Spontaneous finite momentum pairing in superconductors without inversion symmetry

We analyze the effect of magnetic fluctuations in superconductors with strong spin-orbit coupling and show that they drive a phase transition between two superconducting states: a conventional phase with zero center-of-mass momentum of Cooper pairs, and an exotic phase with nonzero pair momentum. The latter is found to exhibit persistent currents without magnetic field in doubly connected geometries such as rings. Surprisingly, the transition temperature into the superconducting state can be increased by applying a Zeeman magnetic field.

Figure 1

The phase diagram as a function of temperature $T$ and $\zeta$ in units of ${\rho }_s$. In the absence of magnetic field (black curve) superconductivity is suppressed to zero ($T_c=0$) at the transition between the states with uniform phase and chiral winding of the phase. The transition temperature is shown to increase under application of an external Zeeman field. The phase difference ${\delta }_{\nu }\mathrm{\Phi }$ as a function of $\zeta$ is plotted in the inset.

Figure 2

The transition temperature as a function of an applied Zeeman field $H$ for (a) $\zeta =0.4{\rho }_s$, (b) $\zeta =0.5{\rho }_s$, and (c) $\zeta =0.6{\rho }_s$. The transition temperature is found from the phase-only model assuming the magnitude of the order parameter $\mathrm{\Delta }$ has a weak dependence on $H$. Panel (d) shows the range of fields for which the phase only approximation holds.

Figure 3

The phase diagram in a ring geometry. While at low temperature the transition lines are functions of $T/T_c$ and $\zeta /{\rho }_s$, at higher temperature they also depend on ${\rho }_s$ explicitly (see Appendix pp4). The corresponding persistent currents at $T\to 0$ are shown in the inset. The current is plotted assuming $n\ge 0$, for negative $n$ the sign of the current is inverted.

Figure 4

The persistent currents as a function of external magnetic field for $\zeta =0.1{\rho }_s<{\zeta }_c$. The magnetic field is expressed in terms of the flux threading the ring in units of the superconducting flux quantum $\phi$. The change in periodicity as a function of $\kappa$ is illustrated for $\kappa /{\rho }_s=0$ (black), $\kappa /{\rho }_s=0.4T^{-1}$ (blue), and $\kappa /{\rho }_s=0.8T^{-1}$ (green). These values of $\kappa$ are chosen to give the correct order of magnitude for a metallic system with a Fermi energy of $1\text{eV}$, a SOC of $10\text{meV}$, and a transition temperature of $0.1\text{meV}$. In addition, we set $N=25$ and $R=0.1\mu \text{m}$.