Valley polarized tunneling through magnetic and electrical barriers in Weyl semimetals

Abstract

Recently, valley polarized tunneling has been predicted in Weyl semimetals with tilted energy dispersion in the presence of magnetic and electrical barriers [1]. It has been shown that Weyl electrons that encounter an electrical potential barrier experience a valley dependent transverse momentum shift, which originates from the coupling of the tilted energy dispersion and electrical potential difference. Despite the electrons being valley-polarized in angle-space, the contribution of each valley to the total conductance remains constant. Additional magnetic barrier breaks the angular transmission symmetry and leads to valley dependent conductance at the barrier interface. Here, we delve deeper into the physics of magnetic barrier structure and provide a detailed quantitative analysis based on more realistic model. We focus on the simplest Weyl semimetal case, where two Weyl fermions emerge with opposite chirality, and reversed tilt vector, whose low energy characteristics can be described by $H_{\mathrm{k,k^{\prime}}}=\eta{h}(\nu_{F}k.\sigma+{w.k})$, where the $\sigma$ is the vector of Pauli matrices in three dimensions, w is the tilt vector, $\eta={\pm}1$ and represents the chirality of the Weyl node. The system consists of a one-dimensional rectangular electrical potential [Fig. 1 (b)], and magnetic gauge potential [Fig. 1 (c)] induced by the ferromagnetic layer deposited on the Weyl semimetal, as illustrated in Fig. 1(a) . The ferromagnetic layer induces two asymmetric spike-like magnetic fields at the interface of the barrier as shown in Fig. 1 (c). The realistic magnetic field profile can be analytically described by $\mathrm{B_{\mathrm z}}(x)$ that is plotted in Fig. 2(a) , where the z is the distance along z-direction [2]. The strongest magnetic field strength $B_{max(z)}$(i.e., the peak value of $B_{\mathrm{z}(x)})$ is experienced by electrons at the Weyl semimetal region that is closest to the top ferromagnetic layer. $B_{max(\mathrm{z})}$ decreases along z, and the field profile spreads out over wider extent in x. We note that the integral of $B_{\mathrm{z}(x)}$ over x is constant at all depth z even though the peak value of $B_{\mathrm{z}(x)}$ reduces along z [i.e., shown in Fig. 2 (c)]. Therefore, the magnetic gauge profiles at different depths exhibit similar characteristics, such that the maximum height of the gauge potential is independent of the field variations along z [i.e., shown in Fig. 2 (b)]. We perform numerical calculations by considering tunneling conductance of Weyl electrons and dividing the whole device into short segments where the magnetic gauge potential is spatially varying along x. The valley dependent conductance profiles are shown in Fig. 2 (d), which exhibit almost the same characteristic at different depths z. Our results revealed that the valley-dependent conductance profile is mostly determined by the maximum point of the magnetic gauge potential, and not sensitive to field variations along the transmission direction, as well as the direction along the B-field.