Mesoscopic behaviour of the threshold voltage in ultra-small specimens of o-TaS 3

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Mesoscopic behaviour of the threshold voltage in ultra-small specimens of o-TaS 3

S. Zaitsev-Zotov, V. Pokrovskii, J. Gill

To cite this version:

S. Zaitsev-Zotov, V. Pokrovskii, J. Gill. Mesoscopic behaviour of the threshold voltage in ultra- small specimens of o-TaS 3. Journal de Physique I, EDP Sciences, 1992, 2 (1), pp.111-120.

�10.1051/jp1:1992127�. �jpa-00246456�

J. Phys. ^I France 2 (1992) ill-120 JANUARY1992, _PAGE iii

Classification

Physics

Abstracts

72.15N

Mesoscopic behaviour of the threshold voltage in ultra-small

specimens of o-TaS~

S. V. Zaitsev-Zotov

('),

V. Ya. Pokrovskii

(')

and J. C. Gill

(2)

(')

Institute of Radio

Engineering

and Electronics, Academy ^of Sciences of the USSR, Marx Avenue18, 103907 Moscow, U-S-S-R-

(2) H. H. Wills

Physics

Laboratory, University ^of Bristol,

Royal

Fort,

Tyndall

Avenue, Bristol BSB ITL, G-B-

(Received J August J99J, accepted J7

September

J99J)

Abstract. In very small (~10~

'~cm~) specimens

of

o-TaS3,

the threshold

voltage V~

for the

onset of Frohlich conduction is found _to vary

irregularly

with the temperature ^{T. This behaviour is}

attributed _to rearrangement of the

charge-density

_wave (CDW) with respect to the

pinning,

_as its

wave-vector varies with T. It is consistent with weak, but not with strong,

pinning

by

impurities.

In

sufficiently

short (<10 ~m)

specimens

the variation of

V~

relative _to its _mean

(V~)

is

substantially independent

of

specimen

volume, _as is

expected

when the

dimensionality

of the weak

pinning

is

effectively

zero. However, the value 0.15 _± 0.05 found for

&V~/(V~),

where

&V~ is the _rms deviation of

V~

from

(V~),

is smaller than for _a

rigid

CDW whose

coupling

to

impurities

varies

sinusoidally

with

phase.

Possible _causes of the

discrepancy

_{are an}

approach

to

strong pinning

through

local elastic distortion _near

pinning

sites, _a departure from zero-

dimensional behaviour due _to

imperfect phase

coherence in _one transverse direction, and the

pining

of widely-separated defects (dislocations) in the CDW structure.

Introduction.

In _some

quasi

one-dimensional

crystals,

the

charge-density

_waves

(CDW)

which

develop

at

sufficiently

low temperatures may be induced

by

an electric field E _{to «} slide _»,

leading

to collective electronic

transport

of the

type proposed by

Frohlich

Ill.

Because the CDW is

«

pinned

»

by imperfections, sliding

in real

crystals

_occurs

only

when E exceeds _a threshold value

E~.

The conventional

view,

introduced

by Fukuyama,

Lee and Rice

(FLR) [2]

^and

developed

in later

works,

is that the continuous CDW becomes

pinned

_to

randomly-distributed

chemical

impurities by deforming elastically

_: its energy of interaction with the

impurities

is

thereby

increased

by

_an amount

W~,

^{but the}

corresponding

decrease of the total energy is

partly

offset

by

the elastic energy

W~.

^{In the} limit of strong

pinning _W~

is much greater ^than

We,

^{and the}

phase

of the CDW is

pinned by

each

impurity separately.

In the

opposite

limit of weak

pinning,

_W~

approaches W~

^{and the CDW is}

pinned

_{over a}

phase-coherence

volume

containing

_many

impurities.

112 JOURNAL DE PHYSIQUE I N° I

Experimental

evidence _as to the character of the

pinning

in real

crystals

is

ambiguous.

Studies of the

dependence

of

E~

_on

impurity

concentration in

large crystals (see Ill

and Refs.

therein)

have led to

contradictory

conclusions.

According

to Tucker and co-workers

[3],

most of the

experimental

results _are consistent with _a modified

strong-pinning

model, in which

depinning

of the CDW

proceeds through phase-slip

at the

impurities,

and which

incorporates

features of both types ^of

pinning.

