On the nature of threshold electric field in quasi-one-dimensional commensurate charge-density-waves

~

Solid State Communications, Vol.55,No.8, pp.691-694, Printed in Great Britain.

ON THE

0038-1098/85 $3.00 + .00 Pergamon Press Ltd.

1985.

NATURE OF THRESHOLD ELECTRIC FIELD

IN

QUASI-ONE-DIMENSIONAL

COMMENSURATE CHARGE-DENSITY-WAVES I.V. Krive, A.S. Rozhavsky Institute for Low Temperature Physics and Engineering, UkrSSR Academy of Sciences, 47, Lenin Ave., Kharkov, 310164, USSR. (Received May 16 1985 by E.A. Kaner)

The theory

of

nonlinear conductivity of commensurate charge-

density-waves based on the soliton-tunneling mechanism is proposed. The well known empirical expression ~ ( ~ ) = ~ N

(~'~')~@

"~-~,/(~-~O(~'~v)

is

+

obtained within the frames

of self-consistent investigation taking into account the coherant Coulomb interaction of charged solitons. The threshold field

~T

is the deconfinement field of bound soliton-

antisoliton pairs. The universality relation ~ T a ( O ) = (~

~O)

- static dielectric susceptibility)

CO~S~

is also obtained.

The phenomenological treatment of localized free charges explains the dependence

~ T ~- a ~

(~L - impurity

It is well known that the dc-conductivity of such quasi-one-dimensional transition-metal

triohalcogenides

as

ctrons and the second nonlinear term is connected with charge-density-wave (CDW) dynamics.In spite of large number of theoretical papers devoted to explanation of nonlinear CDW-conductivity (I) (see for example the review I and also 5,6) no successive theory explaining

NbSe3,TaS 3 depends on the electric fie l d ~ .The empirical expression for conductivity along metal chains is 1,2

field dependence (I) and the characteristics o f ~ was proposed until now.

Here ~ # - Ohmic conductivity, ~ r threshold field. The threshold f i e l d ~ depends on the impurity concentration ~ as~n~ 3 and t e m p e r a t u r e ~ . I ' 2

In this paper w e ~ n e

theory ba-

sed on the CDW-solitons tunneling ideas 6 which explains the enumerated ex-

Also the universality relation~%{0)~emg takes place 4 (E(O)- is the static dielectric susceptibility). The trichalcogenides considered are Pelerls insulators (PI) and the terms in (I) arise due to different conductivity mechanisms,First linear contribution~is formed by normal ele-

concentration).

perlmental facts I-4 in a self-consistent way. The PI is characterized by order parameter ~ e x ? O ~ ) .The modulus ~ defines an energy gap in the spectrum of dielectrilized electrons and the phase governs the dynamics of CDW.The

691

692

QUASI-0NE-DIMENSIONAL COMMENSURATE CHARGE-DENSITY-WAVES

Vol. 55, No. "8

energy scale of CDW-excltatlons is much less than ~ I03K and in weak fi-

mensurability pinning exceeds the im-

elds and at low temperatures ~ < < / k

purity one is:

the

electromagnetic response of PI is due to phase dynamics. The normal electrons exist on the nondielectrilized

lagrangian in the case when the com-

L - " " '°

(2)

parts of the Fermi

surface as in NbSe 3 or they can appear due to impurities as in commensurate TaS 3 2'7.1t seems reasonable to assume

Here ~ - the electron-phonon coupling

that these normal carriers move along the chains in the conditions of localizatinn effect due to different struc-

tm

ture imperfections.For example in TaS 3 at temperatures ~-< 20K the Mott temperature dependence o f ~ e ~ ( - < ~ ) ~ ) 2 and linear temperature dependence of high-frequency conductlvlt~(@~)~T@) ~ 7 is observed.Both these facts indicate on the strong influence of localization.The localized electrons can form gi8 ant dielectric s u s c e p t i b i l i t y ~ ( o ) ~ which reduces when decreasing tempera-

oonst~-t (~ "2~r ¢~pC-tl~) )~ OjQ-the bare frequency of phonons with m m e n ~2kfi~

