On the current conversion problem in charge density wave crystals. II. Dislocations

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On the current conversion problem in charge density wave crystals. II. Dislocations

S. Brazovskii, S. Matveenko

To cite this version:

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

Dislocations. Journal de Physique I, EDP Sciences, 1991, 1 (8), pp.1173-1185. �10.1051/jp1:1991101�.

�jpa-00246403�

J.

Phys.

I France 1

(1991)

1173-1185 AOUT 1991, PAGE 1173

Classification Physics Abstracts

71.45L 71.45J 61.70E

On the current conversion problem in charge density

_wave

crystals. II. Dislocations

S. Brazovskii and S. Matveenko

L. D. Landau Institute for Theoretical

Physics, Kosygina

2, GSP-1, Moscow, 117940, U.S.S.R.

(Received

4

February

1991,

accepted

25 April 1991)

Abstract. Dislocations and

aggregation

of solitons in

Charge Density

Waves

(CDW~

at low temperatures are considered. The

principal

effects of Coulomb interactions _are studied both at zero temperature ^{and in the} presence of residual carriers. The energy and distributions of

phase

deformations and of electric field, the interactions with solitons _are found. The conditions for solitons

aggregating

into

larger complexes equivalent

to dislocation

loops

_are found. The results

provide

an

insight

onto

sequential

steps of the _current conversion in CDW

crystals.

1. Introducdon.

Recently

_a dislocation mechanism for the CDW motion has been

actively

discussed

(see [1,2]).

A

macroscopical picture

for the

phase-slip

nucleation in _a volume _was

suggested already

in

[3].

Later the creation of

phase

vortices at _a

boundary,

which is

equivalent

to the introduction of _an

edge dislocation,

_was considered

[4, 5]

when the role of

junction

effects

[4- 7]

was elucidated. The effects of

preexisting

dislocations _were discussed

recently [8].

^A

systematical interpretation

of CDW

phase

deformations in terms of _a dislocation

theory

in

uniaxially

deformable

crystal

_was

given

in

[9,10].

Based _{on a}

purely phenomenological approach

with _a direct

application

of conventional

crystal concepts,

^the current studies _are

limited

by

two

aspects.

^First

ignoring

^the Coulomb fields

accompanying

the CDW

deformations _one is bound _{to a case} of _a

high

concentration of normal carriers which is observed

probably

in

Nbse~ only.

At present ^{the effects at} low temperatures attract most attention

(see [11-14])

when the Coulomb interactions

play

the essential role. Second _a

microscopical phase slip

nucleation

picture matching

_a

macroscopical

stage ^of dislocation

development

needs _to be considered for the substances under current

experimental

studies. In this _{way a} consistent

picture

for _a

sequential

transformation of the normal _{current to} the

Fr6hlich _one at the

junction

_can be

developed.

So far

microscopic equations

have been

consistently

derived

[6]

for same other instantonic mechanism to describe

phase slips

at the

junction by analogy

^with the _narrow

superconducting

channels

[15].

As well _as for

Eliashberg-Gorkov equations

in

superconductors

the

gapless

case had to be considered

[6]

near the transition

temperature

T~ for _a

strong impurity

II 74 JOURNAL DE PHYSIQUE ^li° 8

scattering. For the _case with _a well

de~eloped

gap ²

Jo

a

moving plane

wall of _an

amplitude

soliton type was considered

[16].

^{This wall is unstable} against ^the

decomposition

into

phase

solitons with a lower energy

(see [17, 18])

for _a system ^of

weakly coupled

_T~ «

Jo

^chains.

At present most

experimental

studies deal with

essentially

_quasi one-dimensional

compounds _[19]

where the gap is

developed

_even at T~ whereas at low temperatures ^there i~

already

a

strongly

correlated 3d CDW

crystal [20].

It follows that both the _{primary con, ersion} of electrons into

microscopic phase slips

and their

subsequent aggregation

into _grouing dislocation in the presence of

weakly

screened Coulomb fields need _to be reconsidered for these materials. The first stage was studied

recently

in _our

preceding publication [21j following

the _some old

[17]

^and

recently

extended concepts ^which have been

partly

described in reviews

[17, 18].

