Solid State Communications,Vol. 41, No. 5, pp. 413-414,1982. 0038-1098/81/050413-02$02.00/0 Printed in Great Britain. Pergamon Press Ltd.

PAIRING IN ONE-DIMENSIONAL ANTIFERROMAGNETIC SUPERCONDUCTORS IN THE PRESENCE OF A HOMOGENEOUS MAGNETIC FIELD*

Y. Suzumurat

and

A.D.S. Nagi

Physics Department, University of Waterloo, Waterloo, Ontario, Canada, N2L 3Gl

(Recieved 23 July 198 1 by M.F. Collins)

Assuming a one-dimensional like electron band and considering the free energy difference 6F (= F, - FN) near the second order phase transition temperature, it is shown that for an antiferromagnetic superconductor in a homogeneous magnetic field (howsoever weak) the minimum of 6F occurs when the state with spatially varying order parameter (having same period as the antiferromagnetic ordering) coexists with the usual BCS state.

THE NATURE of most preferred electron pairing in the anti-ferromagnetic superconductors (AFS) have been discussed, in literature recently. Machida et al. [l] have proposed that in AFS, a modified pairing state which includes spatially varying order parameter, As, is more stable than the usual BCS state. Nass et al

[2], on the other hand, assert that the most favourable pairing is the usual BCS type with spatially homogeneous order parameter, A. In this note we show that for AFS in a homogeneous magnetic field, H,, , (howsoever weak), the minimum of the free energy difference 6F ( =F, -

FN) corresponds to the situation in which the pairing state having spatially varying order parameter coexists with the usual BCS pairing state. In agreement with the results of [2] , we find that for &., = 0, the usual pairing state gives the minimum SF. In our calculations, we consider 6F for temperatures, T, near the second order phase transition temperature, T,. As in [l] , [3] , [4] and part of [2], we have assumed a onedimensional like electron band which satisfies e k I= -en+ Q 1 for k near - kp.

The Hamiltonian of the system is:

H = c ek C;,,Ck,, - AC (C&C&+ + h.c.) k,a k

-AC, CCh++w c-h, + A-o c C& C:kor + h.c.

-4 1% B@‘z)a~~~ii,a~k + CL:: + h.c*) 9 ,

--Ho k;,, (%~~~ii,&k$~ (1)

, I

* Work supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

i’ On leave of absence from Tohoku University, Sendai, Japan.

where el, is the band energy of the conduction electron measured from the Fermi energy, Ci,, is the creation operator for the electron, a and 0 are spin indices, u, is Pauli matrix, A and A,, are superconducting order parameters, HQ is staggered molecular field which is taken temperature independenct as in [ 1 ] , H,, is hom- ogeneous external magnetic field and Q is the wave vector of the antiferroamagnetic state.

The superconducting order parameters are defined by

A = F z(c-k&kCQt), k

‘Q = vl c (C-k& ck+,,), k

‘-e = vl c (c-k-~1 Cd, k

(2)

where V and VI are the coupling constants for the respective pairing states.

Taking ekl= -elh+Qt fork near -kF, equation (1) can be used to write self-consistency coupled equations for the order parameters. Using these equations, the free energy difference SF for T near T, can be written. We fmd that in the presence of H,, , the case A, = + A, gives lower value for the minimum 6F

than the case Acr = - A-, and will only be considered now onward. Writing AQ = A,, we obtain [N (0) is the per spin density of states at the Fermi surface]

6F = N(0) [&,*A2 + 2&,,M, +&,,A$ + . . . ,].

(3)

In equation (3), the next term is of the form Al&jr (j = 0 , - - * , 4). The quantities Be,2, Be,r and Be,e are real and

413

414 ONE-DIMENSIONAL ANTIFERROMAGNETIC SUPERCONDUCTORS Vol. 41, No. 5

1 0.0 HolAo

1 1.0

Fig. 1. The normalized transition temperature T,/T, and the ratio X, =Fm+/A vs HJA, for Ho/A,-, =

0.2 and different valGes:f Xr . Here A, = 20, exp [-l/X],h=N(O)lr,Ar =N(O)Vr where Vand Vr are the couling constants for pairing with spatially hom- ogeneous and spatially varying order parameters, respect- ively .

(4)

where

A = $+I%,

1 0, = n+- --i&i,

2 hr = H(O)Vr,

Ha = Hg/(2rT), &(2aT). (5)

Further, T, = 1.13~ exp [ - l/h] , where h = N (0) V and wD is the Debye cut-off energy. In equation (4) the b_ranch of A’” is defined such that A’O is positive when H,, = 0.

Equation (3) shows that for T sufficiently near T, the minimum value of 6F corresponds to AQ f 0 if

Bo,l # 0 and A0 = 0 if Bo,, = 0 (Bo,o) is usu$ly positive). Now equation (4) gives Bo,l = 0 if H, = 0 and L& is non-zero when H, f 0 (in fact Bo,r ‘v 14{

(3) HOHQ for HO, iQ Q 1). Thus in the absence of H,, the ususal BCS state is more favourable. But, in the presence of a homogeneous magnetic field (howsoever small), the spatially varying order parameter state and the usual BCS state coexist.

By minimizing equation (3) with respect to A and

0.+-----k Ho/&

Fig. 2. The T,/T, and X, vs HJA, for HQ/A, = 0.6 and different values of X1.

A,, the equation for T, and the ratio X = A,lA at T, are given by

Bo, IT = Tc = ;o.l)2 0.0

T = T,,

T-T, A B (6)

0,O T = T,’

Taking H, /A, = 0.2 and 0.6, respectively, the variation

of T,IT, and X, with H,/A, for hr = 0,0.2 and 1 .O are shown in Figs. 1 and 2. The solid (dotted) curve denotes stable (unstable) T, and X,. The stable (unstable) T, and X, are those for which the next term in equation (3) is positive (negative). In the weak coupling limit (i.e. hr + 0), A, + 0 and the present results agree with those given by us earlier [4]. We note from the figures that: (i) T,/T, and Ix, I are enhanced as hr increases and (ii) the enhancement of TJT,, is large when IX, I is large. Our results for h&/A, < 0.49 (> 0.49) are similar to those given in Fig. 1 (Fig. 2).

In conclusion we mention that for a one-dimensional like anti-ferromagnetic superconductor in a homogeneous magnetic field the coexistence of the state with spatially varying order parameter and the ususal BCS state has been established near T,. At T = 0, after complicated calculations, the above coexistence is found in certain regions of the H, -H,, plane.

Detailed results will be published elsewhere.

REFERENCES

1. K. Machida, K. Nokura & T. Matsubara,Phys ROY. Lett. 44,821 (1980).

2. M.J. Nass, K. Levin & G.S. Grest,Phys Rev. Lett. 46,614 (1981).

3. Y. Suzumura & A.D.S. Nagi, Solid State Commun (to appear).

4. Y. Suzumura 8c A.D.S. Nagi,Phys Rev. B (to appear).