The observation that, in thin

specimens

of

o-TaS3 [4]

and

NbSe3 [5], E~ depends

_on the cross-sectional dimensions may be evidence of weak

pinning.

In recent studies of this effect in

NbSe3,

Thome and associates

[6, 7]

have shown that both the behaviour of

E~

and the

phase-coherence

_across the

specimen

thickness _are consistent with the size

dependence being

associated with 2-dimensional behaviour of a

weakly-pinned

CDW.

However,

altemative

explanations

have been

suggested

for the behaviour of

E~ [5, 8],

and

phase-coherence lengths

of the _same order _as the

specimen

thickness _{are not}

necessarily

inconsistent with strong

pinning,

since the concentration of CDW distortion

around each

impurity

allows coherence _over distances much

larger

^than that between

neighbouring impurities [9].

Here _we present ^{the result of} an

experimental study

of very small

specimens

of

o-TaS~.

They

show

clearly

that rearrangement ^{of the CDW with} respect to the

pinning

affects

E~,

and _are thus evidence that the

pinning

is weak.

Specimens

and

experimental technique.

Measurements _were made _on 8

o-TaS3 crystals,

details of which _are

given

in table I. All _were from the _same

batch, large specimens

from which have

E~

₌ ^0.3-0.5 V

cW'

Electrical contacts of

vacuum-deposited

indium _were made _as described in

[10]

_; their

separations

^f

(1-

20 ~Lm) _were

measured,

^with _accuracy ± 0.5 _~m,

using

an

optical microscope.

Cross-sectional

Table I.

No of R

(300 K)

L S _v

iV~)

&V

V/V~ iE~)

Specimen

kohm _~m

~m2

10-'2 cm3 _{mV mV fb} V/cm

0.80 390 780 73 1.0 1.33 1.9

2 a* I-1? 2-3 _x

10-3

0.02 4.1 0.56 13.6 16

2 b* II-? 24 _x 10-3 0.2 18.I 1.4 7.8 7.5

3 11.4 10 _x 10-3 0.035 12.7 0.9 7.3 13

4 1.9 1-2 _x 10-3 4.27 0.76 17.8 32

5 13.0 9 _x 10-3 0.025 14.9 1.62 10.9 17

9.3 1.3 13 37

6 8.0 2-3 1.3 _x 10-3 0.0031

9.8 1.2 12 39

7 13.6 2-3 .2 _x 10-4 17 70

* Two segments ^of the _same

crystal.

N° I MESOSCOPIC BEHAVIOUR OF ULTRA-SMALL

TaS3

SPECIMENS l13

areas s

(down

to 10 ^~

~m~)

_were estimated from the resistance at 300

K, taking

the

resistivity

to be

4x10~~

_{am. The electrical} measurements were made under current-controlled conditions. To reduce the influence of

polarization

and

temperature-induced metastability,

the

current-voltage II-V )

curves were recorded with I

decreasing

in

magnitude.

Results.

Figure

I shows _a

typical

set of records of the

dependence

_on

voltage

^V of the normalised nonlinear

conductivity G(V )/G(0),

where

G(V

=

I/V,

for _a small

o-TaS~ specimen.

It is apparent ^{that the threshold} for'nonlinear conduction

depends irregularly

_on the temperature T.

$

~

o-TaS

1=24pm,

s=8.5. lo

Sri

11

₃ ^~

~

ii

1

160

iso

140

K~,

~~°

q

'Q~ ¹²⁰

~j

110

__

too - _ .

0

lo

Fig. 1. A sequence of normalised nonlinear

conductivity

for _a small

specimen

of

o-TaS3.

Figure

^{2 shows} the variation with T of the threshold

voltage V~

measured in several

specimens

_;

V~

_was taken as the

voltage corresponding

to _a 3 fb increase of G

(V )

over its linear value G

(0) (the voltages corresponding

to a I fb or 10 fb increase behave

similarly).

It is evident

that, although V~

follows _a smooth _curve of the usual form in _a

large specimen (s

=

2

~m~, ^f

= 390 _{~m ;} record

a),

it varies

irregularly

with

T,

in _a

qualitatively

similar

manner, in all the small

specimens (records b-o-

For _a

given

small

specimen

the form of

V~(T)

is

only partially reproducible:

most features _are

reproduced

in measurements

repeated immediately,

but not if

repeated

the

following day.

