Anisotropic order parameters for two-dimensional strong coupling superconductors

ELSEVIER

Synthetic

Anisohopic

order

Metals

parameters

71 (1995)

for t,wo-dimensi0na.l Y. Suzumura

Department

1635-1636

of Physics,

and

Nagoya

T.

hong

coupling

superconductors

Matsuura

University,

Nagoya

464-01

Japan

Abstrnct. By

considering

anisotropic

The tive

the

order

new

interaction

pairing

for

fermion

system,

both

cubic

lat,tice

the

the

strong

with

the

the

coupling weak

der

parameters

has

been

[5, 31. model

nearest

T,

of the

neighbor

, 6 , has been bond

(RVB) and

In t,hr t/J

and

paper,

6 with

the

states

organic

and

conductors

We consider

the

of the

T,

high

concen-

stat.e

s-state,

the

are

fermion

spectively.

= [(Ek The

the order which

and

the

is given

by

A,,k

parameters

by A,,k

k, - cos ky).

correspond

s+id

for the

are defined

Ad k = A(cos

0 = 0, x an’d ~/2

state

=

In eq.(3),

to the extended respectively.

+ iA,k.

The

AV) - F(n,o))/N

k

systems,

where

the

Then

free

energy

is

= $

lAk12

no

= (1 $ c0sB)Aq,~/2

w, is thr Mat,subara

conductors.

= &~~~~~t~y~~~~y

< a_k,akr for

either

the

where

$

s-stat,e

(I - ~0so)Al,k/2.

$

frequency.

extended

A,

>MF,(~

= x,Y),

, Ay and 11 are given

by

and

respectively. tional

to (1 +cos mixed-state

From

T,

t’q (4),

or t,he d-state

and

is given

by

,i.~.,

l-etanhg

and case

number

of 0 5

of the

total

S < 1 is examined

T and

0379-6779/951$09.50 0 1995 Elsevier SSDI 0379.6779(94)02986-9

Science

or the

s-state obtained

d-state propor-

therp

is a possibility be-

rqs.(2)-(5)

are shown

as the

The

(IV).

to the

d-state

(s-state)

We note

t,wo kinds

from by the

the transition of phase

place

that [l l]. the

solid

boundary which

the

decrease

normal

state

the

< t/J

is

int,o the d-state

norma.

take

(III)

of temper-

in tile region

in the region

sep-

d-state

6, - 0.6 for 0.25

mixed-st,ate

transitions

The Curve

II) from By the

from

15~ obtained

curve.

by the (I and

by RIarel

takes

tlourtdary

dashed-solid

mixedstate

is followed II) and

reserved

and

temperature

from

by the

, the transition

(s-state) region

S.A. All rights

the

COllt,aillS a term

at a certain

5 in Fig.1

at, T = 0 is shown t,he the

The

obtained and

is shown

re-

electron due

N,,

s-state

in eq.(4)

(0 < 0 < r)

states

arates already

Quantit,ies

to the

jAk14

B)( 1 - cos O)Az,kAi,k,

of t/J

T,

ature

11)’ + lAk12]‘/2. the

,(2) Ek

correspond

low T,.

from

tanhg,

-

Since

of the

obtained

1-6=$x

temperature

are

d-state

repul-

to the study

(1)

k Ek

A,k

(3)

~CTC/‘Ak’id~w:+cikl_i’ ‘(4)

-

of

k

where

and

(F(A,,

two-

function

of the on-site

Ak

Ak COSk, Ek

sc

c

I.4

Ad.k+

resonating

Several

Ne

lAk(

as

function

N

= [A,\,

t

and the’d-state

with the

expressed

t IAY?),

equations

=

to the halfis rewritten

2

b, + cos ky) and

s+id

by A,

S = 0 corresponds \A,[

k

I: (4J/1%!)xtcosk,

Self-consistency

As,k

the states

Hamiltonian:

= -2t(coskz~cosky),

A,

that

s-state

s-state,

knowledge

heavy

mean-field

++=I?

and

A(cos

surface

anisotropic

as

assumption

kn

whereck

Quantities extended

(TMTSF)zC104

extended

the

generally

the

where

1 + e”

or-

with 6.

