On the phase oscillation spectrum in CDW conductors

Synthetic Metals, 29 (1989) F407 F413

F407

ON THE PHASE OSCILLATION SPECTRUN IN CDN CONDUCTORS

S.N.ARTENENKO and A.F.VOLKOV I n s t i t u t e of

Radioengineering

and

Electronics

of

the

Academy

of Sciences of the U.S.S.R., Mosco, (U.S.S.R)

ABSTRACT By means of

a

kinetic

approach

based on

l o n g - . a v e - l e n g t h phason spectrum i s c a l c u l a t e d . I t

the

Keldysh

technique

the

i s shown that screening of

the CDW charge by q u a s i p a r t i c l e s renders the phason spectrum soundlike with v e l o c i t y depending on q u a s i p a r t i c l e density.

INTRODUCTION The c o l l e c t i v e o s c i l l a t i o n s spectrum in CD# conductors was calculated f o r the f i r s t

time in the pioneering work of Lee, Rice, and Anderson [ I ] .

I t uas

shown there that the P e i e r l s t r a n s i t i o n leads to a modification of phonon spectrum near t h e momentum Q (O i s t h e CDN wave v e c t o r ) two new modes of o s c i l l a t i o n s .

In t h e l o n g - w a v e - l e n g t h

corresponds to amplitude oscillations, the phase of t h e o r d e r p a r a m e t e r . determined,

to a great extent,

the e l e c t r i c f i e l d .

and t o appearance of limit,

one o f them

and the o t h e r one t o o s c i l l a t i o n s

Kinetic

properties

of

o f t h e CDN c o n d u c t o r s a re

by the phase mode (phasons)

interacting

with

For instance, the phase mode is c l o s e l y connected with

F r o e l i c h c o n d u c t i v i t y because the phase o s c i l l a t i o n s at wave vectors k ~ O, and frequencies ~ ~ 0 correspond to uniform s l i d i n g of the CDM. I t was shown in [ I ]

that at zero temperature the e f f e c t s of d i e l e c t r i c screening lead to a

finite

phason frequency. But i f

there i s a large amount of charge c a r r i e r s

(the case of high enough temperatures T ~ A or of incomplete d i e l e c t r i s a t i o n at Peierls

transition)

due t o s c r e e n i n g e f f e c t s [2]

and [ 3 ]

above t h e energy gap, th e phason spectrum i s m o d i f i e d caused by q u a s i p a r t i c l e s .

Such a case was a n a lyse d in

where t h e s e u n d l i k e phason spectrum was o b t a i n e d a t T ~ A : ~ =

kv . But t h e e x p r e s s i o n s f o r

0379-6779/89/$3.50

v

calculated

in [ 2 ]

and [ 3 ] appeared t o be

© Elsevier Sequoia/Printedin The Netherlands

F408 d i f f e r e n t : according to [2] v according to [3] v

~ V~Q/T,

~ vAIt~'7~, that i s , v i.e.,

~ 0 at A ~ O, and

the phason v e l o c i t y is f i n i t e at A = 0

(here v is the Fermi v e l o c i t y , ~ is the dimensionless constant of e l e c t r o n phonon coupling, ~Q i s the bare phonon frequency). Two approaches were used f o r c a l c u l a t i o n of c o l l e c t i v e o s c i l l a t i o n s spectrum.

In the T i r s t o÷ them [ I , 2 , 4 ] Feynean diagrams f o r phonon 6reen°s

functions were summed up; in the second approach, equations f o r the current and f o r the phase ~, derived from microscopic equations, were used. Here we present the r e s u l t s f o r the phason spectrum obtained by means of k i n e t i c approach f o r the case o~ semiconducting CDN conductors l i k e TaS3, blue bronze etc. Ne r e s t r i c t ourselves to the l i m i t of low temperatures T ~ & when the concentration of q u a s i p a r t i c l e s is small. Our r e s u l t s d i f f e r ,

to some e x t e n t ,

from the r e s u l t s obtained in a recent paper [4).

