Scattering on a rectangular potential barrier in nodal-line Weyl semimetals

We investigate single-particle ballistic scattering on a rectangular barrier in the nodal-line Weyl semimetals. Since the system under study has a crystallographic anisotropy, the scattering properties are dependent on mutual orientation of the crystalline axis and the barrier. To account for the anisotropy, we examine two different barrier orientations. It is demonstrated that, for certain angles of incidence, the incoming particle passes through the barrier with probability of unity. This is a manifestation of the Klein tunneling, a familiar phenomenon in the context of graphene and semimetals with Weyl points. However, the Klein tunneling in the Weyl-ring systems is observed when the angle of incidence differs from ${90}^{\circ }$, unlike the cases of graphene and Weyl-point semimetals. The reflectionless transmission also occurs for the so-called “magic angles.” The values of the magic angles are determined by geometrical resonances between the barrier width and the de Broglie length of the scattered particle. In addition, we show that under certain conditions the wave function of the transmitted and reflected particles may be a superposition of two plane waves with unequal momenta. Such a feature is a consequence of the nontrivial structure of the isoenergy surfaces of the nodal-line semimetals. Conductance of the barrier is briefly discussed.

Figure 1

Barrier parallel to the basal plane. (a) Potential energy $U\left(z\right)$. It is finite and constant for $0<z<L$. Otherwise, it is zero. (b) Incident, reflected, and transmitted waves near the barrier. The wave vectors of the incoming and outgoing waves are shown schematically by the dashed lines with arrows. The barrier is represented by the blue hatched area. (c) Relative orientations of the barrier and the isoenergy surface. The isoenergy surface is defined by the equation ${\varepsilon }^2={(m-B{k}_{\perp }^2)}^2+k_z^2$. It is shown by (green) torus in the reciprocal space. The barrier is shown as blue parallelepiped.

Figure 2

Transmission coefficient $T$ for different values of the ratio $\varepsilon /m$ (see legend in the figure). The curves are calculated for $U/m=1$,  $mL=10$,  and $Bm=1$. Top panel: Transmission coefficient $T$ as a function of the dimensionless transverse momentum $k_{\perp }\sqrt{B/m}$. Momentum $k_{\perp }$ is limited by the conditions $\sqrt{(m-\varepsilon )/B}<k_{\perp }<\sqrt{(m+\varepsilon )/B}$, which guarantees that a propagating solution ($\mathrm{Im}{k}_z=0$) exists. When $k_{\perp }\sqrt{B/m}=1$, a perfect transmission due to the Klein tunneling is observed. In addition, the reflectionless tunneling at the so-called magic angles is also possible. Bottom panel: The same curves as a function of the angle of incidence $\theta =arctan(j_{\perp }/j_z)$. The line $\varepsilon /m=0.5$ has a magic angles near $\theta ={70}^{\circ }$. The graph is asymmetric with respect to line $\theta =0^{\circ }$ due to the fact that inside and outside surfaces of torus Fig. 1 are not equivalent.

Figure 3

Transmission coefficient $T$ for different dimensionless barrier widths $mL$ (see legend in the figure). The curves are calculated for $U/m=1, \varepsilon /m=0.3$, and $Bm=1$. Top panel: Transmission coefficient $T$ as a function of the dimensionless momentum $k_{\perp }\sqrt{B/m}$. When $k_{\perp }\sqrt{B/m}=1$, perfect transmission due to the Klein tunneling is observed. In addition, the reflectionless tunneling at the so-called magic angles is also possible. The latter becomes more pronounced for wider barriers. Bottom panel: The same curves as a function of the angle of incidence $\theta =arctan(j_{\perp }/j_z)$. The line $Lm=100$ shows magic angles near $\theta ={70}^{\circ }$. The graph is asymmetric with respect to line $\theta =0^{\circ }$ due to the fact that inside and outside surfaces of torus Fig. 1 are not equivalent.

Figure 4

Barrier perpendicular to the basal plane. Top panel: Incident, reflected, and transmitted waves near the barrier. The wave vectors of the incoming and outgoing waves are shown by the dashed lines with arrows. The barrier is represented by the blue hatched area. The barrier has finite width equal to $L$ in the $y$ direction and extends to infinity in the $x$ and $z$ directions. Bottom panel: Relative orientation of the barrier and the isoenergy surface. The isoenergy (green) surface, defined by the equation ${\varepsilon }^2={(m-B{k}_{\perp }^2)}^2+k_z^2$, is a torus in the reciprocal space. The barrier is shown as blue rectangular parallelepiped. Within the parallelepiped the potential energy is $U$, outside it is zero.

Figure 5

Transmission and reflection coefficients as functions of energy. The curves are calculated for $k_z/m=0.3, U/m=1, mL=5$, and $Bm=1$. Coefficients $T_{\pm }\left(R_{\pm }\right)$ correspond to the transmitted (reflected) waves with $k_y^{(\pm )}$. We assume that $k_x=0$ because nonzero $k_x$ only renormalizes $m$.

Figure 6

Transmission and reflection coefficients vs energy for the scattering confined to the basal plane ($k_z=0$). The curves are calculated for $U/m=1, mL=5$, and $B=1/m$. Coefficients $T_{\pm }\left(R_{\pm }\right)$ describe the transmitted (reflected) waves with $k_y^{(\pm )}$. In this regime both $T_{-}$ and $R_{-}$ vanish. We assume that $k_x=0$ because nonzero $k_x$ only renormalizes $m$.

Figure 7

Dimensionless conductance $mG$ as a function of dimensionless width $Lm$ of the barrier calculated at zero temperature and $Bm=1$. Dotted curve calculated for $U/m=0$ and $\mu /m=0.3$ shows the limiting value of the conductance at vanishing barrier height. Two other curves are calculated for $U/m=1$ and different chemical potentials $\mu /m=0.3$ and $\mu /m=0.1$. Lower chemical potential corresponds to lower conductance due to lower density of states.