[7, 8, 9, lo].

to add some

of superconducting

symmetry By assuming

Ak = As,k-

two-dimensional

of the

the

we examine

implicit

In order

Fermi

states

as

systems

the hole

in terms

state

superconductors

sive intera.r-tion

) J , with

[6] having

case.

superconducting

lattice,

, t , and the hole concentration,

width

particle-hole

, t , and the attrac-

energy

examined states

t,he s+id

present,

dimensional

the

interaction

tration

cl-stat.e

conductor

hopping

band

s-state

[3, 41 where on the

superconductors,

valence the

of zeros

of the

by considering

investigated

a line

) J , on the square

filled

been heavy

extended

Quasi-one-dimensional

was

function

attrac-

has

In the

the

calculated

for the organic

high

consisting

tive

been

21.

having

claimed For

[l,

and

the

sites

of conductors.

interaction as the

from

neighbor

d-state

have

coupling

attractive examined

arises

nearest

kinds the

been

which

the

various

neighbor

have

state

between

studied with

nearest

parameters

state

III (the place

I (the into

region

in the

the IV).

region

Y. Suzumura,

1636

I and the

II. We note mixed-st,ate

treated

that has

in the

the

transit,ion

been

already

mean-field

theory

from

T. Matsuura

thp d-state

found

for

for the

case

the

t-J

I Synthetic Metals 71 (1995) 1635-1636

transit,ion

into

st,ate.

model

of the small

6

1

point The

lower

from

onset,

than

Z’, for

seen

from

the

tion

spectrum,

d-state

both

curve

the

(1)

(s-state) of the

d-state

and

and

(2)

the

mixed-

T,,,,is

Ithe s-state

Thtl

, is calculated

2E,,,

Into

mixed-state, gap

as

of the

as E,,,

is

excita-

= IplA[(l

+

cosH)/(8t2 $ (1 + cos @)A 2 )] ‘I2 in the case of F < 1 and in the case $ 2(1 + cos B)A.2]“2 E m,n = [4t2(2 - (lpl/2t))’ of F > 1 where that

6

qllantitg

or the

d-state

6. The

gap

coupling

the

phase of the

anomaly

1

the

t/J

[l?]

Fig

1 Phase

solid

curve

diagram and

from

on t,he t/J-6

the

solid

T, and

curve

plane denote

where the

the

boundaries

ob-

for

the

:tnisot,ropic

of

strong

such

strong

a state

6) is followed

the

) in the

exists

in

case

mixed-

t,he s-state

gap

and

both

the

the

WC‘conjecture

Y.

the

of Tc: observed

in

from

Kuroda,

S.

Maekawa

Thus

work

for Scierltific

Science

for Scientific

H.

partially

Research

Resrarch from

into

and

was

and Cu!lture

of Superconductivity,

s-

that

the d-state by the CILTVC1;I j in Fig.2.

Grant-in-Aid

and

case

by thp the

between

discussions.

and by Grant,-in-Aid

supercon-

coupling

( t,hc s-state

dependence

of Education,

Science

the

of temperature,

Finally,

to

usefill

by the

Mechanism

A since

the the t,ransition

thankful

supported

tion,

large

as is shown

are

the Ministry

T = 0 respectively.

d-state

temperature

indicates

Fukuyama

dashed-

the

competition

mixed-state We

li and

with

decrease

mixed-state.

of the

YBCO

the

superconducting

the

with

for 6 = 0

the increase

hy

stnall

in the

Ry

ii (the

to the

and

lattice

into

The

state

0.5

J/t.

the small

d-state.

0

square

large

due

compared

we examined

the

transition

state

the

zero

d-state.

In conclusion, with

of both

small

the

with

monotonica1l.v

case

It is found

ant1 the mlxecl-state

) w 1llr ‘_I I 1wconnes

, increases

beside

ductor

$ (1 $ ros B)A2).

E,;,/A

in the

is very

locates

0.5 -

F E 2tlp(/(Bt2

# 0 for hot11 the s-state

E,,,,

li # 0. The

tained

the

temperature

from

No.05640410

on Priority Minrstry

Area,

of Educa-

Culture.

References 1. K.

hliyake,

T.

72

(1984)

Jitrh,)

and

Nagaoka,

2. F.

Ohkawa

(1984)

and

(1987)

11338;

7.

G.

Fig.2

The

T-dependence (0.85,

In Fig.‘?,

typical

of 8 and 6 in the case of (6, t/J)=(O.l,

0.25)(2),

(0.1,

0.5)(3),

(0.8,

0.5)(4)

shown

where

those The

in the

(5)

the

s+id

the

curve

curves

region

curve

shows In

the

(l)>

I, II,

belongs (1)

of T-dependence

state

to for

(curve(2)),

III the all

(2), and

(3)

the

and

(4)

IV in Fig.1

case there

of A and

of

6 =

temperatllres is a clasp

0 are

belongs

to

10

respectively. t/J=0

which

below of A

at

11

and

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Y. Suzumllra,

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