RESULTS Linear response of a quasi-ID conductor f o r the case of small frequencies and gradients (~,kv ~ A) can be calculated with the help of k i n e t i c equations. Neglecting the transverse dispersion of e l e c t r o n i c spectrua ue represent these equations in the form [ 5 , 8 ]

[ ( S / ~ ) 2 ~ / ~ t - V ( G ) ] n z + v a n t / S x = (evE - (v /2)~o/'ax +~b~o)Sntr'd~

(1)

a n l / O t + V~nz/~X + V.L~nl/alr.L +[l~f (E2+ A 2) / (E~) + ~b~/E ) (nl-) = = (V.I.EL) B n l / ~

(2)

+ I e,

where the energy 2 is supposed to be l a r g e r than &, ~2= ~2_ ~ 2 , v = A2vo/~)/~,

) = 2(Ev b +

vo= ~f+ ~ b / 2 , ~ f and ~b are the forward and backward s c a t t e r i n g

r a t e s , r e s p e c t i v e l y ; E is the e l e c t r i c f i e l d , v transverse to the chain d i r e c t i o n ( i . e . ,

i s the e l e c t r o n v e l o c i t y

to the x - a x i s ) , < > means the

averaging over the transverse momenta p . Current d e n s i t i e s along ( j | ) perpendicular ( J )

and

to the x - a x i s and the charge density p are expressed

through n I and nz according to the r e l a t i o n s

j|

J.L = (=r~N/l)((~'b* p

+ ~o) + (EA/4w)~EI/OL

(3)

t J f ) / 2 < v 2 > ) S d e < n l v ~ le I/~>

(4)

= (~NN/l)(2SdElelnz/~

= (O'llN/Vl)(2J'd£1£tnt/~"

- v~0/Bx)

- (F..A/4m')t]EII / ~ x ,

(5)

F409

where ~ I , ~ N

the longitudinal

are

and t r a n s v e r s e

s t a t e , 1 = v / v b is the m e a n - T r e e - p a t h the d i e l e c t r i c

conductivities

in t h e normal

lengh, SA describes a renormalization oT

constant due to the appearance of the Peierls gap. In the most

i n t e r e s t i n g case of low temperatures s A = 4 ~ i N ~ b / 3 ~ 2 ~ 2 / ~ 2 . P The selfconsistency equation f o r the phase @ has a ~orm

2 2 2 = ( v E g / A 2) (2 - S d s ( t / ~ ) a n l / ~ ) (82~o/8t 2 = sj_Vj_q~)/X(og

Here s

2

= D2 ~ ? / ~ q :?

.

In l i n e a r

+2voSdS(t2/~4)n

(6)

z

a p p r o x i m a t i o n the CDM a m p l i t u d e i s not

disturbed and w i l l be considered constant. In equilibrium nz= 0 and nl= tanh(S/2T). The solution f o r deviations of the d i s t r i b u t i o n functions from the equilibrium ones and, correspondingly, the expression f o r the current density x i l l

have the d i f f e r e n t form depending on

the r e l a t i o n between the frequency ~, and i n e l a s t i c and e l a s t i c scattering -I -I rates Te and v o. F i r s t , we consider the case ~ ~ Te '~o" In t h i s case the d i s t r i b u t i o n function keeps i t s equilibrium form but the energy S is shifted by the value of the chemical p o t e n t i a l

n 1 = tanh((S

- p)/2T),

(7)

where ~ depends, g e n e r a l l y that

I e in E q . ( 2 )

Eqs.(1)

for

speaking,

on t i m e and c o o r d i n a t e s .

such a form of n I i s equal t o z e r o .

