Influence of the phonon spectrum and the Coulomb interaction on the jump in the specific heat of superconductors

Volume 42 A, number 1

6 November 1972

PHYSICS LETTERS

INFLUENCE OF THE PHONON SPECTRUM AND THE COULOMB INTERACTION ON THE JUMP IN THE SPECIFIC HEAT OF SUPERCONDUCTORS A.A. COLUB and L.Z. KON Institute of Applied Physics, the Moldavian Academy

of Sciences,

USSR

Received 4 October 1972

The jump in the specific heat (Cs/Cn)7-=~ is evaluated by taking account of the Coulomb interaction as well as the interaction between electrons. For this p&pose the integral equations for the energy gap function A(w) and for the renormalization parameter Z(w) are used.

In the strong-coupling superconductors (for example Pb) the experimental values of the jump in the specific heat is larger than the theoretical values predicted by the BCS theory [l] . To explain this fact the authors of [2] represent the effective electron-electron interaction by a retarded phonon exchange. However in [2] the Coulomb interaction V, is neglected and the integral equation for the renormalization parameter Z(o) [3,4] is not considered. In this paper we evaluate the jump in the specific heat (Cs/Cn)T= Tc by taking account of the Coulomb as well as the phonon interactions between electrons. For this purpose we use the integral equations for the energy gap function A(w) and for the Z(o) [3,4] at a temperature Tnear the transition temperature T,. We assume that the phonon spectrum is described approximately by one Einstein peak with frequency a. To write the approximate function A(o) we use the first iteration of the linearized equations [3,4] . Using the Bardeen-Stephen [5] expression for the critical magnetic field, the jump in the specific heat can be written as (C S /Cn)T=T where

C

= 1 + $ ,4/o;

(0

Table 1 The constants h, /J and the values of the jump in the specific heat for some superconductors.

A

; a2 2 QO

(C&)~=~ (CsICn),

c

(CslCn)exp

Pb

Nb

Va

Ta

Sn

In

1.12 0.10 1.58 1.37 3.73

0.82 0.13 1.12 1.07 2.76

0.60 0.13 1.00 0.98 2.53

0.65 0.13 1.02 0.99 2.56

0.60 0.10 1.02 1.01 2.57

0.69 0.10 1.08 1.04 2.67

3.32

2.66

2.50

2.52

2.54

2.61

2.57 2.58 [ 2.491

2.60

2.73

3.65

3.07

[ 2.871

2(Z(o n T) -Z n (o n )) r12T2 2n + 1

AZC

(P2(o-+rTc) T=T,

1 ;

o=A/IIT,~~;A=A(O,T,); Z,(c+)

= Z(O, T,); ~(a,

Tc) = A(anTc)/A*

In the BCS theory o2 = $ = 0.95. We obtained the following: (1) the coefficient (Y, (2) the coefficient oo, (3) the jump in the specific heat (Cs/Cn)T= T,, (4) the jump in the specific heat (C,/C,), when the Coulomb interaction was neglected All our results, the values of the effective coupling constant h and of the Coulomb pseudopotential ~1, taken from ref [6] , are listed in table 1. The 11

Volume 42 A, number 1 experimental

values of the jump

PHYSICS LETTERS in the specific

is taken from ref. [7] . The values in the bracket taken from

heat are

refs. [8] and [9] respectively.

References

[l] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.

[ 21 B.T. Geilikman and V.Z. Kresin, Fiz. Tverd. Tela 9 (1967) 3111. [3] G.M. Eliashberg, Zh. Eksp. i Teor. Fiz. 38 (1960) 966.

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6 November 1972

I41 D.J. Scalapino, J.R. Schrieffer and J.W. Wilkins, Phys. Rev. 148 (1966) 263. [51 J. Bardeen and M. Stephen, Phys. Rev. 136 (1964) A1485. 161 W.L. Mcmillan, Phys. Rev. 167 (1968) 331. I71 E.A. Lynton, Superconductivity (London, 1969). [81 H.A. Leupold and H.A. Boorse, Phys. Rev. 134 (1964) A1322. [91 R. Radebaugh and P.H. Keesom, Phys. Rev. 149 (1966) 209.