDW)

at low temperatures ^is

developed.

Various types of solitons, which appear in _{a course} of electron's

self-trapping,

_are described. The

adaptation

of CDW media to the formation of _gr- and 2

gr-

solitons is considered. A model

independent approach

is

suggested

to describe interactions of solitons with the Coulomb field and with

phase

deformations of CDW. The effects of

screening, self-screening

and

commensurability

_are considered. Interactions between solitons and with

impurities

are described. Cases of attraction among solitons _are found, which may lead to their

aggregation

into

microscopical phase-slip

_centers.

Inboducfion.

Electronic

properties

of

quasi

one-dimensional conductors with

charge density

_waves

(CDW)

are

peculiar

for two _reasons. The first is related _to the translational

degeneracy

of CDW

ground

state which leads to the

phenomena

of Fr61ich

conductivity [1, 2].

It shows itself in _a

giant

dielectric

susceptibility,

in nonlinear and

nonstationary

effects

(see proceedings

and the reviews

[3-7]).

The second _reason is related to a strong interaction of CDW deformations with normal

electrons,

which leads to their fast

self-trapping

in the _course of conversion to various kinds of solitons

(see

reviews

[8, 9]).

Extensive

experimental

studies _are devoted to the Fr61ich

conductivity (see

reviews

by Fleming, Gill, Grfiner, J6rome, Nad', Ong,

Schlenker _et al, in

[3-6]).

At the _same time the

physics

of solitons is

mainly

studied at another substance-

polyacethylene (see

reviews

[10, iii),

where the effects of CDW

sliding

_are absent for the

case of two-fold

commensurability.

In recent years

experimental

studies of CDW turned to low

temperature and/or

to materials with well

developed crystal

structure of CDW and

corresponding

dielectrization of electronic spectra. ^These are the _so called blue bronzes and

tetrachalcogenides

of transition metals

(see [3-6, 12]).

In this respect and also due to the above mentioned

discrepancy

in theoretical

studies _{we were} motivated _to consider _a

microscopical picture

for the conversion of normal current carriers to solitons and for their

subsequent development

into CDW

phase slips.

It is of

special importance

to trace how the electrons

injected

at the

junction finally

evolve into _a deformation of CDW in the volume. For isolated one-dimensional chain of CDW this _process

270 JOURNAL DE

PHYSIQUE

I M 2

is

principally

clear.

Every

electron _or hole with energy E _near the

edges

_± A of the gap 2 A

spontaneously

evolves into the

nearly amplitude

soliton

[8, 9, 13, 14]

while the

original

electron is

trapped

at the local level _near the gap ^center E

= 0. In this process which goes

through

_a time

wj~'

^{10~ '~} s

(where

_{w~~ is} the

phonon frequency

at the CDW _wave vector

Q

=

2k~)

the energy m 0.3 A is released.

During

_some

large

collisional time the

pairs

of

amplitude

solitons

develop

into 2 _w

phase slips

while

trapped

electrons

disappear

in _an extended continuum below the gap. As _a

result,

the

remaining

_part

m 0.6 A of the

original

energy A is released. But

already

for _a

quasi

one-dimensional system ^of

weakly

bound chains with low 3d

ordering temperature

_T~ « A the

picture

becomes much _more

complicated,

since at T~ T~ ^the very existence of

topological

solitons is

questionable [8, 9, 14, 16-18].

In this

publication

_we will consider the effects of

long

_range structural deformations and Coulomb fields due to the CDW

crystal

media

adaptation

to various

solitons,

which _are born in the process of electronic conversion. We will

study

the static deformations in 3d ordered

CDW media with

point solitons,

interactions of solitons between each other and with

impurities.

We will consider the effects of

screening by

normal carriers and _a commensurabi-

lity.

These results

permit

_us to outline the

microscopical picture

of

subsequent steps

^{for the} conversion of normal current to the Fr61ich current of CDW

sliding,

which is described in Conclusion. A short

presentation

for _some of these ideas _was

given

in

[9].