The initial steps of normal carriers _e conversion process are determined almost

uniquely

in

systems ^{with weak mtercham}

coupling,

characterized

by

a small ratio of the ~d

ordering

temperature _T~ to the _gap 2 Au T~ «

Jo.

The _{process goes}

through

the creation of

amplitude

r-sohtons

(for

all 7~ to the creation of 2 r-sohtons [22~

23]

for T~ T~ and then to their

aggregation

into

macroscopic phase slip

centers

(PCS).

^{The first} stages e _- w, w r - 2 _w

develop irreversibly yielding

_{some energy so} that the process direction does _not

depend

_on

interactions Nevertheless the _rate of the second _{process w + w} -2r requiring pair

collisions

depends

on the interaction

sign.

For the

repulsing

the _{process is} substituted hy, another _{one e +}

w - 2 _w which

requires

_a

high

activation _energy

~o.

The third _proces~

2 _{w + +} 2 _w

- PCS

principally depends

on the type ^of interactions. The

grou>th

^{of PCS} with _a low activation _energy will take

place

for the _case of attraction, which

correspond~

to neu.

experimental

data

[24].

For the

repulsion

_case 2 r-solitons u,ill

play

a role of norm~il

current carriers. The interaction of solitons is mediated

by

deformational and Coulumh

interactions u.hich _are

coupled

to each other due to CDW

polarizability

and the

rc~ulting

interaction type is not known _a prio>.>.

In the

previous

articles

[21]

_we have shown that sohtons of the _{same sign can} be

mutually

attracted

along

the direction _i

parallel

_to the chain

axis. The attraction towards their _common

perpendicular plane

I= (_i, z takes

place

_e~en for the non-screened Coulomb interaction Therefore it is natural _to suppose that solitons will be

aggregated

into clusters whose

grouth

leads _{to a}

macroscopic

^CDW

phase slip.

In this

publication

_we ^{shall find} distnhution, of

deformations,

charge

and Coulomb

potential

in the media around _a soliton cluster And _ue shall

investigate

the cluster _energy

dependence

on the

charge

2 N, _u here A' _is the nuiuber of

aggregated

2 r-sohtons. The interactions with the _gases of

microscopic

^sohtons or of electron~

uill be taken into account. The consideration _is

given

_in the frameuork of macroscopic dislocation

theory

ⁱⁿ terms of which the

phase

? r-soliton _may he considered _{as a} microscrlpic dislocation

loop lying

in the

plane, perpendicular

to the chin _axi~

2. Dislocation

theory equations.

The existence of 2 w~solitons _on the

j-th

chain _{in a} discrete model _can be taken _{into account} hi the condition

~"

di

=

2

w3,,

Iii

~;

where _{~,, is} the CDW

pha;e

_on the n-th

chain,

and the dominant contrihution _to the

integral

(I)

comes from the region of the order of the soliton u,idth /, uherc /,t

(.

t~ is the Fermi

;elocity

(Hereafter Planck _{constant is}

supposed

to be fi I.I

M 8 CURRENT CONVERSION IN CDW CRYSTALS. II 1175

Remember

(see [21])

that the scale is determined

by

the _energy

interplay

between

longitudinal phase

deformations and

phase mismatching

_on

neighboring

chains _n

=

j

^and

n#j provided by (I).

After the

aggregation

of N solitons at the cluster of chains

j)

the scale l is

preserved

for external solitons

only

where the

phase

difference between

neighboring

chains is of the order of _r. The

phase

difference decreases and the

longitudinal

scale increases for internal

chains,

_so that the soliton localization

length

is

proportional

to

N if _we take into consideration the Coulomb

interaction,

and it is

proportional

to

N~

_{without it. Thus the continuum}

theory

is

applicable beyond

the line D which is

determined

by edge

^solitons with the scale I

v~n(x)

_{- q~}

(r),

_r

=

ix, 7)

I--

di

n

~~

where _s is the unit _area per one chain.

