PEIERLS-FRÖHLICH CORRELATION IN A QUASI ONE DIMENSEONAL CONDUCTOR

(1)

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PEIERLS-FRÖHLICH CORRELATION IN A QUASI

ONE DIMENSEONAL CONDUCTOR

T. Tsuzuki

To cite this version:

T. Tsuzuki.

PEIERLS-FRÖHLICH CORRELATION IN A QUASI ONE DIMENSEONAL

(2)

JOURNAL D E PHYSIQUE _{Colloque C6, suppl6menf au}_no_{8, Tome 39, aolit 1978, page C6-701}

PEIERLS-FROHLICH

CORRELATION IN A QUASI ONE DIMENS

l ONAL CONDUCTOR

T. Tsuzuki

Department of Physics, FacuZty o f Science, Tohoku University, 980 Sendai, Japan

RQsumi5.- Nous prgsentons une nouvelle formulation de la corrdlation Peierls-Frijhlich. Nous suppo- sons que le couplage inter-charnes est seulement du aux vibrations du rgseau. La fonction de correlation de la phase intra-charnes et la tempgrature oii la CDW s'ordonne de fagon tridimensionnelle sont obtenues. Nous clarifions la corrdlation entre la fluctuation quantique et le couplage intra- chafnes 6lectron-rgseau.

Abstract.- A new formulation of the Peierls-Frijhlich correlation is presented. Interchain coupling is assumed to be solely due to the lattice vibration. Intrachain phase correlation function and three dimensional CDW-ordering temperature are obtained. The interrelation is clarified between the quantum fluctuation and the intrachain electron-lattice coupling.

We want to present a new formulation of the Peierls-Frijhlich (PF) correlation in a quasi one dimensional electron-lattice system. Due to the one dimensional motion of electrons the thermodynamic fluctuation is so large that the three dimensional charge-density-wave (CDW) ordering temperature Tc is strongly reduced from the mean field transition temperature TcO. It seems to be difficult to describe such strongly fluctuating system by self- consistent perturbations which are constructed on the normal state /1,2/.

We notice the following two points : (i) The mean field is determined solely by parameters in

the microscopic (atomic) stage, and (ii) the fluctua- tions of long wavelength are getting increasingly important due t'o the low dimensionality of the electron motion. So we expect that the field operators themselves are well defined composite operators which can represent the motions of the different stages. This observation may be supported by expe-

The Bose field $. describes the coherent phase field 3

due to the PF correlation. The Boson p is the am-

j

plitude field. The complex Fermi fields

X

and

X

j 1 j 2 represent the electron fields moving with veloci- ties +vF and -vF respectively. If ye put 41. and p

J j

to be constant,

xjl

and

X

describe the motion of j 2

electrons in the mean field approximation. The spatial variations of

X

j l , Xj2, @j and p are much

j slow compared to IlkF.

Our starting Hamiltonian consists of the sum of the Frijhlich Hamiltonian for each chain and the interchain coupling solely due to the lattice vibration. Performing the above transformation and doing the coarse graining of the system over the atomic scale variations, we obtain the reduced Hamiltonian for the semi-macroscopic stage :

H = H. + HI + (1/4)&p2

Q

(M : the total number _{of chains)}

H~ He + Hph + He-ph

riments of optical conductivity / 3 , 4 / , which show

v

ax

the existence of the Peierls gap of electron excita- He=Z dz(xt

1

+

2 %

+~n(xf ,xj2+x~2Xjl) )

J az -Xj2 1

tion above T

.

j

Let us denote the electron field of the j-th chain by

qi

(z, t), and a component 'of the lattice displacement relevant to the Peierls distortion by Q;(z,t). The essence of our physics is represented

J

by a set of' the transformations /5/ :

qj(z,t)=x j l (z,t)exp(i%~+i@~(z,t))+x~~(z,t)

H ~ = 1 J d z ~ ~ ~ ~ ~ , , C l - c o s 2 ($j-$,,)l

exp(-i%z-i$. (z, t)) j ,g

J (1)

Qj(z,t)=p. (z,t)cos(2$z+2$. (z,t)) Here we have neglected the amplitude fluctuation

J 3

(3)

and set p to be a constant p. The operator P is

j @j

the canonical conjugate of @

-

j '

where N.+=X~IXjlfX~2Xj2. Notice that P consists

J - J @ j of not only a@./at but also the electron density

J

difference N We require that the equal time com- j-S

mutators

(X.

(z,t) .X~~(Z' ,t))+=~~ ,R6a,B6(z-z'), and Ja

(@j(z,t),$~~'rt))-= i6 6(z-zl), where

a

= l or 2. j ,R

The last terms of H come from the intrachain electron-lattice interaction, and give rise to a gap A of the electron excitation. H is nothing but the mean field Hamiltonian. The quantity

1

A=

-

flu p is determined by the stability condition 2 q

of p:

where wQ and X=

X

/ m F are the frequency and the 1

dimensionless coupling constant of ?2kF phonons. The bare phasons are described by H where s is

ph'

the velocity of phonons at f2kF. The intrachain coupling among electrons and phasons is given by He-ph. The interchain coupling is nicely rewritten as the interchain phase coupling. This stabilizes the PF state three dimensionally.

long range phase correlation can never establish over the whole range of temperatures. The short range correlation length is naturally estimated by

cph=

vph/rT. (ii) Although

5

ph- as T-+O, the quan-

tum fluctuation prevents the phase transition at cl

T=O. Note that@ (z,?)=exp(-2~x1~1 /cph) in the clas- sical approximation. (iii) since 2a-(1*4~~/Xw~f/: at T=O, tends to zero in the strong

X

limit and one in the weak

X

limit, the effect of quantum fluctuation becomes stronger as the intrachain electron- lattice coupling is weakened.

Let us defined the 3D-ordering Tc by a cempe- rature that <cos 2@ > becomes first non-zero. Tre-

j H

ating H by the mean field approximation we get T from

where JLjQ is assumed to be of nearest neighbour type, and J=Z < j ,k>=n.n.JLj~- For weak IJI S

T~~(J("~('-~). In the strong h Limit ~~+%ial.lI 1 12, while in the weak

X

limit T is suppressed as

I J I

by the large quantum fluctuation.

References

We study the phase correlation. We determine

_{1 1 1}

_Rice,_M.J._{and Str;ssler, S., Solid State} the electron state by H p . Let us first study the Comun.

13

( 1973) 1389

..

one dimensional case JLjR=O. The order parameter 121 Lee, P.A., Rice, T.M. and Anderson, P.W., Phys. Rev.Letters

2

(1973) 462 and Lee, P.A., Rice, correlation function in the Matsubara representa- _{T.M., and Anders0n.P.W..} _{Solid State Comun. 14}

-

tion @(z,T) = <T? exp (2i@j(z,~)-2i@. (o,o)) > is (1974) 703

J

given by H. 131 Tanner, D.B., Jacobson, C.S., Garito, A.F. and

Heeger, A.J., Phys.Rev.Letters

32

(1974) 1301 141 ~rcesch, P., ~trgssler, S. and Zeller, H.R.,

Phys.Rev. BE (1976) 219

20 151 Tsuzuki, T., Osano, K. and Ueno, K., Solid

[

l-exp(-2(

1

z l+ivph') 11 (2) State Commun.

2

(1978) 115 where a=(i/8h2) (vF~~/vphA2), Y=$(l+wc/2nT)-$( l )

,

$(z) is the di-gamma function, and wc is the cut- off frequency of phason. The phason has the disper- sion o=v q, where vph=u/h and

ph

The velocity v agrees with that of ref.?. at T=O, ph