Spin currents and spontaneous magnetization at twin boundaries of noncentrosymmetric superconductors

Twin boundaries are generic crystalline defects in noncentrosymmetric crystal structures. We study theoretically twin boundaries in time-reversal-symmetric noncentrosymmetric superconductors that admit parity-mixed Cooper pairing. Twin boundaries support spin currents as a consequence of this parity mixing. If the singlet and triplet components of the superconducting order parameter are of comparable magnitude, the superconducting state breaks spontaneously the bulk time-reversal symmetry locally near the twin boundary. By self-consistently evaluating the Bogoliubov-de Gennes equations and the gap functions, we find two distinct phases: First, time-reversal-symmetry breaking enhances the spin currents but does not lead to charge current. A secondary phase transition then triggers a spin magnetization and a finite charge current near the twin boundary.

Figure 1

(a) Sketch of the intrinsic metallic interface in a NCSC. If the pairing is dominantly of triplet type, a topologically protected Kramers pair of edge modes carrying a spin current is localized at $x=0$ and $L_x$. Two Kramers pairs of edge modes that are localized near the TB hybridize and thereby lose their topological protection. (b) The spectral function of up-spin quasiparticles at the immediate left of the TB, $A_{x=50,k_y}^{\uparrow }\left(E\right)$, shows the spin polarization of these helical states. The zoom-in (c) reveals their hybridization. [$A_{x=50,k_y}^{\downarrow }\left(E\right)$ and $A_{x=51,k_y}^{\uparrow }\left(E\right)$ are obtained by flipping the figure about $k_y=0$.] Parameters are $J=1.3$, $D=1.75$, $V=1.22$, $U=0.82$, such that ${\Delta }_s/{\Delta }_p\sim 0.425$ (Ref. 19).

Figure 2

(a) Phase diagram of superconducting and magnetic order at the TB as a function of onsite repulsion $U$ and $D=1.8$ (Ref. 19) The parameters $J$ and $V$ are varied to tune the mixing ratio. Continuous lines separate phases of (A) TRS singlet dominated superconductivity, (B,C) TRS breaking phase where the $\phi$ deviates from the values 0 and $\pi$ near the TB, and (D) TRS triplet dominated superconductivity. The color scale measures the spin magnetization near the TB that becomes nonzero in phase (C) via a secondary phase transition in the phase with broken TRS. Dashed lines encircle the region in parameter space where the bulk superconducting gap is nodal, marking the topological phase transition from the trivial (left) to the nontrivial (right) ${\mathbb{Z}}_2^{}$ topological superconductor. (b) Spin magnetization $M$ and phase $\phi$ along the dotted line in (a).

Figure 3

(a) Spin current in the $y$ direction as a function of $x$. The solid lines and the dashed lines are the spin currents with spin polarization in the $x$ and $y$ directions, respectively. Parameters are $J=1.1$, $D=2.0$, $V=1.5$, $U=0.87$, such that ${\Delta }_s/{\Delta }_p=0.41$ (Ref. 19). (b) The $x$ dependence of the orbital supercurrent (red line) and of the magnetization (blue line) near the TB in the TRS broken phase. Parameters are $J=1.3$, $D=2.1$, $V=1.2$, $U=0.87$, such that ${\Delta }_s/{\Delta }_p=0.5$ (Ref. 19).