NASA Technical Reports Server (NTRS) 19920012389: Elliptical flux vortices in YBa2Cu3O7

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by  NASA Technical Reports Server (NTRS)

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 vortices perpendicular to anisotropic planes, even inside
bulk YBa2CuaOv.
The free energy per unit length of an isolated Abrikosov
vortex in a type II superconductor is given by [1] Discussion
H:l_o The London theory solution for the magnetic flux density
+t = _, (1) around a vortex that forms perpendicular to an isotropic
plane indicates that [5]
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 where r is the cylindrical coordinate, A is the London
are isotropic (ab plane) and two planes in which the su- penetration depth, and Ko is a modified Bessel function
perconducting properties are equivalently anisotropic (ae of the second kind. The London theory solution for the
and bc). If _a represents the free energy associated with magnetic flux density around a vortex that forms per-
a vortex that forms perpendicular to the isotropic plane, pendicular to an anisotropic plane indicates that [6,7]
and eTc = e_crepresents the free energy associated with a
(i(')'
vortex that forms perpendicular to an anisotropic plane,
then for YBa2CuaO7 [2]
Bc(K. _ + _ , (4)
e_c H¢*_ O.05T
_ 0.1. (2)
+[' H2_- 0.5T where B is z-directed and A_u, A_x are the London pen-
etration depths in the y and x directions.
Equation (2)indicatesthattheformation of a vortex per- Apart from the fact that the contour lines are ellipti-
pendiculartotheanisotropicplanerequiresonly1/10of cal in shape, the most interesting aspect of equation (4)
theenergyrequiredby theformationofa vortexperpen- is that it predicts an effective interchange of penetration
diculartotheisotropicplane. depths around the vortex. In YBa2CuaO7 for exam-
But inorderfora vortextoform perpendiculartoan ple, measurements by Worthington et al. reveal that the
anisotropicplane,the anisotropicplanehas to be inci- longest penetration depth is in the e direction while the
denton a non-superconductingregion.Thisiscertainly longest coherence length is in the a (or b) direction [2].
thecaseinan a (orb)axisorientedthinfilm,where al- These data should lead to a "cat's eye" vortex model as
most theentiresurfaceisan anisotropicplane.Further, shown in Figure la. (Here z is identified with the c direc-
forbulkYBazCu307, studiesindicatethatasmany as75 tion while y could be either the a or the bdirection.) The
% ofthegrainboundariesina "shakeand bake"sample orientation of penetration depths ought to demand that
containab planes [3],while,depending on the density, B(0, Ayv) = B(A_,0). Instead as shown in Figure lb,
18 to 35 % ofthe bulksample consistsofempty pock- equation (4) predicts that B(0, A_) = B(Avy, 0). This
etsor voids [4]. If many of the isotropic ab planes are remarkable shift has actually been observed by Dolan
incident on other grains, then many of the anisotropic et al., who report oval vortices with penetration depths
ac and bc planes should be incident on the voids. Ev- that are shortest in the c direction [8].
idently there is ample opportunity for the formation of The fact that the most energetically favorable vortex in
199
PRECV.Dfi",IG PAGE 8t.gP_;t_ NOT F!l.lOtCO
a
y 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 diction appears compatible with an argument given by
and is elliptical in shape, ought to have important phys- Clem, in which the entropy flow from the trailing to the
ical consequences. By way of suggesting directions for leading edge of a moving vortex results in dissipation
future study, we now examine two expressions that were [11]. An elliptical vortex should create the least amount
derived from isotropic considerations, and, in lieu of pre- of entropy flow (and hence the least dissipation) when it
senting a rigorous anisotropic treatment, we will quali- moves in the direction of its own elongation.
tatively extend these equations to the anisotropic case. The phase transition of an ultra thin superconducting
Working from the Bardeen-Stephen model, Tinkham film is another phenomenon that might be affected by the
gives the following expression for flux flow resistivity in formation of elliptical vortices. For an isotropic thin film,
an isotropic superconductor [9] the Kosterlitz-Thouless transition temperature, TKT, is
given by [12]
pl <xo,, , (5) 2 d
kBTKT -- 32_.2_, (7)
where Pn is the normal state resistivity, B is the space where d is the film thickness, and Ais the bulk penetra-
averaged magnetic flux, and He2 is the upper critical tion depth. In a c-axis oriented thin film, A = A= = Ab.
field. For YBa2Cu3OT, Pn along c is large compared For an a (or b) axis oriented film, it seems appropriate
to p, along a (or b) [10], while He2 along c is small to replace 1/A2 by an effective value. For example,
compared to H¢2 along a (or b) [2]. Applied to (5), these
differences lead to + (8)
ef.fectlve 2
PJo> PJ,,=PI+. (6)
Since A+> ha, one concludes that
Equation (6) indicates that the flux flow resistivity in
(TKT)a axis oriented < (TKT)c axis oriented"
the a (or b) direction is less than the flux flow resistivity
in the e direction. Equation (4) indicates that the vortex Ifa vortex mediated KT transition does in fact take place
is elongated in the a (or b) direction. Evidently, ,,-aer in high Tc superconducting thin films, then the above
the influence of a driving force, the vortex will move most argument suggests that the transition region in an a-axis
easily in the direction of its own elongation. This pre- oriented thin film would be wider than the transition
20O
region
in a c-axis oriented thin film. , B.B. Goodman, "Type II or London Superconduc-
tors," Reviews of Modern Physics, pp. 12-19, Jan-
Conclusion
uary 1964.
The fact that the most energetically favorable vortex . H. Hickman and T.M. Chen, "London Theory So-
in YBa_Cu30¢ is elliptical in shape should have impor- lutions for the Magnetic Flux Density and the Su-
tant physical consequences. By qualitatively generaliz- percurrent Density Around an Elliptical Abrikosov
ing from the isotropic to the anisotropic case, we note Vortex," scheduled for the January 1990 issue of
for example, that both the Kosterlitz-Thouless transi- the IEEE Transactions on Magnetics.
tion temperature and the flux flow resistivity could be
affected by vortex shape. , V.G. Kogan, "London approach to anisotropic
type-II superconductors," Physical Review B, 24,
Acknowledgement pp. 1572-1575, 1981.
This work was partially supported by DARPA grant no. , G.J. Dolan, F. Holtzberg, C. Feild, and
MDA972-88-J- 1006. T.R. Dinger, "Anisotropic Vortex Structure in
Y1Ba_Cua07," Physical Review Letters, 62, pp.
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