As _{a measure} of the

variability

of

V~,

its standard deviation

3V~

=

(((V~ (l/~) (T))~) )"~

about the smoothed _mean

(V~) (T)

has been

calculated, using

either _a third- _or fifth-order

polynomial approximation

for

(V~) (T).

The values of

3V~,

and of the average of

(V~)

between 120 K and 180 K, are

given

in table I. While the

14 JOURNAL DE PHYSIQUE I N° I

00

1=390pm (a)

___,_..."

"'"""..

... ...,_,:.

50 ")

20

10pm (b)

,_,_,_.,:...___..:""..,::...

lo "...

.,._

9pm (c)

20 .."

_::.,. _... ...._ "'"..." "'~

"..",,

_"

i7 10

8

~ lo

1.spm (d)

~

T ~ ^{.. ._:._} __. :..,

'""."',

"..""' """'

"..

)

^~

_24pm (e)

~

20

..., _:.., .":.._ _{._.;} .. _'

~~

2.spm (f)

lo' "', __..."..,

"..._...._...,, __;., 0

1=2pm,

N=1000

(g)

5°

/~(

' ~/~)

)

0

~

'

loo 120 140 160 160 200

Temperature (K)

Fig.

2. The threshold field _vs. temperature

dependence

obtained for _: (a)

large

; (b), (c), (d) small

specimens

of

o-TaS3

(e), (f~ two segments ^{of the} same

specimen,

which have different

lengths

;

(g) by

computer ^simulation

according equation

I for I

= 2 _~m, and

q(T)

=

q(0)

ii _{+ a.}

exp(-

800 KIT)]

where

q(0)

= 5 _x

l4'cm''

_and

a = 0.6, VT ^is measured in units of wqlevn~.

values of

(V~)

for the small

specimens

differ

widely,

the fluctuations relative _to the average,

specified by 3V~/(V~),

_are

roughly

similar.

Discussion.

In

o-TaS3

the wave vector q of the CDW varies with temperature : a rise from 100 to 200 K results in _a I fb increase in the

component

_{of q}

along

the main

conductivity

direction

II Ii.

The

equilibrium configuration

of the CDW with respect to the

pinning

therefore

depends

on T, and it is reasonable to suppose that the fluctuations in

VT (T)

reflect

changes

^of

configuration.

N° I MESOSCOPIC BEHAVIOUR OF ULTRA-SMALL TaS3 SPECIMENS lls

The threshold

voltage V~

_may be

expressed aiproximately

in the form

V~=E~f+V~~,

where the first term describes the

pinning (bulk

_or

surface)

of the CDW between the

terminals,

and the second is the

voltage

absorbed in

inducing phase-slip

at or near those terminals. The first

problem

^is to

identify

which _term is

mainly responsible

for the fluctuations in

V~(T).

If the fluctuations _were the result of _a

dependence

of V~~ ^{on q}

(which

_may be assumed _to vary

smoothly

with

T~,

one would expect them to be

regular

in

form,

with

amplitude independent

of the

length ^f

of the

specimen (though

a

tendency

to increase in thin

specimens

would _not be

surprising).

In

magnitude

the fluctuations

obviously

could not

exceed,

and

might

be

expected

to be much less

than,

the value of V~~ ^when

phase-slip proceeds independently

at each

terminal,

which in

o-TaS~

in the conditions relevant here is about 1- 3 mV

[12].

In fact the form of the fluctuations is _most

irregular and,

_as table I

shows,

their

amplitude

is correlated with

f (and

_more

strongly

with

(V~) ),

but not in any obvious way ^with s, and in

some cases

(e.g. Fig. 2b)

it exceeds the

expected

_V~~

by

an order of

magnitude.

Thus _we may

conclude that the main _source of the fluctuations of

V~

is the

pinning

of the CDW

expressed by

E~_

In strong

pinning

of the FLR type, with the

phase pinned separately by

each

impurity

and elastic energy of minor

importance,

the threshold field should be almost

independent

of the

equilibrium configuration

of the CDW. Such strong

pinning clearly

is inconsistent with the observed

variability

^of

E~,

and

also,

if the

pinning

is distributed

randomly,

with the

dependence

of

E~

on cross-sectional dimensions when _s ~10 ~Lm~

[4].

