Specific heat of type II superconductors near Tc

Solid State Communications,

Vol. 12, pp. 771—772, 1973.

Pergainon Press.

Printed in Great Britain

SPECIFIC HEAT OF TYPE II SUPERCONDUCTORS NEAR T~ O.L. de Lange Clarkson College of Technology, Potsdam, N.Y. 13676, U.S.A.* (Received 14 December 1972 by R.H. Silsbee)

An explanation is suggested for the anomaly that has been observed in the mixed to normal state specific heat discontinuity of type II superconduc. tors in low fields, near RECENTLY Ehrat and Rinderer1 have measured the

Numerical values for the S Jj andin f2 specific have been 3 The1 ,52, difference heat calculated by Usadel.

mixed to normal the upper critical state field specific in Nb heat discontinuity at 80 Mo20(T0 = 4.16 K). and These are the first measurements very close to 7, they find a marked deviation from the GLAG theory. At lower temperatures there is good agreement with theory. The specific heat, in constant magnetic field H, is uncertain close to 7~,2[42(T~ 2)=JfJ because the transition is broadened by alloy inhomogeneities. Therefore the specific heat jump at the transition2 is determined by linear extrapolation of a C/Tvs. T plot to the critical temperature, 7~.

2 IaT21[t~G,,N(H,7)]H,

t~CMN(H,T)=

—TEa

is, to the first order in 42

~c~(H, ~

=

—

H,

2TXm(d142/d 7)2 +

2Hc 2T[2(dX,,,/d7)(dH~/d7)+ Xmd

2/dT2

—

21(14 3rn(d11’a/dT)

2 —10,

()

and hencç, at the upper critical field,

[a/a(T2)][ACMN/TIH_HC 3

—

Xm (dH~

2

where2

Xm (T)

_Jj)2 =

I ~ *

—

1)2

3].

—

2/dT)

-~

[a/a(T2)J[~CMNIT]HH~,~

=

3.

+~m(TXHc2 fl)3,

—(3/T)4%,, (dHa/d7)

In Fig. 1 the slope of the specific heat jump at the transition calculated from equation (4) is plotted

[8ir~3(2~c~1)1’ —

for Nb 1

90Mo20, as a function of reduced temperature, and compared with the experimental values obtained from reference 1. Below 0.985 7~,the calculated slope

(2)

=

l2irHa(T) S1(24—1)+2S2~ 24{ti(T)+2i4f2(7)}~ (2ac~

2) 4’m(dHc 2/dT

Higher order terms in equation (1) make no contribution to equation (4). As T i,, 4’m(T) diverges, and equation (4) becomes

and, in the dirty limit,3 4)m(T)

2H~ 2IdT)(d

(1)

Xm(TXHc

(4)

(3/T)[dXm/dTXdHc2/dT)

—

t~IGMN(H,T)= H2/8ir

=

2 +

The of purpose of this noteis to examine consistency the extrapolation procedure withthe respect to the GLAG theory. Therefore we consider the dif. ference between the Abrikosov free energy and the free energy of the normal state, to third order in 42 H:2

(2~~ l)~ —

I

Present address: 3835F Miramar, La Jolla, Ca.92037. 771

772

SPECIFIC HEAT OF TYPE II SUPERCONDUCTORS NEAR 13 I

0.5

-

-

I

04

\~

=o.~ss’~~’~

-

—

a reduced field he

I

[

-

-

—2.0

-

0.80

o.~s

o~,s

oJ~o

i.oo

= T,’ T~

FIG. 1. Comparison of the theoretical and experimental slope of the specific heat jump at the upper critical field vs. reduced temperature, t = T/73, for Nb80Mo20. The solid line is calculated from equation (4) for a tnlattice. The experimental points are from

Vol. 12, No.8

H/Hc2(Te), where H is the constant applied field. We expect the extrapolation to be most reliable when he is nearly unity. From reference 1 we find that he ~ 0.97 for the plot with 132 = 0.91 73, H = 685 oe; and he ~ 0.2 for the plot with T~2= 0.997 73, H = 25 oe. For a description of the specific heat well below the transition i.n small fields (732 ~ 73) higher order terms must.be included in equation (3). However when 732 is not too close to 73, equation (3) is expected to be a good approximation; indeed equation (3) is in excellent agreement with the measured specific heat atH= 685 oe (732 = 0.91 73), not only at the transition but also for temperatures with reduced fields between 0.4 and 1. Similar remarks apply to TiM Mo16, where above 0.98 73 there is indication that the extrapolated heat jump exceeds 5 Again,specific the calculated slope of the the~ GLAG values. specific ture, whereas heat jump the slopes diverges of the to extrapolated above thislines temperaare =

—

positive. is positive, and at 0.91 13 there is excellent agreement between theory and experiment. Above 0.985 73 the calculated slope diverges rapidly to at 7,,” and the slopes of the extrapolated lines are about three times larger than the GLAG values in Fig. 1. These resuits suggest that very close to 13 the extrapolation yields an overestimate of the specific heat, although it is satisfactory (within the experimental error of about 1%) at lower temperatures. Below 0.985 13 the extrapolated ~C~/T are in good agreement with the GLAG theory, but are as much as 10% larger than the GLAG 1 The temperature, Te values above this temperature. at which the extrapolation of C/Tvs. T2 starts defines —

*

We conclude that according to the GLAG theory there is substantial negative curvature to the C/Tvs. T2 plot near the transition in low fields. The rounding of the transition would mask this effect, and hence measurement on an alloy with an extremely sharp transition is required. An independent test of the theory near 73 would be provided by measurements of the entropy per flux quantum, S~(S,= i~a2 ~G~/ aHaT). Calculations similar to the above predict that for Nb 80Mo20 the ] slope of S~at(positive) the upperabove critical field [(aSdall),i is negative (below) T= 0.957 73. —

Note that the sign (negative) of 4~,for Nb 85 Mo15 is confirmed by experiment (4). REFERENCES

J. Low Temp. Phys. 7,533 (1972).

1.

EHRAT R. and RINDERER L.,

2.

See for example, FETTER A.L. and HOHENBERG P.C., Superconductivity (Edited by PARKS R.D.), Marcel Dekker, New York (1969). USADEL K., Phys. Rev. B4, 99 (1971). DE LANGE O.L. and OTTER F.A. Jr., Solid State Commun. 9, 1929 (1971). Measurements of BARNES L.J. and HAKE R.R., Phys. Rev. 153,435 (1967), on TiMMol~are comparedwith the GLAG theory in Fig. 1 of reference 1.

3. 4. 5.

Nous preséntons ici une explication pour l’anomalie dans la chaleur specifique d’un supraconducteur de type-Il a H~2prés de 73.