Space-time distributions of solitons for the current conversion problem in charge density waves

HAL Id: jpa-00246579

https://hal.archives-ouvertes.fr/jpa-00246579

Submitted on 1 Jan 1992

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Space-time distributions of solitons for the current conversion problem in charge density waves

S. Brazovskii, S. Matveenko

To cite this version:

S. Brazovskii, S. Matveenko. Space-time distributions of solitons for the current conversion prob- lem in charge density waves. Journal de Physique I, EDP Sciences, 1992, 2 (5), pp.725-740.

�10.1051/jp1:1992176�. �jpa-00246579�

Classification Physics Abstracts

61.70G 71A5L 73.30

Space-time distributions of solitons for the current conversion

problem in charge density

_waves

S. Brazovskii and S. Matveenko

L.D. Landau Institute for theoretical Physics, Kosy@na 2, Moscow, Russia

(Received

3 January 1992, accepted 17

January1992)

Abstract. Dissipative dynamics equations of Charge Density Waves

(CDW)

_are derived for _a homogeneous distribution of solitons and dislocations. Response functions for the CDW

current and for the electric field _are found for

a spontaneous conversion _process of electrons into solitons. A _one- dimensional development of the injection current impulse is studied in

details. The problem is

investigated

for

a purely dissipative ^CDW regime and within _a diffusion approximation for solitons. We find that first the nominal CDW _current jco, which is due to the CDW phase velocity flea

= -xjco, and the electric field Eco _« jco _are established along the sample length in _{a very} short time. Later _on the diRusion front _passes along with _a constant

velocity bEco, where b is _a mobility of solitons. It is followed by the growth of the soliton concentration _ps and by the decrease of local coherent CDW current

j(z, t)

_«

fl(z, t).

At largest time t they _are related _as

j(z,t)

_«

pi~

«

t~~/~.

_{The total electric}

current is nearly additive J _cS j + 2js

being

almost constant. Also _a stationary distribution is studied for generation of

solitons _at the _presence of _a constant CDW _current. It is characterized by _a step-like profile of the defects concentration.

1. The basic

equations.

A

Charge Density

Wave

(CDW)

is _a kind of elastic uniax1al

crystal (see

_{[1, 2]} for

references)

which is characterized

by

distortions

""vi' (I)

Here _w is _a

locally

defined

gradient

^{of the CDW}

phase

_qJ which is restored from

(I)

_{as a many}

valued continuous function. The vector _w is

conjugated

to the stress vector _«

««iW«11 W«1-~«=-ll (2)

U ₌ W

w«d~r

Here

ji

_{is the elastic tensor~ W and U}

are the

thermodynamic potentials

in terms of

w and _«

correspondingly. (~).

At the

adequate

harmonic

approximation

the functional W

acquires

the form

~~~"'~~ 4~s /

_~~~

~"~°~"

_~

~°~~ _c~u2 ~~~~j

~~~

Here 16 is the electrostatic

potential,

s is the unit _area per one

chain,

8~2

2 -2

'~ fins ~~

where _e is the electronic

charge,

_u and _{rD are} the Fermi

velocity

and the

Debye screening length

of _a parent ^metal. In the

following

_we will put h ₌ I.

If _we do not need the electric field

explicitly

then _{we can} exclude 16 from

(3)

to obtain

W(w)

=

~

d~rwAw (4)

4xs

~2

A;k

=

(Ao);k zn;nk,

^Ao =

diag (i,

_«y,

«z),

/h ₌

V~ (5)

Here _{n =}

(1,

0,

0)

is the chain axis _z

direction,

_{ay, az} are the

anisotropies

of elastic modules.

It follows from

(2)

and

(5)

that

« = Aw ₌

Aow

₊ 2nlb

(6)

The field

u ~j

(7)

~~

2x'

^~

is _a force _{per an} element dI of _a dislocation line _[3~

4].

The

longitudinal

stress

V = n« = w~ +

21b/u (8)

is the

potential

_energy for 2x-solitons while

V/2

is the _one for electrons and for

amplitude

x-solitons.

The static

equilibrium

is defined _as

b=0~ b=V«=0w-~En~

E=-Vlb

(9)

where E is the electric field

strength,

Q

=

AOV, VQ

=

I, VW

=

lqJ, ₍₁₀₎

Q~

I

are the

anisotropic gradient

and the

Laplacian. (Hereafter

all notations

correspond

to

preceeding

articles [1,

2]).

The CDW

dynamics

differs from that of conventional

crystals mainly

due to the host lattice friction which is

proportional

to _a

velocity p

=

3qJ/3t

rather to its

gradients.