^{More extensive} studies and the

theory

for the soliton's

aggregation

_or dislocations will be

presented

elsewhere

[19].

Solitons in

quasi

one-dinlensional model of CDW

crystal.

The

quasi

one-dimensional

system

^{of CDW} is characterized

by

the _wave number

Q

m

2

k~,

the gap 2 A in electronic spectrum, ^the

length to

=

VIA,

where

k~

and _{u are} the

momentum and the

velocity

at the Fermi level

(the

Plank constant is assumed to be

unity

fi

=

I).

The CDW deformation has the form

~~(x)

=

Re

A~(x) exp(lox), A~(x)

=

A~(x) exp(iq~~(x))

where

[A~(x)

and

q~~(x)

are

amplitudes

and

phases

_on chains _n and the

dependence

_on the coordinate _x

along

the chain

corresponds

to the disturbed states. In the

ground

state A~ _m A, _{q~ m} ^{0. Here} and in the

following

_we

suggest

for

simplicity that,

at

equilibrium,

all q~~ ^are

equal,

I-e- the CDW _wave vector is

Q

=

(Q, 0,

0

).

In _a

system

^of

weakly

bound

chains,

_T~ «

A, self-trapping

of electrons _on the energy scale A and the

length

scale

to develop independently

_{on any} chain. Then

topological

solitons _are

created which _are

incompatible

with the definition of the

long

_range order.

Adaptational

deformations must appear ^to

equalize

the values of the order parameter on all chains. These deformations _are characterized

by

a low energy T~ ^« A and _a

large length ^I v/T~

»

fo,

which

permits

a

description

in terms of

phases

_q~~ at

given amplitudes

_[A~

m A.

The _presence of ± 2 w-soliton

[15, 16]

_{on a} chain _n

=

j

which

corresponds

to the self-

trapping

of two electrons

(+)

_or two holes

(-)

is taken into account

by boundary

conditions

n

=J

.

wj(+

CO

wj(-

CO

= VW, n

#J

.

wn(+

_CO

)

=

wn(-

CO

) (I)

for _v

= ± 2. The presence of _an

amplitude

soliton

corresponding

to the

self-trapping

of _one electron

(+)

or one hole

(-)

at _a

point _,

is taken into account

by

_a local condition

[14]

Wj(,+o)-pj(,-o)=TW, p~(±CO)

₌

pj(±CO)=o.

This _case is

equivalent [9, 14]

to the condition

(I)

for _v

= ± I. These conditions show that electric

charges

_are

equal

to we.

The energy functional for _a system of chains has the form

3C=

jdxl£((~~)~-£J~~cos(q~~-q~~)+ ~)4i~j- j~~~~dr)(2)

n

'~ X

m

'~ X 8we

where 4i

=

4i(x,r)

is the electric

potential

^relative to _one

electron, 4i~

=

4i~(x,r~),

r ₌ r~ is a coordinate of _an n-th chain. The _presence of _a soliton _{on a} chain

j

is taken into account

by

condition

(I). Independent

of the presence or distribution of solitons defined

by

the set of conditions

(Ii,

the

following

conservation laws for

quantities averaged

_over the

sample's

cross-section take

place

~

~

~

~ ~fi

" Const.

U

~ ^°wn

I

~

W

^{~ ~ " Const.}

(3)

~2_8e~_~_2 _{~jj_} ^ffidr~

US ~ ~

S

where _s is the _{area per}

chain,

_r~ is the

screening

radius in _a

metallic,

without

CDW, phase.

First _we will consider _a model which

permits

_us to

study

the whole

region

of

lengths xi

»

to

for

arbitrary

_n. We suggest ^{that the matrix}

J~

in

(2)

connects _a

large

number of chains. Then _we may consider

phases #~

^of

neighbour

chains at _n #

j

to be close and _we go ^to the continual

description by

substitutions

£

_~

ldr~,

q~

~(x)

~ q~

(r)

,

r ₌

(x, r)

~

S

J=jjJ~~T)/u; 2wJ/us=a.

