I'h~,sica 143B ( 1986l 444446 444 North-Holland, Amsterdam

CASCADE OF THE FIELD INDUCED SPIN DENSITY WAVE TRANSITION IN BECHGAARD gAi,FS

Liang CHEN, Kazumi MAKI a and Attila VIROSZTEK b

Department of Physics, University of Southern California, Los Angeles, CA 90089-0484, ~I;<A

Making use of a two dimensional tlght binding quasi-particle energy spectrwn for organic conductors (Bechgaard Salts), we obtain an analytical expression of the spin density wave (SDW) transition temperature in a high magnetic field along the c* direction. The present theory predicts a cascade of SDW transitions which describes very well the observed transitions tn (TMTSF)2C[O~.

I Following Yamaji we shall consider an

electron system with energy spectrum given by

E(p) : -2ta(COs(ap~)-cosaPF)-2tbCosbp,

-2t cosep3 C

! ! )

-1 -I where we assume that tb/ta:10 , tc/tb:30

Then the single particle Green's function ~:

in the presence of a magnetic field along the

c* direction is determined by

( i ~ + E ( p - e A ) ) g + ÷ ( x , x ' ) : 6 ( x - x ' ) (2)

where m is the Matsubara frequency and we take

.d P~ = PF - l-~x

and

= (o, Hx, O ) (

.m - ~ g, ( x , x ' ) = Sl6.C°( l+ap(x) ' ! -~( !+ap(x~!?

i V

fX

~×p~- I ,~<× - -x ' " l ,'-~, ~I ~ ( x ) d :< i } . , Z t

where

p ( x ) ~ - 2v -1 ( t b e o s (b (p2--eHx))+t b' cos (2b ( p , - e t t x )

a n d

v = 2taaS[nap u ~ ~/2t a

t b ' .-: t b2 / (2~ /2 t ) ÷;

When we neglect the higher order terrors t~

a p ( x ) , Eq. (~) t s t d e n t i c a ~ to the on,:, e b t a i n e l

i,y G o r ' k o v and Lebed '(IL) ) w i t h Y a m a j i ' s

relation (Eq. (6)). HeP, we assumed that ~; :

,~ / a.

J . . .~ Now following llerzt]er ,-~t. aL we tak,

general spin wave veetor

In Eq. (2) we have neglected the Larmor energy

associated with the electron spin, since the

Larmor term is cancelled in the expression of

the transverse spin susceptibility. Equation

(2) is easily solved within quasi-classical

approximation as;

a. Address after July 15: Max-Planck Instltut fur Festkdrperforschung D-7000 Stuttgart 80 West Germany b. On leave of absence from Central Research Institute for Physics H1525 Budapest !14 P.O.B 49 Hungary

Q = (2PF + q x ' ~ /b + qT' ~r'/'?) !

Then the pole of the transverse susceptibi!it)'

at the s p i n wave v e c t o r Q 'is d e t e r m i n e d by

b ['/b fi 2~Tv -I 1 = ~ - - dp 2 x e x p [ i ¢ ( x , o o ) ] A , 2 ~ / b s i n h r ~ ]

0378 - 4 3 6 3 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation

L. (7hen et al. / Cascade o f the field induced spin density wave transition 445

where

~(x,p 2) = qxX+~(sin(2b(p2-eHx))-sin2bp2)

+B(cos(b(p2-eHx))-cosbp2) (9)

with ~n = olin (~'B)I~ We calculate Tc(H) numerically for t a = 2843

K and t b = 265 K appropriate for (TMTSF)2C~O .

and show in Fig. I. In this limit qx takes

= 2t b' cos (bqy)/VbeH

B = 4tbsin(~bqy)/vbeH (10)

and ~ =UN 0 and U is the on site Coulomb

potential and N O = (~vbc) -I is the density of

states at the Fermi surface. Finally d is a

cut-off distance. The integral over P2 of Eq.

