The charge density wave structure near a side metal contact

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The charge density wave structure near a side metal contact

S. Brazovskii, S. Mateveenko

To cite this version:

S. Brazovskii, S. Mateveenko. The charge density wave structure near a side metal contact. Journal

de Physique I, EDP Sciences, 1992, 2 (4), pp.409-422. �10.1051/jp1:1992100�. �jpa-00246496�

Classification Physics Abstracts

61.70.G 71A51- 73.30

The charge density

_wave

structure

_{near a}

side metal _contact

S.Brazovskii and S.Matveenko

L.D.Landau Institute for Theoretical Physics, Moscow, Russia

(Received

29 November1991, accepted 10

January1991)

Abstract. We consider the structure of _a distorted

Charge

Density Wave

(CDW)

_{near a}

side _{contact to a} metal. We show that the electric field screening and the charge penetration

proceed via inhomogeneous distribution ofsolitons

andlor

dislocations. In the _presence of these topological defects

we study selfconsistent equations for elastic and Coulomb fields. For high temperatures we find the soliton density distribution through the sample depth and calculate the capacitance. For low temperatures and concentrations _we study in details the fields and

induced surface charges for _a single dislocation _near the metal surface. At low temperatures and high concentrations _a periodic structure of dislocations emerges ^at some critical voltage

between the metal and the CDW. At small charges _near threshold voltage the dhlocations _are far separated and, unexpectedly, deeply located. So the _contact region naturally provides both generation and storage of topological defects. With the longitudinal current being applied, these

defects will nuceate phase-slips for the CDW sliding.

1 Introduction.

Solitons and dislocations characterize the stable excited _or deformed states of _a

Charge Density

Wave

(CDW) crystal and/or

the

dynamics

of _a conversion _process of _a normal _{current to} the CDW _one

(see [1-4]

for

Ref.).

We will show that these

topological

deformations should _appear

deliberately

due to electric fields at

typical

side contact of the CDW

crystal

with _a metal. In this

way the contact

region

works _{as a} natural generator and accumulator of

topological

distortions

which _can be activated under

appropriate

conditions of _a CDW

sliding.

These concepts _may enable _one to

explain

_some

experimental

data _on dislocations [5, 6] and ^recent observations _on the contact

asymmetry

[7].

Let the CDW

crystal

_occupy the

semispace

_y > 0 with the

(~, z) plane being

in _a contact to a

metal, (~

is the chain

axis).

The metal is characterized

by

a value _4~o of the electric

potential

relative to the _zero

asymptotic

value inside the CDW

crystal. (Hereafter

4~ is the

potential

energy for the electron

charge e).

4lco(~)

+ °>

~(~, °)

₊ 4lo > 0

(1)

The value _4~o is

naturally

defined

by

the materials' work functions _or it _can be monitored in _a field-effect geometry. We will _suppose the temperature to be low

enough

T « 1h, ^when the electron concentration

(2A

is their spectrum _gap is small.

The

necessity

of

topological

defects to _screen the field I.e. to fulfill the conditions

(I)

is clear from the

following

considerations. To create _an

inhomogeneous

_{over y} CDW

charge

concentration

3q7

~

KS d~' ^{~ ~~~~}

(s

is the _area per unit

chain,

_{ag, az are} interchain

spacings)

the _wave number

inirement

_between

adjacent

chains is

required

bq

₌

(d~q7/d~dy)ay

Then at _an

arbitrary

small

charge

redistribution the

3d-ordering

_energy is lost which is _cc

Tf Iv

per chain unit

length

_(Tc is the

3d-ordering

temperature, _v is the Fermi

velocity).

This situation reminds

a distributed

commensurability

effect.

Evidently

the energy lost per chain becomes finite for _a quantum of

charge

2e, if the in-

commensurability

of

neighboring

chains is concentrated in _a form of 2x-solitons

(see

[1,

2]),

their

charge being

2e, ^the

length

I _cc

v/Tc

and the energy Es _cc Tc. At low temperatures the

aggregation

of solitons into threads

along

z-direction is favorable which is

accompanied by

their

y-axis

correlation.

These _{cases are}

naturally

classified in terms of dislocations [3,

4].

One soliton

or their finite

complex corresponds

to _a dislocation

loop embracing

_{one or} several chains. The line of solitons

corresponds

to the

z-elongated

_narrow dislocation

loop.