The system of (~) ^The equilibrium states _are determined by minimization of the potential U providing in this

way a correct sign- the repulsion of similar dislocations. By minimizing W _one would arrive at the

attraction of dislocations similar to electric currents interaction. Somehow this feature is missing in the literature and the sign is corrected afterwards for the forces.

dynamic equations

is formulated in terms of the

density

^{and the} current, _p and I~ of dislocations [3, 4]

VXW=-p (11)

vp=&-I, p=# (12)

Vp=0, #+VXI=0 (13)

V« ₌

ill

+

fl/u~,

Va ₌ b

(14)

where

I

% 3

f/3t.

E.g,

for _a line element dI

= TdI

being

moved with _a

velocity

V _one has

p =

-2xTb((),

1=

VXp, (

_{= r}

(Tr)T (15)

For the sake of shortness _we will omit in the

following

the inertial term

fl/u2

in

(14)j

where

u is the CDW

phase velocity, keeping

in mind that it _can be restored

by

the substitution

7fl

_~

7fl

+

fl/"~ (16)

The kinematic

equations

_are

simplified

if the

connectivity equations (13)

_are resolved in terms of the "dislocation moment"

density

P _[3~ 4] so that

p=VXP, I=-P

(17)

Then

equations (11) (13)

_are reduced _{to a}

single

_one

w + P ₌

VqJ (18)

relating elastic, plastic

and the total deformations.

Remember that

only

_w is

uniquely

defined _{as a} function of state due to its relation with

«. The functions P and VqJ are discontinuous at _some surfaces based _on the dislocation lines. These surfaces _can be

arbitrarily

defined _{at some} initial _moment and there is

only

their

subsequent

evolution which is

uniquely

determined due to

(13) along

the

cylindrical

traces of dislocation lines. We will follow another

approach

_[3~4]

treating

^the

equations

_as

being averaged

over

elementary

volumes which contain the closed dislocation

loops only. Being averaged

the field P becomes _a function of _state related _to the

integra1dislocation loop

_{area vector} dS in _a

volume dV

PdV =

-2xdS/s (19)

For _a system of +2x -wlitons with the linear densities _{p& we} have

P ₌ (P~

0, 0),

P ₌

2x(p- p+) (20)

Now _we will deduce separate

equations

for fields b and V. First

excluding fl

from

equations (12),

(14)~ we obtain

v(v«)

₌

i(w I) (~i)

Applying

the operator ^A to

(21)

we find with the

help

of

(6):

(iV)V«

=

i(b ii) ₍₂₂₎

Finally

_we

apply

V to

(22)

and take

(5), (6)

and

(10)

into account. We arrive at the closed

equation

for b.

KS ₌

i~~(nT7)(nI) i/h(itI) ₍₂₃₎

where

3 32

K ₌

-7~-

+ K, K

=

~(VAN)

= ~~

Jc~j (24)

Generally

the second term in the r.h.s, of

equation (23)

is small relative to the first _{one as}

~/Jc~.

Nevertheless it should be taken into _account for the

charge conserving

dislocation line

sliding along

_n when nI ₌ 0.

Alternatively

_{we can}

multiply (22) by

_n to obtain the

equation relating

V and b:

71h)

= (1h

~C~)(n?)b

+

1(1h

~C~)(nI)

(25)

In the r.h.s. of

(25)

and of the

subsequent equations

_{we may}

always neglect

^~ in _compare to Jc~.

A separate

equation

for V _can be obtained

by applying

operator K to

(25)

and

taking

account of

(24).

^The intirmediate

equation

admits _an order reduction and

finally

_we arrive _at KV = (~C~

-1h)(iQ

_x

nlP)

_{+ i(~C~}

-1h)(nI) (26)

The first term in the r.h.s. of

(26) corresponds

to the static

problem

^while the second _one is in effect for _a

perpendicular

dislocation motion _or for the solitons creation _processes I-e- for

charge growth

_cases. In terms of P

equations (23)

^and

(26) acquire

the forms

KS = 7

(~QP Jc~(nV)(nP)j ⁽²⁷⁾

KV ₌

(~C~ 1h)

((-7)

+

L)(nP)

(nV)(itP)j ⁽²⁸⁾

At P ₌ Pn the last two terms in

(28)

can be written _as

hi _P, hi

=

I al

=

ay3~/3y~

₊

az3~/3z~

which coincides with the static

problem.

Alongside

with

(27), (28)

we can derive _an

equation

for the

phase

_qJ which _can be obtained from

(21)

^{with the}

help

of

(18)

and

(6):

Ki2

₌

-~~(n?)(nP)

₊

/h(itP) ₍₂₉₎

The

equations

for _qJ and 4 _can be written also in the _same form _as the static

equations

in [I~ 2]

(nV) 1- ^7#

⁺

^0w

⁺

~(nV)4j

^{= 0}

⁽³⁰⁾

" "

~4+nw=0

(31)

which should be

completed

with the relation

(18). Remarkably

^these

equations

do not contain

the _source P

explicitly. Excluding

16 from

(30), (31)

_we arrive at

equation (29).