If

E~

in the

present specimens

is due to weak

pinning,

its

dependence

on s

implies

that the

phase

of the CDW is then coherent _over at least _{one transverse}

dimension,

whence the

corresponding

coherence

length L~

in the bulk

crystal

^{is estimated} _as 1-10 _~Lm. Estimates of the coherence in the

longitudinal

direction _can be obtained from the redistribution of

phase

which follows the addition _or removal of _a

wavelength

^{of CDW} in _a process of

phase-slip,

the scale of which is

expected

to be similar _to the FLR

length

Ljj

[13].

Direct

observation,

in

specimens

similar _to those studied

here,

has revealed redistribution _over about 10 _~m

[14].

The redistribution has also been observed

indirectly,

as sudden steps ^{in the resistance}

R(T)

of

o-TaS3 specimens

with

f

₂₅

~Lm, s

~10~~

_~Lm~

_{[4, 12].}

_{The 10 fb}

changes

in R

indicate that Ljj

»0.1f~2.5

_~Lm, consistent with the direct measurement.

Thus,

in the

smallest

specimens

studied

here,

a

weakly-pinned

^{CDW should be coherent in}

phase

over the

length

and _at least _one of the transverse

dimensions,

and therefore appear

effectively

zero- or

one-dimensional.

This conclusion _seems to conflict with the electron

microscope

observation

[15],

in the CDW of

o-TaS~,

of _a strandlike structure with characteristic

length ~0.3~Lm.

The

discrepancy

_may be associated with differences in

specimen

thickness

(L

ii

is reduced when the thickness is less than

L~),

or with the

rapid changes

observed in the strandlike pattern,

presumably

induced

by

the electron beam. A similar

problem

occurs with

Nbse~,

where

recent

X-ray

diffraction measurements

[7]

have

given L~

_~ l.9 _~Lm, Ljj ~ 2.5 _~Lm, whereas

dark-field electron

microscopy [16]

has revealed strandlike domains with

typical

dimensions

respectively

0.2 _~Lm and 2 _~Lm.

In the zero-dimensional limit the CDW behaves _as

though rigid,

so that the threshold field

according

to the FLR model is

given by

N N

eE~

_vn~

= max

£

_{wq cos}

(q _rj

₊ ~P

m wq

£ exp(iq _q (I)

W

j=1 j=1

l16 JOURNAL DE PHYSIQUE I N° I

where _w is the

strength

of the interaction with _an individual

impurity,

_e the electron

charge

and n~ the electron

concentration,

v the

specimen volume,

N the total number of

impurities, q

^their

positions (distributed randomly throughout

the

specimen),

and ~P the CDW

phase.

It is obvious from

(I)

that

E~ depends

_{on q.}

Taking

the root _mean square value of the _sum in

(I)

one obtains

(E()

^"~

=

(wqlevn~ )

N ^'~~

=

(wqle)(n;/n~)'~~ (n~

v

)-

^"2

(2)

where n; is the

impurity

concentration. One _{can see} from

(2)

that

E~

of the small

specimen depends

_on the volume _v rather than the cross-section _s.

Replacing E~ by V~/f

and

f by Rs/p,

where R and _{p are}

respectively

the resistance and

resistivity

of the

specimen,

one

obtains for zero-dimensional _case the

simple

relation

(V ()

^"~

=

(wqlen~) (n; R/p )"~ ₍₃₎

between R and

V~

due _to

pinning.

It is

easily

shown that the _mean threshold

voltage

(V~)

is

(fi)(V()"~

For the one-dimensional _case of

Lj hi, _L~j»tj

but

L~2~

_t~,

w~ere L~j

and

L~~

refer to directions in which the

specimen

thickness is

respectively

_tj and t~, ^{it is}

easily

shown that the FLR model

predicts V~ proportional

to

(f/t))"~

For

specimens having

similar

tj/t~,

so that

(tj)~

_{cc s,} one than has

v ~ ^l/3

(~)

T CC

Figure

3 shows the average

(V~),

observed between 120K and 180K in the small

specimens

of table

I,

_versus the values of R ^"~

(a)

^{and R "~}

(b)

at 300 K.

