Temperature dependent interaction and thermodynamic properties in superconductors

Volume 28A. number 6

PHYSICS L E T T E R S

TEMPERATURE THERMODYNAMIC

DEPENDENT PROPERTIES

30 December 1968

INTERACTION AND IN SUPERCONDUCTOF{S

A. ROTHWARF

RCA Laboratories. Princelon. New Jersey. 08540. USA Received 11 November 1968

The temperature dependence of the interaction in superconductors is used to calculate the relation between A(0)/kT c and (dh/dt) t I as well as the specific heat jump at Tc.

It has been previously demonstrated [1] that the variations in the ratio A(O)/kT c seen experimentally can be understood within the BCS f r a m e w o r k , if the interaction is t e m p e r a t u r e dependent. In view of the e m p i r i c a l relation (dh/dt)t= 1 ~ A(0)/kTc, which Toxen [2] had reported we have considered some further consequences of the t e m p e r a t u r e dependent interaction. The quantity (dh/dt)t= 1 = (TcHoldHc/dT)Tc in the BCS [3] theory is given by I dh l=l T---c-c ½1 ~-= 4~ H2 (C s - Cn)Tc : =

Tc dA2~ ½[ ~ 2 ( 0 ) ~ - /Tc

(1)

where (C s - Cn) T is the specific heat jump, H c is the critical magnetic field, A the energy gap, N(0) the single spin density of states at the F e r m i level, and the weak coupling BCS relation Ho2 -- 4~N(0)A2(0) has been used In this weak coupling limit (dh/dt)t= 1 is determined p r i m a r i l y by ~(dA2/dT)~r^. To see the r e " t; lation between dA 2 /dT and the t e m p e r a t u r e dependent interaction consider the BCS gap ~equation which can be w r i t t e n as V -1 = ~k,(2Ek,(T))-I tanh[Ek,(T)/2kT ], where V is the effective interaction, Ek,(T) = = [e '2 +A2(T)]~ and E' =/~2k'2/2m - ¢ F. Differentiating this equation with r e s p e c t to T and evaluating in the BCS approximations leads to the result at T c _

_ N(0) ~A2~

"

V 21 ~_~)dV = N(0)Tctanh I2 k ~ c +2(2kTc )2 ~T ]Tc Tc . . . . .

0.186

+2 tanh

-

k w ')

~

"

(2)

Setting d V / d T = 0 gives the BCS result for ~ A 2 / ~ T ) T e (in the limit tanh ( ~ w / 2 k T c ) -" I). For d V / d T ) T ~ ¢ ¢ 0 there is clearly an additional contribution to ~ A 2 / ~ T ) T . In principle V could have any temperature dependence; from eq. (5) of ref. i, however, the intrinsic [emperature dependence of the Eliasberg interaction []4 is given2by dV/dT = (~A2/~T), (dV/dA2). This leads to ~A2/~T)T~_ = [~A2/~T)Tr]BCS/(I_ -A), where A = 9.40 (kT c) (dV/dA*)/~'2]T ¢. One thus has for the specific heat jump [(C s - C u ) / C a ] T c ~ = 1.43 ( 1 - A ) -1. Neglecting the repulsive Coulomb interaction and using eqs. (4) and (5) of [1] we nave ~A2~

~2~BCS

~T ]Tc - - ~

ITc

1+

4.70(kTc)2 N(0)Vc

(P/~2-E ' ) - 1

i e ' ( e ' + ~ 2 ) 2 ':

(3)

w h e r e ~ is a c h a r a c t e r i s t i c phonon frequency and e ' is a r e p r e s e n t a t i v e e l e c t r o n energy which is dominant in the interaction. The e x p r e s s i o n for dh/dt)t= 1 for small deviations f r o m the BCS value is

dh\

kTc

°t/t-r-:J = 3.07 ~-~-~) 1 =1

2.35(kTc)2 ( ~ - E ' ) N(0)V c E'(E ..b ~ ~'~) 2

In the same approximation, f r o m eqs. (6) and (7) of [1]

430

(4)

_

•

Volume 28A. number 6

PHYSICS

A(0)

