Physica 143B (1986) 447-449 North-Holland, Amsterdam 447

THE TRANSITION TEMPERATURE IN QUASI-ONE-DIMENSIONAL CONDUCTORS, (TMTSF)2X

Yasumasa HASEGAWA and Hidetoshi FUKUYAMA

Ins t i tu te for Solid State Physics, Universi ty of Tokyo, 7-22-I, Roppongi, Minato-ku, Tokyo 106, Japan

Phase t rans i t ions in quasi-one-dimensional organic conductors, (TMTSF)2X, have been theore t i ca l l y investigated. The spin-density-wave (SDW) t rans i t ion temperature is obtained as a function of the degree of imperfect nesting of the Fermi surface, which is thought to be increased by pressure. The wave vector of SDW at the t rans i t ion temperature is shown to change discontinuously at a c r i t i c a l value of imperfect nesting. The competition between SDW and various types of supercon- duc t i v i t y is also discussed. I t is shown that there are no coexistent state in Ginzburg-Landau region.

I . INTRODUCTION

The organic conductors (TMTSF)2X, X=PF6,

AsF6, etc. undergo the spin-density-wave (SDW)

t rans i t ion at low temperatures, which is

suppressed by pressure, and superconducting

phase appears under higher pressures.l , 2 In

these systems Fermi surface is open and may be

considered to be quasi-one-dimensional (quasi

ID) along the chain axis a. However, the trans-

verse transfer in tegra l , tb, is much larger than

the t rans i t ion temperature TSD W on one hand,

while tb2/ ta, t a being the transfer integral

along the chain axis, which turns out to be the

character is t ic energy of the imperfect nesting,

is of the order TSD W on the other hand. Under

such circumstances the mean f i e l d approximation

may be applicable as long as the dependence of

TSD W on the imperfect nesting is concerned,

although parameters w i l l be affected by

f luctuat ions with the energy larger than t b.

The electronic i n s t a b i l i t y of quasi ID

systems was f i r s t studied by Horovitz et a l . , 3

who showed that Peierls t rans i t ion occurs with

the wave vector Qo=(2kF,~/b,~/c), k F being the

Fermi momentum along the chain axis in the

absence of transfer integrals and b and c being

l a t t i c e constants along the perpendicular

d i rect ions, and that the t rans i t ion temperature

is sharply suppressed at a c r i t i c a l value of

imperfect nesting. Jafarey 4 showed that the

wave vector is d i f fe ren t from QO i f the

t rans i t ion temperature is much lower than t b.

In (TMTSF)2X t b may be varied by external

pressure and Yamaji 5 indicated that the P-T

phase diagram may be understood by means of

quasi ID model. He also obtained the deviat ion

of the SDW wave vecto~ from QO at T=O.

In this paper we f i r s t examine the t rans i t ion

temperature of SDW as a function of the degree

of the imperfect nesting. We next invest igate

the competition between SDW and superconducting

state based on the general form of the Ginzburg-

Landau expansion.

2. MODEL FOR THE BAND ENERGY

Our model for the band energy is given as

c(k) = Vx(Ikxl-k F) -2tbCOSbky ÷2tb'cOs2bky,(2.1)

where v x is a Fermi ve loc i ty and t b' is of the

order tb2/ t a. The transfer integral along c

d i rect ion is neglected for s impl ic i ty . Without

the last term, the Fermi surface is completely

nested with the vector QO. Therefore, t b' rep-

resents the degree of the imperfect nesting, and

is tuned by pressure; t b' w i l l be increased as

pressure is raised. As w i l l be shown in the

next section, the t rans i t ion temperature of SDW

is strongly dependent on tb ' , while i t is inde-

0378 - 4363/86 /$03 .50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundat ion

448 Y. Hasegawa and H. Fukuyama / Transitiott temperature iH quasi-~)tte-dett~e~zsio~zal co~ductor,;

pendent o f t b , i f Q=Qo, or is weak l y dependent

on t b , i f Q~Qo. Then we take t b as a f i x e d

parameter and t b' as a v a r i a b l e pa ramete r .

3. SDW TRANSITION IN THE MEAN FIELD APPROXIMATION

In t h i s s e c t i o n we c o n s i d e r the H a m i l t o n i a n

t t H = '

where c~m (Ckm) i s the c r e a t i o n ( a n n i h i l a t i o n )

o p e r a t o r o f the e l e c t r o n w i t h momentum k and

spin m, and U is the e l e c t r o n - e l e c t r o n i n t e r a c -

t i o n to be assumed o f sho r t range. The SDW

t r a n s i t i o n t e m p e r a t u r e , TSDW, is de te rm ined by

the equa t i on

Xo(Q, TSD W) = l / U , (3 .2 )

where Xo(Q,T) is the s t a t i c s u s c e p t i b i l ~ : y f o r

the s taggered f i e l d Q=Qo+q. For g i ven t b and

t b ' , t r a n s i t i o n to the SDW s t a t e occurs w i t h the

wave v e c t o r Q which g i ves the h i g h e s t TSD W-

We f i r s t examine the SDW w i t h the f i x e d

wave v e c t o r QO. Then we get

N(O)( ~F tanh(m/2TL dE

Xo(Qo,T) = ; 2 l t b , ! v . ~ 2 _ ( 2 t b , ) 2

E q . ( 3 . 3 ) t o g e t h e r w i t h e q . ( 3 . 2 ) is no th i ng but

the BCS gap equa t i on i f we i d e n t i f y 2 t b' w i t h

the o rde r pa ramete r , A(T) , in the gap e q u a t i o n .