oo~ensurability

index

( f o r TaS3 M = 4 ) $ , - ( ~ ' ~ M - 2 ~ I$ ~O= ~ . • N / M ( N - a n integer number N ~ M) - an a r b i t r a r y phase corresponding to minimum of potential energy (2) in the abscence of Coulomb effects;vf,kf,~f-Fermi parameters.Except the first term and the notion about the transverse coherence the lagrangian (2) is the same as studied earlier 6 .The introduced alterations however drastically change the predictions of model (2) comparatively with results 6 . First term in (2) is the field

ture. In real conditions ~ > ~ I T

(T -

the electron-impurity scattering time) Peierls and free-electrons subsystems can be treated as noninteracting and their contributions in c~nductivity are additive.The quasi-one-dimensional nature of PI results not only in the existence of free carriers but also in the interchain coherence of order parameter. Such coherence was experimentally observed when studying high-frequency rest. 9 " ponse of CDW in d c - f i e l d ~ > ~ v We shall study the commensurate CDW in a coherent cluster with size ~A>>~A ( ~ A - the interchain distance). The value ~ A will be treated as an external parameter of theory (in perfect samples it can coincide with a transverse length of a sample).In such coherent volume containing a large number of chains N l ~ n f ( n f - t h e two-dimensinai density of chains) the theory is effectively one-dimensional and the CDW

energy in the medium with dielectric susceptlblllty~(O)+~: where ~ describes the polarization of occupied PeiI0, II. erls valence band~A--~,~VF~,/3~ Lagranglan (2) governs the dynamics of the "slow" phase subsystem in the medium which dielectric properties are formed by the "fast" subsystems - modulus and free electrons (such separation is allowed by PI adiabatic parameters

~l~/~*~,

~I~, ~

).

As far as one-dimensional density of CDW charge fluctuation is ~ = ~ 9 '

it

i s n e c e s s a r y t o t a k e i n t o account

the Coulomb interaction of coherent fluctuations.It results in the selfconsistent determination of electric field ~ in (2).0ne must solve Poisson equation ~ , ~ = 0 where induction is obtained from lagrangian (2) using I2 expre s sion

':j5= 4~n~ a~-ES~

(3)

QUASI-ONE-DIMENSIONAL COMMENSUEATE-CHARGE-DENSITY-WAVES

Vol. 55. No. 8

The independence of K and ~ on tra.sverse coordinates is the consequence of cluster model and is verified by

.

small ratio a.co)/a (o) ~ ~

In the uniform external field parallel to chains the solution of electroneutrality equation is obvious

~ c,<'). 4:~ q.-'k,t-,.to) +

~". ~.,.,+/(,~<.o~+a)

(4>

According %o (4) ~(~) is not an independent variable and we pass to potential ~[q)expressed in terms of ~ ' ~ e only using %he Legendre transformation

I-i: ~@~I~©-

~'

(~)

and relation (4). The lagrangian corresponding to hamiltonian H(~) ( q'~ ~ " ~ e ) is the lagrangian of the Massive Schwinger Model (MSM) of quantum eleotrodynamics expressed in the boson form 13 '

'

'

"

~v~ x ~ l

o0~:5,k~_~,,_

"

~--~ '

_

,

693

pendence of vacuum state on the external parameter ~ .In the region O<~.( I

.

i

cuum is ~ =O.V~aen~ v > ~ >~V the energetically favourable becomes ~=-2~/M.,_ and at low temperaturesT4~EL (El~,/il~ll~__ - the soliton energy) the transition IO> -~ I-2~/~> occures via tunnel creation of s~-pairs with charge density 2 N ~ I ~ .When ~ > ~ > 2~X the additional decay channel l.q~M>-~ t-~,/~

opens = d

so on.So the dis-

crete-- ""number of threshold fields ~ ) = ~ / ~ V appears but the contributions of all the forthcoming decay channels with ~ ) ~ are exponentially small comparatively to the decay rate IO>.l-~/M>. The ~DW-conductivity is proportional to the decay rate Is~ _ ( ~ tun(.~ nf Is~ ,/---~- ). Since ss - pairs create at the distance~Esl~(~-~?)>>~, they can be treated as free solitons and antisolitone and Is~ is calculated on the instanton trajectory ~ - ~ s + + ~_ I4 .The result is

z •

~+~ The principal difference of lagrangi~n

(6) from the one studied earlier 6 is the quadratic term in potential energy.This term leads in particular %o plasma activation of the small oscillations frequency (see also S.N.Artemenko,A.F.Volkov 5 ). If ~ / ~ the nonlinear charged excitations of CDW are the sineGordon solitons (s) and antisolitons i