A(0)-'

kTc = kTc JBCS

LETTERS

30 December 1968

i ( ~ a - e ' ) ( E ' - e') 1 - N(O)V-----~ c (E +/~a)(E' +~/R)

(5)

W h i l e at f i r s t g l a n c e it a p p e a r s that dh/dt)t= 1 i s i n v e r s e l y p r o p o r t i o n a l to A(O)/kTc, on i n s e r t i n g eq. (5) into (4) one h a s dh).BCS t l -

(h-~-E')

-2"35(kTc )2

(E'-E')~

I

(6)

The second t e r m s i n eqs. (5) and (6) a r e of the s a m e o r d e r of m a g n i t u d e , and f o r E' < / / ~ , a s u i t a b l e E' can be found such that * c ~ / d t ) t = 1 = A(O)/kT c. F o r E' > ~ (5) and (6) w o u l d p r e d i c t A(O)/kT c > > dh/dt)t= 1. I f one i n c l u d e s the C o u l o m b r e p u l s i o n t e r m , h o w e v e r , w h i c h enhances * * d h / d t ) t = 1 m o r e than A(O)/kTc, t h e n the a p p r o x i m a t e e q u a l i t y b e t w e e n the two can b e m a i n t a i n e d o v e r t h e e n t i r e r a n g e f o r w h i c h the l i n e a r a p p r o x i m a t i o n s hold. C o r r e c t i o n s f o r s t r o n g c o u p l i n g s u c h a s S h e a h e n [5] h a s s u g g e s t e d can b e i n c l u d e d but do not c h a n g e o u r c o n c l u s i o n s . A s in r e f . 1, we find that f o r t h o s e m a t e r i a l s f o r w h i c h A(O)/kT c i s l a r g e , to o b t a i n n u m e r i c a l a g r e e m ~ n t w i t h e x p e r i m e n t v a l u e s of ~ 2 ~ A(0) m u s t b e u s e d . We c o n c l u d e f r o m t h i s a n a l y s i s t h a t T o x e n ' s p r o p o s e d r e l a t i o n i s not s i m p l y a n u m e r i c a l c o i n c i d e n c e a s s u g g e s t e d by G r u n z w e i g - G e n o s s a r and R e v z e n [6]. I n s t e a d we h a v e found that dh/dt)t= 1 and A(O)/kTc, by v i r t u e of the t e m p e r a t u r e d e p e n d e n c e of the i n t e r a c t i o n , v a r y in t h e s a m e w a y and c a n r e m a i n n e a r l y e q u a l , o v e r t h e i r e n t i r e r a n g e of v a l u e s . T h e i n t r i n s i c t e m p e r a t u r e d e p e n d e n c e of the i n t e r a c t i o n t h u s s e e m s to l e a d to n u m e r i c a l l y c o n s i s t e n t e x p r e s s i o n s f o r the d e v i a t i o n s of A(O)/kTc, dh/dt)t= 1 and ( C s - Cn)Tc/Cn f r o m t h e i r BCS v a l u e s .

• dh/dt)pCS = 1.737 while A(O)/kTc)BCS = 1.764; for the purposes of the discussion here, these are considered to be equal. • * The enhancement factor is such that the second t e r m in eq. (4) is multiplied by ~2 while the second t e r m in eq. (5) by 0~, where ot ~ N(O)Ve/(N(O)Vc- 0.1).

1. 2. 3. 4.

A.Rothwarf. Phys. Letters 26A (1967) 43. A.M.Toxen, Phys. Rev. Letters 15 (1965) 462. J . B a r d e e n , L.N. Cooper and J . R . S c h r i e f f e r , Phys. Rev. 108 (1957) 1175. G . M . E l i a s h b e r g . Zh. Exp. i Theor. Fiz. 38 {1960) 966: Soviet Phys. J E T P 11 (1960) 696; S. H. Liu. Phys. Rev. 125 (1962) 1244. 5. T. P. Sheahen. Phys. Rev. 149 (1966) 370. 6. J . G r u n z w e i g - G e n o s s a r and M.Revzen. Phys. Rev. Letters 16 (1966) 131. * * * * *

431.