A simple Tc equation

Solid State Communications,Vol. 19, pp. 1131-1132, 1976.

Pergamon Press.

Printed in Great Britain

A SIMPLE Te EQUATION J.M. Rowell Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A.

(Received 19 April 1976 by R.H. Silsbee) Based on recent work of Allen and Dynes, and the existing specific heat and tunneling data, a simple Te equation is Tc = 00[20 (X -- 0.25).

NUMEROUS approximations to the Eliashberg strong coupling equations of superconductivity I now exist, the best known being the "McMillan Te equation". 2 This links Tc with an average phonon frequency of the metal (WAve), an average electron-phonon coupling strength (X) and a measure of the Coulomb repulsion between electrons 0a*), which is usually taken to be 0.10 or 0.13. The various more accurate Tc equations a'4 are marked by more complexity, and hence less general usefulness. However, the recent work by Allen and Dynes 4 suggests a particularly simple relationship between Te, WAw, and X. They have shown (see, for example, Figs. 8 - 1 0 of reference 4) that, over a surprisingly wide range of (0.75 < h < 2.6), To/wAve depends linearly on X, particularly when ¢OAve is defined by Wlog. To determine WAve, tunneling or neutron scattering has to be carried out, but one might hope that the Debye temperature would also yield a similar relation, namely Tc/0o would be roughly proportional to X. To demonstrate this dependence, Table 1 and Fig. 1 show data for all materials where 0o is known from specific heat measurements (taken from Table III of reference 2) and X from tunneling. The line drawn, chosen for its simplicity and not necessarily as a best fit to the data, is given by Tc = 00/20 (X -- 0.25). Because of the unusual phonon spectrum of mercury, this point has been discounted somewhat. The point for Nb3Sn, using specific heat data from Harper 5 and the ~ estimated by Allen and Dynes, is satisfyingly close to the line based on the s-p metals and one transition metal.

Table 1

In Sn Hg TI Pb Ta NbaSn

Te K

0o K

Tc/0o

X

3.4 3.72 4.16 2.38 7.19 4.48 17.8

112 200 72 79 105 258 242

0.0304 0.0186 0.0578 0.0301 0.0684 0.0174 0.074

0.805 0.72 1.6 0.795 1.55 0.69 1.67

08

I

]

I

l •

I /Nb~,Sn

II

//

"°71 .06 .05

/./ I I

M~i,~A~

o Hg

p.* = A3--~"

/! _o

.04 / .03

]

T

n

/ .o2

-Tc=~((k-0.25)

Sn

.ol

O

0

1.0

2.0 k

Fig 1. Tc/Oo is plotted vs X for the elements where 0o is known from specific heat measurements and ), from tunneling. The solid line is Tc = 0o[20 CA- 0.25). The dashed lines are given by McMillan's equation with /a* = 0.10 and 0.13. The McMiUan Tc equation, Tc -

0o ( 1.04 (1 + ~.) 1.45 exp \ - - - - - - ~ ~--b~2)t)'}

has been evaluated with/a* = 0.10 and 0.13 and is also plotted in Fig. 1. As is well known, it agrees well with the data for the low Te materials but underestimates X for the high Te materials. I suggest that Tc = 0o/20 (X -- 0.25) can be used as the simplest possible Te equation, and also point out that many more data points could be added to Fig. 1 if specific heat data were available, for example for the T1-Pb and Pb-Bi alloys, which, as tunneling experiments indicate, cover the entire range of ~ between 0.8 and 2.1.

1131

1132

A SIMPLE Te EQUATION

Vol. 19, No. 11

REFERENCES

1. 2.

ELIASHBERG G.M.,Zh. Eksp. Teor. Fiz. 38,966(1960);39,1437(1960) [Sov. Phys.-JETP l l , 696 (1960); 12, 1000 (1961)]. McMILLAN W.L.,Phys. Rev. 167,331 (1968).

3.

LEAVENS C.R., Solid State Commun. 13, 1607 (1973); GARLAND J.W. & ALLEN P.B., Superconductivity (Edited by CHILTON F.), p. 669. North Hollartd, Amsterdam (1971).

4.

ALLEN P.B. & DYNES R.C., Phys. Rev. B12, 905 (1975).

5.

HARPER J., Thesis, Stanford University (1975).