Quantum fluctuation induced diamagnetization in a superconductor

Volume 38A, number 6

PHYSICS LETTERS

QUANTUM

13 March 1972

FLUCTUATION INDUCED DIAMAGNETIZATION IN A SUPERCONDUCTOR

M. FIBICH and M. REVZEN Department of Physics, Technion, Israel Institute of Technology, Haifa, Israel Received 24 January 1972

An expression for the free energy in superconductor that was used to calculate the fluctuations induced diamagnetism is derived from a quanturh theory of fluctuations.

The fluctuations induced d i a m a g n e t i s m in a s u p e r c o n d u c t o r was r e c e n t l y c a l c u l a t e d by Nam [1]. The c a l c u l a t i o n s accounted well f o r e x p e r i m e n t a l r e s u l t s of Gullub et al. [2]. No a d j u s t a b l e p a r a m e t e r was used. We s h a l l show below that the fluctuations d i s t r i b u t i o n function u s e d in the c a l c u l a t i o n [1] is d e r i v a b l e f r o m quantum f l u c tuation t h e o r y [3]. A c c o r d i n g to the quantum fluctuation t h e o r y the p a r t i t i o n function due to fluctuations i s given

by [3] Z = exp ( - ~ F ) =

z=

f

f

d 2 ~ ( r ' s ) exp[-fl./~(~)]

(1)

ff

1

ds f d3r ~*(r,s)(-~-~ ~(r,s)) ÷ 0 1 + ~ dsh(A(r,s),

= ~

k, n

~vkn #kn(r)

w h e r e @kn a r e the eigenfunctions of r e f . [4]. Using eq. (4) we g e t

= ~ (iw v +~Enk)[ ~kknv[ 2 • knv

C'c

1

~"

Fig. 1. The contour of integration C of eq. (7) is deformed into the contour C' around the cut of In z. The cut was chosen to be along the positive real axis of z. the v a r i a b l e s of i n t e g r a t i o n in eq. (1) f r o m A(r,s) to Akn ~ the i n t e g r a t i o n can be r e a d i l y done to yield

-~F = u~E f~_~dkln(iwv + flEnk )-1

.

(6)

In eq. (6) the sum o v e r k was changed into an i n t e g r a l . By the s t a n d a r d p r o c e d u r e [5] we can c o n v e r t the sum o v e r v into an i n t e g r a l ( s e e fig. 1) : ~ln

1

(iwv+ flEkn) =

(3) = ~2~ii

F u r t h e r we expand Av(r) in the c o m p l e t e o r t h o n o r m a l Landau s t a t e s (4)

~Av(r)

Re Z

(2)

w h e r e we adopt the notation of ref. [1]. h is the f r e e e n e r g y d i f f e r e n c e between the n o r m a l and the s u p e r c o n d u c t i n g s t a t e and ~ i s the n o r m a l i z e d [1] o r d e r p a r a m e t e r . Using the p e r i o d i c i t y ~(r,0) = A ( r , 1 ) , r e f . [3], we expand

~(r,s) = ~ exp (iwvs) Av(r) , ~ov = 2~v, v = 0,+1, ±2,....

~ ] Im Z

(4)

8h/~A* = O, (5)

With Ekn a s given in eq. (4) of r e f . [1]. Changing

lnz : l n ( 1 - exp(-flE)). (7) exp(z - ~ E k n ) - 1

The l a s t s t e p is gotten by taking the d i f f e r e n c e of the log z function below and above the cut (fig. 1). Substituting this r e s u l t (eq. (7)) into eq. (6) we obtain

-~F = ~ ~ f

-(dkln(1

-

exp

(-#Ekn)) ,

(8)

eq. (8) i s e x a c t l y eq. (5) of ref. [1]. Thus we have shown that the fluctuation d i s t r i b u t i o n function a s u s e d by Nam i s a consequence of the quantum fluctuation t h e o r y . 385

Volume 38A, number 6

PHYSICS

Reference s [1] S. B. Nam, Phys. Rev. L e t t e r s 26 (1971) 1369. [2] J. P. Go[lub, M.R. Beasley and M. Tinkham, Phys. Rev. L e t t e r s 25 (1970) 1646. [3] M. Revzen, Phys. L e t t e r s 29A (1969) 443.

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LETTERS

13 March 1972

[4] L. Landau and E. Lifshitz, Quantum mechanics (Pergamon P r e s s Ltd., 1958). [5] J. M. Luttinger and J. C. Ward, Phys. Rev. 118 {1960) 1417.