Vortex unbinding transitions in finite superfluid and superconducting films

Vortex unbinding transitions in finite superfluid and superconducting films

IC 4 Physica 107B (1981) 491-492 North.Holland Publishing Company VORTEX UNBINDING TRANSITIONS IN FINITE SUPERFLUID AND SUPERCONDUCTING L. A. Turk...

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IC 4

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

The case of superconducting films is slightly different. In the bulk, the magnetic field of a vortex line is screened over the London penetration depth %; in a film, the magnetic field of the vortex outside the film is unscreened, and the effective penetration depth 9 becomes Affi2%2/d. For small samples RA, the edges will be screened out over the distance A, but the possibility of a real unbinding of vortex-antivortex pairs remains. This effective penetration depth A can be increased by dlamagnetically covering the superconducting film I0, but this situation is then described by the small sample results, which are edge dominated. Neglecting edge effects, the self energy of a vortex in a superconducting film is9 Ev=~kBTKT[No(R/A)-No(a/A)-So(R/A)+S0(a/A)], kBTKTffi(I/A)(~0/4~) 2

(9)

where N O and S O are respectively zeroth order Neumann and Struve functions, and where the core size a%~. For RA, (8) becomes Evffi2kBTKT in(A/a),

N v % (R/a) T/(2TKT - T)

(i0)

(5)

which scales with the perimeter of the sample, is nonvanlshing at all temperatures, and

0378-4363/81/0000-0000/$0250

(8)

with

© North-HollandPublishingCompany

and the number of free vortices finite

is always

491

492

Nv , IRJ R 2 [a)-2TKT/T A

(ii)

Including edge effects, for RA, the boundary layer is confined to a width A from the edge, and in the thermodynamic limit, the edges may be neglected for most of the vortices, as in (ii). If an external magnetic field is imposed, the self energy is modified as in (6). For R
(12)

as derived independently by Fetter II. For R>A, we obtain a modified lower critical field Hcl = (~0/4RA)In(A/a)/in (R/a)

(13)

in qualitative a~reement with the numerical results of Fetter II. We now study vortex-antivortex pair excitations within the lattice gas approach. In the canonical Kosterlitz-Thouless picture (without edges), there are no free vortices for T
(14)

The mean square separation < r2>%a2 (2TKT-T)/~rKT-T)

(15)

diverges as T~TKT. Including edge effects, a pair far away from the edges may be considered to be symmetric about the centre and has energy Epair=4kBTKr[in (r/a)+in(R2-r 2)/(R2+r 2) ]

(16)

The mean square separation ~ R 2 is dominated by pairs in the boundary layer. For superconductors, neglecting edge effects, the energy of a pair is given by Epair-2~kBTKT[N0 (r/A)-N0(a/A)-So (r/A)+S0 (a/A) ] (17) which for small separations rA. For R~R2. For R>A the boundary layer is screened, and there is an unbinding from the logarithmic potential 12 as in (15). The proper treatment of these pair excitations cannot proceed via lattice gas statistical mechanics, since the pairs move in accordance with the Kelvin circulation theorem and hence are inherently dynamic. We thus utilize the Landau theory of elementary excitations. In the superfluid film, the momentum of a pair is given by p = 2~y Osr (18)

where we have blindly neglected edge effects. The normal fluid density couples directly to the number of elementary excitations when an external flow v is imposed On=(i/~R2v2)<$'~> = (i/~v)2fd2p$'~N(p)

(19)

where the number N(p) of elementary excitations (v~ pairs) is given by N(p) = exp -~[E(p)-v'p] The angular integral is easily performed, yielding r -R 3-4T._/TI_ 14TKTVr] _ [4TKTV~].

o. .a r

(20)

(21)

where In is the n th Bessel function of imaginary argument. For zero flow, the integral (21) is dominated by the lower limit, and o n diverges as (TKT-T) -I. At T=0°K the upper limit dominates (21) at a critical velocity v c = ~ in R/a

(22)

Within this unrenormalized treatment an imposed flow v=~Vc(0E~!l) leads to a phase boundary at Tc=(I-~)TKT. This low equilibrium critical velocity seems in conflict with experiment 13. In the superconducting film, the canonical momentum ~=p-eA/C of a pair is given by (18). For R
A we lose the logarithmic interaction at separation r~A, whereupon the unbinding can proceed even more easily. The upper limit A now dominates (21) at a critical velocity Vc=(T/A)in(A/a) corresponding to a critical current J~(~0C/4~%2A)InA/a. This is reduced from the Ginzburg-Landau critical current by the factor (a/A)in(A/a). This depression of the critical current may be responsible for the dissipation seen in microwave surface impedance measurements below the thermodynamic (i.e. Ginzburg-Landau) transition 14. REFERENCES: (i) J.A. Osborn, Phys. Rev. 67, 351 (1945). E.C. Stoner, Phil. Mag. 36, 803 (1945). (2) C. Kittel, Phys. Rev. 71, 270 (1947). (3) M.P. Tosi, in Solid State Physics 16 (Seitz and Turnbull ed.) i (1964). (4) P.P. Ewald, Ann. Physik 64, 253 (1921) (5) S.J. Putterman, L.A. Turkevich, Bull. Am. Phys. Soc. 25, 262 (1980). (6) J.M. Kosterlitz, D.J. Thouless, J. Phys. C6, 1181 (1973). (7) E.H. Hauge, P.C. Hemmer, Phys. Norvegica ~, 209 (1971). (8) W.F. Vinen, Nature 181, 1524 (1958). (9) J. Pearl, Appl. Phys. Lett. ~, 65 (1964). i0) J. Pearl, thesis, Brooklyn Polytechnic, 196~ ii) A.L. Fetter, Phys. Rev. B22, 1200 (1980). 12) L.A. Turkevich, J. Phys. C12, L385 (1979). 13. K. Telschow, I. Rudnick, T.G. Wang, Phys. Rev. Lett. 23, 1292 (1974). 14) J.E. Mercereau, S. Sridhar, Bull. Am. Phys. Soc. 26, 308 (1981). 15) M.R. Beasley, J.E. Mooij, T.P. Orlando, Phys. Rev. Lett. 42, 1165 (1979). Supported respectively by the NSF and ONR.