We p l o t TSD W as a f u n c t i o n o f t b' in F i g . i . The

c r i t i c a l va lue o f t b ' , at which the t r a n s i t i o n

t e m p e r a t u r e to the SDW w i t h QO tends to ze ro , is

g i ven by

(3 .3 )

10 --

isow t ~;o \,.

0 5

t b / t bo

FIGURE l T r a n s i t i o n tempera tu res of SDW w i t h wavE. v e c t o r QO ( s o l i d l i n e ) and Q~Qo (broken l i n e ) .

t t 'o = r~D W / 1 . i S : ~.~- e x p i - i , 'UN( / !~ i , ~: '..4}

where [SDW L) is the ! : r a n s i t i o u tempera tu r~ wnp~

i b ' =b.

Next , we examine the SDW w i t h i i-O0+o. W,~

f i n d t h a t maximum of ,O(@,I is loca ted at q : t

when T ' T ! * . The va ln~ ,~f q, which give!., th~

g loba l maximum oI: >O(Q, : } ~hanges i i s ( : (~r !T im-

o u s l y at T = I I * . Tf i b ' / t b 0 .035. w~ : I : t

O . 2 3 1 < T 1 * / t b ' < 0 . 2 ~ 2 , as shown in F ig . : ' whe~,-

XO(Q,T) is p l o t t e d as a f u n c l i o n o! ....

t b s i n ( b q y / 2 ) / 2 t b ' C O S b q y and v - - vxqx /S tb ' (Osb , : i y .

At lower t empera tu res the niaximum of ' ( Q , :

l oca ted r o u g h l y on the ~u~..': ,,,-u /8+u.. :. We p i~ i

[SOW w i t h DCQo as a br,~kin i i nu in F i q . : . ;

F i g . 3 we p l o t u as a t u r l r t im7 /SDW-

c? - i!.5

> <

2 )-< N-

'2""

:L:. #L'

%b :. J

I ~ I:,

:b} [ :Y . ~i,

FLGUR£ L The s taggered s u s c e p t i b i l i t y • as a f u n c t i o n ( I and v at (a) above and (b) below T I * .

Y. Hasegawa and H. Fukuyama / Transition temperature in quasi-one-demensional conductors 449

The a

2£

IQ

O11 02

FIGURE 3 var ia t ion of the SDW wave vector plot ted as

function of transit: ion temperature.

4. COMPETITION BETWEEN SDW AND SUPERCONDUCTIVITY

In this section we assume, besides the in--

stantaneous repulsive in teract ion U, the at t rac-

t ive in teract ion V(k-k')=Vs+Vtsgn(kx)sgn(kx')

mediated by phonons with the cut -o f f energymD.

With th is in teract ion both s inglet and t r i p l e t

superconductivit ies are possible.

We expand the free energy with respect to the

order parameters A and M for supeconductivity

and SDW, respect ively. In this model i f SDW

exists, i t should be sinusoidal one, since i t

makes fourth order term smallest for f ixed IMI 2.

Then we get

explained by the geometry of the Fermi surface. 5

In this case we should take QO as a so-called

optimal wave vector for the Fermi surface and

our results are va l id also for th is case.

Comparing Fig.1 with the experiment in

(TMTSF)2X,8 we may conclude that the SDW with

QCQo is real ized at 6kbar. The jump of the wave

vector at the c r i t i c a l value is calculated to

be (aqx,bqy)~(-10-3,±2×10 -2) for tb'/tb=O.035

and tb/ta=0.1. The sign of aq x is changed when

the sign of t b' is changed. I t w i l l be possible

to detect the var ia t ion of the SDW wave vector,

though i t may not be easy.

We f ind that the f i r s t order t rans i t ion bet-

ween SDW and superconductivity occurs in the

Ginzburg-Landau region. In our framework we

cannot dist inguish whether superconductivity is

s inglet or t r i p l e t . While the f i r s t order tran-

s i t ion temperature is strongly dependent on t b'

i f Q is f ixed to QO, i ts dependence can be

weaker when the var ia t ion of Q is taken into

account. This feature seems to be consistent

with experiment. 8

F-F 0 =2[A[AI2+~IAI4+A'[MI2+~'IML4+ClA21[M21] (4.1)

where F 0 is the free energy of the normal state.

We f ind C 2 >BB' always for the present model,

and th is corresponds to the strong coupling case

defined by Imry et a l . 6 Thus the coexistent

state is not the state with the minimum free

energy, and the f i r s t order t rnansi t ion occurs

between superconductivity and SDW~

5. CONCLUSION

We have investigated the low temperature

properties of quasi ID system within the mean

f i e l d approximation. The wave vector of SDW is

shown to vary discontinuously at the c r i t i c a l

value of imperfect nesting, Recently, the wave

vector of SDW at ambient pressure has been

observed to be incommensurate, 7 and i t was

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