Evidently

D is the dislocation

line,

_so that condition

(I)

takes the form

I°~dx~=2w. ⁽²⁾

%x~

For the

general

_case the CDW

phase acquires

_an increment 2 _w

along

_any closed

path

L

encircling

the dislocation line D. The direction _on the circuit L is defined

by

_a

tangent

unit vector _T on D as shown in

figure

24 in

[25] Chapter

4. In the dislocation

theory

the

phase

q~ has _a

discontinuity

_3q~

= ± 2 _{w on some} finite surface F

resting

_on the circuit D _so that q~ - 0 at

large

distances. In terms of the soliton

problem

the

phase #

is _a continuous function

of _x which has the finite increment

(I)

between ₊ and

infinity.

This

discrepancy

in

definitions

corresponds

to the transformation of the surface F into semiinfinite

cylinder along

x

resting

_on the

~ircuit

D.

At

macroscopic

scales

ix ii hi, [/- _i~j ^»s'/( _{(x~, i~}

_are the soliton

coordinates)

condition

(I)

is

equivalent

to the differential

equation

%q~~

_ _

=2w3(x- I) 3(r -r~)s. (3)

%x

In _terms of the dislocation

theory equation (2) corresponds

to a case, when the

displacement

vector u and the

Burgers

vector b _are directed

along

the chain axis _x

u=~n, _b=-2wn,

n=

(1,0,0).

In the differential form

equation (3)

reads

jvxwj =2«~3(t) (4)

where _w

= Vq~ is an

analogue

of a distortion tensor w~

~,

f

^is the 2d radius-vector from the

dislocation line at _a

given point

on the

plane pirpendicular

_{to the tangent vector}

T.

At the continual limit _{we can use a}

phenomenological

model

independent

_energy

functional.

Jc

=

dxd/~

v

~9'

j~

^{+ « °9'}

j~j

^{+ * ~9' ~~~}

(va)2) ⁽⁵⁾

s 4

w %x %7 _W %x 8

we~

l176 JOURNAL DE PHYSIQUE K 8

where

a is the

anisotropy

constant

(a

«

),

W is the electric

potential~

_{E~ is} the dielectric

constant without Coulomb effects of CDW.

The Coulomb interaction for CDW has been considered in many

publications [26-29j.

Note (see

[27])

that

opposite

_{to a} rather _common

opinion

the

quantity

_E~ has _no dielectric

contribution from the CDW gap

proportional

to w

)/A~ (w

~

is the

plasma frequency)

unlike the

case of _a twofold commensurate Peierls dielectric

(e.g. polyacethylene

_or

Nbsm).

By varying

functional

(5)

_we obtain the

equilibrium

conditions which allous for the

following

form

VB

= 0, B

=

(q~

₊ ^{~ ~~} (6)

~~Alfi(.~, /)+

_nw

=

0 (7)

K~

(

=

~

,

&

, K

~

=

~~

= >'d ^~

~X ]/ VSE~

where r~ is the

Debye screening length

^for the parent ^metal.

We must derive the differential

equation

for _~

jr

and

air

from

(6)

and (7) _in such _{a uaj} that condition

j4)

is satisfied.

By constructing

the _vector

product

of

equation (4)

and _{vector n}

we obtain

(nV)w -V(nw)=

^~

w[T xn] 3(f).

(8)

Substituting equation (81

ⁱⁿ

(6)

differentiated _{over x we}

finally

obtain

i(r~v)~g

₊

_jr~v)2

_m

~

i

«<j3(I)jr~x _{rjj (9)}

j ~

l

=

~

~ + c1

~

~

=

~

~

f

= ~. ?<'a ^~' _~)

lx II' df~

From

(7)-(9)

follow the model

independent

equations

showing

that the fields

a~eraged

_over

a

sample

cross-section _{are not} sensitive _to the presence of solitons and dislocations.

~~

^4l _K ^{~ 4l} ₌ COnSt.

~K~

I jj ^~~"

₊

= const. In j

f

=

f(x)

= ^~ Ii-,,

/~

^~~~

~

Note that it is necessary ^to redefine the

phase

to transform from

(10)

to the point ^soliton

picture equations [21]

~

~D(-K,

/)

- ~g ^* j.;~

Ii

_{+ pj_v,} /

d_I

where _p~ is the soliton

density.

