The most energetically favorable vortex in YBa2Cu3O7 forms perpendicular to an anisotropic plane. This vortex is elliptical in shape and is distinguished by an effective interchange of London penetration depths from one axis of the ellipse to another. By generalizing qualitatively from the isotropic to the anisotropic case, we suggest that the flux flow resistivity for the vortex that forms perpendicular to an anistropic plane should have a preferred direction. Similar reasoning indicates that the Kosterlitz-Thouless transition temperature for a vortex mediated transition should be lower if the vortex is elliptical in shape

Chen, T. M.

Dekker, A. J.

Hickman, H.

English

NASA Technical Reports Server

N92-21632
ELLIPTICAL FLUX VORTICES IN YBa2Cu30_
H. Hickman, A.J. Dekker, and T.M. Chen
Department of Electrical Engineering
University of South Florida, Tampa, Florida 33620
Abstract- The most energetically favorable vortex in YBa_Cu307 forms perpendicular to an anisotropic plane. This
vortex is elliptical in shape and is distinguished by an effective interchange of London penetration depths from one
axis of the ellipse to the other. By generalizing qualitatively from the isotropic to the anisotropic case, we suggest
that the flux flow resistivity for a vortex that forms perpendicular to an anisotropic plane should have a preferred
direction. Similar reasoning indicates that the Kosterlitz-Thouless transition temperature for a vortex mediated
transition should be lower if the vortex is elliptical in shape.
Introduction
The free energy per unit length of an isolated Abrikosov
vortex in a type II superconductor is given by [1]
H:l_o
+t = _, (1)
where H¢1 represents the lower critical field and 4'0 rep-
resents a single flux quantum. Single crystal YBa2CuaO7
can be viewed as a type II superconductor that con-
tains one plane in which the superconducting properties
are isotropic (ab plane) and two planes in which the su-
perconducting properties are equivalently anisotropic (ae
and bc). If _a represents the free energy associated with
a vortex that forms perpendicular to the isotropic plane,
and eTc = e_c represents the free energy associated with a
vortex that forms perpendicular to an anisotropic plane,
then for YBa2CuaO7 [2]
e_c H¢*_ O.05T
_ 0.1. (2)
+[' H2_- 0.5T
Equation (2)indicatesthatthe formation of a vortex per-
pendicularto the anisotropicplane requiresonly 1/10 of
the energy requiredby the formation ofa vortexperpen-
dicularto the isotropicplane.
But inorderfora vortextoform perpendicularto an
anisotropicplane, the anisotropicplane has to be inci-
dent on a non-superconducting region.This iscertainly
the case in an a (or b)axisorientedthinfilm,where al-
most the entiresurfaceisan anisotropicplane. Further,
forbulk YBazCu307, studiesindicatethatas many as75
% ofthe grain boundaries in a "shakeand bake" sample
containab planes [3],while,depending on the density,
18 to 35 % of the bulk sample consistsof empty pock-
ets or voids [4]. If many of the isotropic ab planes are
incident on other grains, then many of the anisotropic
ac and bc planes should be incident on the voids. Ev-
idently there is ample opportunity for the formation of
vortices perpendicular to anisotropic planes, even inside
bulk YBa2CuaOv.
Discussion
The London theory solution for the magnetic flux density
around a vortex that forms perpendicular to an isotropic
plane indicates that [5]
where r is the cylindrical coordinate, A is the London
penetration depth, and Ko is a modified Bessel function
of the second kind. The London theory solution for the
magnetic flux density around a vortex that forms per-
pendicular to an anisotropic plane indicates that [6,7]
(i(')'Bc(K. _ + _ , (4)
where B is z-directed and A_u, A_x are the London pen-
etration depths in the y and x directions.
Apart from the fact that the contour lines are ellipti-
cal in shape, the most interesting aspect of equation (4)
is that it predicts an effective interchange of penetration
depths around the vortex. In YBa2CuaO7 for exam-
