Heat capacity studies of high Tc superconductors

Journal of the Less-Common Metals, 62 (1978) 127 - 136 0 Elsevier Sequoia S. A., Lausanne - Printed in the Netherlands

HEAT CAPACITY

STUDIES

127

OF HIGH T, SUPERCONDUCTORS*

G. S. KNAPP Argonne National Laboratory, Argonne,

III. 60439 (U.S.A.)

(Received June 20,1978)

Summary This paper reviews a number of heat capacity studies of superconductors. We show that by analyzing the data over a wide range of temperature much information about the electronic, lattice and superconducting properties of a material can be derived. We first review how McMillan’s strong coupling parameter h can be determined solely from heat capacity data. We then discuss detailed studies of the A15, Cl5 and ternary molybdenum chalcogenides. These studies provide information about electronic screening effects on superconductive and lattice properties of the Al5 and Cl5 materials and about the distribution of phonon modes in the molybdenum alloys. It is an honor to have been asked to contribute an article to this special issue commemorating the sixtieth birthday of Dr. B. T. Matthias. I was one of Dr. Matthias’s early students (1962 - 1967), and I aided in the setting-up of some of the first equipment used in his laboratory. I studied magnetism for my thesis research, but became interested in superconductivity after coming to Argonne. It is this latter work which I would like to discuss. In 1972 a student of mine, Ronald Jones, and I became interested in determining the magnitude of h, the McMillan strong coupling parameter [l] , from heat capacity data. At that time there were no experimental determinations of h for transition metals, yet almost every paper on strong coupling superconductivity discussed this parameter. The parameter h is related to the measured low temperature specific heat coefficient y. by To-

lim C

2

T-O~=yn2h;N(8f)(l

+h)=-&s.+Yl

(1)

where N(E,) is the band structure density of states at the Fermi level, ha the Boltzmann COnStad and +rb.s. and yi the band structure and electron-phonon renormalization contributions to y, respectively. At that time few band calculations were reliable enough to calculate N(E,), so eqn. (1) was not useful in determining h. However, it was known [2] that, at temperatures above the Debye temperature 0 n, yr + 0; an experimental determination of the high temperature electronic specific heat coefficient, extrapolated back *Dedicated to Professor B. T. Matthias in celebration of his 60th birthday.

128

to T = 0,could therefore be used to measure IV(&); eqn. (1) would then yield the magnitude of A. We found that it was possible by careful thermodynamic analyses to separate the high temperature electronic contribution from other contributions to the heat capacity that were of similar or larger magnitude. Figure 1 shows examples of such determinations for Nb, V and Ta. The curved broken line is the theoretical line describing yl(T) and the straight broken line is the extrapolation to yield 7b.s. (0). Within the limits of experimental error, the results for iV(&) and h are consistent with band calculations or with the McMillan expression for T, T,

=

1.04 (1 + h)

!k 1.45 exp -

h -j~*(l

+ 0.62X)

(2)

where p * is taken from Bennemann and Garland [ 31, Knapp and coworkers have determined h for SC, Y, Pt, Pd and V [ 41, for Nb and Ta [ 51 and for a-U [6].

8-

VANADIUM

6--_-_--_-_-----_D 4-

0 P

0

0

O-

z-

TANTALUM

TEMPERATURE,

K

Fig. 1. Plot of 7 us. temperature for vanadium, niobium and tantalum. Dotted lines are the theoretical temperature dependences calculated by Grimvall [ 2 1. Broken lines are the constrained extrapolations of the unenhanced specific heat coefficients to 2’ = 0.