&, - ~ /oo<> The Coulomb interaction (quadratic

term in (6)) confines solitons and antisolitons in homogeneous neutral condensate of CDW.However the striking feature of the MS~ is the threshold de-

i.e. we obtained the field dependence of nonlinear term in conductivity (X). In the papers 6 where the tunneling mechanism was first proposed the elecctric field [ in (2) was treated as external one and as result the decay rate was thresholdle s s. %.= The threshold field ~ x is inversely proportional to ~ + ~ ( 0 ) .The measured dielectric susceptibility is~(O)=~m+~(O)+~ where ~%p is the phase contribution.In weak fields ~ v when s~ - pairs are in confinement they form neutral bound states breathers I5 with size ~ . and ~ can be estimated using simple expression for dipole oscillator with frequen-

694

QUASI-ONE-DIMENSIONAL COMMENSURATE-CHARGE-DENSITY-WAVES

cyLOoand

number of l e v e l s

We obtaina TaS~

n~ E$/~ o •

(%f ) .=.For

chain coherence the electric field creates ss - walls with area ~ I analo-

and

~ ( O ) ~ I06~8.0n t ~ e o t h e r

hand

gous to two opposite charged capacitor plates.The deconfinement field ~ v is

I0 " and ~ IO ~ and it is natural to consider ~ ( O ) ~ ( O ) ~ 4 ~ " .A Then and e plain the characteristics of the threshold field~

Vol. 55, No.

one-dimensional Coulomb field inside the capacitor. We wish to acknowledge S.N.Arte-

.

menko,S.A.Brazovskii,I.O.Kulik,A.M.Ko-

The physical meaning of ~ v quite transparent.Beaause

is

sevich, F.Ya.Nad and A.F.¥olkov for numerous helpful discussions.

of the interR E FE}~E

N C E S

I. G.Gruner , Comments Solid State Phys., IO , I83 (I983) 2. S.K.Zhillnskii et al., Zh.Eksp.Teor.Fiz.,

8~,

362 (I983) (Soy.

Phys. JETP) 3. J.W.Brill , N.P.Ong et al., Phys.Rev. B2~ , I518 (198I) 4. W.W.Fuller , G.Gruner et al., Phys.Rev.B23 , 6529 (I981) 5. S.N.Artemenko 8I,

, A.F.Volkov

, Zh.Eksp.Teor.Fiz.,

80 ,2018 (I981~;

I872 (I98I~ (Sov.Phys.JETP);

P.A.Lee , T.M.Rice,

Phys.Rev. ~ ,

6. K.Maki , Phys.Rev.Lett.,

~,

J.Bardeen , Phys.Rev.Lett.,

3970 (X979)

46 (I977); 45,

I978 (I980);

I.V.Krive , A.S.Rozhavsky , Fiz.Nizk.Temp.,

6 ,1272 (~980) (Sov.

Phys. Low Temp. ) 7. S.K.Zhilinskli

, F.Ya.Nad , M . E . I t k i s . ,

367 (1984); M.E.Itkis , F.Ya.Nad.,

Phys.Stat.Solidl

, (a)81,

Proceedings 23 All Union Con-

ference on Low Temp.Phys.,Part 2 , I30 , Tallin= , I984. 8. I.M.Lifshitz

, S.A.Gredeskul

, L.A.Pastur , Introduction to the

theory of disordered media , Moscow , Nauka , I981. 9. J.Bardeen , E.Ben Jacob et al., Phys.Rev.Lett., R.M.Pleming , Solid State Comm., ~ I0. P.A.Lee , T.M.Rice , P.W.Anderson.,

49 ,493 (i982);

,I67 (I982) Solid State Comm., ~

,714

(1974) II. I.V.Krive , A.S.Rozhavsky , Sov.Phys.JETP., I2. E.M.Lifshitz

, L.P.Pitaevskii

54 ,95~ (I98I)

, Relativistic

quantum theory ,

Part 2.,Moscow , Nauka , I97I I3. S.Coleman , Ann.Phys.,(N.Y.),

!QI ,239 (I976)

Y4. N.K.Pak , Phys. Rev. D I6 ,309o(I977)

(see also I.V.Krive,A.S.Ro-

zhavsky in ref. 6) I5. R.Rajaraman , Phye.Repts., 2IC ,227 (I975)