We arrive at the

following

form for the energy functional

where

H~

₌

£

s6

(r _r~

^{~ °~~ ~} J _cos

(

q~~ q~

)

₊ ^~ 4i

°~~

(5)

4 _w ax _w ax

The _sum in

(5)

_{goes over} those

chains,

which contain solitons.

A variation of the energy functional

(4), (5)

over q~, q~~, 4i

gives

the

equilibrium

conditions for the system ^with solitons

u 0~

q~~ 04l

(x,

_r~

)

j ⁺ wJ _sin

(q~~

(x)

_q~

(x,

_r~

)

= 0

(7)

2 _ax ax

~

~

A ~fi

(x,

_r

)

+ ~ ~'

~~'

~

+

£

s6 (r

r~ )

^~~~

^~~'

^~~

=

0

(8)

UK dx ax

272 JOURNAL DE PHYSIQUE I N 2

where

Let _us consider _one soliton _on the chain

j

₌

0,

_r~

=

0. Before

taking

into _account the electric field

4i, equations (6)-(8) acquire

the form

3~q~o

Aq~ + s6

(r)

=

0

(9)

3x~

f2 ^~~~~ (10)

+ Sin

(9'0

_i~~~~~ ^~

~

~

~~~

The solution of

equation (9)

relates

functionally q~(x, 0)

and

q~o(x)

as

~~°'~~ ~l ~t'~

ii

_x

-1'+12ji/2

^~~~~

The

system

^of

equations (10)-(11)

defines

self-consistently

the functions

q~o(x)

^and

q~(0, x)

which describe the

phases

_on the central and _on the

surrounding

chains. Then the

phase

_q~

(x, r)

is found in the whole volume with the

help

of

equation (9).

We _{can see} that

equations (10)

and

(I I)

indeed have _a solution for the

topological

soliton which satisfies the conditions

(I).

Unlike _a conventional sine-Gordon

equation

the

asymptotic

at

ix

»

I

_{is not}

exponential

but rather has _a power law. Indeed the

major _conterpart

to the

integral

in

equation (I I)

_comes from the

region y[

_w

^I. By partial integration

in

(I I)

and

taking

into account

(i)

_we find at

(x( ^hi

q~o(x)-2w0(x)~q~(0,x)sgnxm- ~~~+~~~~~~~~~~~~ ⁽¹²⁾

4x 8 _ax

The second term in

(12)

is _a correction which _comes from the

region

_y~x.

Taking

the Coulomb interaction into account _we would find

analogously

that

wo(x)

m w

(o, x) i/x. (13)

A

comprehensive study

for the field's distributions _at

large

^{distances will be}

given

in the next

chapters

_on the basis of _a model

independent approach.

Interactions of sofitons at

large

^distances.

Now _{we are}

mostly

interested not in the local structure of solitons which is model

dependent

but in the distributions of fields _q~, 4i at

large

distances and in solitons' interactions at

jr

_»

^I.

For these kinds of

problems

the _presence of solitons at a

given

chain _may be taken

into account

by using asymptotic

conditions

(I)

which _can

by

written _as

3q~~

-dx

= v~ ^w

(14)

lx

where vj ^{is an index v} from

(I)

for the soliton _at the

point _j.

The main

counterpart

to the

integral (14)

_comes from the

vicinity

of _a soliton

to

«

ix _-,

~

i,

_{since the power laws} of

(12)

and

(13)

_are fast

enough

for convergency of

(14).

At

large

distances the condition

(14)

may be written _as

3q~j

= v

~

6

(x ~) (15)

w lx

The system is described

by

the model

independent

Hamiltonian

(4)

where the solitonic part 3C~ should be

presented

_as a Hamiltonian for _some

point

_sources.

3C~ =

d~r p~(r) V(r)

p~ =

£

_v~ s3

(r _r~)

V

=

~ ~~'

+ 4i

(16)

2 lx

where _{p~ is a}

weighted density

of

solitons, equal

to

injected charge density,

Vis _a

potential

of solitons' interaction with

long

_range fields _q~ and 4l.

The Hamiltonian

(16)

is derived in the

following

_way. The term 4l

analogously

to

(4)

describes the Coulomb interaction of the soliton's

charge,

which is considered _{now as a}

point charge

in

(15).

The first _term in

(16) provides,

_{as we} will _see, the

rigorous

conservation law

(3)

and

permits

us to take the

charge

distribution

correctly

into account in the system. Indeed

integrating equation (3)

term

by

term

£ 6Wn"~£3Wj, 3W

"

(+C°)~W(~C°)

n#J J

we arrive at the

paradox

_{: every} term in the L-H-S- part ^is

equal

to zero, while the R-H-S-

part

is finite. The

compensation

_comes

evidently

from _a

divergency

at

large

n and _x, where the continuum

approximation

is valid.

Any

finite number of terms in the _{sum over n may} be

neglected,

_so the definition of the

charge density

in the continuum limit is _not violated.

We notice that the

potential

V for the interaction of solitons _appears to be

exactly

the _same invariant combination of fields _q~ and ~fi _as the

potential

for normal electrons _near the CDW

edge,

where V

plays

the role of _an electrochemical

potential.

By variating

the functional

(5)

and

(16)

_over

q~ and 4i _we arrive at the

following equations

u fiq~ + 2 ~~

+ vu

~

p~(r)

₌ ⁰

(17)

~

~

A4i ₊

~

+

wp~(r)

₌ 0.

(18)

UK lx

Integrating equations (17)

and

(18)

_{over a}

perpendicular plane

_we obtain the

equations

for

quantities averaged

_over a

perpendicular

cross-section defined _as in

(3) by

~

~

