Non-integer flux quanta for a spherical superconductor

A thin film superconductor shaped into a spherical shell at whose center lies the end of long thin solenoid in which there is an integer flux NΦ0 has been previously extensively studied numerically as a model of a two-dimensional superconductor. The emergent flux from the solenoid produces a radial Bfield at the superconducting shell and N vortices in the superconducting film. We study here the effects of including a second solenoid (carrying a flux f) which is inserted inside the first solenoid but passing right across the sphere. This Aharonov-Bohm (AB) flux does not have to be quantized to make the order parameter single valued. The Ginzburg-Landau (GL) free energy is minimized at fixed N as a function of f and it is found that the minimum is usually achieved when the AB flux f is half a flux quantum, but depending on N the minimum may be at f = 0 or values which are not obvious rational fractions. PACS numbers: 74.20.De, 74.60.-w Typeset using REVTEX 1 A spherical shell with a magnetic monopole at its center provides a useful geometry for numerical studies of a thin film (two-dimensional) superconductor in a perpendicular magnetic field. Furthermore, an investigation of the ground state of the vortices in this system revealed an interesting geometric effect. While the ground state of vortices penetrating an infinite type II superconductor plane is the well-known Abrikosov vortex lattice, where vortices form a triangular array, on a spherical surface a perfect triangular lattice cannot form without the presence of at least twelve disclination defects and when N , the number of vortices on the surface is large other defects appear. In this paper, we descibe a situation where again geometry and topology play key roles. We consider as in Ref. 1 the ground state of vortices in a spherical superconductor in the presence of a radial magnetic field generated by a monopole at the center of the sphere, but the vector potential describing the magnetic field contains also an additional AharonovBohm (AB) flux. It should be possible to realize this system experimentally by inserting a solenoid into the center of a spherical superconducting shell as the field which emerges from the end of the solenoid approximates to the field from a magnetic monopole. One may visualize this system as a spherical superconductor with a couple of very thin solenoids one of which ends at the center of the sphere and the other lies along the z-axis. (See Fig. 1.) The quantum mechanical system which consists of a magnetic monopole and an AB flux is known to have many unusual properties as discussed in Ref. 4 in detail. For instance, there exist solutions to the Schrödinger equation for which the Dirac quantization of monopole charges does not hold even if the wavefunction is required to be single-valued. In our system, the monopole charges should be quantized, since it is determined by the number of vortices penetrating the superconductor as discussed below. One of the main results of this paper is that, in the ground state, the system organizes itself such that a nonvanishing AB flux is induced with its strength given by a fraction of the fundamental flux quantum with the actual value related to N , the number of vortices, in a way which is obscure to us. This ’quantization’ of the AB flux differs in its origin from other types of flux quantizations which are usually determined from a topological consideration such as the requirement that the 2 order parameter remain single valued as one passes round a circuit. In the present system, the flux is quantized dynamically in the sense that it is determined by minimizing the free energy for a superconductor. We consider a spherical type II superconductor of radius R and width d ≪ R in the presence of a radial magnetic field H = H(r)r̂. This magnetic field is produced by a magnetic monopole at the center of the sphere. The system can be described by the following Ginzburg-Landau free energy for a (complex) superconducting order parameter Ψ(θ, φ): F [Ψ,Ψ,A] = dR ∫ dΩ [ h̄ 2m |DΨ(θ, φ)| + α|Ψ| + β 2 |Ψ| + 1 8π |∇ ×A−H|], (1) where dΩ is the solid angle element, α, β and m are phenomenological parameters, and D = −i∇− (e/h̄c)A with the vector potential A and the charge of a Cooper pair e = 2e. Physical properties of the system including the effect of fluctuations are described by the partition function given by