Magneto-optical Kerr effect and signature of the chiral anomaly in a Weyl semimetal in magnetic field

One striking property of the Landau level spectrum of a Weyl semimetal (WSM) is the existence of a chiral Landau level, in which the electrons propagate unidirectionally along the magnetic field. This linearly dispersive level influences the optical properties of WSMs. For example, it was recently shown that a complete optical valley polarization is achievable in a time-reversal symmetric Weyl semimetal placed in a magnetic field [S. Bertrand et al., Phys. Rev. B 100, 075107 (2019)]. This effect originates from inter-Landau level transitions involving the chiral Landau level and requires a tilt of the Weyl cones. In this paper we show how the magneto-optical Kerr effect (MOKE) is modified in a WSM with tilted Weyl cones in comparison with its behavior in a normal metal and how a valley polarization can be detected using MOKE. We study both the Faraday (longitudinal) and Voigt (transverse) configurations for light incident on a semi-infinite WSM surface with no Fermi arcs. We use a minimal model of a WSM with four tilted Weyl nodes related by mirror and time-reversal symmetry. In the Voigt configuration, a large peak of the Kerr angle occurs at the plasmon frequency. We show that the blueshift in frequency of this peak with increasing magnetic field is a signature of the chiral anomaly in the MOKE.

Figure 1

Toy model of a WSM with time-reversal symmetry and a mirror plane perpendicular to the $\widehat{\mathbf{z}}$ direction. The $y$ axis passes through the cross. Blue and red circles indicate Weyl nodes with opposite chiralities.

Figure 2

Dispersion of the first Landau levels of the four nodes for $\mathbf{t}=t\widehat{\mathbf{z}}$ and $\mathbf{B}=B_0\widehat{\mathbf{x}}$. The chiral Landau level is represented by the dashed line. The length of each arrow indicates the lowest-energy interband transition under right circularly polarized light. When the tilt vectors of the nodes are perpendicular to the static magnetic field, all nodes have the same interband absorption threshold. For clarity, the position of the different nodes has been shifted in $k_{\parallel }\ell$.

Figure 3

Dispersion of the Landau levels for the four nodes for $\mathbf{t}=t\widehat{\mathbf{z}}$ and $\mathbf{B}=B_0\widehat{\mathbf{z}}$. The chiral Landau level is represented by the dashed line. The length of each arrow indicates the lowest-energy interband transition under right circularly polarized light. When the tilt vectors of the nodes have nonzero components along the static magnetic field, different pairs of nodes have different interband absorption thresholds in the quantum limit, thereby leading to a valley polarization. For clarity, the position of the different nodes has been shifted in $k_{\parallel }\ell$.

Figure 4

Real part of the relative dielectric function ${\varepsilon }_{\parallel }={\varepsilon }_{xx}$ as a function of $\omega$ for an external static magnetic field $\mathbf{B}=B_0\widehat{\mathbf{x}}$ with $B_0=0.3$ T. The zero of ${\varepsilon }_{\parallel }\left(\omega \right)$, which gives the plasmon frequency, is pushed to a lower frequency when inter-Landau level contributions are considered. Parameters: Total electronic density $n_e=1\times {10}^{20}{\mathrm{m}}^{-3}$, momentum relaxation time ${\tau }_0=10$ ps, Fermi velocity $v_F=3\times {10}^5$ m/s, and tilt $t=0$.

Figure 5

Frequency dependence of the Kerr angle ${\theta }_K$ and real part of the dielectric functions ${\varepsilon }_{+}$ and ${\varepsilon }_{-}$ in the Faraday configuration, where $\mathbf{q},{\mathbf{B}}_0,\mathbf{t}\parallel \widehat{\mathbf{z}}$. The interband transitions contributing to the main peaks are indicated. The $0\to 1$ and $-1\to 0$ transitions are each split into two as a consequence of valley polarization; the Weyl nodes $\tau$ contributing to each of the subpeaks are indicated. Parameters: $B_0=0.2$ T, $n_e=1\times {10}^{20}{\mathrm{m}}^{-3}, v_F=3\times {10}^5$ m/s, $\mathbf{t}=0.5\widehat{\mathbf{z}}$.

Figure 6

Kerr angle as a function of frequency for different values of the magnetic field $\mathbf{B}=B_0\widehat{\mathbf{z}}$ in the Faraday configuration. Parameters: $n_e=1\times {10}^{20}{\mathrm{m}}^{-3}, v_F=3\times {10}^5$ m/s, $\mathbf{t}=0.5\widehat{\mathbf{z}}$.

Figure 7

Kerr angle as a function of frequency for different magnetic fields in the Voigt configuration. The blueshift of the large peak in ${\theta }_K$ is related to the chiral anomaly. Parameters: $n_e=1\times {10}^{20}{\mathrm{m}}^{-3},{\tau }_0=10$ ps, $v_F=3\times {10}^5$ m/s, $\mathbf{t}=0.5\widehat{\mathbf{z}}$.

Figure 8

Frequency dependence of the dielectric functions ${\varepsilon }_{xx}$ and ${\varepsilon }_v$ and of the Kerr angle ${\theta }_K$ for a magnetic field $B=0.3$ T. Parameters: $n_e=1\times {10}^{20}{\mathrm{m}}^{-3},{\tau }_0=10$ ps, $v_F=3\times {10}^5$ m/s, $\mathbf{t}=0.5\widehat{\mathbf{z}}$. The first four peaks in $Re\left[{\varepsilon }_{xx}\right]$ come from the transitions: $0\to 1,-1\to 0,0\to 2,-2\to 0$.

Figure 9

Kerr angle in the Faraday configuration (left $y$ axis) and dielectric function $Re\left[{\varepsilon }_{+}\left(\omega \right)\right]$ (right $y$ axis) as a function of frequency for an external magnetic field $B_0=0.2,0.3,0.4$ and an electronic density $n_e=0.25\times {10}^{20}{\mathrm{m}}^{-3}$.

Figure 10

Kerr angle in the Faraday configuration (left $y$ axis) and dielectric function $Re\left[{\varepsilon }_{+}\left(\omega \right)\right]$ (right $y$ axis) as a function of frequency for an external magnetic field $B_0=2,3,4$ and an electronic density $n_e=0.25\times {10}^{22}{\mathrm{m}}^{-3}$.

Figure 11

Kerr angle ${\theta }_K$ in the Voigt configuration as a function of frequency for $B_0=0.2,0.3,0.4$ T. Parameters: $n_e=0.25\times {10}^{20}{\mathrm{m}}^{-3}$ and ${\tau }_m=10$ ps.