We define _a coherent CDW current

j

via the

phase velocity #

_as in the absence of defects. The total electric current J and the _current of solitons

is

are determined

by

virtue of

corresponding

conservation lows via the

charge density

_p and the

density

of defects _ps. Then _we have _a set of relations

~'

~

~~'

^~~

lx ^"

~~~~

J=

j+2js (34)

2. Local

perturbations.

As a first

application

_we will consider the solutions of the derived

equations

which

correspond

e.g, to instantaneous _processes of soliton creation. First _we should find _a fundamental solution

£ for the operator K

Kg =

b(r)b(t) (35)

At the _space Fourier

representation

_we have

£(k,t)

=

~)/

exp

1- ^{~$~t) (36)}

7 7

where

K(k)

=

k~+Jc~k]

is the Fourier transform

oft

_in

(24)

at up = az = I which is

supposed

henceforth. The results for ay = az = a < I _can be restored with the

scaling

~ at, _z ~

«i/2z,

_{y ~}

y, z ~ z, ^{~b ~}

«1'2~b,

_~ ~ ~.

At the coordinate

representation

_we obtain from

(36)

£ =

1 (il

^~~~

f~,~ ^dmexP I- ^~lll~~ 7(nm)~ i~

(37)

At

large perpendicular

distances

r(

_»

^z~

₊

4t/1»

_we find

~ _~

,3/2($~/2~cr

~

'

~~

^~

~~~~~12

~~~~

Notice that at _zero order of

(Jcri)~~the

function £ does not

depend

_on z. At distances small

versus time

r21/4t

_{« at} tJc2

Ii

» I _we find

Along the _axis

ri

" ₀ the _ntegral in (34) is lculated

exactly to

give

us

The

solution

of

~ =

-(~c2 ~h))£

*

P(r,t) (41)

where

f

_{* g}

=

f(r ^r',

t

t')g(r', t')dr'dt'

For the CDW current _we obtain from

(32) (39)

~2 _{a o}

I

"

PG~

*

R~

~~~~

The most

important

local _process is _a fast spontaneous ^conversion of electrons to _x- solitons

(see

_{[5~ 6] for}

references).

The _process time

wj/

_(wph is _a

phonon frequency)

_can be considered

as an instant _so that within _a diffusion time scale _{we can} write

P _{= -KS}

~j _{b(r r;)0(t t;), (43)}

Here _r; and t; are the I-th soliton creation coordinates. Then the irradiated current is

given by

I

"

~

~~

)£(~

_r, ii

t,) (44)

At the Fourier

representation

lc~k(

.cnv(~ ~) ₍₄₅₎

J(~i

_~°)

Jc2k] +

17k2w

+

kl

^~

where

f~~

is the electronic current converted into solitons _or dislocations. The

corresponding

electric field is found from

(28)

as

E(k,W)

~tS

(7rn/2)(7

+

iki/W)J(k,

_W)

₍₄₆₎

As _we will discuss later the time scale to _" Jc2

/7

is

extremly

small. Then _{we see} from

(45)

and

(46)

that the current conversion _{goes on}

nearly identically only

for _a

fairly homogeneous

_over

the

perpendicular

crossection _process when L~ < JCL[

,

L~ and Li

being

the characteristic

length

scales. Also _{we see} from

(46)

^{that the} Ohm low is valid for the CDW motion when the time scale is also small _versus

perpendicular

scale t «

(71a)L [.

We conclude that the reactive time

dispersion

is

ultimately

related to

perpendicular

finite size effects. For

homogeneous

_over

perpendicular

crossection distributions the time

dependence

is due to _a soliton

mobility

after their creation. These effects will be studied in the next

chapter.

3. The

injection impulse

^evolution.

Consider _{now an}

averaged

evolution of the CDW fields due to a

homogeneous

over

sample

cross-section

injection

of solitons. Our

example corresponds

to a current

injection by

_{a narrow}

contact in _a thin

sample,

Integrating equations (21), (30), (31)

at P ₌

(P,0,0),

_w

=

(w,0,0)

_over

perpendicular

coordinates _we arrive _at the

following

system

16'+#=0 (48)

where

z z

~ p~ ⁱ ~

l'

~

II

~

=

~co

Z, j2=

~co

W Xl, = Z.

The functions F and _are the

plastic

and the elastic contributions

correspondingly

to the total

phase

_qJ. The CDW current

(32)

is

given

_as

I

₌

-)#

=

-)(fl

₊

)) ₍₄₉₎

The electric field is found from

(48)

to be

~2~

E ₌

-@ (50)

At the

dissipative regime

_u ~ cc~,

equation (47)

determines the CDW _current via its elastic deformations

~~;

_~

~2p p,,

Generally equations (47), (48)

^describe an excitation of the CDW

plasma

mode

by

the

injection

_or the motion of defects which _are characterized

by

_a difference of

corresponding

currents

bjs

₌

F/2x (see Eq. (53) below).