At the soliton

approach [21]

^the

phase

_~ does not take _into account the central chain~ _i-e- in terms of the dislocation

approach

_u-e miss the chains crossing

M 8 CURRENT CONVERSION IN CDW CRYSTALS. II 1177

the

loop. By excluding

_q~ from

(9)

with the

help

of

(7)

_we obtain the

equation

for the field W

Ka

=-«~2uinx~j<&(t) (ii)

K

=

16

K

~(nV

_)~

(12)

The

equations

for %q~/%x ^{and for the}

potential

_energy

[21]

of solitons V _are obtained from

(11)

with the

help

of

equation (7)

v~~ =-2~~4l,

_V=4l+

^ih= (I- _~~j

_4l.

₍₁₃₎

@x K 2 %x

K

it follows from

(11), (13)

that all fields _q~,

4l,

V have

singularities

_on the dislocation line D. Introduce the Green function

D(r)

for the

operator

^{K related with} one soliton solutions

~g~ and 4l~ which have been

investigated previously [21].

KD

(r)

=

s&

(r)

««UK

~Ai D(r)

=

2

4l~(r) (14)

2

ws(K~-A)

^~

D(r)

= 2 ~~.

(15)

Now _{we can} obtain the

general

solution of

(11)

in the form of _an

integral

_over the dislocation line D and then _{over an}

arbitrary

surface F based _on D. We find

4l

=

waK~idl[nxv]D(r-r'). ⁽¹⁶⁾

With the

help

^of

(14)

_we obtain from

(16)

the solution in terms of the surface

integral

4l

=

~

df'n4l~(r r')

_{+ waK}

~G (17)

S

where

G

= I

df'(V (nV

n

)(nV )

D

(r

r'

)

The first term in

(17)

has _a

simple physical meaning

_{as a}

superposition

of the fields of

point

fictitious

solitons, occupying

the normal to n

projection

of the dislocation surface. The second term vanishes for _a

loop lying

in the transverse

plane.

The fields %q~/%x and ^V are obtained from

(16)

with the

help

of

(13). Taking

into _account

(15)

from

(11), (13)

we find

~~'=

"~

dl[n

_x

Vi

_q~~

=

"~

df'(nA (nV) V)

_q~~(r r'

). (18)

%x s

~ s

~

By integrating (18)

_{over x we} arrive at

explicit expressions

for V

V

=

"

df'vq~~(r r')

=

"

ld~r'hi

_q~~(r r'

). (19)

S

F~

~ V~

117X JOL'R~AL t)E PHh'SIQLL " x

Here F'~ ~ind

i'~

are the,urface and the volume of'a semiintInite

cylinder

resting ontrl the

circuit D

parallel

to n,

dt[

= [Lll x n di _i~ the

cjlindei

~uil~ice element

Simil~irli _to i19) ,ic obt~iin tram Ill. II ~) an

e,plicit

e,pro,~irIn for the

ph~i,c

~lri

= tlt'~~~/~lr~r i?ii

I

Finallj

_tie fifid _{cl con,enient} e,pro,,ion ti~r the

ph~i,e

_{~c in} the f~riii at'cl ~uperp<),ition ⁱⁱⁱ ,o!item field; ,in~ilJrl, _to 1171

~

~c'jr

=

T Lit" _n~ jr r~ Ill

I

For this purpow u-e,,ill _{use an} ezhil,

pro,ed

identit,

ni ain~li~-K~)n+a~i~n-

in~)~)-n~(n~)i~- (n~)ni Ill!

i~ _c

Jpplj

the operator (??I to Dir r~j cind

peifori~i

inicgration o, cl F Thc LHS ol Ill gi,e, ~i fi-function

;ingulaiitj

_on F~ the Iii,t. ;econd and ihiid _{termh in} Rlis cl iii gi,c ii~c iii, ii~ ii, _n ~ iG ~i

ieipecti,el,

iconipaie to II 51. (?I _I. II 7)j Wc ~~htciin _{~i, ~i} ie,lilt

~irl _" w~lrj-~ mfi

jfj

_(I ~l'

