Some superconducting films
display an increase of Tc in the parallel magnetic critical field. This
strengthening of superconductivity calls for an explanation involving both spin-orbit coupling
(SOC) and magnetic field. With this motivation Karen introduces a continuous
free-electron model in the presence of
a Rashba SOC. When a magnetic field B along x is turned on, the Zeeman
field generically shifts the chiral bands in opposite directions along y.
However, for small momenta q=2mu_0B/v_F the lower Rashba band no longer depends
linearly on B and therefore identifies a circular Fermi surface centered at
zero momentum, while the second chiral band is shifted by q giving a SC order parameter Delta(r ) =Delta
exp(iqr) similar to the FFLO case. Since the pairs in the first band don't
depend on B, the decoherence effect of B only arises from the pairs in the
second band. Disorder has a non
trivial effect on this finite-momentum SC state: at low disorder, pairs scattered
in the smaller B-dependent branch of the Fermi surface stay there for long time
and suffer a strong pair breaking. This leads to rapid decrease of the critical
field Bc on the disorder scattering. Increasing disorder pairs in the
B-dependent branch scatter more frequently in the B-independent branch and
suffer less pair-breaking leading to a strengthening of SC and a recovery of
the critical field with disorder. This justifies the choice of a model where
disorder is assumed to kill
triplet SC, while the singlet finite-momentum pairing is mildly affected.
The SC state in the
presence of B is also characterized by a finite magnetization which enters the
free energy via the SOC. According to the Edelstein magnetoelectric effect, the
supercurrent is accompanied by a transverse magnetic moment, which also
acquires a monopole structure when a supercurrent vortex is present.
The (Gibbs) free energy is
then transformed passing to a lattice
XY model having additional terms arising from the Rashba-like
magnetoelectric terms. Once the magnetic degrees of freedom are integrated out
one obtains a free energy for a classical spin model with nearest-neighbor
ferromagnetic coupling (favoring uniform SC) and a frustrating term
proportional to the Rashba SOC leading to an helical magnetic solution
corresponding to finite-momentum SC. Finally one sees that the presence of the
external magnetic field enhances the superfluid density and it extends the
region with helical magnetization. This indicates that the increased stability
of SC under parallel B might be related to an exotic finite-momentum
superconducting state.
Blogged by Marco Grilli
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