The final

adaptation

of the electronic

charge

down to _some

depth

L

corresponds

to the

single

dislocation line

ii ^z

passing through

_a

point

_{y =} L at _{some ~.}

Following

results of [14] we can

readily

know the threshold value _4~o for the

charge

penetra- tion _or in another words for the onset of

screening.

For the _cases encountered above it

equals

to

Es, ks

_and

po>

correspondingly.

The

energies

Es _cc

u(a/s)~/~,fls

_cc

v(&/s)~/~

_are deter- mined

by

the maximal _a

= da and the minimal &

= ay among the parameters ^{of structural}

anisotropy

_{[1, 4].} The value of the dislocation chemical

potential

_{po cc}

a~/~wp,

^where _wp is the

plasma frequency,

is allowed to be both

larger

and smaller than Es [4].

2. The

equflibrium

conditions for the CDW _near the side

junction.

Distributions of dislocations

(see

[8,

9])

_or solitons _are described

by

_a vector of dislocation

density

_{p or a} vector of their

dipole

moment

P,

^which _are ^related _{as p =} VXP. For _an element of _a dislocation

pdv ₌

2xS(()dl,

PdV

= 2xds

(2)

dl =

rdl,

_r

(= I, iv

₌ 0.

>vhere dl is the

length

element of _a dislocation line

D,

ds is _an element of _a

phase discontinuity

surface based _on the line D. For the ~2x- soliton at the

point

_r

= 0, we have

P

=

~2xsnb(r) (3)

where _n is the chain axis vector. Hereafter _we define the

phase

_{q7 as a}

single

valued

(and consequently discontinues)

function which

corresponds

to the definition of the field

q~* in [2].

It is related to the

locally

defined continuous

phase gradient

_as

w~Vq~, Vq~=w+P (4)

The fields

P =

Wnl'> j

₌

-#I'

define the

charge density

_p and the CDW current

density j.

^The

equilibrium equations

_{[3, 4]}

in terms of

q~ and P

acquire

the form

(nV)jt(Vq7 _P)

₊

(2/u)(nV)~j

₌ 0

(5)

2/vK~A4l

₊

n(V~ P)

= 0

(6)

'~

us ^'

~'~ "P' ~

d~' ^"~

$'"~

_dz

@2 @2 @2

~ ~~

d~~ ^~ ~~

dy~

^{~ ~~} dZ~

where _K and wp are the inverse

Debye

radius and the

plasma frequency.

Remember that in static

problems

the constraint P ₌ nP

always

holds for soliton and the _{same may} be chosen for dislocations in the

perpendicular plane

which is

supposed

^below. Remember also [1,2] that the field

where

ir

is ^the

_stress ^plays

the

roleof the 2x- soliton

3.

High temperature:

^{the soliton} _gas.

In this

chapter

_we

study

the _case of _a

high density

soliton _gas when P may be considered _{as a}

homogeneous

function of _y

only.

^Then

equations (5), (6) acquire

the form

)£9'

"

0,

_{q2 =}

q2(z, y) (8)

( _~~

+

~)

^P =

0,

P

=

P(y),

~b ₌ ~b(y)

(9)

To

complete

the

equation system

_we need to add _an

equilibrium

condition with

respect

to the distribution

P(y).

We will suppose that temperature is

high enough and/or

the Coulomb

interaction is strong

enough (see

[3,

4])

so that _~o > Es. Under _any of these conditions _we

can

neglect

the counterpart ^of dislocations and of their

loops

which characterize the many electronic clusters. Then P is related to the statistic distribution of solitons

only.

P =

2ir(P- P+);

_{p~ =}

^d~~ exp(-fl[Es

+

(V ~)]), fl

₌

I/T (10)

Here p~ and +~ are the densities and the chemical

potentials

of the +2ir solitons.The

length

d

depends

_on the

configurational averaging

type. For quantum solitons d

-J

(MT)~~/~,

M

-J Tc

/u~

where M is the soliton _{mass, u} is the CDW

phase velocity.

Otherwise for

trapped

solitons d is

an average distance between

impurities along

the chain. The shift _~

#

0 is

always

present due to _a

microscopical charge nonconjugation

which is

mainly

due to the 2kF-

Phonon dispersion

[10]. ^{It results in}

~ «

fl,

~

i<

E~

(ii)

where _c and ~ _are the

velocity

and the

frequency

of the 2kF

Phonons,

_g is the

electron-phonon coupling

constant. The

inequality

in

(11) provides

the

ground

state

stability.