129

Encouraged by the success of this work, a postdoctoral fellow, Samuel Bader, and I attempted to apply this method to determine h for some vanadium-based Al5 compounds. We found that some of these materials had unusually large anharmonic contributions to the specific heat, making the procedure developed earlier less accurate. However, by paying careful attention to anharmonic and temperature-dependent electronic contributions to the heat capacity, we were able to find a number of interesting relaticnships between the superconducting, electronic and lattice properties of these materials [ 7, 81 . Heat capacity data reflect certain averages of both the electronic and the lattice densities of states. In the past, most data on superconductors were taken at low temperatures and only y and a low temperature Bo were determined. This Debye temperature gives information about the very long wavelength phonons and does not reflect any properties of the optic modes. Much more information can be obtained by measuring the specific heat as a function of temperature and analyzing the data properly. For harmonic solids, all moments of the phonon spectra are temperature independent and can be determined if the electronic contribution to the specific heat is known. For an anharmonic solid the phonon spectrum is temperature dependent and only the “zeroth” moment or the geometric mean frequency wg can be determined as a function of temperature, where wg=

(?l,w,

3N

1/3N

1

(3)

Here s labels the phonon mode and N is the total number of modes. The anharmonic contribution to the heat capacity is approximately linear and its coefficient A is related to w g by A=--3Rz

1

aa

-2~_-3R??2 s w, aT

wg aT

(4)

We analyzed the heat capacity data for a number of Al5 compounds in this way [B] . The results are illustrated in Fig. 2 as an effective Debye temperature 8 (T) = e1’3A w,(T)/k,. (The values of 0 (T) for Nb3Sn, VsSi and V,Ga are within 5 - 10% of the values obtained from inelastic neutron scattering experiments on polycrystals [ 91,) Nb,Sb behaves normally, indicating phonon softening with increasing temperature owing to lattice expansion. In contrast, the high T, compounds show considerable softening of 0 (T) with decreasing temperature. We found an interesting correlation between the anharmonicity and N(E,), which is illustrated in Fig. 3. The correlation of (l/w g) (ao,/a T) with N(E,) could be a manifestation of strong selective electronic screening. The high value of N(E,) can cause the frequencies of certain phonon modes to decrease significantly. These frequencies can be temperature dependent for two reasons. Firstly, electronic screening could cause the effective second-order term in the lattice potential energy to be reduced relative to the third- and fourth-order terms. This reduction enhances the anharmonicity. Secondly, electronic screening by

130

480-460

5

440

-

380

-

360

>

--,440

li

e

0

360 340

-

:

=

v, Sn ; ; .--.z360

-

-340 .-

34ot-

340

340’-

o-o-;;

3oa7

-,

Nb, “.

280-

I

100

c;:--:;cI;;

Nb,

Sn ________YII - -

NbsAI S”o,7 _

I

I

200

300

Sbo.3 _2 c

340 7, F-300

-

280

400

TCKf Fig. 2. The temperature dependence of the effective Debye temperature associated with the geometric mean phonon mode frequencies for the indicated Al5 compounds.

N(EF)

(states

/eV-atom)

Fig. 3. The phonon frequency shift parameter (-A/3R) of N(Ep) for the Al5 compounds.

z (l/w,)

(?k+/aT)

as a function

near-Fermi-energy electronic states can be temperature dependent because of sharp structure [7] in N(E) near E,. Table 1 shows X, Mw ,“, XMG:,” and the density of states for these compounds. Note the strong correlation between N(E,) and X for the vanadium but not the niobium compounds. The parameter hilfw~, which in ~c~ill~‘s original theory is equal to N(E,)12 where I2 is an electron-phonon coupling parameter, does not show any simple relationship to N(E,) or Z”,_Note, how-

Nb&l Nb3Sn NbBSb

2.4 2.3 0.4 0.1

0.2 0.8 -1.0

-_= 3R

dlno, dT

superconductors

V,Si VaGa V3Ga0.5Sno.5 V3Sn

of Al5

A

1

Compounds

Properties

TABLE

(1O--4 K-l)

1.6 2.4 0.4

3.8 4.8 2.7 2.0

N(EF) (states

eV_’

atom-l)

1.07 1.17 0.3

0.86 0.91 0.62 0.56

h

7.82 7.18 10.85

8.61 6.43 7.30 8.14

MO: (eV Aw2)