It

can be shown

Substituting

(7) in

and (2), one can easily obtain nz and a correction to nI determining

the transverse current density. Doing so, we get

jn= ~#B(-i

J.= ~ ( - ~

+ ~)/~x

+ ~lN(~/el)

181

* V.p)

p = -(2O'lN/Vl)(Nn/J

(g) + v~O'/2) -

where • i s the e l e c t r i c a l + ~f)/vb]exp(-A/T),

(SA/4w)a2t/ax2

potential,

~! = ~|NNnl~'~/~o

meaning. For i n s t a n c e , current,

- (SA/4~l~2t/~x~t

the first

the last

(10)

Nn=lF2~A/Texp(-A/T) . Eqs.(8-io)

term in E q . ( 8 )

~1,

~#= 2 ~ j . N [ ( V b +

have t h e e v i d e n t p h y s i c a l

d e s c r i b e s the d i s p l a c e m e n t

and second terms are the ohmic and d i f f u s i o n

currents,

n e x t term i s t h e CDM c u r r e n t . The p o t e n t i a l

~ must obey t h e c o n t i n u i t y

E x c l u d i n g ~ from t h i s .

= i ~k2/(;k2,

For q u a s i p a r t i c l e

equation,

i~)

currents

equation 8p/~t

+ divJ

we have f o r F o u r i e r components of

one has

= O.

the

F410

• . ~ l.~ / ( D^k 2 + i ~ ) , JqpH = ~NlkN 2^ where Dk

=

Jqp

= or ik ~ i ~ / ( D^E2 + ~ ) ,

(11)

2 D k , Da, =~D, v]/2Nn~DN. Dikl÷

To obtain the necessary equation for free o s c i l l a t i o n s of the phase ~, we exclude the e l e c t r i c a l potential • from the selfconsistency equation (7) using the Poisson equation. Supposing again kv ~ A , we gel iki

£&k|

(Sw~rgNNn/Vl)Dk2/(Dk + ~ ) ] - i ¢ ~ '

= 0,

~ = (12)

where E

i s the t r a n s v e r s e d i e l e c t r i c

constant.

An analogous c o n t r i b u t i o n

from

the l a t t i c e to E i is neglected because i t is small in comparison with E& ~ 1, the c o e f f i c i e n t ~ ~ ~bexp(-A/T) describes CDM damping due to quasiparticles. For s i m p l i c i t y , we w i l l drop ~ f u r t h e r . EquaLing the expression in braces in Eq.(12) to zero, we obtain the dispersion r e l a t i o n for the phase 2_

s2k 2_ °~okll/[kll+ 2 2 2 E.Lk 2 / £ A + (6NnA2 /v2)Dk2/(1)k ^ ^ 2 + i~)] .L .k

where the notation

2=o ( 3 / 2 ) ~

(13)

= 0,

is introduced.

We investigate now the phase o s c i l l a t i o n s , propagating along the x - a x i s , putting k

=0. In this case, Eq.(13) is reduced to

(14)

co2- O~2o/[I + Oal/(1)k2+ io))] = 0 2 where ~1 = 3~IA l~jNVb ' and Ne use here (and below) the notations k2 and 2 instead of k I and OI. In Eq.(14) there is the denoeinantor with a c h a r a c t e r i s t i c diffusion term which is contained also in corresponding equations of the paper [4]. From Eq.(14) i t follows that when ~o ~ 1

the

damping is s l a l l and ~ = ~ low temperatures.

(see F i g . l ) . This case is realized at s u f f i c i e n t l y o F i n i t e value of phason frequency at k ~ 0 is caused by the

displacement current in Eq.(8). In opposite l i m i t ~o ~ I '

the phason spectrum

is influenced e s s e n t i a l l y by screening action of q u a s i p a r t i c l e s , and we can omit 1 in the denominator in Eq.(14) noN. Thus, ue obtain 2

~=O i.e.,

~2(Dk2+ i ~ ) / ~ 1 , the d i s p e r s i o n r e l a t i o n

(15) has t h e s o u n d l i k e c h a r a c t e r .

phasons i s v = ~oV/2A'P~-_,__ and t h e damping c o t f f i c i e n t

The v e l o c i t y

i s equal t°~z-/C°l ' u

of that

F411

(D

(=)

