Physica 107B (1981) 491-492 North.Holland Publishing Company
VORTEX UNBINDING TRANSITIONS
IN FINITE SUPERFLUID AND SUPERCONDUCTING
L. A. Turkevich
FILMS
and S. J. Putterman
Department of Physics University of California Los Angeles, California 90024 For superfluid and small sample superconducting films, edge effects dominate and obscure the unbinding transition. Edge effects can be screened out, so that the unbinding transition may be observed, only for large superconducting films.
The effects of sample shape in three dimensional problems concerning long range potentials are well-known in the context of the depolarization factor for dielectrics I and the demagnetization factor in magnetic resonance 2. The conditional convergence of Madelung sums 3 can be traced to the possible contributions to the energy from charges at the surface, and rearrangements for their convergence 4 are attempts to minimize such contributions. It is thus not surprising that "edge effects" become important for problems with logarithmic interations in two dimensions. In this communication we study these effects for the vortex unbinding transitions in superfluid and superconducting films 5.
diverges only at the higher Hauge-He~mmer transition 7. The sudden appearance of free vortex excitations 6 at TffiTKT is lost. If the sample is rotated with angular velocity ~, (4) becomes Evf2kBTKT[In(R/a)+In(I-r2/R2)-(~/T)(R2-r2)] The last (centripetal) vortex to the centre. layer dominated in the the lower limit begins ical angular velocity ~cl ffi (T/R2)
(6)
term tends to force the Whereas the boundary integral leading to (5), to dominate at the critin (R/a)
(7)
which is the Vinen critical veloclty 8. We first study the effect of edges on isolated vortices within the lattice gas model. Neglecting edge effects, the self energy of a vortex in a superfluid is given by Ev = 2k B TKT In (R/a)
(I)
k B TKT = (~/2)y2Os
(2)
with where R is the sample size of a circular film, a is the vortex core size, Osffi0sd is the surface superfluid density with d the film thickness, and y--~/m. The probability of a vortex excitation in a cell of size a is exp-SE. The total number of thermally excited vortices
vanishes in the thermodynamic and dlverges 6 for T>TKT .
limit for T
Including edge effects, the vortex sees an image of opposite sign due to the edge, and the self energy is modified to Ev = 2kBTKT [in(R/a)+in(l-r2/R2)]
(4)
where r is the distance of the vortex from the centre of the film. Physically, there is no cost in energy when a vortex is inserted within a core size a from the edge. The number of thermally excited vortices is now dominated by contributions from a thick boundary layer