ii here

~o ^' and ~ are dellncd _in (~l ~ifid ₁₁71. _(,), ~ _{i, ~i} unii ~tep ^l'unction on ihc _,ui l'ace F

a8,[Fj

=

df' nb jr r

~i

,

C

hoosifig

the ,urf'acc F

,i, ihe cj lindci F

o c oh,er,e th,it the fir,t and hecond _ICI

iii, ~it the RHS ill j

di~appear

J; ,,elf _a, the ,econd teriu in the definition ot ~ II 71 J~~

~i ic,ult _oe

arri,e ,it

equ~ition

1201

i~"e _coin al,u choo~e the ,unlace F _to he

plane

^{for the} iuosi inipori~int

speci,il

c~i,c al _ii

plcine loop

^{Then tile} term G _in Iii and

(~?l

,anishes In this _{ca,e j} iifid ~'" d;ffei

only hi

the _{it ~ii} the di;continuit, _i; dcflneLi At

large

di~tznee, froiu _J

loop encircling

~i ch~iin, jibe du~ter

it ith the

ch~irge

? I') _tie ~,lie Llue to ill

<

lr)i~.i~,jrj, q7'ir)-1)

_{iii i-cc} Since the function

q7 dccrea~es _at

large

di~tancc, then _it h~i, _cl

Lh;continuity

(5 ~ ^' Ii ₌ ^? m

ConseLjiientl,

Llue ti~ iii the function _{~c jr i~ Lontinuou~ o, cl i} and iuonotonicall,

ch~iige,

l'r~~n~ ii to 2

m

along

_i II',,e _cros; ihe ~uiface F bi _{,ar,ing 1}

~ri~J5./I=li ~r(JJ./I=?m

Thu, the definition at'the

pha~e

~.~

corre,pond;

to the concept of the

di~placen~ent di;continuity

_in the disloci:tion theon uherea~ the definition ~f the

phase

_q7

corre~pond,

to the

phj~ical

picture ^{ol'l r-~oliton} aggregate. ^The continuity al'the

pha,e

~oiri oiei

i m~cl it, ini,irimwe

reg~irding

the choice iJf _~i ,urfiice ~ l'ollo,i; froiu _it, rel~ition _to

4S _in Iii

~ '<

~r(~~ ii

=

-Z fit ~4l(1 I)

I

since the

charge

^den,iti ~4l 4 _m has _no

;ingularitie,

at ~in, ,urfiice

M 8 CURRENT CONVERSION IN CDW CRYSTALS. II l179

3. The energy of

aggregation.

We consider _now the energy in the presence of _an

arbitrary

dislocation

loop.

We transform the last term in energy functional

(5) using

the

equation (7).

As _a result the energy can be

written in the form

3C

=

$ ld~r

^wB

⁽²³⁾

4 _"S

where B is defined in

(8). Equations (4)-(6)

and

(23)

show that _our

system

^is

equivalent

to some

magnetic

media with «the

magnetic

induction» B and the

magnetic

field»

H

= 2 _w in the _presence of the unit _current in the circuit D.

Equation (7) plays

_a role of nonlinear

anisotropic

nonlocal

susceptibility. Introducing

a

vector-potential

A

by

definition B

=

V _x A after standard transformations _we obtain from

(23)

3C=iiAdl ^=I

^Bdf.

⁽²⁴⁾

2

D

2

F

By choosing

F to be the

cylinder

surface F~ we obtain

3C

=

d _{df~ V~ (25)}

~

F~

For the _case of _a

plane loop

_we have

3C

=

V

df. (26)

F

We _see that for this _case the energy is

interpreted

_{as a sum} of

point

soliton

potentials

in the dislocation

loop plane. Obviously

_we must not _use in

(24)-(26)

the

representation

with _a

discontinuity

_on the _same surface.

4. The

plane

circular dislocation.

It is natural to _assume that the axial

symmetric configuration

defined

by

_a circular dislocation

of radius

Jio: rR(=Ns

in the transverse

plane

has the lowest energy for

given

N. We will _use

equations (17), (21), (19)

to determine fields _~,

4l,

V. The

single

soliton solution ~~ and 4l~ ^{in these} formulas has been

investigated

in

[21].