½ (D

o/

(o

o

"

kv/Jl

is it

increases exponentially

At s m a l l ,

k

2o (Dk ^2 + i~)/~l

i s worth t o emphasize t h a t

oT phason v e l o c i t y

articles

is

, we have i n s t e a d oT Eq.(15) (16) t h e s c r e e n i n g by q u a s i p a r t i c l e s

t o an i n c r e a s e oT damping (see [ 4 ] )

renorealizes

The spectrum of these o s c i l l a t i o n s

(lower branch).

but T i n i t e

2 = (D|k : / Dkl) It

kv/A

a t T ~ O. The damping i s weak p r o v i d e d t h e

Trequency i s not t o o low: ~ > o ~ / ~ ] . sh~xn in F i g . 2

n

too.

The l a s t

the rigidity

but,

l e a d s not o n l y

a c c o r d i n g to E q . ( 1 5 ) ,

t o an i n c r e a s e

c i r c u e s t a n c e means t h a t t h e s c r e e n i n g

oT the CDM in an e q u a t i o n Tot ~ used in eany

(see, Tor e x a e p l e , [ 9 , 1 0 ] ) .

e

Consider now the case oT high frequencies: ~ ~ T - I . drop the c o l l i s i o n term I

e

In t h i s case, xe can

in Eq.(1,2) and Tind the s o l u t i o n of these

equations in e x p l i c i t Tore without i n t r o d u c t i o n oT the p o t e n t i a l ~. For

simplicity

we

restrict

t o t h e case kj.-- O. An e x p r e s s i o n Tor the

ourselves

d i s t r i b u t i o n Tunction n d e t e r o i n i n g the q u a s i p a r t i c l e c u r r e n t along x has the z the Tore

n = - ~ ) k v i / [ k 2 v 2- i ( ~ ( e )

-

212/~2]

(17)

z

At = ~ v o , one can n e g l e c t t h e l a s t

term in the denominator in E q . ( 1 7 ) .

t h e h e l p of E g . ( 1 7 )

the current

me can c a l c u l a t e

of phase o s c i l l a t i o n s .

T h i s can be done e a s i l y ,

t h e denominator in E q . ( 1 7 ) largest

find

neglecting all

e x c e p t t h e term k2v 2, because t h i s

xhen t h e damping i s small

expression

and, t h e n ,

(Dk 2 ~ ~ ) .

Finally,

Mith

the spectrue

the terms in term i s t h e

xe come again t o t h e

(15).

IT ~ ~ ~, one can n e g l e c t t h e second term in t h e denominator in E q . ( 1 7 ) . Then, f o r q u a s i p a r t i c l e

current

we o b t a i n t h e f o l l o x i n g

expression

F412

jqp= ( ~ N / 1 ) v ~ k ~ A ,

A = 4Nn/(~T:t)Idx

x2e-X/[x 2 _(2

/~)(A/2T)]

+

o

+ [i~2~(~)/(Tkv~3/2llexp[_lA/T)lkv/~)l,

~ = k2v 2_ 2

(18)

The second term in Eq.(18) describes the Landau damping. When k = O, we can express i

through ~ from the equation j = 0 and use the s e l f c o n s i s t e n c y

equation. Ne f i n d f o r phason spectrum

~2-~2/(1

+ 3A2A)

= 0

(19)

O

o

Consider a g a i n two l i m i t i n g

cases. At v e r y low t e m p e r a t u r e s ,hen ~

= (48/)r~)NnAT , we have ~ = ~o ( F i g . l ) . that ~ ~2,

then E q . ( 1 9 )

Landau damping i s absent)

I f the t e m p e r a t u r e i s n o t v e r y low, so

has two s o l u t i o n s .

At small k (kv ~ ~ ,

have the s o u n d l i k e c h a r a c t e r ,

by a n u m e r i c a l

factor

only

i. t .

the

t h e r e i s a high f r e q u e n c y mode: 2 = ~ 22- 2 4 1 T / A ) l k v ) 2 .