~~~[~~

+

~#~~~

+ gr

z

_v~

(x ,)

=

o

UK )X ~X

~

~ ~

~~

K

~fl~

(X)

= COnSi.

(19)

~

)X

We _{can see} from

(19)

that in the volume

= const., the average electric field is _zero, while for the

junction problems

_We should omit

exponentially growing

solutions and

neglect

exponentially decreasing

solutions _{on a}

microscopical

scale

K

'

Equations (19)

_are exact for the discrete model

(2)

_as it has been assumed for _our construction of Hamiltonian

(16).

274 JOURNAL DE

PHYSIQUE

I M 2

Without Coulomb field the solution of

equations (17)

and

(18)

for _a soliton at the

origin

Would have the form

~~~~~ 4~a

3' ~~~~

It _can be

easily

_seen that

~~~~

i

~~~ ~

so that the first condition of

(19)

is satisfied. A distribution of the

charge density

in the media is

given by

p

(r)

=

e6

(r)

₊ ^~

i~~~~

=

e6

(r)

~

~

l~

~

~

~

(21)

w x wa

tj

t

For the interaction energy of two solitons with indexes _{vi, v~ we}

easily

find from

(16)

to

(20)

W(r)

= ±

~~'

= vi v~

~~

l-

+

~

~

(22)

°~ _~

(t( (t(

We notice that the interaction of solitons

changes

the

sign

at a cone surface

r/[x[

=

(a/2)"~,

r =

jr

For solitons of the _same

charge

_{vi v~}

~

0)

which appear due to the

self-trapping

of _a similar kind of

carriers,

the attraction takes

place

in the

perpendicular

sector of the _cone:

r ~

xi

_a

/2)

"~.

The account for the Coulomb interaction

essentially changes

^the distribution of the fields.

For the function 4i

(x, r),

_q~

(x, r)

_we

easily

find from

(17)

and

(18)

the

following equations

l~q~~(x,

r ₌ ws

(K

^~

A)

^~ 6

(x,

_r

) (23)

ax

k4i~(x,

r

)

= 4 _we ^~ _a

A~

6

(x,

_r

) (24)

where

k=fib-K~ ^~~j

_h

~ ⁼

~~.

0X 0r

Equations (23)

and

(24),

_can be solved

explicitly

in two cases when the operator

l~

can be factorized : a

region

of smooth

dependences

and for the

special

_{case a}

= I. For

gradients

small in

comparison

to K, I.e. in the

region

where the continuum model is

applicable,

we obtain

~fi~ m

~~

exp

~~

(25)

2

ix

d

ix

~ ~2

~~~~'

~

fi

^~~~

@

^~~~~

where d

= 4 _a ~'~/K. Notice _a very unusual inverse

exponential dependence

_{on x} in

(25)

and

(26)

which is evident from the

analogy

between the

longwave

limit

oft

_in

_{(23), (24)}

_{and the}

operator

^for a two-dimensional diffusion in _« time _{» x.} A

peculiar

feature of this

dependence

is that

gradients

in

(25)

and

(26).

