Superconducting state of an electron gas in a homogeneous magnetic field

Volume 20, number 6

PHYSICS LETTERS

SUPERCONDUCTING STATE OF AN ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD

1April 1966

GAS

A. K. RAJAGOPAL *

Physics Department, University of California, Riverside, California and R. VASUDEVAN **

Physics Department, University of California, Berkeley, California Received 16 February 1966

The superconducting instability of an electron gas in a homogeneous magnetic field is here investigated taking full account of quantization effects using a variational method. A Ginsburg-Landau type equation with anisotropic field dependent effective masses is derived for the gap near the transition. The s t a b i l i t y of the n o r m a l state of an e l e c t r o n gas in the p r e s e n c e of an e x t e r n a l u n i f o r m magnetic field (directed along z - a x i s , say) towards the s u p e r c o n d u c t i n g state will be d e s c r i b e d here using a v a r i ational method of solution for a c e r t a i n s u m of ladder p r o p a g a t o r s which c o r r e s p o n d to s c a t t e r i n g of p a r t i c l e s with holes. The c o r r e s p o n d i n g p r o b l e m in the a b s e n c e of the m a g n e t i c field was c o n s i d e r e d by T h o u l e s s [1]. E v a n s and Rickayzen [2] i n v e s t i g a t e d in a s i m i l a r m a n n e r the effect of the m a g n e t i c field on the n o r m a l state by including only s e m i - c l a s s i c a l effects of the field in the e l e c t r o n p r o p a g a t o r s . The u s u a l r e a s o n for o m i t t i n g the q u a n t i z a t i o n of the single e l e c t r o n l e v e l s in the p r e s e n c e of the field is that only for v e r y high fields these effects become significant. However a need for a f u l l e r i n v e s t i g a t i o n is e x p r e s s e d r e c e n t l y by s e v e r a l a u t h o r s as for example, Douglas and F a l i c o v [3]. In this p r e l i m i n a r y r e p o r t we p r e s e n t the r e s u l t s which include the full q u a n t i z a t i o n effects. Following E v a n s and Rickayzen [2], A ( r l r n ) , the s u m of l a d d e r p r o p a g a t o r s c o r r e s p o n d i n g to the s c a t t e r i n g of p a r t i c l e s with holes, s a t i s f i e s the equation

h(rlr n) = ~ G(rlrnco1)G(rlrn-co 1) V ~ f d3r2G(rlr2co1) G(rlr2-CO1)A(rlrn). Wl + [~ co1

(1)

Here G(rlrn+ co1) i s the u s u a l one e l e c t r o n G r e e n ' s function in the p r e s e n c e of an e x t e r n a l m a g n e t i c field, V is the potential of i n t e r a c t i o n which p r o j e c t s out only those p a r t s of the field o p e r a t o r s whose m o m e n t u m l i e s in a s m a l l r a n g e n e a r the F e r m i s u r f a c e given by (l~(k) - k2/2rnl) < COD, where COD is the c h a r a c t e r i s t i c Debye frequency. V i s otherwise a s s u m e d to be a delta function potential in space. Also ~ = 1/kT. Now, any i n t e g r a l equation of the f o r m A(XlXn) = Ko(XlXn) + V f Kl(XlX2)A(X2Xn)d3x2 ,

(2)

where Ko is some inhomogeneity and K 1 is a s y m m e t r i c k e r n e l , can be solved by c o n s t r u c t i n g a v a r i a tional functional of A, J(A) = f A2(XlXn)d3x1 - 2 f Ko(XlXn)A(XlXn)d3x 1 - v f f A(XlXn)Kl(XlX2)A(X2Xn)d3Xld3X2,

(3)

whose f i r s t v a r i a t i o n tort A gives eq. (2). In p a r t i c u l a r , if we s c a l e A(XlXn) in the f o r m X(xn) A(XlXn) the e x t r e m u m d e t e r m i n e d by v a r y i n g X(Xn) m u s t be independent of such s c a l i n g of A. This gives * Supported in part by the U. S. Air Force Office of Scientific Research. ** Supported in part by the U. S. Air Force Office of Scientific Research, Grant No. 132-63 and 130-63. On leave of absence from the Institute for Mathematical Science, Madras, India.

585

Volume 20, number 6

PHYSICS LETTERS

1 April 1966

- ( f K o C X l X n ) A(%1%n) d3%1)2

Je×t (A) = f A ( X l X n ) d3Xl {A(Xl%n) - V fK1(%1%2) A(x2xn) d3%2} "

(4)

When K 1 is not s y m m e t r i c , one m a y s y m m e t r i z e it by the usual techniques. It is e a s y to see that the instability in the solution a r i s e s whenever the denominator in (4) goes through zero and changes sign. F r o m eq. (1), we m a y identify K o and K 1 of eq. (2) and p r o c e e d to examine the instability in the ladder sum by studying the homogeneous equation corresponding to (1). It m a y be pointed out here that the instability of the n o r m a l p a r a m a g n e t i c state towards spin density wave state was investigated by one of us r e c e n t l y [4] by a p r o c e d u r e s i m i l a r to the one outlined above. There the k e r n e l r e l a t e s to a sum of bubble d i a g r a m s unlike in the p r e s e n t p r o b l e m where ladder d i a g r a m s a r e involved. Using the complete G r e e n ' s function for the electrons in a homogeneous magnetic field in the f o r m given by one of us [5], in the gauge A = ~rl x H w i t h / / = (0, 0, H), we have (r = (~, z)) K l ( r l r 2 ) = V exp [i T e H" (r2