(8) yields

-I iqxX 2elnkx -I ( dx 2~T-~v e ZJIn(~,B) j (11)

: Id sinh[~] n

where k = beH and qx : Nk

In(~,B) = inL J~(~)Jn_2~(B) (12)

and J£(z) is the Bessel function. Finally the

integral over x gives;

-I ~Iin(~,B)12{~n(2_~)+~(~)_Re~i~+ i(N+n)s]} = 2~T "

n (13)

where

s = ~vk , • = 1.78 .....

and ~(z) is the di-gamma function and m0 is the

cut-off frequency related to d by

7~o = vd -1 ( 1 4 )

As the magnetic field is decreased, Io, I,, I2,

... take successively the largest value. If we

take only the largest value I n for a given

magnetic field, we obtain

Tc(H ) = 2Y~ exp(_~1) (15)

0

~ = 6 0

o: Expe r . N= ,,

/ | I

4 8 12 16 H ( T )

FIGURE I The transition temperature of T (H) is shown as function of magnetic fields. O~en circles are taken from [4]. We obtain an excellent

agreement for N ~ 4. The full curve is calculated from Eq. (13), which is exact.

the commensurate value qx = -nk with n integer.

The corresponding sin (~bqy) is shown in Fig.

2.

We find an excellent agreement with

experimental result for (TMTSF)2C~O ~ by

Naughton et al.4 as shown by open circles. For

the present fit we have chosen ~o and ~ as 2Y~ o

= 85 K and ~ : 0.8092, which suggests that

perhaps we are not in the weak-coupling limit.

It is possible to include all terms in Eq.

(13) to determine T . We find an equally c

excellent agreement with experiments as shown

by full curve in Fig. I by readjusting ~o and

~, which are slightly (by a few percent)

different from the previous values. On the

446 L. Chen et al. /' Cascade o/ the ]~eld induced spitz densiO' wal, e transitio~

other hand the quantizatlon of qx and qv are

hardly affected; necessary change in qy is

smaller than tenth of percent for example.

Therefore the two dimensional mode] proposed

well. On the other P,~,nd t~e presenl mod(~

appears to have no r'o<xr, wJti~ nega t iw~ t~, w~i

is suggested From -t :~ign reversal <A the Ha!

i 9

c 'oe f f i c i e n t r~ J h igh magnet i , f i e ] i

or) '< 4 3

O 5 I0 IS H ( T )

FIGURE 2

sin(bqy/2) is shown as function of magnetic field. For H > 10 Tesla sin(bq /2) = 0 corresponding to the SDW predicted by Gor'kov and Lebed with N = O.

by Yamaji I describes the observed field induced

SDW transition in organic conductors extremely

ACKNOWLEDGEMENT

We would l i k e t o <h:usk brs. ! )er l is J e r , ~ ] l t ,

, r .F. Kwak ar] l ~(, 'famuj:~ f o r :sdnJ;nF r l ~ , ~

pr, e p r i n t s p r i o r t ~ljb] i ~ t i )i]. ]'he ",r~-,,5,,~

wor'k i s s u p p o r t e d by ! , i~t , l )na] ,',,"i~n ,

Foundat~or] under" grant uumber DHflS-'-~ F-!5.

RBFERENCBF,

!. K. Yamaji, S y n t h e t i , : MetaLs i3

!;.

'4.

5.

19 ~ 6 ~ ~ [.. P. ( }o r , kov and . ( ] . Lebpd , ,T. p,,,¢.<. ( P a r i s ) b e t t , 45 1~84) i,a<~, M. H e r ] t i e r , q, Montamb~ux .~nJ i . , e d e r e J, Phys. ( P a r i s ] Le t t , . ;~E (198a) ;o4~ . M.J. Naughton, J.S. Brooks, L.Y. Chiang, R.V. Chamberlin, and P.M. Chaikin, Phys. Rev. Lett. 55 (1985) 969.

M. Ribault, D. J@rome, J. Tuchendier, Weyl and K. Bechgaard, 4. Phys, (Ps is L e t t . 4~ (1983) L95~; M. l { / b a u i t , Mo] , ' r L i q C r y s t . I19 r ~ 9 ~ 91. J . F . Kwak, J .E . Sch r i b ,~ r , f . M . ,Sha~kJa, Williams, H-H. Wang and L.Y. Ch/an~, Phv Rev. bett. 56 {1986 ~' 9'<?.