CALCULATION OF THE SUPERCONDUCTING TRANSITION TEMPERATURE IN REFRACTORY COMPOUNDS

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CALCULATION OF THE SUPERCONDUCTING

TRANSITION TEMPERATURE IN REFRACTORY

COMPOUNDS

H. Winter, H. Rietshel, G. Ries, W. Reichardt

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, aoat 1978, page C6-474

CALCULATION OF

THE

SUPERCONDUCTING TRANSITION TEMPERATURE IN REFRACTORY COMPOUNDS

++ ++

H. winter+, H. Rietshel

,

G. ~ i e s + and W. Reichardt

++

Kernforschungszentm KarZsruhe GhrbH, + ~ n s t i t u t fur Technische Physik, I n s t i t u t fiir Angewandte Kemnphysik, Postfach 3640, 0-7500 KarZsruhe Federal Republic of Germany.

Rbsum6.- Nous calculons la temperature de transition supraconductrice Tc pour les composds ZrC, YS, NbC et NbN. Les propri6tds Clectroniques ont 6t6 obtenues par un "cluster approach" tandis que la dynamique du rdseau a dt6 ddcrite par un modele en couches. Nos resultats signalent une forte influ-

ence des phonons optiques sur T

.

Abstract.- We calculate the superconducting transition temperature T for the refractory compounds ZrC, YS, NbC and NbN. The electronic properties are obtained by mean$ of a cluster approach while the lattice dynamics is described by double shell models. Our results indicate a strong enhancement of T by optical phonons.

INTRODUCTION.- Refractory compounds (RC) are metal- lic compounds with rock-salt structure and of compo-

sition MX, where M stands for a transition metal and

X for C, N, 0 or S. They are notable for their hard-

ness, high melting points and in some cases high su-

perconducting T

.

These high T values are accompa-

nied by phonon anomalies, whereas in superconducting

RC no phonon softening is observed / l / . For this

reason the RC are prominent examples for the empiri- cally found correlation between lattice instabili-

ties and high T

-

values in superconductors.

Much theoretical effort has been undertaken to clarify the origins of high Tc superconductivity

in the RC. Hanke et al. / 2 / proposed a microscopic

model in which a simultaneous resonancelike increase of the electron-phonon coupling and decrease of pho-

non frequencies lead to a substantial enhancement of

Tc. Several authors applied the Gaspari-Gyorffy (GG) -theory to the RC, thereby attaining rather good agreement between theory and experiment / 3 , 4 / . As the GG-theory is based on the rigid muffin tin approximation which does not allow for the mechanism

proposed in 121, these results seem to contradict

/ 2 / . On the other hand, in the papers quoted the GG-

ally rather crude estimates are used for<w2>. Since

T depends exponentially on the uncertainty in

<w2>renders any quantitative result based upon this approach dubious.

CALCULATIONS.- In this paper we present calculations of Tc for the RC ZrC, YS, NbC and NbN where the phonons are treated on the same level of accu- racy as the electrons. To this end we used a non- local extension of the GG-theory / 5 / to calculate the

Eliashberg function a2(u)~(u) which within this

theory is given by

theory has been used in local approximation provi- ₊

Here, e+ (a) is the amplitude of a phonon with wave

ding X Gn the form _qv

vector

4

of the

v

-th branch and for the ci -th ion I-'@

(1) 3

X = C

-

with ~osition p in the unit cell, G is the

. C% M$& CL -+W

corresponding phonon frequency and T 1s the reci-

3

In eq. ( 1 ) ?la denotes the scattering power of the procal lattice vector wich reduces q to the first

a

a-th atom in the unit cell and <W"> is a mean squared Brillouin zone. n(gf), n

"

and 6 a are the

l * n O 1

phonon frequency. Whereas the electronic quantities band structure quantities commonly used within the

r) _anowadays a calculated with considerable care usu- GG-theory and defined in e.g. / 3 , 4 / . Tc was termi-

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ned by solving the Eliashberg equations numerically using the program of 161. For all examples the cou- lomb repulsion p* was assumed to be p* = 0.13.

As a model for the lattice dynamics we chose the double shell modelof Weber /7/ which presently provides the most reliable phonon frequencies and amplitudes in the RC. The electronic quantities were obtained using a cluster approach based on the Lloyd formalism. A short ouline of it can be found in /8/ where it has been applied to Th H

.

The

4 1 5

main impro-.ement in comparison to this earlier work consists in the exploitation of the full symmetry of the crystal. In the present calculations 9 shells have been taken into account (a total of 123 atoms). Selfconsistency has been obtained after 6 to 8 ite- rations starting from the Mattheiss construction. The muffin tin radii and exchange parameters were cho-

sen according to /h/ and kept during the iterations. The self consistency procedure resulted in a cHarge transfer metal -t non-metal and in a corresponding

shift of the non-metal S- and p-resonances to higher energies. The results of these calculations are given in Tab. 1.

limits our theoretical T

-

values may lie systema- tically to low as a consequance of the omission of the resonance mechanism proposed in / 2 / , though this effect must be much less pronounced than suggested by those authors.

Tab. cons ref.

I1 transition temperatures T and coupling tants

X

for 4 refractory compo&nds, last column :

for the used shell models.

On the other hand it is evident that coupling of electrons to optical phonons plays an important role for T in the RC, a fact which already has been stated in / g / . This is clearly shown by the third column of Tab. 2 which lists the T

-

values obtained by omitting the optical parts in a2(w)F(w). Thus we believe that the key to the high Tc of some of

the RC lies in their increased atomic density resul- ting in additional vibrational degrees of freedom and an overall increase of the

na.

(For free electrons

n

can be shown to be Q Q

,

where V is the atomic volume. For transition metals, this re- lation is preserved as a trend /10/). Our assumption is supported by the results for YS. The volume of the unit cell in YS is about twice that in NbC. In spite of its pronounced phonon anomalies 1111, YS has a low Tc ('L 3 K).

Tab. I phase shifts 61, partial DOS n l and n

0 1

'

Fermi energy E and total DOS "(cf) (per spin and atom) for 4 regractory compounds.

In Tab. 2 our theoretical values for Tc are given together with

X

decomposed in acoustical and optical contributions. References for the shell models used are given in the last line. The deviations of the theoretical T 'S from experiment are of the order of 30 % or less. For a superconductor with

X

% 0.7, p* % 0.13 and T % l0 K (e.g. NbC) this can

be traced back to an uncertainty in

X

of about 10 %. This is as good as can be expected in view of the

*

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References

/ l / Smith, H.G., Glzser, W., Phys. Rev. Lett.

2

(1972) 3 5 3

/ 2 / Hanke, W., Hafner, J., Bilz, H., Phys. Rev. Lett.

37 (1976) 1560

7

/ 3 / Klein, B.M., Papaconstantopoulos, D.A., Boyer,

L.L., Proc. of the Sec. Roch. Conf., Plenum, N.Y.

(1 976)

/4/ Schwarz, K., Weinberger, P., J. Phys. C

8

(1975) L573

/5/ Rietschel, H., to be published in Z. Physik

/ 6 / Bergmann, G., Rainer, D., Z. Physik

143

(1973) 5 9

/ 7 / Weber, W., Phys. Rev. BE (1973) 5093

/ 8 / Winter, H., Ries, G., 2. Physik B 2 (1976) 279

/ g / Rietschel, H., 2. Phys.

B 2 2

(1975) 133 / 1 0 / Butler, W.H., Phys. Rev. B E (1977) 2567 / l l / Roedhammer, P. Reichardt, W., Holtzberg, F.,

Phys. Rev. Lett.

60

(1978) 465