Surface density of states and topological edge states in noncentrosymmetric superconductors

We study an Andreev bound state (ABS) and the surface density of state (SDOS) of a noncentrosymmetric superconductor where spin-singlet $d$-wave pairing mixes with spin-triplet $p$ (or $f)$-wave pairing by spin-orbit coupling. For $d_{\mathit{xy}}+p$-wave pairing, the ABS appears as a zero-energy state. The present ABS is a Majorana edge mode preserving the time-reversal symmetry. We calculate the topological invariant and discuss the relevance to a single Majorana edge mode. In the presence of the Majorana edge mode, the SDOS depends strongly on the direction of the Zeeman field.

Figure 1

(100) surface of square lattice. White and black circles show the sites without and with potential $V$, respectively.

Figure 2

Local density of states for a $s+p$-wave in the bulk (solid lines) and at the surface (dashed lines) for $\lambda =0.5$ and $r_s=1.0$, $0.8$, $0.2$, and $0.0$.

Figure 3

Angle-resolved local density of state of $s+p$-wave pairing is plotted as a function of $k_y$ with $\lambda =0.5$. (a) ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$, (b) ${\Delta }_t={\Delta }_0$ and ${\Delta }_s=0$.

Figure 4

Local density of states for $d_{x^2-y^2}+f$-wave in the bulk (solid lines) and at the surface (dashed lines) for $\lambda =0.5$ and $r_s=1.0$, $0.8$, $0.2$, and $0.0$.

Figure 5

Angle-resolved local density of state of $d_{x^2-y^2}+f$-wave pairing is plotted as a function of $k_y$ with $\lambda =0.5$. (a) ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$, (b) ${\Delta }_t={\Delta }_0$ and ${\Delta }_s=0$.

Figure 6

Local density of states for $d_{\mathit{xy}}+p$-wave in the bulk (solid lines) and at the surface (dashed lines) for $\lambda =0.0$ and $r_s=1.0$, $0.8$, $0.2$, and $0.0$.

Figure 7

Angle-resolved local density of state of $d_{\mathit{xy}}+p$-wave pairing is plotted as a function of $k_y$ with $\lambda =0.0$. (a) ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$, (b) ${\Delta }_t={\Delta }_0$ and ${\Delta }_s=0$.

Figure 8

Local density of states for $d_{\mathit{xy}}+p$-wave in the bulk (solid lines) and at the surface (dashed lines) for $\lambda =0.5$ and $r_s=1.0$, $0.8$, $0.2$, and $0.0$.

Figure 9

Angle-resolved local density of state of $d_{\mathit{xy}}+p$-wave pairing is plotted as a function of $k_y$ with $\lambda =0.5$. (a) ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$, (b) ${\Delta }_t={\Delta }_0$ and ${\Delta }_s=0$.

Figure 10

Fermi surfaces of spin-split bands due to the Rashba spin-orbit coupling (solid lines) and cutting lines $k_y=0.20\pi$ and $0.45\pi$ (dashed lines).

Figure 11

Polar plot of $\theta (\mathit{k})$ for ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$ as a function of $k_x$ with fixed $k_y$ for a $d_{\mathit{xy}}+p$-wave NCS superconductor. (a) $k_y=0.2\pi$ and (c) $k_y=0.45\pi$. The white circle corresponds to $k_x=\pm \pi$, and arrows show the direction in which the values of $k_x$ increase. The corresponding $\theta =\tan {}^{-1}(m_2/m_1)$ is plotted as a function of $k_x$ for (b) $k_y=0.2\pi$ and (d) $k_y=0.45\pi$.

Figure 12

Polar plot of $\theta (\mathit{k})$ for ${\Delta }_t={\Delta }_0$ and ${\Delta }_s=0$ as a function of $k_x$ with fixed $k_y$ for $d_{\mathit{xy}}+p$-wave NCS superconductor. (a) $k_y=0.2\pi$ and (c) $k_y=0.45\pi$, where the white circle corresponds to $k_x=\pm \pi$ and arrows show the direction in which the values of $k_x$ increase. The corresponding $\theta =\tan {}^{-1}(m_2/m_1)$ is plotted as a function of $k_x$ for (b) $k_y=0.2\pi$ and (d) $k_y=0.45\pi$.

Figure 13

Local density of states at the surface for $d_{\mathit{xy}}+p$-wave with ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$ for $\lambda =0$ and (a) ${\mu }_{\mathrm{B}}\mathit{H}=0$, (b) ${\mu }_{\mathrm{B}}{H}_x=0.1$ ($\mathit{H}\parallel \mathit{x}$).

Figure 14

Local density of states at the surface for $d_{\mathit{xy}}+p$-wave with ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$ for $\lambda =0.5$ and (a) ${\mu }_{\mathrm{B}}\mathit{H}=0$, (b) ${\mu }_{\mathrm{B}}{H}_x=0.1t$ ($\mathit{H}\parallel \mathit{x}$), (c) ${\mu }_{\mathrm{B}}{H}_y=0.1t$ ($\mathit{H}\parallel \mathit{y}$), (d) ${\mu }_{\mathrm{B}}{H}_z=0.1t$ ($\mathit{H}\parallel \mathit{z}$).

Figure 15

Angle-resolved local density of state for $d_{\mathit{xy}}+p$-wave pairing in the presence of the Zeeman magnetic field is plotted as a function of $k_y$ with $\lambda =0.5$, ${\Delta }_s={\Delta }_0$ and ${\Delta }_t=0$. (a) ${\mu }_{\mathrm{B}}{H}_x=0.1t$ ($\mathit{H}\parallel \mathit{x}$), (b) ${\mu }_{\mathrm{B}}{H}_y=0.1t$ ($\mathit{H}\parallel \mathit{y}$).

Figure 16

Local density of states at the surface for $d_{\mathit{xy}}+p$-wave with ${\Delta }_t={\Delta }_0$ and ${\Delta }_s=0$ for $\lambda =0.5$ and (a) ${\mu }_{\mathrm{B}}\mathit{H}=0$, (b) ${\mu }_{\mathrm{B}}{H}_x=0.1t$ ($\mathit{H}\parallel \mathit{x}$), (c) ${\mu }_{\mathrm{B}}{H}_y=0.1t$ ($\mathit{H}\parallel \mathit{y}$), (d) ${\mu }_{\mathrm{B}}{H}_z=0.1t$ ($\mathit{H}\parallel \mathit{z}$).