This mode is described

by

^the

following

_set of

parameters: Frequency

Q

= uJc, relaxation time _To

"

7/Jc2

₌

4xa,

where

a is the CDW

conductivity,

the

dissipation

to inertial

regimes

_crossover time _ri

"

1/(iu2)

=

1/(roo~).

For

high

_gap CDW materials the

following typical

^values _may be deduced

(see [7,8]

^for

reviews):

~c _-~

10~cm-~,

_u

-~

10~cm/

_s, Q

-~

10~~

s-~,

Q

~w 10wph

The value

rp~

should be

placed

between wph -~

10~2s~~ where attenuation is small and the

experimental

_range ^{10~ -10~°} Hz. One _{can suppose see} [8] for

discussion)

^that

ri~

-~

10~° -10~~ s~l

w that

rp~

=

Q~ri

'~' 10~~ s~~ which is far

beyond

the

highest

admissible CDW

frequencies

~

-~

10~~s~l.

Our estimations of _To

correspond

to the CDW

conductivity

a -~ 101~ s~l

-~

10~(Qcm)~~

^The parameter _i is estimated _to be

1-~ roJc~ ~w

10~~ s/cm~,

_i

-~

I/(riu~)

-~

10~~ -10~~ s/cm~, ro/ri

-~

10~~ 10~~

We _see that the scales _To and Jc~~ for time and

length

_may be considered _as

infinitesimally

small and the derivatives _can be

neglected respectively.

Then

equation (47)

is reduced to _a local _one

xii

_{= Jc~@ =}

(2/u)E (51)

which tells _us that both the the CDW _current and the CDW elastic

displacement

_{are propor-}

tional _to the electric field. So the field reaction of the CDW resembles both _a metal and _a dielectric with respect to its total and elastic

polarizations correspondingly.

Also _{we can} write

equation (47)

at _u ~ cc~ as

#

+

To(#

+

))

"

b,

to _see that _one may

neglect

the elastic counterpart to the current _« in _compare to the

plastic

counterpart _cc

#

At last _we find that the

potential

_energy is determined

mostly by

it's Coulomb counterpart

V =

uq7'+

21b ₌

2(1- Jc-~3~/3z~)16

m 21b,

analogously

to static

problems

_[l~

2].

Now consider the

dynamics

of the function P. It is related to _a CDW matter

injection

in the

course of electrons conversion _to solitons and _to the

subsequent

redistribution of P due to the

soliton

mobility

and diffusion. We will

study

the

simplest

_case of the continuous creation of solitons in _a

sample

cross-section _z

= 0 to model

a current

injection by

_{a narrow} side

junction

in _a thin

sample.

In this _{way we}

neglect

the diffusion of electrons while their _current

I(t)

is converted to solitons. So _we have

$

=

-2xb(z)I(t), I(t)

₌

Io0(t)

where b and 0 _are the Dirac functions and the current is

supposed

to be switched

on at t ₌ 0.

The redistribution of P is

provided by

the solitons diffusion _even if P is stored

mostly

in the form of dislocations since the

adsorption

of

~2x,wlitons

is

required

to

change

_a dislocation

loop

_area. There is _a

variety

of soliton

mobility

mechanisms which

are

additionally complicated by

the twa-step process mediated

by

x-solitons

(see

[5,

6]).

^{We find} an upper

boundary

for the

longitudinal mobility

b

considering

the _energy

dissipation

due to the CDW friction _i. For the wliton diffusion coefficient _we find

D ₌ bT

=

i'~~

_"

)1-~, i'

> 1

(52)

where 2~

-~ Es is the 3d transition temperature ^and

Es

is _an activation _energy of solitons.

In the diffusion

approximation

the soliton

density

_{ps =}

-P/2x obeys

_a

continuity equation

F ₌

)

(VW +

24l) (54)

We will limit ourselves to _a

unipolar

_case when

only

_one

sign

of solitons is taken into account.

This situation is realized either at low temperatures ^{when the}

injection

level exceeds _a volume

concentration Pco or for the

majority

^carriers in the volume

having

the _same

sign

_as the

injected

carriers

(2).

