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Submitted on 1 Jan 1978
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ON THE DISTRIBUTION OF TRANSPORT
CURRENTS IN THE MIXED PHASE OF AN IDEAL
MATERIAL
V. Kogan
To cite this version:
ON THE DISTRIBUTION OF TRANSPORT CURRENTS IN THE MIXED PHASE OF AN IDEAL MATERIAL
Kogan, V.G.*
Phys. Dept. 3 Techn-ion, Haifa, Israe I.
Abstract,- A weak inhomogeneous mixed phase of a material without pinning is considered in GL-region. It is shown that deviations from homogeneity are described by a London-type equation with a characteristic length A that diverges at H „. Transport currents flow in a layer of thick-ness A near the surface ; out of this layer the vortex system remains uniform. This contradicts the "force-free" model.
Let us consider the inhomogeneous mixed phase in the GL-regxon. Suppose the induction B is close to H . everywhere in the sample: H --B « H _. It is clear that such an assumption implies that only weak inhomogeneities are under consideration. We also assume the sample size L to be large enough, so that (a) the sample can be divided into small regions ("subsystems") with linear size of order of
H « L ; (b) the induction B is almost constant in
each one of these subsystems and (c) the distance
SL is still large enough with respect to an
intervor-tex spacing ao(&>>ao). These are the usual assump-tions in the transition from the "micro-" to "macro-description".
The order parameter |T| is small everywere, and we can solve the linearized GL-equations in each subsystem, considering B as a constant parame-ter. The periodic solution of the GL-equations can be obtained on the basis of the fixed B as the lo-west approximation to the microscopic magnetic field
(rather than on the basis of H „ as was done origi-nally by Abrikosov); this was shown first by Eilen-berger /l/. It was found later that this approach also allows nonperiodic solutions (for the fixed B) Brandt used this method to investigate distortions of the flux lattice /2/. Perhaps the simplest non-periodic solutions were those proposed by the au-thor /3/.
Let us denote such a solution in a subsystem as tpo(r, r, , B) , where r, are locations of the
vor-tex cores in a plane normal to B. For the fixed B (i.e. the fixed number density of vortices) one can choose the locations r, arbitrarily.
Suppose now that the locations of the vortex lines are known exactly throughout the whole sample under some given inhomogeneous external conditions. Then these locations are also fixed in all of the subsystems. Let us label the subsystems by the "ma-croscopic" radius-vector R. We can say now that the order parameter in each subsystem is given approxi-mately by <f)0 \j> B (R)j o r * M r , R ) , where R and B(R)
are parameters (which are constant in each subsys-tem and change from one subsyssubsys-tem to another).
We look now for the order parameter in the whole system, of the form
¥ = \Jj0(r,R") Uj(f) explle^R")] , where Ui (R) and
Q\(R) are slowly varying functions;
the factor Ujexp (iS^) is supposed to "sew together" the solutions i|ig in different subsystems. Uj (S) is close to 1 and has no zeros because i|ig(r,R) in our procedure already has the correct zeros of ¥ (and
similarly 0i(R) has no singularities). Denoting ^Q
as U Q exp(iGo) we can write :
V = U0(?,R) U!(R) exp|i0o(?,R") + i0i(R)J.
The correction Gi to the phase of the order parameter must be accompanied by some correction
Jl(R) to the current J0(r,R) and by a corresponding
correction Aj(K) to the vector potential. All these corrections have to be connected by the GL current
equation 4irJ/C = U2(v0 - xl)/x. Here ¥ = U exp(i0);
the dimension-less operator v = X3/3r contains the * Addr.: Max-Planck-Institut fur Metallforschung,
I.Ph., 7000 Stuttgart 80, Biisnauerstr. 171,W. Germany.