(5)

x rl) ] c(lei - ~21, (~i - z2)),

where

rrl,n=O

/eH 2~ eH . f e l l 2\ e x p k - ~ - p ] L n ( ~ - p 2) Lmk-~- p ] X

(6) 1 £

x ff dkzldkz2 exp{i(kzl kz2)Z}~tanh ½flClln(kzl)+tanh ~/3 llm(kz2)] (21r)2 + e l l n ( k z l ) + ~Tl-£(kz2--~ ] in the s a m e notation as in ref. 5. Evans and R i c k a y z e n [2], omitting the Landau level s t r u c t u r e (the s u m s on m, n in (6) above), used only the G r e e n ' s functions for f r e e e l e c t r o n s in the a b s e n c e of the magnetic field and r e t a i n e d only the phase t e r m in (5) above. The condition for instability c o m e s about by looking f o r n o n - t r i v i a l solutions of the equation A(r)= V fG(r~

exp ( i ~ -

(r' x r ) / A ( r + r ~ d 3 r ' .

(7)

As in ref. 2, a ( r ) can be a s s u m e d to be s m a l l and slowly varying n e a r the transition t e m p e r a t u r e and expanding the exponential, we m a y a r r i v e at a differential equation for it. { i o + ! / (1 ) ( V + 2 i ~e- A ) ±2+ ~' [ (°2) 3--~0 2 ] A(r) = __2~ A(r) Here

[~1)

(8)

p2

Here the - sign implies the components perpendicular to the field direction of H. This is an effective Schr(Jdinger equation for the Cooper p a i r in the magnetic field with different effective m a s s e s along and perpendicular to the field. This f e a t u r e d i s a p p e a r s when the Landau level states for the electrons a r e r e p l a c e d by f r e e e l e c t r o n states (when /2(1) b e c o m e s 2/2(2)) as was done by Evans and Rickeyzen [2]. It is interesting to point out that the usual Ginsburg-Landau equations as derived by Gorkov f r o m the superconducting state is also obtained in this limit. The supercooling field obtained now f r o m (8) is

The details of these calculations which include Landau levels and the investigation of the s u p e r c o n ducting state will be p r e s e n t e d elsewhere. 586

Volume 20, number 6

PHYSICS

One of us (R.V.) thsnks Professor Callaway for discussions.

1 April 1966

LETTERS

K. M. Watson for kind encouragement.

We thank Professor

J.

References 1. D. J. Thouless, Ann:Phys. (NY) 10 (1960) 553. 2. W. A. B. Evans and G. Rickayzen, Proc. Phys. Sot. (London) 63 (1964) 311. 3. D. H. Douglas Jr. and L. M. Falicov, Progr. in low temperature physics, ed. C. J. Gorter, Vol. IV, (North-Holland Publ. Comp., 1964) p. 97. 4. A. K. Rajagopal, Phys. Rev. (to appear in Feb., 1966 issue). 5. A. K. Rajagopal, Physics Letters 5 (1963) 40. *****

SURFACE NUCLEATION IN A SUPERCONDUCTOR COATED WITH A NORMAL METAL *

Laborutoire

J. - P. RURAULT de Physique des Solides associk au C. N. R. S., Facu.lt& des Sciences, 91-Orsay Received 19 February 1966

We show that the surface sheath [l] usually present in a type II superconductor S under high fields is completely destroyed (HII= Hc2 exactly) when a normal metal N is deposited on top of S, provided that: (1) the electrical contact between N and S is good; (2) the conductivity 0 of N is larger than the normal state conductivity as of S. For the opposite case UN
We restrict our attention to dirty metals and assume a perfect contact between N and S along the Oyz plane (Hr to Oz), the N region corresponding to x > 0. For H = H,, , the pair potential A(%) (in a gauge where A is real and where Ay = H(x-x0)) is ruled by the equations deduced from refs. 2-5. s:

d2,+(F)” H2(x

-ax2

0

with A(x .+-c9)=0;

- x~)~A = A, t2S

(Y2(X)(X-Xo)A22=0

(3)

With

(1)

_“““+(F)2

H2(x - X~)~A = -A, (2) 0 dx2 t2XT L. with A (x -+m)=O. 4s and
+oO

s-co

The optimum x0 corresponds to zero total current in the sheath. From refs. 3 and 5, this can be written

* Supported by the Centre National d’Etudes Spatiales. ** From notations of ref. 4, tN = K-i.

In this last relation, V is the BCS interact&i, N is the state density at the Fermi level, $2 is the derivative of the digamma function and D is the diffusion coefficient such that a = 2 NDe2 . In practice, the N side.

a2 is also of the order of unity on

We assume again the boundary conditions

of

ref. 6

(G)o_

=;(E)

0+

=;

(7 =3

.

(4)

The extrapolation length b must be deduced from 587