~ e s i d e s , t h e r e i s a low f r e q u e n c y mode 2 2 k2v2/(k2v2+ ~A/T), ~ = ~0 which i s w e a k l y damped at a l l k. In t h e l o n g - w a v e - l e n g t h oscillations

~=

their

(20) limit,

velocity

mhen t h e

differs

from E q . ( 1 5 )

(see F i g . 2 ) .

DISCUSSION Thus, i t

turns out that the phason spectrum changes e s s e n t i a l l y with

temperature. At T = 0 and k ~ 0 the frequency of phase o s c i l l a t i o n s is f i n i t e ( F i g . l ) due to the displacement current (the l a s t term in Eq.(12)). Nhen the temperature increases, the q u a s i p a r t i c l e s screen out space dependent e l e c t r i c field.

This leads to the strong modification of the phason spectrum which

becomes l i n e a r at small k. The phase v e l o c i t y of o s c i l l a t i o n s increases on cooling e x p o n e n t i a l l y . Note that s i t u a t i o n here d i f f e r s from the case of superconductors, mhere weakly damped phase o s c i l l a t i o n s e x i s t at ~ ~

near T only [ 6 , 7 ] . c We have obtained the large phason damping assuming that the wave vectors k

are small but f i n i t e .

Damping c o e f f i c i e n t y f o r s p a t i a l l y uniform phase

o s c i l l a t i o n s , which determines the v e l o c i t y of the CDW at high E or ~, is e x p o n e n t i a l l y small at T < A. In order to obtain the c o n t r i b u t i o n to ¥ ,

d e c r e a s i n g on c o o l i n g have t o t a k e i n t o due t o t h e i r

not so r a p i d l y

account,

anharmonic i n t e r a c t i o n

These p r o c e s s e s

(as i t

i s observed e x p e r i m e n t a l l y ) ,

f o r example, the s c a t t e r i n g

caused by t h e e l e c t r o n -

phonon c o u p l i n g .

can be a n a l y z e d by means of diagram t e c h n i q u e as i t

in t h e paper [ 1 1 ] .

The c o n t r i b u t i o n

we

of phasons by phonons was done

of t h e s e processes t o y depends on

t e m p e r a t u r e w e a k l y (power law dependence) i f

the t e m p e r a t u r e i s h i g h e r than

F413

the e n e r g i e s of s h o r t - w a v e - l e n g h phonons because in t h i s processes ensure r e l a x a t i o n of t o t a l

case

Umklapp-

momentum of electron-phonon system o n l y .

The s i t u a t i o n here i s the same as in a normal metal with closed Fermi s u r f a c e . When T ~ T = h o (~ i s the phonon frequency near the 3 r i l l o u i n zone o m m boundary) then the phonon c o n t r i b u t i o n to y i s small: Yph ~ e x p ( - T / T o) because the r o l e of Umklapp-processes becomes n e g l i g i b l e . I t means t h a t at T < T the o CDW v e l o c i t y and the C)W c o n t r i b u t i o n to the c o n d u c t i v i t y above the t h r e s h o l d field

should be very l a r g e . Presumably, a sharp increase of c u r r e n t observed

in blue bronze [12] can be e x p l a i n e d by t h i s circumstances.

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2

Y. K u r i h a r a , J. Phys. Sac. J a p . , 49 (1980) 852.

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4

K.Y.M. Nong, S. Takada, Phys. Rev., B36 (1987) 5476.

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S.N. Artesenko, A.F. Volkov, in "Charge-Density-Naves in Condensed B a t t e r "

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S.N. Arteaenko, A.F. Volkov, Usp. F i z .

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6. Schon, in " N o n e q u i l i b r i u m S u p e r c o n d u c t i v i t y " ed. by A . I .

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"North-Holland'.

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S.N. Artesenko, A.F. Uolkov, A.N. K r u g l o v , Sov. Phys. JETP 64 (1986) 906.

9.

L.

Sneddon, Phys. Rev., 329 (1984) 719.

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