Enhanced chiral edge currents and orbital magnetic moment in chiral $d$-wave superconductors from mesoscopic finite-size effects

Abstract: Chiral superconductors spontaneously break time-reversal symmetry and host topologically protected edge modes, supposedly generating chiral edge currents which are typically taken as a characteristic fingerprint of chiral superconductivity. However, recent studies have shown that the total edge current in two dimensions (2D) often vanishes for all chiral superconductors except for chiral $p$-wave, especially at low temperatures, thus severely impeding potential experimental verification and characterization of these superconductors. In this work, we use quasiclassical theory of superconductivity to study mesoscopic disc-schaped chiral $d$-wave superconductors. We find that mesoscopic finite-size effects cause a dramatic enhancement of the total charge current and orbital magnetic moment (OMM), even at low temperatures. We study how these quantities scale with temperature, spontaneous Meissner screening, and system radius $\mathcal{R} \in [5,200]\xi_0$ with superconducting coherence length $\xi_0$. We find a general $1/\mathcal{R}$ scaling in the total charge current and OMM for sufficiently large systems, but this breaks down in small systems, instead producing a local maximum at $\mathcal{R} \approx 10{\text{--}}20\xi_0$ due to mesoscopic finite-size effects. These effects also cause a spontaneous charge-current reversal opposite to the chirality below $\mathcal{R} < 10\xi_0$. Our work highlights mesoscopic systems as a route to experimentally verify chiral $d$-wave superconductivity, measurable with magnetometry.