Landau levels with magnetic tunneling in a Weyl semimetal and magnetoconductance of a ballistic $p\text{−}n$ junction

We study the Landau levels (LLs) of a Weyl semimetal with two adjacent Weyl nodes. We consider different orientations $\eta =\angle (\mathbf{B},{\mathbf{k}}_0)$ of magnetic field $\mathbf{B}$ with respect to ${\mathbf{k}}_0$, the vector of Weyl node splitting. A magnetic field facilitates the tunneling between the nodes, giving rise to a gap in the transverse energy of the zeroth LL. We show how the spectrum is rearranged at different $\eta$ and how this manifests itself in the change of behavior of the differential magnetoconductance $dG\left(B\right)/dB$ of a ballistic $p\text{−}n$ junction. Unlike the single-cone model where Klein tunneling reveals itself in positive $dG\left(B\right)/dB$, in the two-cone case, $G\left(B\right)$ is nonmonotonic with a maximum at $B_c\propto {\mathrm{\Phi }}_0{k}_0^2/ln\left(k_0{l}_E\right)$ for large $k_0{l}_E$, where $l_E=\sqrt{\hslash v/\left|e\right|E}$, with $E$ for the built-in electric field and ${\mathrm{\Phi }}_0$ for the magnetic flux quantum.

Figure 1

Magnetoconductance $\frac{G\left(B\right)}{G\left(0\right)}$ vs dimensionless magnetic field $\zeta \frac{vB}{cE}$ plotted from Eq. (18) with parameter $\zeta k_0{l}_E=2$. The dashed line represents the result of Ref. [20] and the black line depicts first term of the sum (17).

Figure 2

Landau levels for $\eta =\angle (\mathbf{B},{\mathbf{k}}_0)=0$ plotted from Eq. (6). The magnetic field is supposed to satisfy $l_{B}a<l_E^2$ and electrochemical potentials from different sides of the junction are noted as ${\mu }_{\pm }=\mu (\pm \infty )$.

Figure 3

Energy vs momentum along the magnetic field $\mathbf{k}\parallel \mathbf{B}$ dependence (computed numerically) with parameters $\zeta =1.6$ and $k_0{l}_B=1.4$ for different angles $\eta =\pi /6,\pi /3,\pi /2$. Material parameters are taken for TaAs, ${\hslash }^2{k}_0^2/2m\simeq 12$ meV, and the magnetic field is set to $B\simeq 34$ T to make the gap visible at $\eta =\pi /2$.

Figure 4

Effective gap ${\mathrm{\Delta }}_{\text{eff}}$ as a function of orientation $\eta$.