Quasiparticle tunneling between two weakly coupled quasi-one-dimensional charge density waves

Synthetic Metals, 41-43 (1991) 3891-3894

QUASIPARTICLE

TUNNELING

3891

BETWEEN

TWO

WEAKLY

COUPLED

QUASI-ONE-DIMENSIONAL

CHARGE DENSITY WAVES

K.M. MUNZ

Abteilung fur Mathematische D-7900 Ulm, Bundesrepublik

Physik der Universitat

Ulm,

Deutschland

ABSTRACT The dc and ac conductivity of a tunneling junction between two charge density wave (CDW) systems is calculated. Temperature ordinary

Green's

impurities

tunneling

current

consistent

Born

analogous The between

functions

across

the

approximation

to Cooper-pair

linear

ac

the

linear

corresponding

in

the

mean-field

and the transfer Hamiltonian junction. are

leading

approximation

Impurities

treated

to

of

breaking

breaking in superconductors

conductivity ac

is obtained

response

results of Re0(~)

and

the

by the dc

for

CDW

with

method are used to evaluate the the

self-

electron-hole

in

pairs

with magnetic impurities. well

case:

known

scaling

relation

0dc((eo/~)V=m)=Re0(~).

The

are displayed.

INTRODUCTION In contrast

to the

superconducting

case

case has as yet found limited attention.

of

quasiparticle

Artemenkov

and Volkov

1984 the theory for the dc tunneling current

using the Keldysh

[2]

calculated

in

1985

tunneling

[I] developed technique.

the

thermodynamic

properties

impurities

in the mean-field

approximation.

The

purpose of the

to combine

these

i.e.

the

influence

two works,

to study

of

CDW's

of

the CDW

with

in

Roshen

ordinary

present

work is

impurities

on the

tunneling current between weakly coupled CDW systems.

THEORY Consider

two CDW

phase

domains

which

are separeted

by an insulating

The current can be calculated

in this quasi-one-dimensional

of

as

the

transfer

0379-6779/91/$3.50

Hamiltonian

formulated

by

Artemenko

layer.

system by the method and

Volkov

[I].

The

© Elsevier Sequoia/Printed in The Netherlands

3892 H~lltonian K of the system can be written as K=KL+KR+HT=Ko+HT , where KL and KR are the left ~ d

right hand full many-body Hamiltonians for the two CDW's. HT is

the transfer operator. The current is given by the change of the electrons in ^^ one of the CDW systems: I=-eoNL=-(ieo/~)[HT,NL] . In the present case, HT can be written as [3]

HT=k,~,0 --°e+(k){TO+TQc°s(6)~1-TQsin(6)~2}B-°(q) + h.c. ,

(I)

where TQ and TO are constant transfer matrix elements with and without "Umklapp" from one of the Fermi planes to the other. The two component "spinor" e+(k) is --o defined as ~+(k):=(c+(Q/2+k),c+(-Q/2+k)) for the left hand side of the junction, ---0 0 0 < ( q ) is defined for the right hand side in an analogous way, and IkI,lql<
denote the Pauli spin matrices, 6 is the phase difference between the 1 direct TO and backscattering TQ transfer matrix element, Q designates the wave vector of the CDW soft phonon with Q=2k F (kF Fermi wave vector) and 0 is the spin state of the electron.

In order to calculate the tunneling current

linear response theory is used with respect to

the

HT [33.

THERMAL GREEN'S FUNCTION To obtain the tunneling current,

it is necessary to know the Green's function

of the CDW with ordinary impurities [2]. The Green's function is definded by the thermal average over the imaginary time ordered dyadic product of the ~ (k,T') -,0 operators in the Heisenberg picture: G(k,k',~'):=-
(~__o(k,~')®~ (k',O))>. ConT~ --0 sequently, the thermal Green's function is a 2 by 2 matrix. The interaction with impurities

gives rise

to a self

energy

~(k,iVm),

here

given in

the Fourier

transform with the Fermi frequencies v =(~/~8)(2m+I). This self energy is given m in the self-consistent Born approximation by [2] • x 2 1 [(k, iVm)=~IUQl ~k[, G(k' ,iVm).

In

(2), G(k,i~)m)

is

the

impurity

impurity concentration,

L

boundary condition

IUQI is

(dimension:

considered in shift

and

energy.length).