~~x~ ~~~ _)~~j~

do not

drop

at

large

distances _{as one} would expect ^{but rather decrease}

only

in the

longitudinal

sector _a ~'~[

xi

_~

jr

where solutions

(25)

and

(26)

_are

applicable.

In this

region

the Coulomb

potential

^{dominates in} the total interaction of solitons

(16),

V

m 4i.

We _see that the interaction

changes

from _a power law

repulsion

~

e~/2 xi

at

ix

_~

r~/d

to an

exponentional

attraction at

rla"~< _xi <r~/d

and reaches

exponentially

small

positive

values

exP

(- rid)

at the

validity

limit.

In the

perpendicular

sector r

<xa"~

_at

r »d _{we can}

neglect

all interactions with the

exception

of the _case of weak Coulomb interaction with

relatively

strong interchain

coupling d~

_»

i~

=

sin

,

I-e- _a ^~ _»

e~/u8

at small distances

I

< r < d.

The field distributions in the whole

region

of _{x, r are}

conveniently

demonstrated

by

_an

exact solution of

equations (23)

^and

(24)

at _a

= I. In this _case the operator

l~

_{is factorized}

and _we find the

following

exact solutions of

equations (23)

^and

(24).

4i~(x,

_r

)

=

~~ ~ ~~~

'~ ch

fl ₊ sh fl

( ₍

_I

+

F) (27)

" 4 F F

K

~s exp(- ?) ~~

~ ~u k ^~

(i

₊

f)j ^(~~~

Ws(x,

r

)

~ i ^~ 16 f

~

P

where fl

=

Kx/2,

F

= K

jr /2.

The interaction _energy of solitons with indexes _{vi, v~} is found from

(16), (27)

and

(28)

_:

W(r)

= vi v~

~ _~

exp

(- F)

2 fl ₊ sh fl ₊

F~ F~

+ + +

~£

+

(

ch

lj

(29)

F F F F _r F

At

xi

» r the

expressions (27)-(29) correspond

to

dependences (25)-(26)

at _a

=

I _as it should be. In _a whole

region

of

large

distances F _» I the main factor in

(27)-(29)

is

exp([fl[ F) (30)

F

We _see that interactions fall down

exponentially

_on

going

from the chain _axes

r=0 but

they

_grow

exponentially

in the sector

(x(

<r

by going

from the

plane

x =

0. The interaction

(29)

becomes

negative

at very short distances

jr

_{< K} ^' where the free

276 JOURNAL DE PHYSIQUE I M 2

phase

interaction

(22)

is recovered _as it should be. Out of this spot we

always

have W _~ 0 and the

repulsion along

_r takes

place.

We may

speak

about the short range

binding

of

solitons

only

at weak Coulomb

coupling e~/u

« I when the width I

/K

is

larger

than the soliton

length ^I s"~

_{We should}

keep

in mind that interactions in the

perpendicular

sector which decrease

exponentially

at a

microscopical

d scale may be restored

by

_many factors. For

example

there is _an attraction due to the

exchange

of localized electrons for solitons with

v =

I. This process is

specially important

for the irreversible conversion _{w + w}

~ 2

w.

