Temperature dependence of a generalized Ginzburg-Landau coefficient for weak and strong coupling superconductors

PHYSICS

Volume 17, number 3

LETTERS

it follows that the leading term in the temperature series for the impurity magnetization is identical to that for the host. This is in marked contrast to the exponential behaviour expected if the molecular field theory were to persist to zero temperature. The above discussion and the results shown in fig. 2 are based on a direct calculation of A(EJ, utilizing the equations of motion of the two- time temperature-dependent Green functions, decoupled by RPA. For simplicity, the results are given in fig. 2 for SI = S, simple cubic structure, and a Heisenberg Hamiltonian. The latter fact nominally restricts it to insulating ferromagnets, but the authors believe that above the very low temperature region calculations based on the Heisenberg Hamiltonian are at least semi-quantitatively applicable to metals as well, although for more subtle reasons. More quantitative details will be given in a subsequent paper.

15 July 1965

References 1. R. Pauthenet, Ann. Phys. 3 (1958) 424; J. Phys. Radium 20 (1959) 388. 2. V. Jaocarino, L. R. Walker and G. K. Wertheim, Phys.Rev. Letters 13 (1964) 752. 3. Y.Koi, A. Tsujimura and T.Hihara, J. Phys.Soc. (Japan) 19 (1964) 1493. Phys. 4. J.G.Dash, R.D.DunlapandD.G.Howard, Rev., in press. 5. H. Callen and S. Shtrikman, Solid State Communications 3 (1965) 5. 6. V. L. Bench-Bruevich and S. V. Tyablikov, The Green function method in statistical mechanics WorthHolland Publ. Comp.) 1962. 7. R. A. Tahir-Kheli and D. Ter Haar. Phvs. Rev. 127 (1962) 88. 8. T.Wo1fra.m and J.Calloway, Phys.Rev. 130 (1963) 2207.

*****

TEMPERATURE COEFFICIENT

DEPENDENCE FOR WEAK AND

OF A GENERALIZED STRONG COUPLING

GINZBURGLANDAU SUPERCONDUCTORS

*

A. PASKIN Brookhazlen

National

Laboratoyl,

Upton,

New

York

Received 28 June 1965

Although the detailed explanations of various superconducting properties must ultimately rely on a microscope formulation, in the past various phenomenological formulations such as those of Ginzburg and Landau [l] have given considerable insight into superconducting properties. It has recently been found [2] that a two-fluid formulation [3] also will generate a temperature dependent set of differential equations of the form of the linearized Ginzburg- Landau equations [l]. The two-fluid calculation is equally valid over the entire temperature range instead of only near T, as is the case of the Ginzburg-Landau formulation. We designate as K the entire coefficient in the linearized equations which corresponds to the Ginzburg-Landau parameter KGL in the Ginzburg- Landau equations. This identification allows us to replace KGL by K in results obtained previ* This work was performed under the auspices of the U.S.Atomic Energy Commission.

ously by solving the linearized Ginzburg-Landau equations. The earlier two-fluid relationship for K was found [2] to yield KB, whose temperature dependence is in approximate agreement with experiment. It has since been pointed out by Rosenblum and Cardona [4] that there is a variation in the temperature dependence of K m Hc3 / H, for weak- and strong-coupling superconductors. We show here that the generalized form of the two-fluid model [5], which was found to match the temperature dependence of the critical field for strong- and weak-coupling superconductors [6], can also be incorporated into a formulation that explains the observed temperature dependence of K. Using the (Y two-fluid model of Gorter and Casimir [5] and following Marcus and Maxwell [6] we write f, the difference in free energy between the superconducting and normal state, as f(u) = (Hf/8n){(t2/a)[1

- (l-i~~1@,/~)*]

-

U2i+u’2}1 (1) 235

Volume 1’7, number 3

PHYSICS

LETTERS

15 July 1965

1.f ,STRONG

COUPLING

(0=0.55)

1

I

I

I

I

I

1

I .4

W

I

COUPLING

u 9 ‘,I I”

I.2

I.0 -0.03 WEAK

COUPLING 0

’

-0.04 O.@

I

I

I

0

0.2

0.4

0 I 0.6

I 0.6

I 1.0

I

Fig. 1. Two-fluid and experimental deviations from the parabolic temperature relationship for critical fields for strong- end weak-coupling superconductors. where Ho is the critical

field at t = T/T, = 0, U=*/*,, no1*j2 andn, I‘k, I2 are the concentrations of superconducting electrons with and without a field, respectively, and a is an adjustable parameter which takes into account the variation in properties with the type of coupling. The parabolic behaviour of the critical field (i.e. H /Ho = (1 - t2)follows for (Y = 0.5. Introducing (ly into the Bardeen two-fluid formulation [3] for the variation in concentration of superconducting electrons in the x directions with a field along the z direction, we obtain the linearized equation

d2U/dt2 = V2U + 2zKGL(t)2Hc-2df/dU,

(V2U- [*KGL(oJ2 lee 12](1 -

,n\ W

G)}U.

