Influence of a chiral chemical potential on Weyl hybrid junctions

We study the transport properties and superconducting proximity effect in NSN junctions formed by a time-reversal symmetry broken Weyl semimetal (WSM) in proximity to an $s$-wave superconductor. We find that the differential conductances and induced pairing amplitudes strongly depend on the angle between the junction direction in real space and the axis separating the Weyl nodes in momentum space. We identify the influence of a chiral chemical potential, i.e., the electron population imbalance between Weyl nodes of opposite chirality, on the transport characteristics of the junction. Remarkably, we observe a net spin polarization of Cooper pairs that are generated via Andreev reflection in the two WSM regions. The spin polarization is opposite in the two WSM regions and highly sensitive to the chirality imbalance and excitation energy.

Figure 1

(a) An NSN junction formed by a Weyl superconductor sandwiched by two WSM regions (or leads) along $\widehat{z}$ direction. The superconductor is grounded while the two WSM regions are connected to the voltage sources $V_L$ and $V_R$, respectively. (b) The junction direction $\widehat{z}$ deviates from the axis ${\widehat{e}}_3$ separating the Weyl nodes by an angle $\alpha$. The blue and red colored cones, separated in ${\widehat{q}}_3$ direction, denote the Weyl fermions of positive and negative chirality, respectively.

Figure 2

Contour plots of the (a) local and (b) nonlocal conductances ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}$ with $\mathrm{\Lambda }\in \left\{\mathrm{L},\mathrm{R}\right\}$ as functions of the angle $\alpha$ and bias voltage eV. The green dashed curves indicate the effective superconducting gap ${\tilde{\mathrm{\Delta }}}_0\left(\alpha \right)$. (c) Local and (d) nonlocal conductances ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}$ as functions of eV for different choices of $\alpha$. For different $\alpha (\notin \{0,\pi \left\}\right)$, the curves of ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}\left(\mathrm{eV}\right)$ are qualitatively the same. ${\mu }_S={10}^6{\mathrm{\Delta }}_0, {\mu }_N={10}^3{\mathrm{\Delta }}_0, \chi =0$, and $L_s=\xi$ for all plots.

Figure 3

(a) Local and (b) nonlocal conductances ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}^{\pm }$ as functions of the bias voltage eV. The black dashed curves are for the absence of a CCP, $\chi =0$. The blue and orange solid curves are ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}^{+}$ and ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}^{-}$ with $\mathrm{\Lambda }\in \left\{\mathrm{L},\mathrm{R}\right\}$ in the presence of $\chi =0.7{\mathrm{\Delta }}_0$, respectively. The CCP shifts the curves of $G_{\mathrm{L}\mathrm{\Lambda }}^{\pm }\left(\mathrm{eV}\right)$ in the eV axis oppositely. Here, $\alpha =\pi /2$ and other parameters are the same as those described in the caption of Fig. 2.

Figure 4

Total (a) local and (b) nonlocal conductances ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}^{\mathrm{\Sigma }}$ as functions of the bias voltage eV. The black dashed and red solid curves are for the absence and presence of a CCP $\chi =0.7{\mathrm{\Delta }}_0$, respectively. ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}^{\mathrm{\Sigma }}$ are always even functions of eV. Total zero-bias (c) local and (d) nonlocal conductances ${\overline{G}}_{\mathrm{L}\mathrm{\Lambda }}^{\mathrm{\Sigma }}$ as functions of the CCP $\chi$. Here, $\alpha =\pi /2$ and other parameters are the same as those described in the caption of Fig. 2.

Figure 5

Local pairing amplitudes as functions of the angle $\alpha$ and energy $\varepsilon$. (a) opposite-spin amplitudes $f_0^{+}\left(z_L\right)/f_0^0=f_z^{+}\left(z_L\right)/f_0^0$, equal-spin amplitudes (b) $f_{\uparrow \uparrow }^{+}\left(z_L\right)/f_0^0$ and (c) $f_{\downarrow \downarrow }^{+}\left(z_L\right)/f_0^0$ at the left interface $z_L=-L_s/2$. All amplitudes are normalized by $f_0^0$, the zero-energy value of $f_0^{+}\left(z_L\right)$. We choose ${\mu }_S={10}^6{\mathrm{\Delta }}_0, {\mu }_N={10}^3{\mathrm{\Delta }}_0, \chi =0,$ and $L_s=\xi$ for all figures.

Figure 6

Position dependence of the equal-spin pairing amplitudes $f_{\uparrow \uparrow }^{+}$ and $f_{\downarrow \downarrow }^{-}$ in the left WSM region for various energies $\varepsilon$ and vanishing $\chi$. For all except the zero energy, the amplitudes decay exponentially to zero in the WSM region. Here, $\alpha =\pi /4$ and other parameters are the same as those described in the caption of Fig. 5.

Figure 7

(a) Equal-spin pairing amplitudes $f_{\uparrow \uparrow }^{+}$ and $f_{\downarrow \downarrow }^{-}$ in the left WSM region at $z=-5\xi$ for vanishing (black and dashed line) and a finite (blue and orange lines) CCP $\chi =0.7{\mathrm{\Delta }}_0$. (b) The finite CCP results in a large net spin polarization of Cooper pairs at $\varepsilon \approx \pm \chi .$ Inset: If the CCP is present only in the leads, while absent in the superconductor, then the peak and dip are slightly skewed. Here, $\alpha =\pi /4$ and other parameters are the same as those in Fig. 5.

Figure 8

Spin polarization $f_{\uparrow \uparrow }-f_{\downarrow \downarrow }$ at $z=-5\xi$ as a function of (a) the CCP $\chi$ for different choices of energy $\varepsilon$ with $\alpha =\pi /4$ and (b) the angle $\alpha$ with $\varepsilon =-0.5{\mathrm{\Delta }}_0$ and $\chi =0.7{\mathrm{\Delta }}_0$. The other parameters are the same as those described in the caption of Fig. 5.