The

equatio6s (53), (54)

and

(49)

_can be summarized _as

j'- _2xjs

=

j'-

_bTF"

~~~(F

+

))F'

=

-Io0(t)[0(~) f(t)] (55)

Here _an

arbitrary

^function

f(t)

is the the

integration

constant of

equation (53)

_{over ~.} Com-

monly

_we will consider the _case

f

+ 0 when the currents vanish

asymptotically

to the left from the contact

~ ~ -cc~ :

E(-cc~,t)

₌

0, j(-cc~, t)

= 0.

Otherwise for

symmetrical

current exhausts _{at z ~} +cc~ _we should substitute at the r-h-s- of

(55) 0(z)

~

(l/2)sgnz,

I-e-

f(t)

₌

1/2. Following

_an earlier discussion _we

can

neglect

the term -~ in

(55)

to obtain _a closed

equation

for the

plastic phase

F

F bTF"

~(~

^{FF' =}

-Io0(t)0(z) (56)

At the _same

approximation

other

quantities

_are

expressed

in terms of F _as

P ₌

-2Kps

₌

~~,

m -To

~~, ^~~

"

2Kp

_{m -To}

~~,

_E

m

-lu

^~~

~ "

K fit ^{' ~~ "} 2K fit ^~

2~'

^{~ " ~°}

2x~~~~~~~~'

~ l

3~F

~

~° x~'~ x~

_3t2 ^{~ ~°}

We _see that

beyond

the

injection layer

variations of electric _current J and of the

density

_{p are}

negligible.

The _zero order

approximation

in _To

corresponds

to the local

electroneutrality.

The variations of the CDW current and of the soliton's _current

js nearly

compensate ^{each other.}

It is _very

important

^that

equation (56)

^does not contain

microscopic

scales _To

=

i/Jc~

and Jc-~ To

analyze

the

macroscopic

scales _we consider first _a

homogeneous equation (55)

at

Io _" ^{0. It describes} _e-g- ^the CDW current ~elaxation after the

injection

is switched off _or the

distributions at _z < 0 I-e-

beyond

the in-out interval. This

equation

is characterized

by

the

following

scales for the

length

_To, time to and'the

velocity

_no1

~o "

7bu

₌

lTDi, IT

= u

IT;

to _"

~( ID;

_{no "}

~o/to

"

T/ui

₌

I/ilT

If the estimate

(52)

takes

place

then _To

-~ I

=

u/Tc

is the soliton

length

and to is the soliton diffusion time _over its

length

which is measured

directly by

the NMR.

In _terms of dimensionless variables

z/zo, t/to, equation (56) acquires

the form F F" F'F

=

u0(t)0(z),

_{v =}

Ioto (57)

(~ A temperature distribution asymmetry of +2x- solitons is due the charge conjugation breaking.

It's major _{sours comes} from the phonon dispersion _{[9] which} splits the activation _energy E&

= Es + (,

( cc

b~c/vwph

where _c and wph are the 2kF sound velocity and hiquency.

Notice that the condition

(52) implies

that _v « I.

Equation (57) depends

_{on a}

single

dimensional parameter _v which is the

injection intensity.

We _can exclude it in terms of other scales for the

phase G,

time ii and

length

_xi

G =

vF,

ii ₌

I/(I(to)

=

to/v~

_xi

=

(Dti)~'~

=

l/(Io71T)

₌

zo/v. (58)

Then

P _"

~i~G'>

i~J _"

(~/~0)G (59)

It is

important

to notice that while the scales ii and _xi shorten with

increasing injection

level v, there _are invariant combinations: the diffusion ratio D ₌

z(/ti

and the defects

density

P ₌

G'/zo.

In variables X

=

z/zi,

T ₌

t/ti

_we have G"

G'h

"

~°(~)°(~)

+

f(~)

(Go)

with the

boundary

conditions

G'(+03, T)

₌

Pco/zi

= Pi

G(+03, T)

= F& ₌

7r(to/v)J(+O3i T) (61)

It follows from

(61)

^that

f=©+(I-p)+I=©-(I-p)

In such _{a way can we} know scales and

magnitudes

in their

dependence

on v even before

a

detailed

study

of the

equations.

In the

following

_we will

study

the solutions of

equation (60)

in various

regions spanning

the

(z, t) plane

with

exception

of _{some crossover} lines.

First _we will

study equation (57)

in the linear

regime

when the nonlinear _term

F'fi

can

be

omitted,

which

corresponds

to the

negligible

contribution from

mobility

in _a selfconsistent field.

F'F _< F, F"

(62)

Then _we find

evidently

that

1 ~2

~

~~~~2(xjl)1'2

^~~~ 4Dt ~~~~

There _are two

regions compatible

to

(62)

where _we obtain from

(63)

1. At z2 _« Dt and t «11

~ ~

~~~~°(il~l'2'

~

"'~~°~ _~ " "