Recall that all fields decrease

exponentially

on a

microscopic length

d in _{a transverse sector}

~ ²

d«cv2[xj~ iii(

d= ^"

(27)

K

and

they

have _a

specific

combined

exponential dependence

in the

longitudinal

sector

"~ixi

j

~

iii

»d

~

~2

~

(2g)

~P~ =

j _(f(r)(

,

l' ~

W ^~~~~

1_ ^f2

g(r)

= -exp

,

f(r)i jg(r)j

fl

2j,tj

l180 JOURNAL DE PHYSIQUE ^{1 li° 8}

where

f

=

(I, 0= _r/d;

_f

=

(f(

We shall _{use exact} solutions

[21]

of

equations (14), (15)

at _a

=

I valid for all

r.

f (r )

=

ch I sh I ^~ h

(r ) (29)

%i

g

(r

= sh I ch I ^~ h

(r ) (30)

%i where

h

(r)

_m

( _{exp(- f)}

For _a

physically important

_{case a «} I

approximate

solutions

(27), (28)

_may be obtained from

(28-30) by

the scale transformation

i

4l(x,/)~a24l xa2,/

j

q~

(x, /)

_{~ q~}

(xa ^~,

^/

⁽³¹⁾

By substituting (28)-(30)

in

(17), (21)

we obtain

4l

(r )

=

~ ch I sh I ^~ I

(r (32)

8 off

#

^*

(r

= sh I ch I ~

I

(r ) (33)

4 off

~"i _(~~~(~+$ ~~~~~(j~~~~

^~~~~

where

1

=

2

d2fl, ~~P[~ (i~

₊

(I _ij)~)~/2j

~'( _Ao

[i~

₊

(/ / _)2jl/2 ⁽³⁵⁾

~ '~ l

The

investigation

of relations

(32)-(35)

is facilitated

by

observation that

I(r )

is

proportional

to an effective _« electric

potential

» of the

charged

disk in _an

isotropic

medium with _a unit

screening length.

In the

vicinity

of the

plane

_x

=

0 _we find

q~=w8(Jlo-jij), V=4~mjuK8(llo- iii) (36)

with

exponentially

small corrections. At the dislocation axis /

=

0 _we find

exactly 1(x, o)

=

4 «

(exp (- ii )

_exp

i- (P

₊

fi)"21)

Consider _now the field distributions _near the dislocation line.

(x( «Ro, lyl«Ro, y=Ro- III

li° 8 CURRENT CONVERSION IN CDW CRYSTALS. II l181

We _can consider the internal part ^of the

loop

_{as an} ^infinite

semiplane

_{y ~} 0 to obtain

I(x, y)

=

4

w&(f)

_exp

(- (i( )

4 sgn

~y)

^~

Ko (fl

+

f~)

df

=

lil

2wexp(-(I()+4 ~Ko~(d+f~)~)df ⁽³⁷⁾

o

where

Ko

^{is the modified Bessel function.}

In addition to

(36)

_we find that _{on a}

cylinder

_y

=

0 the functions

q~ and 4l take half of their values for _r

~

Ro.

Derivatives of _q~, 4l and

consequently

of the field V have

diverging singularities

_on the line D. It is easy ^to demonstrate for transverse components ^{of the} electric field

Ei

and the force

Fi

that

Ei

₌ ^~~ ₌

~ _ch

k- sh I ^~

Ko(p (38)

°Y ² ii

Fi

=

~~'

=

~ ch

I l ₊ ^~~ 2 sh

it _Ko(p

) (39)

°Y

⁴

ad

_ok

where

p2=P+i~.

At the minimal distances _{p «} I _we find _from

(38)

and

(39)

1 = 2 _w 4

f

In _{p +} I arctan Y

x

Vm~~~+~VK.

8~2

₄

Formula

(40)

show that the attraction to the dislocation

cylinder

takes

place

outside in the sector

xj

~ y. The

potential

_energy is

negative

outside at y < 0. The electric field is

always

directed outside. The formula for

Fi

at the first term in the

expression

for V in

(40)

corresponds

_{to a} free

phase

vortex when the Coulomb interactions _as should be for r~rD are

neglected.

