HAL Id: jpa-00217641
https://hal.archives-ouvertes.fr/jpa-00217641
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
ON THE FLUCTUATION
MAGNETOCONDUCTIVITY OF AN ANISOTROPIC
DIRTY SUPERCONDUCTOR
E. da Silva, B. Gyorffy
To cite this version:
JOURNAL DE PHYSIQUE Colloq~~? C6, supplhment au no 8, Tome 39, aoat 1978, page C6-494
E.Z. Da Silvami and B.L. ~ y o r f f y ~ *
Brookhaven National Laboratory, Upton, L.I., New York 11973, USA
RBsum6.- On calcule la conductivit6 due aux fluctuations en presence d'un champ magn6tique applique H, en fonction de l'angle entre H et le courant, dans le cas d'un supraconducteur anisotrope "sale". Abstract.- We calculate the fluctuation conductivity in presence of an external magnetic field H as function of the angle between H and the current for an anisotropic dirty superconductor.
As is well known /l/ the upper critical field Hc2 of a dirty type I1 superconductor depends on the lowest eigenvalue of the differential operator
-
&DaBaavB where a@ denote cartesian coordinate axis X y z, is the diffusion tensor for theD a ~
electrons and a is a-component of the canonical a
momenta for a cooper pair in a magnetic field
H
(if H is represented by a vector potential A then-
2 ie
-
a
-
=gV
+ -A). Also, it has been noted that super-C
-
conducting fluctuations in presence of the field
of (T,H,8) were measured and the results differed a6
markedly from the theory presented here, the con- clusion of Citvak et a1./5/ that superconducting fluctuations are dominated by reduced dimensionali- ty due to the fibrous morphology of this metallic polymer would be considerably strengthened.
To second order in the order parameter Y the generalized Ginsburg-Landau free energy for an anisotropic dirty superconductor in a magnetic field can be written as
H above T (H) could provide information about the
-
whole spectrum of
-
a%DaBaaaB. Consequently, when ( 1 )the interpretation of measurements on HcZ are dif- where f, from macroscopic theory /6/ is given by ficult, as in case of anisotropic superconductors
like layered compounds /2/ and (SN)x 131, it should be useful to study these fluctuations.
To facilitate such an investigation we pre- sent explicit formulae, valid for anisotropic dir- ty superconductors, for the fluctuation magneto-
f
conductivity tensor o (T,H,8) where 8 is the an-
a0
gle between the-measuring current and 8 . Our dis- cussion follows closely that of Mikeska and Schmidt /4/ and, in the isotropic limit treated by them, our results for
ob
= ozZ(~,~,0) andf f a
c J ~ = UZZ(T,H,?) agree with theirs. As a by-product
of our calculation we obtain an expression for Hc2 (T,8) which agrees with that of Tilley /l/.
Studying of along the chains in (SN) without X
a magnetic field Citvak et al. /5/ found one-dimen- sional fluctuations. By contrast the measurement of other properties of (SN)x /3/ suggests that it is a three-dimensional anisotropic semi-metal. If
W
Research supp0rted.b~ the Div. of Basic Energy Sciences, Dept. of Energy, under contract No EY-76-C-02-0016 and NSF Grant DMR 76-82946 K X Permanent address
: H.H. Wills Physics Labora- +
tory, Univ. of Bristol, U.K.
Supported by Fapesp
-
Grant 06-Fisica-76-0223Here
9
=-
H-'
$D a a ,$ denotes the digammaa
a B a Bfunction, TcO si the critical temperature in absen- ce of the magnetic field. For a dirty superconduc- tor D is proportional to the normal state con-
aB
ductivity tensor 171.
In this theory, the cc-component ofthe current is
where $' is the trigamma function.
For T>Tc the order parameter B fluctuates about its mean value W O . To discuss these fluctu- ations it is useful to expand $ in terms of the eigenfunctions 4 corresponding to the eigenvalues
U
E of the operator
9:
1.1
and
The dynamics of the fluctuations can be ob- tained from microscopic theory / S / , and for small frequencies the relaxation approxim=tion gives (iw +
r
)Y (W) = 0u
!J (6)where the relaxation rate
r
near T is given byv
P +E(]-?! +*(1/2+Zu))v
I
(7) with T-Tc (H)Z
= E /47ikT and E =- v ! J B Tc (H)Consider the case where the diffusion tensor is diagonal D =D =DI, DZZ=D and H makes an an-
xx YY
gle 0 with the axis of anisotropy. The eigenvalue problem of eq.(5) may be solved for the gauge choi- ce A = (Hzsine
-
HycosB,O,O) if we make the fol- lowing coordinate transformationIn the primed coordinate system eq.(5) sepa- rates and the eigenvalues are
E qn =Mwc(n+l/2) +HqIg(8)q2 (9)
where w -
Futhermore, the eigenfunstions are
with being the normalized harmonic oscillator n
solutions.
We shall calculate U using eq.(ll) entirely in the primed frame of reference
.
Since we imposed the boundary conditions in this coordinate system we have changed both the shape and the volume ofthe sample. The former is irrelevant and we correct for the latter.
The fluctuation conductivity along the ani- sotropy direction is given by the "Kubo formula"/4/
where p = (k,q,n), ltU, are the matrix elements of the current onerator in eq.(3) and from the eaui-
partition theorem 2$'(1/2)kBT
.
A full<l'ul 2> = + l (1 /2+zu)ru
discussion of eq.(ll) will be published elsewhere. Here it is evaluated in three limits. Defining the pair breaking parameter p(8)=p 0 g(8)
'l2
where(a) p<<~<<l (vanishing magnetic fields)
as one should expect for zero magnetic field there is no angular dependence.
(b) €<<p<<] (small magnetic fields)
4M*( TT
j
D_L Ev2 g(e)(c) p>>] (large magnetic fields)
where a = 314 Rn3.
Evidently for an arbitrary orientation of the magnetic field the conductivity has a component whose temperature dependence is characteristic of
one-dimensional fluctuations and another
whose temperature dependence is three dimensional
(E-"). Their relative contribution varies rapidly with 8.
Measurements of Hc2 (T,B=O) and H (T,B=a/2) c2
can be used to determine D and D I I
.
Thus the abo-I
ve theory is free of adjustable parameters.
References
/2/ Prober,D.E.,Schwall,R.E. and Beasely,M.R. to be published
/3/ Azeuvdo,L.J., Clark,W.G., Deutscher,G., Greene, R.L., Street,G.B. and Suter,L.J., Solid State Comun.~(1976)197
/4/ Mikeska,H. and Schmidt,H.,Z.Phys.230(1970)239 /5/ Civiak,R.I., Elbaum,C., Nichols,L.F., Ka0,H.I.
and Labes,M.M. ,Phys.Rev.~(1976)5413
/ 6 / Maki,K.,Phys. Rev.
148
(1976) 382171 Caroli,C., DeGennes,P.G. and Matricon,J.,Phys. Kom. Mat.
