Anomalous contribution to the fluctuation conductivity of a superconductor in a magnetic field

Volume 35A, number 4

PHYSICS LETTERS

14 June 1971

ANOMALOUS CONTRIBUTION TO THE FLUCTUATION CONDUCTIVITY SUPERCONDUCTOR IN A MAGNETIC FIELD

OF A

G. E. CLARKE Department of Physics, University of Essex, Coichester, Essex, UK

Received 20 April 1971

The “Maki diagram” is calculated for a dirty superconductor in the presence of a magnetic field and pararnagnetic impurities.

There has recently been considerable discussion concerning the dominant contribution to the fluctuation conductivity of a superconductor above its transition temperature [1-4]. In particular, Thompson [1] has proposed that a contribution of the type considered by Maki [2] should in general be added to that originally proposed by Aslamazov and Larkin (AL) [5] which is equivalent to that obtained using time dependent Ginzburg-Landau (GL) theory [9]; he has shown that for thin films in the presence of pair breaking agents, the Maki contribution [2] should die out. Experiments on thin films in parallel magnetic fields [4] tend to support this hypothesis. We here consider the dirty limit and evaluate the Maki diagram shown in fig. 1, specifically in

Fig. 1. 3ir T) and find the following contribution to the static conductivity (w = 0) 1. Bulk material: 2T

—

-

‘~~X —

-

3e

rvl(2eH)1/2

(

Hc2

=

~~1/2

\H H 02 /

the presence of magnetic fields and paramagnetic impurities. The straight lines represent electron Green functions while the wavy line represents the Tmatrix which is calculated in ref. [6]. Performing calculations similar to that of Makifor [2] the andcontribution introducing of thethis magnetic field, we find diagram to the response function: t~Q(rr‘w)

=

z~(_k~~. io iw

2e2Tvl 3 ~

+

-

-

1

+ ~VlE~

) ~

~

\2 X

~

-

~

—~-~--E ‘~

As mentioned by Maki [2], this contribution is isotropic. 2. Thin films: (i) Ilparallel to film surface 2H2d2~2\ /3 1 (~je2H2d2~2 ~‘ a 11 — 8ds ~ \ ~e ~ being the GL coherence length at O°Kand s = (T - T~o)/T~.o. This result agrees with that of Thompson [12] with his pair breaking parameter = le2H2d ~2, (ii)fl perpendicular to the film surface

2rT~12,rT~ \2+l2rT fi * E~(w) 4~(r)~~(r) ~

-

3eT

(Hc

2 aj~ rrvldH VIHc2 correspondwhere ~Pn(r) are the complete 2 setwith of eigenfunctions of the operator (V+2ieA) ing eigenvalues E~, E~(~) is given in ref. [6] and we are working in the gauge A = (0,Hx, 0) with the magnetic field If, parallel to the Z-axis. We finally consider the GL region (2eHvl <<

(2)

4

We have also calculated the contribution from this diagram when perpendicular impurities are present using the results of Abrikosov and Gorkov [10]. We obtain the following results: 1. Bulk materials. 233

Volume 35A, number 4

=

e2 ~

where

=

PHYSICS

=

(5)

1 +~/4TTS)1/2±(~/4TTS)1/21

LETTERS

14 June 1971

time dependent GL theory will not be valid. Finally we note that the AL contribution parallel to the field in bulk material is proportional to 3/2 and will therefore be the dominant (H_H~ 2) term. In contradiction to eq. (6), Danner and Baumann [11] have obtained results in agreement with the

T

5 is the pair breaking parameter introduced in ref. [10]. 2. Thin films. 2 = ln[(+ Ex),/Ecj (6) e

(~~)

in agreement with Thompson where Cc

=

AL theory for films in perpendicular fields and further experiments, particularly on bulk material, would seem to be worthwhile. 1 would like to thank Dr. D. R. Tilley for useful discussions and the SRC for a maintenance grant.

r/4TT 8.

We conclude from the above calculations that the contribution does not diverge when paramagnetic impurities are present nor for a thin film with a magnetic field parallel to its surface. The AL contribution in magnetic fields has been calculated in refs. [6-8] and is singular in all geometries. Hence, in these three cases the AL contribution will be dominant and the results of time dependent GL theory [7,8] would be expected to hold. However, in the magnetic field case in bulk material and for films in perpendicular fields the Maki term is also singular. In the bulk case, the contribution is isotropic and proportional to (H_Hc2) 1~~2 while the AL contribution perpendicular to the 2 and fieldhas is the alsosame proportional magnitude. to For a (H-H~2)” film with the field perpendicular to the surface, both contributions are proportional to (H - H~ 2)~ and the “Maki term” has twice the value of the AL term. In these two cases, therefore, the two contributions must be added together and simple

234

References [1] H. S. Thompson, Phys. Rev. Bi (1970) 327. [2] K, Maki, Prog. Theoret, Phys. 39 (1968) 897; 40 (1968) 193. [31W. E. Masker and H. D. Parks, Phys. Rev. Bi (1970~ 2164,

E. Crow, R. S. Thompson, M. A. Klein and A. K. Bhatnogar, Phys. Rev. Letters 24 (1970) 371. [5] L.G.Aslamazov and A. 1. Larkin, Phys. Letters 26A (1968) 238; Soviet Phys. Solid State 10 (1968) [4]

~.

875. [6] K. D. Usadel, Phys. Letters 29A (1969) 501; Z. Phys. 227 (1969) 260. [7] D. R. Tilley and J. B. Parkinson, J. Phys. C2 (1969)

2175, [81 E.Abraharns [9] G. E. Clarke and andU. J.W. R. Tilley. F.Woo,J.Phys. Phys.Letters C3 (1970) 25A 2448. (1968) 117.

[10[ A. A. Abrikosov and L. P. Gorgov, Soy. Phys. JETP

12 (1961) 1243. [11] S. Danner and F. Baumann, Phys. Letters 33A (1970) 82.