Thilo Kopp: h/e periodicity in loops of nodal superconductors.

Thilo discussed the periodicity of the ground-state energy and
the supercurrent as a function of the magnetic flux threading
a superconducting ring. He presented a joint work by F. Loder,
A. Kampf, J. Mannhart, C.W. Schneider, Yu.S. Barash and himself.
He first reviewed what is historically
known: flux quantization, periodicity of the ground-state energy,
and of the supercurrent in units of phi=h/2e. He recalled for us
that the states corresponding to q times phi, where q is an even
integer (London states) are related by a gauge transformation.
However, there is no such a relation between states corresponding
to q even and odd. The degeneracy between q even and odd is lifted
in s-wave superconductors when the diameter of the ring is smaller
than the coherence length of the system, since in this case, the
discrete nature of the electronic states becomes relevant showing
in general differences between half-integer and integer flux
quanta. The aim of the work by Thilo and collaborators was to look
for a mesoscopic superconducting system where h/e periodicities
become observable.

The theoretical work consisted in the numerical solution of the
Bogoliubov-de Gennes equations for a BCS-Hamiltonian with a Peierls
phase factor corresponding to the coupling to a vector potential
for a magnetic field threading a 100x100 lattice through a 30x30 hole.
The superconducting order parameter was chosen to be a d-wave one.
The idea is that while the main contribution to the supercurrents
comes from the states closest to the Fermi energy (E_F=0), most of the
condensation energy comes from the lobes. In such a way a d-wave
superconductor is protected from reaching the critical value of
the superfluid velocity by the Doppler shift, in contrast to s-wave
superconductors.

Due to the nodal character of the order parameter, discrete states
very close to E=0 are present. Thilo discussed first the evolution
of the eigenenergies as a function of flux close to q=0. As the
magnetic flux is increased, supercurrents are present and the discrete
states of the finite system shift accordingly (e.g. the states closest
to zero increase their energy). However, an abrupt change takes place
very close to h/4e (where the parabolas in the infinite case cross).
From there on, one enters the regime with q=1. Both the ground-state
energy and the supercurrent show an h/e periodicity, and the change
from states with increasing q takes place at odd integer multiples
of h/4e. At such points, the condensate reconstructs.

The finite size effects discussed by Thilo vanish as 1/R, where R
is the radius of the ring. Claudio Castellani asked whether an estimate
can be given for the sizes required to see the effect. Thilo said
this should be the case of rings in the micrometer range where a
percent effect should be still observable. Zlatko Tesanovic pointed
out that in an s-wave superconductor presumably a length scale should
exist, where the effect essentially vanishes.