Broken Translational and Time-Reversal Symmetry in Unconventional Superconducting Films

We demonstrate that films of an unconventional ($d$-wave) superconductor at temperatures $T\lesssim 0.43T_c$ can exhibit unusual superconducting phases. The new ground states beside the broken gauge and the point group symmetries can spontaneously break (i) continuous translational symmetry and form periodic order parameter structures in the plane of the film, or (ii) time-reversal symmetry and develop a supercurrent flowing along the film. These states are the result of the strong transverse inhomogeneity present in films with a thickness of several coherence lengths. We show a natural similarity between the formation of these states and the Fulde-Ferrell-Larkin-Ovchinnikov state.

Figure 1

A $d$-wave superconductor confined to a film with pair-breaking surfaces. (a) The SC-normal (N) transition as a function of temperature for several $q_x$-modulated states. At low temperature, transition into a uniform ($q_x=0$) SC state shows reentrant behavior and is preempted by a transition into a state with finite $x$ modulation; (b) the envelope of various $q_x$ curves gives the maximal confinement vector $Q_y^{*}(T,Q_x)=\pi /D^{*}$, plotted together with the corresponding modulation vector $Q_x(T)$. The coherence length ${\xi }_0=\hslash v_f/2\pi T_c$.

Figure 2

Free energy density of a $d$-wave superconductor confined to a film, as a function of the inverse thickness for $T/T_c=0.1$. In films with $D<D^{\star }=\pi /Q_y^{\star }$, the states with superflow have lower energy than the $q_x=0$ state. The inset shows self-consistently calculated order parameter profile $\Delta (y)$ and the current density $J_{s,x}(y)$ for small $q_x$ and $D\lesssim D^{\star }$. The current has anomalous (paramagnetic) contributions near the edges due to Andreev bound states, and $Q_y^{\star }$ marks the point where the average superfluid density in the film ${\rho }_s\equiv \underset{q_x\to 0}{lim}{J}_{s,x}^D/q_x$ vanishes.

Figure 3

The phase diagram of a $d$-wave superconductor in a film. The shaded region marks the ground state with flowing current. Suppression of this state would open access to the state with broken translational symmetry.

Figure 4

(a) Inhomogeneous state of FFLO type, when the pair-forming states with opposite spins belong to different Fermi surfaces, split by a magnetic field; (b) state with spontaneous current along $x$. The Fermi surface is shifted by fixed center-of-mass flow $\pm Q_y$. Paired states can have additional $x$ momentum $Q_x$ to minimize pair breaking.