Here the units are as previously given [2] with

“CL(t) = (4@c) Hc52, where 6 is the London penetration depth. We designate the term containing KGL(0) as the generalized Ginzburg-Landau COefficient K. It can be shown that the explicit temperature dependence of K is /4\

Witha = 0.5 we obtain our previous results for K(a = 0.5) = KB. It has been found [6] that 01= 236

I

I

I

0.2

0.4

0.6

0.6

H%¶E

C%.o,

I I.0

J

Fig. 2. The temperature dependence of H /H 0~K in two-fluid model and from experiment 93 or sirongc and weak-coupling superconductors. For convenience the experimental values of K (in arbitrary unite) are normalized tc the value of K(O! = 4) at t = 0.4. the

I2

=

I

Pb

0.38 and a = 0.58 approximate the range in behaviour of the critical field of weak- to strong-coupling superconductors, respectively. In fig. 1, we exhibit the manner in which the (Y two-fluid model reproduces the deviations in the critical field from the parabolic behaviour. In fig. 2, we show how the same range in (2 reproduces the temperature dependence of K. For convenience, the theoretical and measured values (in arbitrary units) are normalized to the same value at t = 0.4. The agreement between the theoretical and experimental temperature dependence of K is good. It is apparent that there is a systematic trend which correlates in the same manner that the deviation from the parabolic critical field varies with the details or “strength” of the coupling. The strong-coupling superconductors have a more temperature dependent K than do the weak-coupling superconductors. It might be notes that Hg which has less of a deviation from the parabolic relationship than Pb seems to have a more temperature dependent K. This result seems anomalous in terms of the two-fluid formulation. The two-fluid results suggest that a microscopic calculation [‘I] also ought to yield a variation in the temperature dependence of K with different coupling and further that the temperature

PHYSICS

Volume 17, number 3

dependence of K should correlate with deviations from the parabolic relationship for critical field. It would be of value to test these conclusions in detail. I thank Drs. P. M. Marcus and M. Strongin for their helpful discussions of the two-fluid model.

LETTERS

15 July 1965

1. V.L.GinzburgandL.D.Landau, Zh.Eksperim.i Teor . Fiz. 20 (1950) 1064. 2. A.Pazbkin, M.Strongin, P.P.CraigandD.G. Schweitzer, Phys. Rev. 137 (1965)A1816. 3. J.Bardeen, Phye.Rev.94 (1954) 554. 4. B. Rosenblumand M. Czrdona. Physics Letters 13 (1964) 33. 5. C.J.Gorter and H.B.G.Czzimir, Physik.Z.35 (1934) 963. 6. P.M.Marcus and E.Maxwell, Phys.Rev.91 (1953) 1035. 7. J.C.Swibart, D. J.Scalipino and Y. Wada, Phys. Rev. Letters 14 (1965) 106.

*****

CATION

DISTRIBUTION

IN TITANOMAGNETITES

(1-x)Fe304

-xFe2TiO4

W. O’REILLY and S. K. BANERJEE

Department of Physics, University of Newcastle, Newcastle upon Tyne, 1, England

Received 26 June 1965

Existing models for the titanomagnetite solid solution series (l-x)Fe304_xFe2Ti04, as suggested by Akimoto [l] and Nobel[2] and Chevallier et al. [3] have not yet been confirmed experimentally. Akimoto et al. [4] measuring saturation magnetization at room temperature in moderate fields (3kOe), concluded that the distribution followed the Ne’el-Chevallier model as modified by Gorter [5] in which a certain amount of Ti4+ ion occurs in tetrahedral sites. More recently the neutron diffraction data of Ishikawa et al. [S] have shown that Ti4+ ions always occupy octahedral sites, conflicting with that of Forster and Hall [?I, who found 0.08 Ti4+ ions on tetrahedral sites for Fe2TiO4. The Ne’el-Chevallier model is based on the empirical preference of Fe3+ for tetrahedral sites. The most favourable configuration from consideration of electrostatic energy places Fe2+ in tetrahedral sites [8]. Besides saturation magnetization, another physical property which is sensitive to cation distribution in substituted magnetites is electronic conductivity which has been used by Verwey et al. [9] in the case of the Fe304 - MgCr204 solid solution series. Polycrystalline samples of titanomagnetites were made over the entire range, using the method Barth and Posnjak [lo], by sintering intimate mixtures of Fe, Fe808 and TiO8, pressed into bars. Saturation magnetization was measured using a vibrating sample magnetometer operating

at 77’K. The maximum magnetic field available was 30kOe. The observed moment (fig. 1) coincides with the Neel-Chevallier model for O
Fig. 1. 237