ple, measurements by Worthington et al. reveal that the
longest penetration depth is in the e direction while the
longest coherence length is in the a (or b) direction [2].
These data should lead to a "cat's eye" vortex model as
shown in Figure la. (Here z is identified with the c direc-
tion while y could be either the a or the b direction.) The
orientation of penetration depths ought to demand that
B(0, Ayv) = B(A_,0). Instead as shown in Figure lb,
equation (4) predicts that B(0, A_) = B(Avy, 0). This
remarkable shift has actually been observed by Dolan
et al., who report oval vortices with penetration depths
that are shortest in the c direction [8].
The fact that the most energetically favorable vortex in
199
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https://ntrs.nasa.gov/search.jsp?R=19920012389 2020-03-17T13:07:38+00:00Z
ay ylL
c
m
x
(a) (b)
Figure 1 An Abrikosov Vortex in the Anisotropic Plane of YBa2CuzOr
(a) Anticipated Structure Based on Measurements of Hcl
(b) Actual Structure Based on London Theory and Observation
YBa2CuaO7 forms perpendicular to an anisotropic plane
and is elliptical in shape, ought to have important phys-
ical consequences. By way of suggesting directions for
future study, we now examine two expressions that were
derived from isotropic considerations, and, in lieu of pre-
senting a rigorous anisotropic treatment, we will quali-
tatively extend these equations to the anisotropic case.
Working from the Bardeen-Stephen model, Tinkham
gives the following expression for flux flow resistivity in
an isotropic superconductor [9]
pl <xo,, , (5)
where Pn is the normal state resistivity, B is the space
averaged magnetic flux, and He2 is the upper critical
field. For YBa2Cu3OT, Pn along c is large compared
to p, along a (or b) [10], while He2 along c is small
compared to H¢2 along a (or b) [2]. Applied to (5), these
differences lead to
PJo> PJ,,= PI+. (6)
Equation (6) indicates that the flux flow resistivity in
the a (or b) direction is less than the flux flow resistivity
in the e direction. Equation (4) indicates that the vortex
is elongated in the a (or b) direction. Evidently, ,,-aer
the influence of a driving force, the vortex will move most
easily in the direction of its own elongation. This pre-
diction appears compatible with an argument given by
Clem, in which the entropy flow from the trailing to the
leading edge of a moving vortex results in dissipation
[11]. An elliptical vortex should create the least amount
of entropy flow (and hence the least dissipation) when it
moves in the direction of its own elongation.
The phase transition of an ultra thin superconducting
film is another phenomenon that might be affected by the
formation of elliptical vortices. For an isotropic thin film,
the Kosterlitz-Thouless transition temperature, TKT, is
given by [12]
2 d
kBTKT -- 32_. 2 _, (7)
where d is the film thickness, and A is the bulk penetra-
tion depth. In a c-axis oriented thin film, A = A= = Ab.
For an a (or b) axis oriented film, it seems appropriate
to replace 1/A 2 by an effective value. For example,
+ (8)
ef.fectlve 2
Since A+ > ha, one concludes that
(TKT)a axis oriented < (TKT)c axis oriented"
Ifa vortex mediated KT transition does in fact take place
in high Tc superconducting thin films, then the above
argument suggests that the transition region in an a-axis
oriented thin film would be wider than the transition
20O
regionin a c-axis oriented thin film.
Conclusion
The fact that the most energetically favorable vortex
in YBa_Cu30¢ is elliptical in shape should have impor-
tant physical consequences. By qualitatively generaliz-
ing from the isotropic to the anisotropic case, we note
for example, that both the Kosterlitz-Thouless transi-
tion temperature and the flux flow resistivity could be
affected by vortex shape.
Acknowledgement
This work was partially supported by DARPA grant no.
MDA972-88-J- 1006.
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201



Elliptical flux vortices in YBa2Cu3O7

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