For

asymptotic

values far in the volume _we find from

(1), (7)-(9)

y~oo:

1b~0, P~Pn~, ~~'~Pn~,

^wz~0.

⁽¹²⁾

The last condition in

(12)

means

electroneutrality.

We also find from

(10)

Pm ₌ -A

sh(fl~),

A ₌

(j) _{exp(-flEs) (13)}

The

only nonsingular

^solution of

equation (8),

which satisfies conditions

(11),

is the

homoge-

neous one:

q~ = Pmz

(14)

It _means that in

spite

of the _presence of defects the absolute

displacements

of the CDW

crystal

do not

depend

_on the

depth

_y. Then the final system of

equations

follows from

(9), (10), (14)

for functions

4l(y)

and

P(y):

21b ₌ ~ vPm + VP +

Tsh~~

^~

(16)

Generally equations (7), (15)

have _an

integral

which

provides

the conservation of _a

properly

defined _energy

density:

~

)~ +

Q(~ 2ib)

+

)P~

= const

(17)

~C V

Q _m

Qs(~

21b

E~) Q~(-~

+ 21b

E~) (18)

Qs(()

_c~

(T/d) exp((/T) ₍₁₉₎

Equation (17)

where Q is the

thermodynamic potential

of solitons is model

independent

within

our framework of local

equilibrium.

^The

expression (18)

where Qs ^is a

partial potential

for _one

kind of solitons

corresponds

to the dilute _gas

approximation.

^Formula

(19)

which is

equivalent

to

equation (16),

is valid for the Boltzmann statistics. Furthermore _we will _suppose the +2ir- solitons to be the

majority

carriers _so that P < 0.

Integrating equations (16) -(19)

_we obtain the

following

results:

Near the surface at P »

T/v,

A _we have

P _CS

2v~~lb

_CS

2v~~iPo exp(-icy) (20)

We conclude that the initial surface

potential

lbo

'~ A induces _a

high

concentration of the order of

limiting

_one P _oc

fp~

which

drops

_very fast at

microscopical

distances down _to

a thermal concentration

PT

~w

T/v.

Then in the range A _« P «

T/v

P

+~

~j~j~~,

~ _« in_y

(20

At _even smaller P <

A,

there is _{a crossover} from the _power law to the weak

exponential

_one

P _oc A

exp(-6y),

6

=

1c(vA/2T)~/~

« 1c

(22)

Finally

_we find _a

Debye

linear

screening regime

_near the

equilibrium

concentration of solitons

P Pm _oc

exp(-ly),

1

=

1c(P/T)~/~ (23)

Here I coincides with the residual carrier

screening

parameter ⁱⁿ

[1- 4].

The obtained relations allow _{us e.g. to} calculate the contact

capacitance C,© _(per

_unit

area):

C=

(lb~~lb'j

,

©= (bib'/blbj ⁽²⁴⁾

~r Y=o ir y=o

Depending

_{on PO "}

P(0)

_{or lbo " 16(0) we} obtain:

I)

High

concentration

lPo(m2(lbo(/v»T: Cm©m~c~~

I-e- the

capacitance

^is due to _a

microscopic layer

oc

~c~l,

where the

charge

is

mainly

_concen-

trated.

ii)

L6w concentration

PO~ « Po « T _, c _{c~ ~} ⁱ(I Po

v/T)~/~

/~

i ~~

ln(Po/PO~)

^{' " '~~}

('

~°

/~') (25)

In this

region

Po

*

Pm exp(-ibo/T).

Now consider the

stationary perpendicular

current

j regime.

^The

perpendicular mobility

b and the diffusion coefficient D ₌ bT of solitons

being

small the CDW deformations _can be treated

statically.