JOURNAL DE PHYSIQUE
Colloque
C6,
supplément au n°
8,
Tome
39,
août
1978,
page
C6-625
Résumé.- Une phase mixte faiblement inhomogène d'un matériau sans pinning est considérée dans la région GL. On montre que les déviations par rapport à l'homogénéité sont décrites par une équation du type London avec une longueur caractéristique A qui diverge à H ». Des courants de transport sont concentrés dans une couche d'épaisseur A près de la surface ; en dehors de cette couche le système de vortex reste uniforme. Ceci contredit le modèle "force-free".
usual ("microscopic") penetration depth A .
-+
Let us now substitute
3
=rfO
+ J1,
2
=;So
+
,
= O o+
01 and U = UoUl in the GL- equation and take an average over the subsystems of the equation which we have obtained. In this avera--+
gingallthose functions that depend only on R should
-+
be considered as constants, whereas <Jo> =<u$
+- +-
(VOO - x A o ) > = 0 obviously. We therefore get
+-
4 d l / c = <u;> u:(ifOl-x Al)/x, where all quantities
-+
are functions of R only.
-+
In the limit B+-Hc2, U1+l and <u:> (R)approa- ches the constant value <u$> of the homogeneous case. Then the equation
-+ +- +-
determines how the corrections to Jo
,
O o and A.tend to zero in this limit. Taking the curl of equa-
+
+ +-tion (I), we have finally Ahl = <u:> hl(where h l is slowly varying correction to the magnetic field),
or in the conventional units :
-+
~21
=< u f >
h1/12 (2) From the macroscopic point of view equation (2) can be considered as an equation for the inhomo-+ - + -+-+ +
geneous correction b (R) = B(R)
-
BO to the origi-+
nally uniform induction Bo :
This is a London type equation with a characteristic depth A=A
<uf>-l
,
that diverges at Hc2 as
( l-H/Hc2)-
$
This length has been introduced by Clem / 4 /
as a distance on which some vortex contributes es- sentially to the field near another one. In the dy- namical theory of elasticity of the vortex system,
h arises as a radius of nonlocality (Brandt 151).
Equation (3) shows that deviations from homo- geneity (e.g., macroscopic currents) can penetrate into the uniform phase only down to distances of
order h, which can be large near Hc2. If-I-H/Hc2 is
nos small,
A %
A, i.e. on a macroscopic sca1e.k 0.Although the assumption of "slow variations" does not hold in this region, one can say qualitatively that macroscopic currents tend to be concentrated near the sample surface, as was observed by Kroeger and Schelten 161.
It should be noted that the basic equation
-+
+of the "force-free" model AB =
-
L X ~ B is different from (3). In particular, for the density of the-t.
transport current J in a wire situated in a longi-
tudinal field, the "force-free" model gives /8/ : j z % J O (ar), jJI % Jl(at) (Jn are the Bessel func- tions of the first kind ;,r,$,z are cylindrical coordinates); the magnetization of the wire due to the twisting of current lines 8M % 1' (for a
small total current I); 6M is independent of the
+
wire radius a and of the relative orientation of I
-+
and the external field Ho 191.
However, in the theory based on equation(3),
j, % Io(kr), jJI % Il(kr) (In are modified Bessel functions), 6M%I, 6M%a for ka >> 1 and depends
-+ -+
on the sign of I.Ho. These results are obtained by the author in reference 171.
References
/I/ Eilenberger,G.,Z.Phys.E (1964) 32
/2/ Brandt,E.H.,Phys.Stat.Sol.d (1969) 381 and 393
/ 3 / Kogan,V.G. J.Low Temp.Phys.,g (1975) 103.
141
Clem,J.R.,LT 14,2 (1975) 285 Amsterdam.161 Kroeger,P.M. and Shelten,J.,J.Low Temp.Phys., 25 (1976) 369.
-
/7/ Kogan,V.G.,J.Low Temp.Phys.,z (1978) (to be published).
/8/ Walmsley,D.G. ,J.Phys.F : Metal Phys.
,2,
(1972)510.
/9/ Bergeron,C.J.,Williams,M.W. and Haubold,A.D.,