Except

for the

longest specimen,

^for which

f (24 ~m) probably

exceeded

Lj,

the data _are consistent with _a linear relation between

(V~)

and R

"(

_{and thus} with zero-dimensional weak

pinning. Extrapolation

to R

= 0

gives (V~~)

^small

^compared

^with

(V~),

in agreement ^with

previous

measurements on

o-TaS3. Nevertheless, though

the data do _not fit _a linear relation between

(V~)

and

R"~

_with

positive (V~~),

one-dimensional weak

pinning

with

L~~

~ t~ ^remains a remote

possibility,

_as

tj/t~

was not measured and may have varied between

specimens.

20

~

(a)

~

(b)

o o

o o

o o

A A

/

8

/

8

o o o o

0

0 50 loo 150 0 lo 20 30

R(300K)~~~ (Ohm~~~) R(300K)~~~ (Ohm~~~)

a) b)

Fig.

3.- Mean threshold

voltage averaged

_over 120<T<180K _{as a} function of the

specimens'

resistance _{at room} temperature R(300 K) (a), _versus R(300

K)W

(b), _versus R(300

K)'J3

N° I MESOSCOPIC BEHAVIOUR OF ULTRA-SMALL

TaS3

SPECIMENS II 7

The

hypothesis regarding

the

origin

of the fluctuations in

V~

is

supported by comparing

the

experimental

data

(e), (f~

ⁱⁿ

figure

2 with the

computer

^simulation

(g)

for the _same

specimen length ^f

_; this shows a

representative

behaviour of

V~,

calculated

using equation (I)

with q

(T) roughly

_as observed. The

temperature

^scale on which

V~ varies,

which is determined

by

the

change

in

qf

_and is _not

peculiar

to zero-dimensional weak

pinning,

is similar in the calculated _curve and the

experimental

data. The

similarity

is evidence that the

change

_{of q}

with T is indeed the _source of the observed fluctuations. Further evidence is

provided by

the

correlation,

_apparent in the data for other

specimens,

between

f

_{and the}

rapidity

of variation of

V~

with T.

The relative

amplitude

of the fluctuations

depends

_on how

closely

the system ^behaves as a

zero-dimensional sinusoidal CDW. In that

limit,

described

by equation (I),

the standard

deviation

3V~m ~/((V~ _{(V~) )~)}

_of

_V~

about its _mean

(V~)

is

easily

shown _to be

(4/gr -1)"~ (V~),

I-e-

0.52(V~).

As its fluctuations _are

proportional

to the _mean, the

general

form of

V~(T)

does not

depend

_on the total number of

impurities

^N

(in analogy

with the fluctuations of resistance in usual

mesoscopic systems).

Figure

4

shows,

_{as a} function of

specimen length f,

the relative fluctuation

3V~/ (VT)

found

experimentally.

The measured values, whose decrease with f accords with the

expected longitudinal

coherence

length

_L11 10 _~m,

provide

an estimate of 0.15 _± 0.05 for

3VT/ (VT)

when

f

= 0. This is

considerably

less than the value 0.52

predicted

for the zero-dimensional limit

(which applies

also to a

rigid

CDW

interacting

with _a distribution of

pinning strengths w).

Because of the statistical

inadequacy

of data from _a restricted

temperature

range, values of

3VT/ (VT) actually

measured in _a zero-dimensional _{case are}

likely

to be somewhat smaller than 0.52.

However, they

would be

significantly greater

than _was observed _: 100

independent

simulated _curves gave

3V~/ (VT)

between 0.4 and 0.6.

One concludes from this that the

experimental results,

_even when f _«

Ljj,

_are not

wholly

consistent with the

pinning

of _a

rigid

sinusoidal CDW

by

fixed

impurities. Clearly, departures

from the ideal can be of three types,

respectively involving flexibility

of the

CDW,

_non- sinusoidal

dependence

of the

pinning

force _on its

position,

and

pinning by

defects not fixed in the lattice. Each tends to reduce

3VT/(VT).

o-z

o

o.15

o

~~

~

o V

'

~ 4O

o

°

o,05

0

0 lo 20 30

Length

of

specimen (pm)

Fig.