~~~~ ~

~

(xl~t)l'2i

~~~~

2. At z2 _{» Dt} and

z/t

» _{vi "}

Tilt

i

J~l/2

_13/2 ~2 ₁

~ _" ~~~~~° xl'2 ~2 ^~~~

Dtl

'

"

~' ~

_" ~~° _~~~~

Consider _{now an}

opposite strongly

nonlinear limit when _{one can}

neglect

the diffusion _term j~j/"~

DF" _«

F, (ibu/2)F'F. (66)

Then

equation (60) acquires

^the form

G(I

_G~)

=

-°(X)°(T) (67)

which _can be reduced to the

ordinary equation by

the

assumption

G =

Tg(X/T),

where

g(f)

satisfies the Clero

equition

9 "

f9'

+

(g' 1)-~ (68)

The

appropriate

solution of

(68)

is found to be

-2ti/2

₊

_t,

_{o < <} i

g =

(69)

-1,

f

> I

which

gives

_{us one}

region compatible

with

(66):

3. At

(z/zi)~

_{> t}

_Iii

(ioi/rz)1/2 +1/r,

o <

~/i

_{< vi}

P ₌

(70)

o, ~/i

_{> vi}

(io~/n)1/2,

_{o <}

_~/i

_{< vi}

7rj ₌

ioi ~/i

_{> vi}

where r ₌

(ibu)~l

Finally

there is _a

region

where the local current of solitons

fi

may be

neglected

_so that _we arrive at the

equation

G" ₊

G/d

=

0(X)S(T) (71)

Again

^it can be reduced to _an

ordinary

equation

by

the substitution

G =

T2/3 ~(

~

/~ni /3)

where

g(q)

satisfies the

equation

9"

₊

9'(29 9')/3

"

o(Q),

_"

x/T~/~ ₍₇₂₎

We need the

region

_q « I when the solution of

(72)

is

given by

_a series

g m A+ ~~

+

(73)

Here the constant A < 0 is defined

by matching

conditions for

adjacent regions

3.

Finally

we obtain

4. At t »11,

(~/~i)~

_«

t/ti

12/3 ~i/311/3

~ _"

ri/)Di/3~~~~' ^J

^*

_,r2%3 @ ^(~~)

At _{~ <}

0,

_on another side of the contact, ^{there is} _a ^nontrivial

region

which is studied

similarly

to the _case 4. At intermediate time _we find:

5. At _{~ <} 0, _~

l»

_~ii

(l~l/~i)~

»

t/ti

»

(z/zi)~

P -~

-t/z~iT7

,

j

-~

1/jzjiT7

In summary the

regions

1- 4 _cover the whole

plane

_(z~

t)

with _crossovers

taking place along

the lines

(X~

₌

T,

T _>

I)

and

(X~

=

T,

T > 1)~ and

(T

= l~ X <

I)

_as shown in

figure

I.

The

regions

2 and 3

actually overlap

_over the sector X _> T where solutions 2 and 3 coincide within

exponential

_accuracy.

X=T /

IT

3

~ ~

/ _~

iT

~

i T

I

~

~ s

Fig. 1. Characteristic regions _on the plane

(X, T).

Generally

_our consideration shows that _{at any} finite time T for all X » T _we have

j

rtS

Io/x,

P _rtS 0. It _means that

everywhere

before the front which _moves with constant

velocity

_vi

= bE there _are the nominal values of the electric field and of the CDW _{current as} for

a

homogeneous

^motion. Let _us follow _now the time

development

_{at an}

arbitrary point

_z » Jc~~

Instantly

at _a

microscopic

time _To the nominal values of currents _are established at the

negligible

concentration of solitons: J _rtS

j

_St~

Io/x,

P _rtS 0. The

following

time

development

differs for _a distant X _> I and for _{a nearest} X < I

regions.

For X > I at T < X the CDW current is still

nearly

constant

j

₌

Io/x~

and P is

exponentially

small. At

an intermediate time X « T « X~

j

decreases while P increases

following

_{a power} ^low

j

_cc

(X/T)~/~~

P _cc

(T/X)~/~~

while at

large

time T » X~ both

j

and P

change

_more

slowly: j

cc

T~~/~,

P _cc

T~/~ independent

of X.

For X _< I at T < X~

we have

again j

rtS

Io/x~

at P _St~ 0. Later at X~ _{< T < I the}

current is

nearly

constant

being equal

to _a half of the nominal value

j

rtS

Io/(2x)~

while P

increases _as P _cc

T"~ independent

_on X.

Finally

at T » I the behavior is similar to the

region

X > I _: P _cc

T"~, _j

cc

T~~/~

_{The results of numerical} calculations for T _or X

dependences

of P and

j

_are

presented

in

figures

2 and 3.

o

~ T40

-3

-IO 0 X 20

0,5

T4 T=30 T=50

j

0

-IO 0 30

Fig. 2. The computed plots of

P(X)

and

j(X)

at various time moments T.

We conclude in

genera1terms

that the defect

populated region expands

from the

injection layer

with _a constant

velocity

of the soliton

mobility

_{vi "} bE. While their concentration is

growing

the current and the field decrease in such _{a way} that at

large enough

time

j

_cc I

IF.

Consider _now

briefly

the effects of _{a nonzero} volume concentration of solitons Pco < 0. Now

we must define the

plastic phase

as

Fi =

/~ ^{(P Pco)dz, j j-co}

= -Fi

(75)

-co

Fi DF"

rfi(F(

₊

_Pn~)

₊

rj-co

=

-Io0(t)0(z) (76)

We _see that _at

j-co

₌ ^{0 the solutions of}

(76)

^lead us to the results which differ from those for the _case Pco ₌ 0

only by

the transformation

P~P-P~, D~D/(i-rP~), io~io/(i-rP~), r~r/(i-rP~) (77)

Notice that the total electric _current J values do not

depend

_on the transformation

(77)

since it is not related _to the function _{F at} the _zero order

approximation

J _rtS Jo when it is determined

through

the nonrenormalized value of lo ^Then we obtain that the _current of

solitons

is

does not vanishes at X _> T but has _a finite limit

2js(z

»

t)

rtS

-rPco j

₌

-xrPcolo/(I rPco)

o

x=Io

P x=o x=5

-3

0 T 60

o,5

X=20

)

x=~ x=Io

=2

x=-z o

~ ~

Fig. 3. The computed plots of

P(T)

and

j(T)

at various points X.

Reminding

that E is

uniquely

related _to

j,

_we find that the coherent part of the CDW _con-

ductivity j/E

does _not

depend

_on Pco ^{whereas the total}

conductivity J/E

is

proportional

to

(I rPco)

with accordance _to the _current

additivity (41).

4.

Stationary

solutions.

We observed that _a

sharp injection impulse

evolution continues to be

nonstationary

_even behind the diffusion front

passing.

Nevertheless _we cannot exclude that _a

stationary regime

for currents

and/or

^for concentrations is stabilized under _some other conditions.

Consider

again equation (76)

_or the

complete

system

(47), (55). Among truly stationary

solutions

3j/3t

_{= 0~}

3ps/3t

= 0 _one can find

only

the trivial _one

j(z,t)

=const which _corre-

sponds

to invariable local CDW velocities. We have

~'

=

~ E

=

~~

ji

₌

j

_%

jo

" const, _p

-~

ji'

_% 0

(78)

x x7u xi

For the concentration and the current of solitons _we obtain from

(76)

at

ii

" 0

P ₌

-(

