Anomalous Andreev Bound State in Noncentrosymmetric Superconductors

We study edge states of noncentrosymmetric superconductors where spin-singlet $d$-wave pairing mixes with spin-triplet $p$ (or $f$)-wave one by spin-orbit coupling. For $d_{xy}$-wave pairing, the obtained Andreev bound state has an anomalous dispersion as compared to conventional helical edge modes. A unique topologically protected time-reversal invariant Majorana bound state appears at the edge. The charge conductance in the noncentrosymmetric superconductor junctions reflects the anomalous structures of the dispersions, particularly the time-reversal invariant Majorana bound state is manifested as a zero bias conductance peak.

Figure 1

Andreev bound state ${\epsilon }_b$, effective pair potentials for each Fermi surface ${\Delta }_1$ and ${\Delta }_2$ are plotted for the $d_{xy}+p$-wave case as a function of $k_y/k_2$. $\beta =0$ for panels (a), (b), and (c). $\beta =0.5$ for panels (d), (e), and (f). ${\Delta }_t={\Delta }_0$, ${\Delta }_s=0$ for (a) and (d). ${\Delta }_t=0.8{\Delta }_0$, ${\Delta }_s=0.2{\Delta }_0$ for (b) and (e). ${\Delta }_t=0.2{\Delta }_0$, ${\Delta }_s=0.8{\Delta }_0$, for (c) and (f). In all panels, curves $a$ (solid line), $b$ (dotted line), and $c$ (dashed line) denote ${\epsilon }_b/{\Delta }_0$, ${\Delta }_1/{\Delta }_0$, and ${\Delta }_2/{\Delta }_0$, respectively.

Figure 2

Similar plots to Fig. 1 with $\beta =0.5$ for $d_{x^2-y^2}+f$-wave (a) and $s+p$-wave case (b) with ${\Delta }_t=0.8{\Delta }_0$, ${\Delta }_s=0.2{\Delta }_0$. In all panels, curves $a$ (solid line), $b$ (dotted line), and $c$ (dashed line) denote ${\epsilon }_b/{\Delta }_0$, ${\Delta }_1/{\Delta }_0$, and ${\Delta }_2/{\Delta }_0$, respectively.

Figure 3

Tunneling conductance ${\sigma }_C$ for $d_{xy}+p$-wave case. (a) $Z=5$, $\beta =0$, (b) $Z=5$, $\beta =0.5$, (c) $Z=1$, $\beta =0.5$, (d) $Z=2$, $\beta =0.5$. $a$ (solid line): ${\Delta }_t={\Delta }_0$, ${\Delta }_s=0$, $b$ (dotted line): ${\Delta }_t=0.8{\Delta }_0$, ${\Delta }_s=0.2{\Delta }_0$, $c$ (dashed line): ${\Delta }_t=0.5{\Delta }_0$, ${\Delta }_s=0.5{\Delta }_0$, $d$ (thin solid line): ${\Delta }_t=0.4{\Delta }_0$, ${\Delta }_s=0.6{\Delta }_0$, and $e$ (dot-dashed line): ${\Delta }_t=0.2{\Delta }_0$, ${\Delta }_s=0.8{\Delta }_0$.

Figure 4

Tunneling conductance ${\sigma }_C$ with $\beta =0.5$ and $Z=5$. (a) $d_{x^2-y^2}+f$ wave and (b) $s+p$ wave. $a$ (solid line): ${\Delta }_t={\Delta }_0$, ${\Delta }_s=0$, $b$ (dotted line): ${\Delta }_t=0.8{\Delta }_0$, ${\Delta }_s=0.2{\Delta }_0$, $c$ (dashed line): ${\Delta }_t=0.5{\Delta }_0$, ${\Delta }_s=0.5{\Delta }_0$, $d$ (thin solid line): ${\Delta }_t=0.4{\Delta }_0$, ${\Delta }_s=0.6{\Delta }_0$, and $e$ (dot-dashed line): ${\Delta }_t=0.2{\Delta }_0$, ${\Delta }_s=0.8{\Delta }_0$.