Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals

We study nonlinear magneto-optical responses of metals by a semiclassical Boltzmann equation approach. We derive general formulas for linear and second-order nonlinear optical effects in the presence of magnetic fields that include both the Berry curvature and the orbital magnetic moment. Applied to Weyl fermions, the semiclassical approach (i) captures the directional anisotropy of linear conductivity under a magnetic field as a consequence of an anisotropic $B^2$ contribution, which may explain the low-field regime of recent experiments; and (ii) predicts strong second harmonic generation proportional to $B$ that is enhanced as the Fermi energy approaches the Weyl point, leading to large nonlinear Kerr rotation. Moreover, we show that the semiclassical formula for the circular photogalvanic effect arising from the Berry curvature dipole is reproduced by a full quantum calculation using a Floquet approach.

Figure 1

Semiclassical picture of CPGE and SHG induced by a Berry curvature dipole. The distribution of Berry curvature in momentum space is indicated by the color scale, with the red region corresponding to ${\mathrm{\Omega }}_z>0$ and the blue region to ${\mathrm{\Omega }}_z<0$. (a) CPGE arises from circular motion of the electron wave packet in momentum space driven by circularly polarized light. The dipole structure in $\mathrm{\Omega }\left(k\right)$ induces an anomalous velocity ($\dot{\mathit{k}}\times \mathbf{\Omega }$) in the $x$ direction denoted by green arrows. (b) SHG arises from oscillation of the electron wave packet driven by linearly polarized light in the $x$ direction. The Berry curvature dipole leads to an anomalous velocity that undergoes two oscillations in the $y$ direction in one driving period. The configuration of Berry curvature shown preserves $C_{2v}$ point-group symmetry (which is present for typical polar crystals that support CPGE and SHG), where the $y$ axis corresponds to the polar axis and the $yz$ plane to the mirror plane.

Figure 2

Angle dependence of longitudinal magnetoresistance (LMR) for Weyl semimetals derived from the semiclassical approach [Eq. (59)]. Blue lines are polar plots of the LMR $[{\rho }_{xx}\left(B\right)-{\rho }_{xx}(B=0)]/{\rho }_{xx}(B=0)$ as a function of the relative angle $\theta$ between $\mathit{E}$ and $\mathit{B}$. We show angle dependences of contributions to the LMR from (a) the Berry curvature, (b) the orbital magnetic moment, (c) the interplay between the Berry curvature and the orbital magnetic moment, and (d) the total angle dependence of the LMR.