~~~

(_

^2E_~ ^{~ ~}

^~~~~

^~

^{~~°°' Js}

⁼

)°(z)

_~~g~

So _we found that the

density

of solitons

approaches

the thermal

equilibrium

one p-co =

-P-co/2x following

the Boltzmann law for the

potential

_energy 2Ez when the drift and the diffusion currents compensate each other. At _z > 0 there is _a constant concentration of solitons with _a different value

pco =

P+co /2x

_{= p-co} +

Io/(2xbE)

Notice that the CDW

conductivity

_acdw does not

depend

_on the

injection

level while the

partial conductivity

in the

injection

interval is

proportional

to the ratio of currents

~2

j

^{~2 e~ J} _j

~)s,~~~

~

) (80)

acdw "

j£ _4,~'

^~'

s E _{) ~}

Finally

the CDW

phase

is

given

as

qJ =

-xjot $

exp

(- ~)

z

S(-z)I- ^(zS(z)

^{+ P-co~}

⁽⁸¹⁾

Suppose

_now that the CDW _current is still time

independent

while the soliton

density

_can

change.

Then _we find

just

_one nontrivial solution

r-iE(z)

=

j(z)

=

-jo

tarn

((( _z)

,

((~

⁼

()ch-2 ((( _{z) (82)}

This solution

corresponds

to the

symmetrical configuration

with _a reversed direction of current which is convergent now towards _z

= 0 at any

sign

of

jo.

The local concentration of solitons

near the contact decreases

linearly

in time which

corresponds

to the

minority

carrier

injection.

Within _our

unipolar

model _we cannot trace this solution

beyond

some time t* _cc

jp~

when _ps

crosses zero level.

5. Conclusions.

We have derived the

dissipative dynamics equations

for the CDW in the _presence of solitons and dislocations

(Ch,I).

The _response functions _were found for the white noise conversion _process of electrons into solitons

(Ch.2).

The results of _an

experimental significance

_were obtained for _a

problem

of _{a current}

injection impulse development (Ch.3).

The

simplest

one-dimensional for- mulation

corresponds

to a narrow contact at a thin

sample

side surface. The

purely dissipative

CDW

dynamics altogether

with _a diffusion

approximation

for solitons _were

employed.