8.4 8.4 3.3

7.4 5.9 4.5 4.6

hMog2 (eV

Ae2)

132

ever, from Fig. 3 that there is a strong correlation between an electron property and a phonon property, the anharmonic specific heat coefficient. Later Hafstrom et al. [lo] applied these same techniques to the Cl5 materials V2Hfl _xTax. These materials show structural lattice transitions (Fig. 4) and are reasonably good superconductors (T, = 10 K). By varying x, large changes in both the electronic properties and the lattice properties can be studied. In addition, since Hf and Ta are adjacent atoms in the periodic table, only minor changes of atomic mass or atomic volume occur as a function of x, so that changes in phonon frequencies due to mass variation can be ignored. The Hf-rich VsHfr -.*.Ta, compounds have very large temperaturedependent magnetic susceptibilities and very large values of the electronic density of states as measured by heat capacity. Figure 5 shows both the magnetic susceptibility and the mean phonon frequency as a function of temperature for these Cl5 materials. The similar-

40

,0

35

30

-

-

V,

Hf

O v2

HfO.98

Tao.02

o

V2

H10.95

Too.o5

A

V2

Hfo.93

TOO.07

l

v2

Hfo.90

Tao.10

o

v2

HfO.80

Tao.20

A

V, To

s E ,o P 9 3

25

F u” 20

T PK)

Fig. 4. Heat capacity us. temperature in the vicinity of the lattice transformation perature 2’~ for V~Hfl-+Ta, compounds.

tem-

H

RANGE

OF XT MAXIMUM

T CK)

Fig. 5. Magnetic susceptibility and geometric mean frequency (expressed as Bo = 1.4 Ew,/kB) us. temperature. The lattice transformation temperature TL and the range of the maximum in the corresponding x(T) data are shown.

ity in the shapes of these curves and the variation with composition suggest that electronic renormalization of the phonon frequencies is very important. Using a simplified theory which elucidates the role of electronic screening in softening the phonon frequencies we showed that for this class of compounds [lo] 4

= a,2 (1

--F[X(nI)

where fl;2, is a mean bare phonon frequency characteristic and IQ(T)] is a function of the measured susceptibility

(5) of the materials which increases

134

monotonically with increasing x(T) and is independent of temperature or composition. The Q2, parameter is expected to be almost independent of composition because all the alloys have approximately the same mass and atomic volumes. Figure 6 shows a plot of the square of the effective Debye temperature 8, (which is associated with tip) versus x(T) for all Cl5 compounds. Note that eqn. (5) seems to be verified, confirming that electronic renormalizations of the lattice frequencies play a dominant role in the determination of lattice properties including the observed phase tr~sitions. The correlation between lattice softening and electronic properties in the V,Hfl _xTa, alloy are similar to the correlations (Fig. 3) observed in the Al5 materials. However, the effects are larger and there is no problem with mass variation, so the physical relationship given in eqn. (5) is more easily verified. The last study I would like to mention concerns the high T, Chevrel phase materials which Matthias et al. [ 111 discovered to be high temperature superconductors. These materials have the general formulae XMosSs and XMoeSes where X can be many different elements. Samuel Bader and I became interested in these materials and measured the specific heats of several different compounds. We found that most of these materials had very large heat capacities at low temperatures; this indicates that there must I

12

I

I

\

II

-

-

7

6

-

\

[II

V2Hf

’

‘2

O

Vz Hfo.95

Too.05

l

V2 Hfo.90

TQ0.m

’

‘2

ToO.20

A

V2 To

Hf0.98

ToO.O2

Hf0.80

5 I

2 X (Tf t10v4

3

4

emu/g-atom)

Fig. 6. Plot of e$, i.e. w:(T), us. x(T) with temperature as an implicit parameter. Data are plotted for the temperature range 100 - 300 K which includes the extremes in C+(T) and X(T) and, in two cases, the lattice transformation temperature.

