Double Andreev reflections in type-II Weyl semimetal-superconductor junctions

We study the Andreev reflections (ARs) at the interface of the type-II Weyl semimetal-superconductor junctions and find double ARs when the superconductor is put in the Weyl semimetal band tilting direction, which is similar to the double reflections of light in anisotropic crystals. The directions of the double (retro and specular) ARs are symmetric about the normal due to the hyperboloidal Fermi surface near the Weyl nodes, but with different AR amplitudes depending on the incident electron. When the normal direction of the Weyl semimetal-superconductor interface is changed from parallel to perpendicular with the band tilt, the double ARs gradually evolve into one retro-AR and one normal reflection, resulting in an anisotropic differential conductance, which is a unique observable signature for the double ARs in experiments.

Figure 1

Schematic diagrams for (a) retro-AR in normal metal-superconductor (NM-SC) interface, (b) specular AR in graphene-superconductor interface, and (c) double ARs in type-II WSM-superconductor interface. The solid black arrows indicate the directions of the incident and reflected electrons, and the dashed red arrows indicate the directions for the reflected holes. In type-II WSM-superconductor interface, normal reflection is prohibited and the holes are reflected in the retro and specular directions of the incident electron.

Figure 2

Hyperbolic equienergy lines ($E_{e\pm },E_{h\pm }$) for electrons and holes in the $k_x-k_y$ plane in the WSM side. Here $k_z=0$. The green dashed lines are the asymptotes for $E_{e\pm }$. The intersections of the horizontal dashed line with equienergy lines A, B, C, D denote the incident modes for electrons and reflected modes for holes. The black arrows at A and B denote the directions of the incident electron, and the arrows at C and D are the directions for the double ARs. Here we choose point A as the incident electron mode with an incident angle $\alpha$, the holes are then reflected in the retro-direction $A_1$ and the specular direction $A_2$.

Figure 3

(a) Dispersion relations for electrons and holes in the type-II WSM with finite $k_y$ or $k_z$. $E_{e+}$ ($E_{h-}$) denotes the tilted conduction band for electrons (holes), and $E_{e-}$ ($E_{h+}$) is the tilted valence band for electrons (holes). The intersections of the black dashed line with the bands denote the incident modes ${\mathrm{\Psi }}_{e\pm }$ for electrons and the reflected modes ${\mathrm{\Psi }}_{h\pm }$ for holes. (b) Dispersion relation at normal incidence ($k_y=k_z=0$) with the incident angle $\alpha =0$. The intersection of $E_{e-}$ with the $k_x-$axis is $k_{x0}=\frac{\mu }{\hslash (v_1-v_2)}$. The holes are reflected into the modes that are symmetric with the incident modes of electrons about $k_x=k_{x0}$.

Figure 4

AR coefficients $A_1$, $A_2$, and $A$ as functions of the incident energy $E$. The parameters are (a) $\mu =0.5$, $v_1=2$, $k_y=0$; (b) $\mu =0.5$, $v_1=2$, $k_y=0.2$; (c) $v_1=2$, $k_y=0.2$; and (d) $k_y=0.2$, $\mu =0.5$.

Figure 5

The incident energy $E$ and incident angle $\alpha$ dependence of the retro-AR $A_1$ [(a) and (c)] and specular AR $A_2$ [(b) and (d)] with $v_1=\sqrt{2}$ and $\mu =0.5$ [(a) and (b)] and 0 [(c) and (d)].

Figure 6

AR coefficients $A_{1e+}$, $A_{2e+}$, and $A_{e+}=A_{1e+}+A_{2e+}$ as functions of the incident energy $E$ for the conduction band $E_{e+}$ incidence. The parameters are $\mu =0.5$, $v_1=2$, $k_y=0$ in (a) and $\mu =0.5$, $v_1=2$, $k_y=0.2$ in (b). One can see that $A_{1e+}$ and $A_{2e+}$ have the same values as $A_2$ and $A_1$ in the $E_{e-}$ incidence [see Figs. 4 and 4].

Figure 7

Double AR coefficients $A_1$ and $A_2$ vs the incident energy $E$ for different barrier width $d$. The barrier strength $\chi$ is fixed to 1. The other parameters are the same as Fig. 4.

Figure 8

(a) Schematic diagrams of the AR $A_1$, $A_2$, and the normal reflection (NR) when changing the type-II WSM-superconductor interface orientation angle $\theta$. (b) The AR angles ${\alpha }_{A1}$, ${\alpha }_{A2}$, and normal reflection angle ${\alpha }_{\mathrm{NR}}$ as functions of $\theta$. (c) The coefficients of AR $A_1$, $A_2$, $A$, and normal reflection vs $\theta$. Here the incident angle $\alpha$ is fixed to be ${30}^{\circ }$, the incident energy $E=0.5$, $\mu =0$, and $v_1=1.1$.

Figure 9

(a) Conductance $G\left(eV\right)$ vs the bias $eV$ with the interface orientation angle $\theta =0$ and (b) the zero-bias conductance $G\left(0\right)$ as a function of $\theta$. The parameter $v_1=2/\sqrt{3}$ and the cutoff value $\tilde{q}$ is set to be 30.