The value

Ei

and the second term in

expression

for V

provide

corrections due to Coulomb interactions. Note _we should also

keep

in mind that the

region

Kr « I is available

only

if the minimal

longitudinal

width I of the dislocation line is smaller than the Coulomb

length

d. For

arbitrary

_a « I it

requires

that

l~

^~

,

l~

«

d~, d~~

~~

a ~

which _means that

y~

2

=

~

~

a~ (41)

hUe~

l182 JOURNAL DE PHYSIQUE ^{li° 8}

So the

singularity region (40) and, correspondingly,

the attraction of solitons to the dislocation line _are available

only

for systems where Coulomb interactions _are weak

enough

and _an interchain

coupling

is strong relative to the Coulomb interactions.

At

larger

distances _K

Ro

» p ^» I functions

Ei

and

Fi

decrease fast in the horizontal sector.

E~ ~Fi ~exp(- p), KRO

»

(f(

~

Ill.

In the vertical _{sector we} have _a combined

dependence

j

Ei

₌

Fi

=

I

K~

~

^~

exp

~- ^{~ (42)}

2

2(k( 2(k(

K~IO ^»

iii

_~

iii

which is the 2d

analogy

of the 3d soliton solutions

[21].

We _see that in the

region (42)

the Coulomb interaction is

prevailing,

the forces _are directed outside D and the energy is

positive.

Recall that these results for the

special

_{case a}

=

I _are transformed to the results for the

general

_{case a <} I

by

the

scaling equations (31).

5. Effects of _a weak

screening.

The

screening by

the residual carriers

(electrons

_or

solitons)

is relevant if the

screening length

A ^' is smaller than the

loop length (A

^' «

Ro).

The

screening

is taken into _account

[21] by

the substitution in

equation (12), (13), (15)

A

- A ^~ while

I

_and

ii

_{should be left intact.}

As _a result at

large

distances _{we must use the}

single

soliton solutions

[21]

of the type

~'~~~~

4

jj

_*

lx

(x~

₊

~la _*)"~'

^{~ ~~~~}

i~* lx

~'~ ~~~~

£Y*=

i§K~4£Y

where a* is _an effective

anisotropy

_parameter which is very small. Thus the

phase

distribution is determined with the

help

of _a effective

problem

for the dislocation without the Coulomb field but with _a

greatly

enhanced

anisotropy

_a ^* From

(21)

_we obtain _an obvious result

~~~~~~~~'

^~~~

~'

_~

l

(~y*)2 jr*j»dla*; iii »A~~;

where D is _a solid

angle

for the

loop

from the observation

poinr

r in terms of reduced coordinates _r. It follows from

(42)

that the main contribution to the

integration

due _to the parameter K~/A^~ » l _comes from the electronic

potential

4l. This

potential

has _a

quadrupole

type

so that there _are sectors with

negative ((x( <(7(/(a*)"~)

and

positive

ix

~

ii /(

_a

*)~'~) energies.

In this _{way we} find fields 2

Nq~~,

^{2 NW}

~

far

apart

from the

loop

r »

Ro.

The

screening

effects _near the

loop

_are

important

when its size is

large enough R~A~',

_N

~s/A~.

_{The fields}

near the

loop

D _{are more}

easily

determined

by solving equations (I I) directly.

After the substitution A

- A ^~

equations (11), (12)

take the form of _a

dipole

filament

equation.

li° 8 CURRENT CONVERSION IN CDW CRYSTALS. II l183

lS+«*Sl 4~=-«»«1&(x)&(Y)> ^{YRo- i~i (44)}

From

(44), (13)

_we find

~ ~

~) ^~y~

)"2 y2

₊

)2

~y * ^'

~ ~~~~~

~~~i

^{~~~~ ~~~~}

At the

validity

fiInit

we obtain

~max

" ~~~ ~K

We obtain the result similar to

(40)

but with _a

sharply

narrowed vertical _sector

(x

~

(/(a*)~'~). _Except

this sector the attraction of solitons to the

loop

D takes

place

outside the

loop.