^Then

equation (15)

^{still holds but}

equation (16)

is _to be

generalized

to the diffusion _one:

J

₌

-Dt _bPll

_P

=

-27rP (26)

Also the condition

(I)

is to be

changed

to bib

~ ~

i

~ ~ ~

(27)

~

by

^°'

bpm'

As well

as before _we consider

only

one type ^{of carriers. We arrive} at the

equation

for E:

TE"

₊

(E"K~~ E)(2E'

₊

Tl~)

₊

2Tl~Em

= 0,

(28)

p/p~

=

E'/(T>2)

For small Em _oc

j

the formulas

(20), (21)

are recovered. But at

large

_{y we} find instead of

(23)

another decrement

E Em _c~

exp(-ky),

k

=

(Em /T)~

+ l~

)~/~ Em IT (29)

We _see that the effect of finite Em is essential at

Em »

IT,

k _m

l~T/(2Em ₍₃₀₎

I.e. when the

voltage change

_over the

screening length

^l~l is greater ^{than the} temperature.

In this _case the diffusion

penetration length

^k~~ is smaller than the

thermodynamic

_one l~~

4. Low concentration and low temperature:

single

dislocations.

As we discussed in the Introduction

(see

also the

Conclusions)

the

aggregation

of solitons into dislocations is favorable _at low temperatures

provided

^that _{[3, 4]} the Coulomb interaction

constant is weak

enough

or the residual

screening

^is still effective. In this

chapter,

we will

study

_a

single

dislocation _{near a contact} to _see that the side surface _{y =} 0 effects

deformations,

electric fields and the dislocation _energy. The results _are different for _a metal and for _an

insulator,

the latter

being

considered _as the _vacuum hereafter. To

modify

the

equilibrium

conditions for the CDW volume

(5), (6)

_we write down the _energy functional [2] as

7i "

/ _d~~) ^~~~

~

~~~

+

(l~) P~l

₊

~l>~4~ (v~b)~l

+

(~)6(Y)j ₍₃₁₎

Y>°

Here P is the dislocation moment

density,

^l~l is the residual

screening length,

and

p(x) b(y)

is _an external surface

charge

at the metallic surface. Consider _a dislocation

loop

in the

plane

z = 0 extended

along

the _z-axes

(the length

Lz _~

oo).

It is embraced

by

the two dislocation lines at _{z =} 0,_{y =} Yi and _{at z}

= 0,_{y =} Y2. At

(

_"

0,

there is

only

_one dislocation which has been

splited

off from the surface _{y =} 0. For the

pair

_we have

P =

-21r9(y _Yi)9(Y2 v)b(z)n.

Varying

the functional

(31)

with respect to the fields

q~ and ib _we find

equilibrium equations

~~

_~

_~~ ~' _~ ^~~

~ ~~

~~

~~~~

A l~ _{ib +}

~~'

= P~

p(z)6(y) (33)

They

_can be written for the fields 4l and

q~

separately:

t~

=

-~r~2v«) ^{jJ(y yi) J(y y~)j J(z)} (ip(z)J(y) (34)

k)

=

-21r(~~

+

l~ A) )6(z)16(y ^{Yi) 6(y}

Y2)1 + ~~

)P(z) _)6(y) (35)

where

k =(-l~

₊

A)I _~C~~

fi2

At _{« =} I, ¹ ₌ 0 there is _an exact solution of

equations (34), (35)

for _one dislocation in the volume [3, 4]:

~Po =

j~ ^ch> sh>j j~ I<oi(>~

+

z~)~/21dz (36)

aq~o ~~

(37)

/~sh>i<o (£~

+ i~~~~~

Here E ₌

lcz/2,§

₌

icy/2

and Ko is the modified Bessel function.

Approximate

solutions for

arbitrary

_«

~bo ₌

~'~~i~~~

Erf

1(~«ii'l

x

)~/~l

sign

_Y

(38)

and

~ 4«ii~(

z

()

~~~

~~~

«l~~~

z

()

^{~~~~ " ~~~~}

are valid for 1 ₌

0,

at y »

«~/~z.

_Here

Erf(

is the Error function. The solutions for

1#

0:

lbo ₌

~~(~~~

~

~~

_~ ~,

«* =

«l~/~c~ (40)

y « z

t°

=

~l~~

~~

+i~~ (41)

are valid at _x

»1c/(«~/~l~),

y »

l~~,

where «* « _« is _an effective

anisotropy

constant.

Taking

the surface into account the variation of

(31) provides

_us with the

boundary

condi- tions:

~)

lY=o "

0,

_ly=o to

(42)

for the _vacuum

boundary

and

for the metal

boundary.