135

be a large number of very soft phonon modes. We proposed that these materials could be thought of as molecular crystals on the basis of these data [ 12, 131 and subsequent inelastic neutron data [ 131 obtained in collaboration with Sinha at Argonne and Schweiss and Renker at Karlsruhe. The MO,& units could be considered to be quasi-rigid units which were weakly bound to each other and to the X atoms. A full lattice dynamical calculation has been carried out by Bader and Sinha [ 141 and their calculations confirmed that the molecular crystal concept is indeed useful for viewing the lattice dynamics of these materials. I have attempted to illustrate in this short paper that important information can be derived for superconductors from heat capacity measurements that are extended to high temperatures. Unfortunately, most measurements in the past were only made at low temperatures so much information was lost. Measurements above 70 K are not easy to make and perhaps this has limited the use of the technique. In this paper I have not attempted to cite the extensive literature (see Douglass [ 151) concerning the subjects discussed. I have even largely ignored other measurements made by my colleagues at Argonne. These measurements are discussed by Fradin et al. [ 161. Most of the work discussed above would not have been possible without the pioneering work of B. T. Matthias and his colleagues. Virtually every system I have discussed here had been either discovered or extensively studied by Dr. Matthias and his coworkers. Being a former student of his, I owe an extra debt to Bemd, for I benefited both from his wisdom and from having the privilege of working and learning in his laboratories.

Acknowledgments I wish to thank my many collaborators including A. T. Aldred, S. D. Bader, F. Y. Fradin, Z. Fisk, R. W. Jones, T. E. Klippert and C. W. Kimball. This work was supported by the U.S. Department of Energy.

References 1 2 3 4 5

W. L. McMillan, Phys. Rev., 167 (1968) 331. G. Grimvall, J. Phys. Chem. Solids, 29 (1968) 1221. K. H. Bennemann and J. W. Garland, AIP Conf. Proc., 4 (1972) 103. G. S. Knapp and R. W. Jones, Phys. Rev. B, 6 (1972) 1761. G. S. Knapp, R. W. Jones and B. A. Loomis, in W. J. O’Sullivan, K. D. Timmerhaus and E. F. Hammel (eds.), Proc. 13th Int. Conf. on Low Temperature Physics, Boulder, Colorado, 1972, Vol. 3, Plenum Press, New York, 1973, p. 611. 6 S. D. Bader and G. S. Knapp, Phys. Rev. B, 11 (1975) 3348. 7 G. S. Knapp, S. D. Bader, H. V. Culbert, F. Y. Fradin and T. E. Khppert, Phys. Rev. B, 11 (1975) 4331. 8 G. S. Knapp, S. D. Bader and Z. Fisk, Phys. Rev. B, 13 (1976) 3783.

136 9 B. P. Schweiss, B. Renker, E. Schneider and W. Reichardt, in 0. H. Douglass (ed.), Proc. Conf. on Superconductivity in d- and f-band Metals, 1976, Plenum Press, New York, 1976, p. 189. 10 J. W. Hafstrom, G. S. Knapp and A. T. Aldred, Phys. Rev. B, 17 (1978) 2.892. 11 B. T. Matthias, M. Marezio, E. Corenzwit, A. S. Cooper and H. E. Barz, Science, 175 (1972) 1465. 12 S. D. Bader, G. S. Knapp and A. T. Aldred, Ferroelectrics, 17 (1977) 321. 13 S. D. Bader, G. S. Knapp, S. K. Sinha, P. Schweiss and B. Renker, Phys. Rev. Lett., 37 (1976) 344. 14 S. D. Bader and S. K. Sinha, Phys. Rev. B, in the press. 15 0. H. Douglass (ed.), Proc. Conf. on Superconductivity in d- and f-band Metals, 1976, Plenum Press, New York, 1976. 16 F. Y. Fradin, G. S. Knapp, S. D. Bader, G. Cinader and C. W. Kimball, in D. H. Douglass (ed.), Proc. Conf. on Superconductivity in d- and f-band Metals, 1976, Plenum Press, New York, 1976, pp. 297 - 312.