Due to a very

large anisotropy

of the diffusion coefficients for solitons _one should first consider the

dependence lIj (x)

=

W(x,

_r

= const. at _a

given

r. We _see that the function

lIj

^has a minimum _W~~~ at _x

=

0 Which is

exponentially

shallow in _r. There is _a

large region

of attractive forces d

W/0x

_< 0 for the

positive

_energy W _~ 0 at _{x ~} x~~~

r~/d,

which is

separated by

a

large

barrier W~~~

e~d/r~

from the

repulsion region

at x ~ x~~~. The minimum

depth

and the barrier

Weight

_grow with

decreasing

_r. At _r

= 0 extrema are absent and the

repulsion

of _a non-screened Coulomb type ^{twice weakened}

(the

first term in

(29))

dominates. At small

finite _r W~~~ is

negative.

Effects of

screening

and

commensurability.

At

large

distances the

principal

role is

played by

Coulomb

screening

due to residual carriers and

(in

commensurate

case) by

the

pinning

_energy. Both effects do not influence

essentially

the structure of solitons but should be taken into account for the

long-range

fields ~fi and _q~.

In real CDW

systems

we may find two mechanisms for

screening.

One of them is related to a redistribution of the soliton gas

density

which interacts due to

(26) simultaneously

with the fields ~fi and _q~. Another _one is related _to the residual carriers. For

example

_a

weakly

filled band like in

Nbse~

_{or a} low activated electronic band _may exist Which do not

participate

in the

formation of CDW I-e- do _not interact With _d

q~

/dx. Along

With the

screening

_we shall consider

possible

effects of

commensurability ([2, 7])

of CDW with

underlying

lattice.

Taking

both effects into _{account we} will find _new terms q~~ and ~fi~ in the Hamiltonian

(16).

Thus

effects of electronic

screening

^and

commensurability

are taken into _account

by

the

substitution in

equations (17)

and

(18)

like

A~A-A~, i~i-8~; A~=4vre~0n~/dp; _A«K;

_8«K

where _~~ and _{p are} the concentration and the chemical

potential

of free electrons and 8~ _is

proportional

to the

pinning

_energy. A solution for the modified

equations (17)

and

(18)

in Fourier

representation

takes the form

4

we~(ak (

₊

_8~)

~fi~ =

l~(k)

~~~

~~~

_~

~~

~ ~ _~~

w~ ~

(31)

K(k)

where

l~(k)

=

(k~+A~) ((~+ 8~)+K~kj/mK~kji+ak( +p~k( +K~q~;

k

=

(kjj,k~); p

^~=

8~+ aA~;

_q

=

8A/K. (32)

Taking

into _account

only

lowest order terms in

kjj, k~,

_8, as shown in formula

(32)

_we find

from

(31)

the

following

solutions

R*

=

(x, ri(a *)"~)

_;

a *

=

p

~iK~

(34)

Formulas

(33)

are valid if the FouRer transform of

(31)

is convergent at small

enough k~

^when we can

neglect

the term ak

(

in

(32).

It takes

place

at distances

R* ~

dla

*

(35)

Otherwise _we retum to

dependences (25)

and

(26).

It follows from

(33)

and

(35)

that the Coulomb part ^{in the} interaction

potential

V dominates

over the

phase part following

the

parameters

_K

~/p

^~ _» l. At

qR*

_< I _we obtain from

(33)

and

(34)

v~r) ~i~ Ii

l~ ~l~li~~li)'(1/~ ⁽³⁶⁾

We see from

(33)

and

(36)

that the distributions of fields

q~ and ~fi _are similar to the fields of

dipole

and

quadrupole

_sources

correspondingly

for _an effective Coulomb

potential.

There _are two

significant

differences in

comparison

to similar results

(20)-(22)

for the model without

physical

Coulomb interactions. First of all the effective

anisotropy

is

strongly

enhanced.

£Y~£Y*=fl~/K~@a_

Secondly

for ₈ # 0 the finite radius of

screening

_q~ ^' _appears. Both of these effects extend the sector of solitonic attraction at

large

distances

(35) practically

to the whole sector

xi

_~

rla

^"~

_except

for _{a narrow}

longitudinal

_one at

xi

_~

r/(a *)"~

In the sector

ix

<

rla "(

The effects of ₈ and A _are not

important

in the _presence of fast

(exponential

_on the scale

K

') dependences (30).