Under these constraints _we

ignore

both _a

preliminary

^diffusion of electrons and the intermediate stage of _x- solitons formation

(see

_[1,2] for

discussions).

These _{processes smear} the

injection layer effectively.

Also _we constrained ourselves with

unipolar regime

when the

injected

carriers _are of the _same

sign

as the

equilibrium

_ones. For this _reason

we are not

considering

effects of

depletion

with respect to the volume _or to the contact _area

(see [10]).

We find that first the nominal CDW _current

jco

and the electric field Eco _cc

jco

_are estab- lished

along

the

sample length

in _{a very} short time _t. Later _on the diffusion front _passes

along

with _a constant

velocity

_{c =}

bEco,

where b is the soliton

mobility.

It is followed

by

the

growth

of the wliton concentration _ps and

by

the decrease of the local current

j(z,t). Typically they

are related _as

j(z;t)

_{cc pi}^~ _cc

^t~~/~

_{or cc}

(z/t)~/~

Also _we considered _a

possibility

of

stationary

distributions. The time

independent

solutions

are found to exist for _two

special

_cases: for

generation

of solitons

by

the constant CDW current and for _a time limited

depletion

due to _a

minority

carrier

injection.

It is worth

mentioning

that out of the current

passing

interval I,e, at _z < 0 there is _a

growing region

of

increasing field,

current and soliton concentration. This result correlates with

experimental

observation [11] ^{of nonlocal effects.}

As

expected

_our

equations

_are not able _to describe the oscillations in the _course of the current conversion. The _reason is that _we did not allow for _a

possibility

of soliton

aggregation

into dislocations _so that _we did not put a limit for their

increasing

concentration. To discuss these effects _{we can} encounter the

following

two scenarios.

I. Let the Coulomb constant and

/or

_a minimal

anisotropy

coefficient _am;n be

large

with respect

to a maximal

anisotropy

coefficient _«mm

(see [1,2,10])

so that the

inequality

~o "

a$;$wp

< Es _rtS

(amax /s)~'~, (83)

holds which is

equivalent

to

~2 _~y~~

~

^~

txmin

Then the

phase diagram

must show _a critical line which resembles the

wetting

line of _a

liquid-vapor

coexistence. It is determined

by

_a local chemical

potential

of the soliton _gas

»(Ps, T)

_{= ~o}

E~,

_{p «} Tin _{p~ <} o

(84)

When this line is

being passed

with

increasing

_ps the condens ation of soliton _gas into the dislocation

loops

_or into the dislocation lines

split

from the surface takes

place (see [1,2,10]

for references and

discussions).

ii. The second scenario takes

place

either if the

inequality (83)

does not hold _or

(with

_a small

probability)

_even in the _gas

region

with respect to the line

(84).

It is related to the nucleation of

largely separated

dislocations _or of their extended

loops

when the

perpendicular

sizes exceed the

screening length

R _> l~~. More

precisely

_{a stronger}

inequality

has to be fulfilled in terms of the screened dislocation chemical

potential ~d(R)

~d(R)

_«

~o/(~R)

<

Es

₊

~(Ps)

This condition _can be

always

satisfied but

actually

^the _process

depends essentially

_on ^the

nucleation facilities. The rate _can be slow because of

large

sizes

required

and also due to the soliton

repulsion

^from the dislocation

loops

within

screening

distances. For both scenarios the

periodicity

is

expected

to be

explained by

the alternation of the accumulation and the

depletion

stages ^{within the soliton} _gas ^cloud near the

junction.

References

[1] ^Brazovskii S., Matveenko S., Zh. Eksp. Tear. Fiz. 99

(1991)

887.

[2] ^Brazovskii S., Matveenko S., J. Phys. I France1

(1991)

269.

[3] ^Landau L-D, LiI§hitz E. M., Theory of Elasticity

(Pergamon

Press,

1980).

[4] ^Kosevitch A-M-, Uspekhi Fiz. Nauk 84

(1964)

579.

[5] ^Brazovskii S., ^Matveenko S., Zh. Eksp. Tear. Fiz. 99

(1991)

1539.

[6] ^Brazovskii S., Matveenko S., J. Phys. I France 1

(1991)

1173.

[7] ^Monceal~ P., Electronic Properties of Inorganic Quasi-One-Dimensional Materials. Part II.

(Reidel,

Dordrecht,

1985).

[8] ^Grliner C., Rev. MJd. Phys., 60

(1988)

1129.

[9] ^Brazovskii S., Matveenko S., Zh. Eksp. Tear. Fiz. 87

(1984)

1400.

[10] ^Brazovskii S., Matveenko S., J. Phys. I France 2

(1992)

409.

[11] Saint-Lager M-C-, Monceal~ P., ^Renard M., Europhys. Lett. 9

(1989)

585.