4.

Dependence

of the relative _rms fluctuations of the threshold

voltage

&V~/

(V~)

_on

specimens

length.

II 8 JOURNAL DE PHYSIQUE I N°

The effects of

flexibility

_{appear on} two

length

scales. The

larger

scale is that of the coherence

lengths L,

which refer to an average

phase

~P of the

CDW,

taken _{over a} volume

containing

_many

impurities. Although

the evidence

suggests

^{that this}

macroscopic phase

_was

substantially

constant

throughout

the shortest

specimens,

^variation across one transverse

dimension

(I.e.

one-dimensional weak

pinning)

remains a remote

possibility. Obviously,

such

variation would lead to a reduced value of

3VT/(VT)

for weak

pinning, effectively by

averaging VT

_over several

independent

zero-dimensional domains. With about 10 such

domains, 3VT/ (VT)

would be reduced to the value exhibited

by

the shortest

specimen

here.

Phase variation _{on a} much smaller scale _{occurs, as} Abe

[9] pointed

_out, in the

vicinity

even

of

weakly-pinning impurities.

The

adjustment

of

phase

from its value at an

impurity

to the

macroscopic

_average ~P _{occurs over} distances of the order of the

Ginzburg-Landau length, giving

_a

large

local distortion _{on a} scale

usually

much less than L.

To take _some account of such distortion,

equation (I)

_can be re-written _as

N

eET

_{vn~ = max}

£ _F~ (5)

W j j

where F~ ^{= wq cos}

(qq

+ ~P ₊

3~P~)

^{is the}

pinning

force exerted

by impurity j,

~P is the

macroscopic

_average

phase,

and the local distortion is

specified by

_3~P~

= ~P~ ~P, where

~P~ is the

phase

at

j.

The

phase

shift 3~P~

depends

on ~P, and is available from

3~P~ =

aF~/wq (6)

where _a is _a dimensionless _measure of the local

flexibility

of the CDW. Because of

3~P~ the

pinning

force _no

longer

varies

sinusoidally

with _~P, but its average with respect to ~P remains _zero

provided

that

a ~

l,

which is the condition for weak

pinning.

The

departure

of F~ ^{from a sinusoidal}

dependence

_on ~P leads to lower values of

3VT/(VT).

This

happens, essentially,

because the correlation between values of

jjF~

for different ~Pis reduced, _so that the maximum in

equation (5) is,

in effect, taken from _a

larger sample

of random values.

However,

_even when _a

=

I

(representing

the limit of weak

pinning

within the conventional FLR

model),

numerical solution of

equation (5) gives

3VT/ (VT)

=

0.32,

still about twice the

experimental

value.

When _a is increased

beyond I, corresponding

to the onset of strong

pinning,

the

F~

acquire

_{non-zero average} values when ~P varies

monotonically,

and show

hysteresis

when the direction of

change

_reverses. As

VT

includes now a term

proportional

to the number of

impurities N,

whereas

3VT

remains

approximately proportional

to

N"( _{though slightly}

reduced, 3VT/(V~)

decreases _as

a rises. While in

principle

this

might

_account for reduced values of

3VT/ (VT),

those observed would

require

_a to be

improbably

close _to the critical value I. Thus if

n/n~

₌

10~~ _(as

_{the abundance of Nb in the Ta}

starting

material

suggests), N~10~

_{in the} smallest

specimens, requiring

a~1.25 to account for the observed

3VT/ (VT)

=

0.15. It is

unlikely

that the

flexibility

of the CDW would

fortuitously provide

a

value _so close _to

I,

over so

large

a range of temperature.

Further,

the small

specimens

in

figure

2 show

roughly

the _same

3VT/ (VT), despite having

volumes

(and

hence

N) ranging

over almost two orders of

magnitude.

It _seems

implausible, therefore,

that

strong pinning

of the conventional FLR

type plays

_a

significant

role in

determining (VT).