Consider first the _vacuum

boundary

_case when

p(z)

₊ 0. As well _as for _a conventional elastic

theory

the

boundary

conditions

(42)

_are satisfied

by introducing

the

image

dislocations of the

opposite sign

at,

symnJetric points

_{z =} 0, _{y =} -Y. The solution for _one dislocation takes the form

16 ₌ lbo(Y

Y)

_lbo(Y +

Y) (44)

and

similarly

for _q~.

For the metal surface the

image

method does not work

apparently

since _now the

opposite parity

is

required by

^the first and the second

boundary

conditions

(43)

unlike the conditions

(42).

So the

problem

is to be solved for the

semispace

_y > 0

taking

into account the surface

charge explicitly.

For _{a no}

screening

case

(1

₌

0)

we find from

(34), (35)

16 ₌

lbo(z,

_Y

Y)

_lbo(z>v +

Y) (K~v/2)£2

_x

P(z)6(v) (45)

where

£~=)ch>i<o((>2

+

12)1/2)

is the solution of the

equation

k£2

=

A6(z,y).

For q~ ^we obtain

sinfilarly

bi' bi'o(z, Y) _bq~o(z,

_{y +}

y)

3

$ _{by by}

^{+ ~l ~}

^~~Pl') fib(Y) ⁽⁴⁶⁾

~~~~~

~~ = _21rK

shit<o ((£~ +§~)~/~)

is the solution of

equation

t£i

=

alazd(z, y), i

_{x g +}

/ _{i(z z',}

y

y')g(z', y')dz

Formulas

(45), (46),

etc.' _are

general12ed

for _a

pair

of dislocations _as

f(Y)

_~

f(Yi) f(Y2).

The solution

(46)

satisfies the first

boundary

condition

(43)

at

arbitrary p(z).

Hence the surface

charge density p(z)

is to be defined

by

the second

boundary

condition

(43)

~ (y=o # 0.

Substituting

the solutions

(38)

to

(45)

and

taking

account of

(43),

_we obtain the

expression

ifi~yl/2

C3

~~~yl/2

p(z)

₌

~y

/ _exp(-t)

_Sin

t _CDS y2~

~)

dt

(47)

o

Asymptotically

we find from

(47)

8 _~

~l/2

p(Z)

~K --, ~

« l

ICY Y _lt

(~~)1/2y2

z ~yl/2

p(Z)

_~K

~/~ y2 ^~

Z a lC

The total induced surface

charge Q exactly

compensates for the _access CDW

charge

2N due to the presence of _a dislocation. It is distributed

over a

length

X _sd 1cY~

/«~/~.

Q

₌ ^~~

/~~ p(z)dz

=

-2~~

_Y

= -2N

(48)

s

_m s

Here N ₌

LzY/s

is the number of chains between the dislocation line and the surface which may be viewed _{as a} number of

aggregated

solitons.

At the _same

approximation

the

potential

16 can be found from

(45)

_as

16 ₌

lbo(x,

_y

Y) lbo(z,

_{y +}

Y)

+ 61b

(49)

where bib is _{an excess} part ^due to

p(z):

bib =

-21cva~/~ /~

^~~

_exp(-(I

₊

~')t) sin(~'t)(cost

_sin

_t)

cos

(~'"j~ ^{~) (50)}

o t Y Y ~CY

An exact

explicit expression

is available from

(49),(50)

_on the line _z

= 0

(y

>

0):

16 ₌

~~)~~~

9(Y y) lcv«I/~tg~l( ~')

+

)

_In

_{~§~ )~~}

+ It follows that

at y ~

0,

_16(y) ~ 0; at y ~ +oo, 16 c~

-1cvY~/y~.

At various

regions

we find from

(49)

