Equilibrium current in a Weyl semimetal–superconductor heterostructure

A heterostructure consisting of a magnetic Weyl semimetal and a conventional superconductor exhibits an equilibrium current parallel to the superconductor interface and perpendicular to the magnetization. Analyzing a minimal model, which as a function of parameters may be in a trivial magnetic insulator phase, a Weyl semimetal phase, or a three-dimensional weak Chern insulator phase, we find that the equilibrium current is sensitive to the presence of surface states, such as the topological Fermi-arc states of the Weyl semimetal or the chiral surface states of the weak Chern insulator. While there is a nonzero equilibrium current in all three phases, the appearance of the surface states in the topological regime leads to a reversal of the direction of the current, compared to the current direction for the trivial magnetic insulator phase. We discuss the interpretation of the surface-state contribution to the equilibrium current as a real-space realization of the superconductivity-enabled equilibrium chiral magnetic effect of a single chirality, predicted to occur in bulk Weyl superconductors.

Figure 1

Mixed momentum/real-space illustration of the SN heterostructure considered in this article. It consists of a Weyl semimetal slab of a finite width $W$ with two Weyl nodes separated along the $k_z$ axis and counterpropagating Fermi arcs on the top (solid blue) and bottom (dotted blue) surfaces. The Weyl semimetal slab borders on a superconductor (light blue) at the bottom surface and it is capped by a trivial insulator at the top surface. Because of this built-in spatial asymmetry of the heterostructure, the superconducting proximity effect acts asymmetrically on the two Fermi arcs.

Figure 2

(a,b) Reflection phases ${\varphi }_{\pm }(k_y,k_z)$ at chemical potential $\mu \to 0$ and energy $\varepsilon =0$ (after first taking the limit $W\to \infty$) for parameter choices corresponding to the (a) trivial and (b) topological regimes. (c,d) Factors $s\left(\varphi \right)sin(\varphi /2)$ (blue), $(t/d_{+}-t/d_{-})$ (red, dashed), and $\partial {\varphi }_{+}/\partial k_y$ (red, solid) for the same parameter choices as in (a) and (b), respectively. (e) $k_z$-resolved equilibrium current ${\mathcal{I}}_y\left(k_z\right)$ as a function of $k_z$ from Eq. (58) (solid curve). The sign of the current changes if $k_z$ goes from the topological region ($k_z$ between the Weyl nodes at $\pm k_0$) to the trivial region. The dashed line shows the result at ultrasmall chemical potential within the finite-size gap of surface states, see Eq. (65). The parameters are $m_0=0.5t, \beta =1.5t, t=t^{\prime }=t_z^{\prime }=1$. In panels (a) and (c) we further set $k_z=1$; in panels (b) and (d) we set $k_z=2.6$.

Figure 3

$k_z$-resolved equilibrium current $d{\mathcal{I}}_y/d\mu$. The superconducting gap $\mathrm{\Delta }=0.01t$; the other parameters are the same as in Fig. 2. The solid blue and dashed red curves are obtained from Eq. (41) with finite chemical potential $\mu =0.01t$ and $\mu ={10}^{-6}t$, respectively. The width of the normal region is $W=300$ and $W=5$, respectively. The solid and dashed black curves are obtained from Eqs. (58) and (65), respectively. The inset shows a closeup at the Weyl node at $k_0\approx \pi /2$. The discontinuity in the derivative of $d{\mathcal{I}}_y/d\mu$ vs $k_z$ near $k_0$ is a finite-size effect and disappears upon further increasing $W$.

Figure 4

Equilibrium current $d{I}_y/d\mu$ as a function of the exchange field $\beta$. The solid blue curve is for finite chemical potential $\mu =0.01t$ and width $W=300$, which meets the condition $\mu \gg T_{-}{d}_{-}$ for most of reciprocal space. The dashed red curve is for ultrasmall chemical potential $\mu ={10}^{-6}t$ and width $W=5$, which meets the condition $\mu \ll T_{-}{d}_{-}$ for most of reciprocal space. The black dashed curve shows the difference of these two cases, which is the contribution to $d{I}_y/d\mu$ associated with the Fermi arcs. Other parameters are same as in Figs. 2 and 3.