We _can

easily

_see it with the

help

of _{a more}

general

exact solution available for _a

= I, A

= 8. For this

special

_case the kemel

l~

_factorizes

_again

_and

we arrive at exact

equations

of

type (27)

and

(28)

^with the substitution of

exp(- RF)

to

exp(- F),

where

p~

= + 2

p

~/K~.

Eventually

it

brings

_a

screening

factor

exp(-

_q

xi

to formulas

(25)

and

(26).

SELF-SCREENiNG IN THE GAS oF soLiToNs.- We consider _now the _case of _a finite

concentrations of solitons n~ ^«

i~

For the scales

large

in

comparison

to the _mean distance between

solitons,

the

density

_p~ in

(16)

_may be considered _{as a} continuous variable

p~(r)

_{~ n~} + 6n

(r) (37)

The average concentration of solitons n~ is

compensated

due to

(4) by

the average

gradient

(~'

^~

(~'

⁺

(~'

^{= wsn~x +}

(~' ⁽³⁸⁾

x x x x

which is related to our definition of the

ground

state as a

homogeneous

one in the absence of solitons. Local

change

in the soliton concentration due _to

perturbations

from the fields 4i and

q~ is

equal

to

6n

(r)

=

( V(r) (39)

278 JOURNAL DE

PHYSIQUE

I M 2

where _p is the chemical

potential

of the soliton _gas.

Expressions (37)

and

(39) give

us the additional term in the functional

(16)

an a ²

63C

=

~ _4i

+

it _{d~R. (40)}

2 0p ² ax

In

taking

_{average over} the fluctuations of the solitons _an anomalous term may appear. To take into account a control of this

problem,

the derivation of variational

equations

which

correspond

to

(16)

and

(40)

_was done

independently

from the first

principal

of many electronic

problems. (See Appendix

in

[14]

_or the Review

[8].)

We _see that

expression (40)

contains the term

4i~ analogously

to the _case of electronic

screening.

Other two terms

give

an

unimportant

renormalisation of coefficients in the combination of terms from

(15)

Which contain

0q~/0x. Consequently self-screening

of solitons may be included

additively

to the

definition of the parameter ^{A~ for the electronic}

screening

A~~A~+4wv~0nJ0p~.

Here _we should

put

^index v = 2 since the most real effect _comes from the 2 w-solitons.

Their concentration _may be enhanced due to their low activation _{energy or} it may be defined

by

the

proximity

to a

commensurability point.

Interaction of solitons nith defects.

Let _us consider _a

long

_range interaction of solitons With defects which

pin

the CDW. We consider _an isolated defect at the

point

_ro with _a minimum

pinning

_energy at _q~

(ro)

= 0. The

term which describes this _energy must be added to the functional

(16).

6W=

)dxdr( (q~(r)- 0)~6(r-ro). ₍₄₁₎

The variation of the

functionil (4)

with the additional term

(41) gives

_us another local _source at the

right

hand side of

equation (17)

_:

(- «Iv ) Cs(q~ (ro)

0

)

6

(r ro) (42)

The energy for _a bilinear continuous functional with the

point

_sources

(16)

and

(42)

_can be calculated in _a

general form,

Which does not

depend explicitly

_on the functional form

(16).

We arrive _at

W

"

~'(rj) j

_°

(W

(ro)

0

(43)

and after the self-consistent definition of

q~(ro)

We obtain

W=((q~~(ro-rj)-0)~ (44)

Here

q~~(r)

^is the

corresponding

solution for

single

soliton

problems

Which have been considered above. The constant

©

_{differs from C in}

_{(40) by}

some renormalisation due to self- energy of the

long

_range deformations introduced

by

^{the isolated defect} without soliton.