~ _~

~~~~~

~~"~~~~'~~~~~~~~'

«l~~~

z

~ ~'

«~~~

z

~ ~~~~

~b >

«~/~(~r1~)~/~v''l~'~

exP

1-~«ifi

~

ii II

^ch

l~«i'l~ _ii

+

Shl~«i'l~ ill

~'~~"~~~~~ ~~~

~

i

_~~

_$~~2

'

401~~

z

~ ~~~~

An

important

result is that

additionally

to the

expected positive

maximum _CS

«I/~Kv

_at

z =

0,

_y < Y the function

~b(z, y)

has _a

negative

minimum _m

-0, 5tv~/~Kv

at _{z =}

0,

_{y >} Y. We conclude that there is _an attraction

region

for electrons and solitons which

provides

favorite conditions for

a dislocation _{to grow} towards the volume. The

plot of16(0,y)

is

presented

in

figure

I. For _z

#

0 the extremas over y decrees and smoothen.

Analogously

we obtain for the function q~

q~ ⁼ q~o(Y

Y)

_q~o(Y +

Y)

_{+ bq~}

(53)

where

bq~ = 8

/~

^~~

_exp(-(I

₊

~')t) sin(~'t)

_{sin t sin}

(~'"j~ _t~)

o t Y ~CY

~KV

o f 4j

Y

y

Fig. 1. The potential dependence _on the depth _y in the plane z = 0 for

a dislocation line _at the point

(0, Y)

At _z # 0 the _extrema get smooth and their amplitudes decrease.

Again

_on the line _{x =} 0 the

integral

in

(23)

is calculated to

give

b6q~

8«~/2Yy

~i'~ ~'

fiz

~(y2

+

y2)2

Otherwise at

large

_{z we} obtain

~~'*

4

/~~i/2 '"~~~/(Ky~)

_»

max(I, Y/y)

Now _we consider the effects of residual

screening

when 1

#

0, l~ _« K2. We find from

(34), (35), (40)

and

(41)

bib =

-~lD

x

p(z) (54)

6(j

₌

K2$D

x

p(z) (55)

where

~~~'~~ 7r(o°)1/2~2

~~(Y~ ⁺

a°z~)~~~

is the solution of

equation

KD ₌

6(z, y)

at _z »

K/l~,

_y » l~~ where K _CS

-al~b~/by~ _lc~b~/bz~,

_a°

=

al~/1c~

Unlike the nonscreened _{case now} the

boundary

condition

bq~(z, y) Iffy

_(y=o + 0 is not satisfied

by

the

expression (55)

at y ~ 0 at

arbitrary p(z),

which would be able _to make _our

probleni

incorrect. A _more detailed

analyses

of the basic

equations

shows that

a correct _crossover

happens

_near ^the surface

layer

_y < l~~ _{where the solution}

(55)

is not valid. From

equation

(35)

_we ^obtain

at y ^~ 0. The

purely

^screened

approximation corresponds

to « ₌ 0 when the

integral

is

divergent

and _we would have

sgn(y)f(z)

instead of

yf(z).

So _{we are} left

again

with the

equation (42)

to obtain

4Y(o°)V~

jy _» I

p(Z)

_"

j(y2

+

~y°Z~l'

So _at

large

Y the total surface

charge

Q

=

~~

/ _p(z)dx

=

-4irLz/ls (56)

s

saturates at _an

independent

of Y value

(56)

_so ^that

only

_a finite part of the dislocation

charge

is

compensated by

the surface and the rest of the

screening proceeds

in the volume.

Solutions for fields ₁₆ and

q~ ^can be written in the form

(49), (53)

^{where functions} lbo and q~o are defined

by (40), (41).

At the _nearest

possible region

lY > I,_y >

l~l,

_{z »}

(o° )~~/21~~,

we obtain

» ₌

~~il/~ ~)liii~±iii(~ ⁽⁵⁷⁾

Jli

=

i~/~~~y

ill~[[.~~~~ ⁽⁵⁸⁾

At distant

regions ztv*l'~

»

Y,y

_or at y » Y,

tv*l'~z

_the counterparts _{6q~ and} 61b from

p(z)

can be

neglected.

5. The dislocation _energy.

A

general expression

for dislocation _energy [3, 4] is not affected

by

the _presence of _a metallic surface at the

potential

lbo ₌ 0. For _a dislocation

pair

it reads

W ₌ ~~

/~

(v

~'

+

1b) dy (59)

S Y, ^z

Substituting

the

expressions (36), (37), (44), (40), (41),

we find for the _vacuum

boundary

_case

the

following.

For the nonscreened

region lYi,2

« the result is the _{same as} in the volume:

W =

~~~~(Y2 ^{Yi)~~, Yi,}

Y2 <

l~~ (60)

For the screened

region lYi,2

» 1, ¹ Yi Y2

IS

I

W ₌

~~)(~~ 21n((Y2 Yi)I)

In

~~'())~~j ⁽⁶¹⁾

1 2

we obtain _a conventional elastic

theory

result which shows the attraction to the surface

by

the

image

forces.

Consider _a

single

dislocation _near the metallic surface.

Substituting expressions (49), (50), (53)

to

(59)

we obtain the

following.

For the nonscreened

region

lY _< I

W = in

2«~/~~CvYLz Is

_< Wo ₌

)«~/~~cvYLz Is (62)

so that the low _area is

preserved.

With respect to the volume the dislocation _{energy is}

numer-

ically

lower _near the metallic surface. For the screened

region

lY _» I _we find from

(40), (41), (57), (58) only

corrections _to

(61):

~~'~

~~~

~~~~'~~

(a°)~/21Yi

~~~~

6.