Near the dislocation line inside the

screening

_area

y~

₊

(A x)~

< A ^~

(46)

We find in addition _to

(42)

~

~2

2

(47)

&4l

=

V~UK (~Y~ %

K

~ ~'

This contribution is small due to

(46).

Nevertheless it becomes dominant _over the

exponentially decreasing

nonscreened solutions if _K

y~/x

is

large enough

inside the

region (46).

6. The

aggregation

_energy of the

plane

cwcvlar

loop.

The energy of the dislocation

loop

is calculated

following (26)

with the

help

of _exact results

(34), (35)

combined with the scale transformation

(31).

The

principle

difference between formula

(20)

and the similar standard

expression

for _{a vortex}

loop

_energy is that the value V is finite

everywhere

in the

loop plane being nearly

constant

(36), (40) beyond

the _narrow

region

_near the

loop

_y « I. For this _reason the _energy is

proportional

to the _area rather than to the

perimeter

of the

loop. By substituting

solution

(40)

in

(26)

_we obtain

3C

=

I

a

"~w~

^N +

CE~ N"~,

_N

=

d (48)

Wp = UK

,

C

~ nlaX

(I,

In

(Y/£Y))

where w~ is ^the

plasma frequency

for the parent ^{metal. The last} term in

(48)

describes the

intemal

loop

_energy. If the

loop

size is

larger

than the

screening length

the value

V vanishes in the

plane

inside the

loop

in accordance to

(45). Again

the main contribution

comes from the Coulomb field which is distributed _{on F} siInilar to

phase gradient (45).

With _a

logarithmic

_{accuracy we} obtain the result that differs from standard vortex energy

by

_a

large

coefficient

~ K

IA

3C _a ^"~ _w

~

j~~

~~~ln

j~~

,

N *

=

(49)

N N _A

s

The

aggregation

of solitons is

energetically

favorable if their chemical

potential

p relative to the

loop

is smaller than the activation _energy of _a

single

soliton.

JOURNAL DE PHYSIQUE I T I, M8, AOOT lwl 47

l184 JOURNAL DE

PHYSIQUE

li° 8

j

p =

~~<E~~u(I _)~ (50)

We _see that for the nonscreened

loops

N _< N*

)

v

IN-1'2). _psi)

We conclude that the

aggregation

is favorable for weak Coulomb interactions

independent

of the

anisotropy parameter

a. Moreover if condition

(41)

is

satisfied,

then the adhesion of the

single

soliton to the

aggregate proceeds

without _a barrier _near the

loop.

For

large loops ARC

~ l the

screening

^{is effective and the}

aggregation

of solitons is

always

favorable

being

also stimulated

by

^{the attraction} sectors

iii

² _~

^x~

_a ^*

Large loops

_are favorable in _compare to intermediate

loops.

7. Conclusions.

We

investigated

the dislocation

loops

in CDW

crystals arising

_{as a} result of soliton

aggregation.

The

emphasis

_was

given

to the effects of the Coulomb interactions.

Equilibrium

distributions for the Coulomb

potential

4l and for the CDW

phase

_{q~ are} found to be concentrated in the

highly elongated

volume V

= L _x

R(

with L

R(/d.

In the

loop plane

the fields 4l and _{~ are} constant

q~'~

4l

= po~

$w~ _beyond

_the

narrow dislocation line

vicinity.

The

screening

effects become

important

if

Ro

_~ ^{A ~~ in the} presence of residual current carriers

(electrons, solitons)

characterized

by

their

screening length

A

'.

_{In this}

case at

large

distances the

phase

distribution resembles _{one as} for the _case without Coulomb interactions but with _a

highly

enhanced

anisotropy (43).

It results in _a

narrowing

of the

longitudinal repulsive

sector. The

plane loop

_{energy was} calculated.

Contrary

to the conventional

dislocations line the energy has _a term

proportional

to the dislocation

loop

_{area or} to the number of solitons

(46).

It _was shown that the

aggregation

of solitons into the dislocation

loop

is

energetically

favorable for weak Coulomb interactions

independent

of the

anisotropy

coefficient. If the

loop

size is

larger

than the

screening length

the energy has _a standard form

(47)

and the

aggregation

of solitons is favorable at least for

large

N.

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