(2)

averaged

is the length

Green's

of the

the absolute

function,

system

value of

x

used in

denotes the

the impurity potential

Only backscattering with the wave vector Q=2k F is

(2) since forwardscattering with q=O merely produces

and without

loss

the

periodic

of generality

it

can be put

equal

to zero

an energy [2]. The

Green's function obeys the Dyson equation G -1(k,ivm)=GO1(k,i~m)-[(k,i~ m), where Go1(k,iv m ) -

denotes the thermal Green's function without impurities in the mean-

field approximation: ~exp(i~)

is the

Go1(k,iVm)=iVm-Ek~3-~exp(i~)T+-~exp(-i~)~

complex order

parameter for

the

. In Go1(k,ivm) ,

clean system

and ~k is the

3893 energy ~

of

the

free

electrons

which

fulfill

the

necessary

nesting

condition:

~

~k:=eQ/2+k=-~

Q/2+k

.

A possible "Ansatz" for G-1(k,i~ m) is

G-1(k,i~m):i~m-~k~3-Amexp(i~)~+-Amexp(-i¢)~ - ,

where ~m and ~m are functions

(3)

of i~ m. The Dyson equation together

completely determine the thermal Green's

function.

with

The Dyson equation --

(2,3)

contains

~

two dependent equations for ~m and Am' and with the new quantity ivm/A m and the analytic

continuation

i~ +w+iE

these

two equations

are corresponding

to

(JA)

m

=W(m)-a(T)(W(~)/(I-W2(~))I/2)),

where W(~) is generally a complex function of ~.

In the above equation for W(~) the quantity a(T) is defined as e(T):=(~A(T)) -I (T denotes

the temperature) with

I/T=(2z/~)XlUQI2No , where ~ is the relaxation

time of the electrons in the metallic state and N the density of states for o free electrons at the Fermi surface. Note that a(T) is the important parameter of the tunneling current-voltage

characteristic.

The density of states for the

CDW quasiparticles

is given by N~DW(~)=~N a-1(T)Im(W(~)) . For a(T)
state.

DC RESPONSE, CONDUCTIVITY Consider

a tunneling junction where the left hand side is grounded and the

right hand side is held at fixed voltage V [3]. Consequently, Hamiltonian for the CDW,

in addition to the

there is an extra term on the right hand side of the

junction which is given by H=-eoVN R. The external perturbation H merely produces a phase exp(iCt) in the interaction representation of B (q), where ¢ is defined --O

as ¢::(eo/~)V.

In the formula for the the linear response only terms which are

bilinear in ~,a + and

8, B ÷ are considered,

this is the consequence

that KL,R conserves the number of particles.

of the fact

After a longer but straightforward

calculation, for details in the pure case see [3], the time independent current can be brought into the form ^

.

,CDW, ~ ,,CDW, +'~ *~imp~mJL *~impT M ~)R

~

(*)=e-~NId~(f(~)-f(~+~))~NoL

~NoR (4)

2 TO

~ ~+¢

COS(¢L-~ R)

{ I ~ 2- +- ~ 2 AL AR *O *Q In (4),

RN

is the resistance

IWL(~)I21WR(~+¢)I

2 }.

of the junction

above

the critical

temperature,

f(~) is the Fermi distribution function and CL-¢R is the phase difference of the two CDW relation:

phase

domains.

The linear

0dc((eo/Z)V=~)=Reo(~).

ac conductivity

is obtained

Results are displayed in Fig.1.

by the scaling

3894

eoRN/AL(T)

[a)

,

I

,

I

,

T/TooL 1 2

/"-

Reo'(w)/oN

Odc/O' N

i

1

{b) ,

i

T/Tc°L=0"999

0.999 0.98

3 0.9 ~, 0.78 5 0.62

,,

,o1,,

2 -

TL=TR o

" 0 i 0

~

AR(0I=2AL(0) '

Odc/O N 1

i ' i 2 eoV/AL(T]/,

Reo(u)/o ,

I

N

0

(c) '

I

o

.- ~ " ~ ~0 "J6/ 2 " ~

2eoV/AL[T)/, U/~L[T )

OdJO N 1

1

AR{O)=2AL(~) 0.1

Reo(u)/o ,

I

N

(d) *

f L

f

o

,

0

,

2eoV/AL(O )/,u/XL(O)

,

0

[-

/

2eoV/AL(T]4 W/~L(T)

Fig.1. (a) DC current vs. scaled potential difference across the junction. All parameters for the right hand side of the junction are expressed in quantities 2 for the left one. Parameter ~:=(To/(T O2+ T ))cos(~L-~ R) measures the ratio of the direct to backscattering tunneling processes. In (b,e,d), the dc (real part of the) scaled conductivity vs. scaled voltage (frequency) is displayed. Parameter o N is the conductivity above the critical temperature.

REFERENCES I S.N. Artemenko, A.F. Volkov, Soy. Phys. JETP, 60 (1984) 395. 2 W.A. Roshen, Phys. Rev., B31 (1985) 7296. 3 K.M. Munz, W. Wonneberger, Z. Phys. B, 79 (1990) 15.