High

concentration and low temperature: ^the _array of dislocations.

We have shown that the dislocation _energy is

always

lower _near the surface I.e, the disloca- tions _are accumulated

nearby

it. At _some

high enough

concentration _a

periodical

structure of

dislocations is

expected

to arise.

Suppose

that _a

long

_sequence of N » I dislocations is formed at the _same

depth

_yo below the metallic surface with the

length

Nxo _" L~. Their energy is

go +°°

W _oc N

lbdy,

ib ₌

£16(nxo,

_Y)

(64)

_~

Here i$ is the total field _at the dislocation surface

(x

₌

0,y)

,

created

by

the _sequence of dislocations and

by

the induced surface

charge.

Consider first the shallow _structure when

Ky(/xo

« I _so that every dislocation

primarily

interacts with the surface. With the

help

of

(51), (64)

we obtain

W _c~ N

wo

+

~'~j~~~ )I)( Ill ^~~) ₍₆₅₎

where Wo is the energy of _an isolated dislocation

Wo _"

(a~~~Kvyo

ⁱⁿ 2 +

Ca~~~v) ~~,

C _cc I

s

Here the second term is _a self _energy of the dislocation line [3,

4].

The functional

(65)

^is to be minirni2ed at the

given charge Q

=

2yoNLz Is

I.e. at the

given charge density

_{oc q =}

yo/xo. Finally

_we obtain

~° "

IA

~° =

l

C~

IA,

(~Po

~o)

_«

q3/s

_~~~~

we _see that

Ky]/xo

_cc

q~/~,

_so ^that our

assumption Ky]/zo

< I is satisfied at q < I. We conclude that at small _{q near} the critical

voltage

dislocations _{appear as a} dilute _{array at} the

large depth,

the last result

being unexpected.

Our

neglect

^of

screening

^{is valid}

provided

that both

inequalities lyo

« I and

l~zo/K

_« I _are satisfied. We find that at

decreasing

_q the second condition is invalidated first which _means that the interacjion between dislocations

starts to be screened before their interaction with the surface is affected. We conclude that the concentrations under _our studies _are bounded from below _{as q »}

(1/1c)~~

Consider _{now an}

opposite

_case when

icy]/zo

_> I. The summation in

(64)

_can be

adequately

transformed with the

help

of the Poisson formula.

Taking

account of formulas

(38), (52)

_we

find at yo > _To

~~~~

~~

^~

^~

^{~~~° ~~~~~°} ~~ ~ ~o

~i~ 2~li

^~

x(exp(-6(y

+

yo)m~/~) (sin(6(y

⁺

^yo)m~/~) 2sin(6yom~/~)(cos(6ym~/~) sin(6ym~/~))j

⁺

+

exP(-6(y yo)m~/~) Sin(6(y yo)m~/~)l (67)

where

1/2

$ _# ^~~

20

Calculating

the

integrals

in

(67)

_{we can}

neglect

the small _{ternw c~}

exp(-icy(/4z)

tb obtain from

(64), (67)

~ _~