Let _us find the distribution of the fields

q~ and 4l in the _presence of _a defect. Variation of the functionals

(5), (16), (41) gives

_us in

analogy

to

(23)

and

(24)

the

following equations

1~4l

= 4

we~

a

A~

6

(r rj) (8 we~/u) C(q~

0 ^~ 6

(r ro) (45)

ax

kw

= grs

(K

² A

) ) _{(r ro)}

+

(2 grs/u)

c _w o

)

AS

(r ro) (46)

Comparing

the r.h.s. of

equations (45)

and

(46)

to the r.h.s. of

equations (23)

and

(24)

_we

are able to express the solution for the

problem (45), (46)

via the solutions _q~~, ~fi~ for the

single

soliton

problems

without defects

~P

(r)

=

~P~(r r~)

C (q~

(ro)

0

) _q~~(r ro) (47)

(r)

=

q~~(r rj) £

(q~

(ro)

0

) 4l~(r ro)

+

(

(q~

(ro)

°

) ) _{Ws(r ro) (48)}

It is

interesting

to notice that the main contribution _to the solitonic

potential

V _comes from the Coulomb field ~fi

inspite

of the fact that _a defect interacts

explicitly

with the

phase only.

Meanwhile the field ~fi induced

by

_a defect coincides with the field 4l~ induced

by

the soliton which

provides

the

correspondence

of results

(43)

and

(46)

at

large

distances when q~~(ro

r~)

« 0.

To

analyze

the interaction

(43)

it is natural

(as

well _as for the _two soliton

problem

of Ch.

3)

to underline the

dependence

JJi (x)

=

W(x,

_r

= const.

)

Remember that q~~(x, r

)

is _an odd function of _x and it has the _{extrema at x}

= ±

x~(r ).

^{In the}

general

_case 0 _w formula

(44)

describes the attraction towards

x~(r)

^from the

region J~~(r)<x<co (we

_suppose ^here that

0~0).

The

perpendicular

forces in the

valley

x =

J~~(r)

_are directed to the defect. At small 0 the absolute minima extend from the

microscopical vicinity

of defect to _a closed line £ which is defined

by

the

equation q~~(r ro)

m 0

For _a

special

_case 0 =0 the attraction towards the

plane

x=0 from the

region ix

_{< x~} and the

repulsion

in

perpendicular

directions take

place.

This case can be found

only

for coherent

crystallographic positions

of defects relative to CDW which _may

probably happen

for _a

special

_case of mobile

impurities (see [20]

and references

therein).

At

large

distances the interactions of both

w- and 2 _w- solitons with _a defect differ

only by

_a coefficient

v = 1, ² in

q~~ and the effect of attraction is

general.

However the bound state at the defect will be

principally

different since the w-soliton contains _a nuclei where the

amplitude

_goes

through

_zero, and

consequently

the

binding

_energy will be

~

A.

Probably

at low temperatures every

dopant

forms _a neutral

complex

with _a w-soliton which

plays

the role of the neutral defect studied above. Bound states of

charged impurity

with solitons _were

considered

recently

in

[21].

This article also contains

important

references _on the Coulomb interaction in CDW.

Conclusions.

In the

present publication

_we have solved _a number of

stationary problems

which _are

required preliminary

to describe the conversion of _a normal current to the Fr61ich current of CDW. We

considered the formation of _w and 2

w solitons and _we constructed the model to describe the structure of solitons in

crystallographically

ordered media of CDW.

Then _we found the model

independent approach

to describe the interactions of solitons

280 JOURNAL DE PHYSIQUE I M 2

with

long

_range Coulomb and deformational fields. On this basis _we calculated the

interactions of isolated solitons with each other and with

impurities.

Also _we studied the combined effects of weak

commensurability

and

screening

_or

self-screening by

the residual

concentration of electrons _or solitons. For all _cases the

regions

_are found where interactions between solitons _are attractive to their _common

perpendicular plane.

In the _case of

screening

we may also trace _an attraction between solitons in their _common

plane.

These results show _a

tendency

towards the

aggregation

of solitons in _some

complexes

which _are

equivalent

to dislocational

loops.

These

problems

will be described elswhere.

Acknowledgements.

One of the authors

(S.B.) acknowledges

the

hospitality

of the Institute for Scientific

Interchange

Foundation in

Torino, Italy.

We _are

grateful

to N. Kirova for the

help

in

editing

the

manuscript.

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