Similar

objections apply

to the reduction of

3VT/(VT) being

attributed to

pinning by

defects not fixed in the lattice. If

capable only

of elastic

displacement,

such defects have effects

equivalent

to those of the deformation of the CDW

just

discussed. If able to diffuse

through

the lattice, _{so as} to remain _near

positions

of minimum energy as q

varies,

the defects

N° I MESOSCOPIC BEHAVIOUR OF ULTRA-SMALL

TaS3

SPECIMENS l19

make _a

strong-pinning

contribution to

VT proportional

_to their total number.

This,

like other

strong-pinning

terms, would be

unlikely

to reduce

3VT/(VT) merely

to the values observed.

The

remaining possibility

is that the

departure

of

F~

^{from sinusoidal}

dependence

_on ~P is greater ^than can be accommodated within the FLR model without

introducing

strong

pinning.

The modification of the FLR model

proposed by

Tucker and collaborators

[3]

offers _a

possible

_way of

avoiding

the effects of strong

pinning,

even

though

_{a ~} l. In this

model,

the local

phase

~P~ is assumed to be

strongly pinned by

each

impurity j,

but processes of

phase- slip, whereby

~P~ is

changed suddenly by

± 2 _{gr, are} introduced _{to ensure} that

3~Pj

never

exceeds _gr. As

regards ET

such

pinning

is

weak,

in that the average of

Fj (~P ) (=

a 3

~P~

)

with respect to ~P is _zero. Numerical calculation with this

F~(~P) gives 3V~/(VT)

m

0.26.

Inadequate

statistics could have caused this to appear

experimentally

in the range 0.3 _to 0.2.

The lower value is not far from

agreement

^{with what} was observed.

There _are,

however,

other

possibilities.

An

arbitrarily

small

3VT/(V~)

can be made

consistent with zero-dimensional weak

pinning, by arranging

that the maximum in

expression (3)

^is taken from _a

sufficiently large

number of

independent

random

samples.

This may be achieved

by abandoning

the FLR model

completely,

and

assuming

_F~ ~P

)

to be _non-zero

only

within _narrow

well-separated

_ranges of _{~P, not}

necessarily arranged periodically.

The

observed

3VT/ (VT)

₌ 0.15 _can be accounted for in this way

by allowing F~(~P ⁾

to be _non-

zero over about 2 fb of _~P, with _mean value _zero. A

physical

mechanism which

might

lead to such F~ ^{is the}

pinning

of

widely-separated

dislocations in the CDW. These would have to be

an intrinsic feature of the CDW if the

resulting VT

were to be

reproducible

after nonlinear

conduction,

and show correlation _{over an}

appreciable

_range of temperature. ^{A variation of}

VT(T)

on a temperature ^{scale similar} to that

predicted by equation (I)

would result from

dislocations which remain fixed in the CDW _as q

changes,

and interact with _a

given impurity

over a distance of the order of _a CDW

wavelength. Apart

from the value of

3VT/ (VT),

however,

the

experiments provide

_no evidence that such dislocations did contribute _to

VT

in the present

specimens.

Concluding

comments.

The

phenomenon reported here,

of

specimen-dependent

variations of threshold field with temperature, ^{is in} a sense a new kind of

mesoscopic

^behaviour

specific

to

quasi

_one-

dimensional conductors. Each

VT(T)

_curve,

being dependent

_on the

particular

distribution of

pinning (and perhaps

also _on the distribution of CDW

dislocations),

is

peculiar

to the

specimen

observed. The

temperature-dependent

wave vector of the CDW

plays

a role similar to that of _a

magnetic

field in usual

mesoscopic systems.

Although

_some details remain

unresolved,

it is clear that the

pinning

is

essentially

of the weak

type.

^The

magnitude

of the

variations, although conceivably explicable

in terms of

imperfect phase-coherence

_over the

specimen cross-section,

_seems more

readily

accounted for if the effective

pinning potential

is _not sinusoidal. This suggests ^{that local} distortions of the

CDW may be limited

by phase-slip

at the

pinning sites,

_or

perhaps

that the

pinning

is of well-

separated

dislocations in the CDW structure.

Acknowledgments.

We would like _to thank V. Frolov for

providing

the batches of

high-quality o-TaS~,

and F. Ya. Nad' for discussions. One of _us

(SVZZ)

is very

grateful

for the kind

hospitality

of the

University

of Bristol.

120 JOURNAL DE

PHYSIQUE

I N°

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