~~~ ~~~ _~

~

~~~~ ~l12 ~ ~/2 l~~~~

° ~0 ₁

In _an attempt to minimize

(68)

at _a

given

_{q, we} would arrive at the values

TO tK

~~,

y0 tK

(69)

ltQ ltq

which _are

beyond

^the

physical

constraints To, Yo < 1/1c as well _as

they

are at the limit of _our

assumption

since

icy]/zo

_c~ I. We conclude that the _case of

large icy]/zo

_> I does not take

place

and the

physical region

is covered

by

formulas

(66)

at q < 1.

7. Conclusions.

We have considered the deformed structure of the CDW _{near a} side metallic surface which

corresponds

to the

typical experimental

contact. We have shown that the

charge penetration

and the electric field

screening proceed

via _an

inhomogeneous

distribution of solitons and dislocations. We derived and solved the self consistent

equations

for

deformations,

electric field and the defect distribution.

For

relatively high

temperatures ^when dislocations _are

evaporated

in favour of the soliton gas we found _e.g. the soliton

density

_versus the

depth

_{p~ c~}

T/y2

and the _contact

capacity

C _c~

Q

_versus the surface

charge Q.

We notice the invariance

bq~/by

₌ 0 of the

geometric

CDW

phase

_q~ ^{which determines the structural}

displacements

_c~

cos(2kFz

+ q~). ^In other words there is _an

equilibrium compensation

between elastic deformations and the discomnJensurations due to solitons.

For low temperature ^and concentration _we considered the fields and the induced surface

charge density

created

by

_an isolated dislocation _near the surface. Unlike the _{vacuum case} the solution for _a metal contact cannot be obtained

by

the

image

construction and the

explicit

consideration nf the surface

charge p(z)

is

required.

For the dislocation

depth

within the

screening length

_{l~ ~, a} wide

longitudinal region

is affected _{z <}

lcl~~,

_{z < 1cY~,} where ₁₆(m _~i,

p(z)

_oc

1/(s1cY~).

Within this

depth

lY _< I the _area low confinement takes

place

^similar to dislocation

loops

in the volume

[1,2].

A dislocation is attracted _to the surface

by

_a constant force F _{= ~i}

Is (per

unit

length Lz)

where _~;

= ~o for _a volume and for the _vacuum

boundary

while _~i

= ~i for the metal

boundary.

An

important

result is th at

numerically

_~i < ~o so

that the metallized part of _a side surface

provides

_a

potential

well for dislocations. Another

important

effect of the metal is the

arising

of _a

negative

^minimum for the

quasi particle potential

V CS16 at _some y > Y _{near a} dislocation. So the

possibility

_appears for _a dislocation

growth directly by

the

injected

carrier

binding

_or

by

accumulation of solitons in the

vicinity.

At low temperatures ^there is _a critical

voltage

16 ₌ ~i ~S

«~/2wp

^for a

periodic

dislocation _{array to} emerge. Near the threshold _at small

charge density

the dislocations _are

widely separated

and at the _same time

deeply

located which is

unexpected (see (69)).

The _crossover between

regimes

of the soliton _gas and of the dislocation structure may

happen

to be _a

phase

transition of _a

liquid-vapor

kind. Consider the low

screening

_case under such conditions when the dislocation _{energy per}

charge

2e ~d = ~i <

Es

_so that _a dislocation is

preferable

at _zero temperature

(see

the

Introduction).

Due to the _area low for their energy W _oc N the dislocations

pin

the chemical

potential

of solitons like the

liquid droplets

for

thi

saturating

_vapor. Hence the

phase boundary

for the first dislocation to appear is defined

similarly

to _a

wetting point ~(ps)

_{= ~d} Es <

0,

_{ps c~}

d~~ exp(fl~).

In the _presence of

screening

the low

perimetric

W _c~

N~'2

_{In N}

implies

that _~d

" 0. Then it

seems that _at finite temperature the far

separated

dislocations _may coexist with _an

arbitrary

low concentration of solitons. Nevertheless the

screening

^condition

imposes

_a constraint

(see

the discussion after formula

(66))

1/lC

CS

(pslT)~/~

> q~/~

This

inequality

bounds the surface

charge density

_q from above _or the soliton concentration and the temperature ^{from below.}

Our

general

conclusion is that the contact

region naturally provides

the

generation

and the storage for

topological

defects which will nucleate the

phase slip

centers _{as soon as} the CDW

current starts to flow.

References

ii]

Brazovskii S., Matveenko S., Zh. Eksp. Tear. Fjz. 99

(1991)

887.

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

(1991)

269.

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

(1991)

1539.

[4] Brazovskii S., Matveenko S., J. Phys. 1France1

(1991)

1173.

[5] ^Csiba T., Kriza G., Janossy A., EtJrophys. Lett. 9

(1989)

163.

[6] ^Gill J-C-, Fisika 21

(1989)

89.

[7] ^Itkis M-E-, private _commun..

[8] Landau L-D, Lilbhitz E. M., Theory of Elasticity

(Pergamon

Press,

1980).

[9] Kosevitch.A.M., Uspekhi Fiz. NatJk 84

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