Electric-field induced nonequilibrium orbital magnetization and resulting nonlinear charge and valley Hall effect

We provide deeper insights into the nonlinear transports in strained monolayer graphene and find that both nonlinear valley and nonlinear charge Hall effects can be interpreted with the orbital magnetic moment (OMM). Since strain induced anisotropic velocities and band-warping terms break the inversion and rotation symmetry, the nonlinear valley and charge Hall effect emerges. We demonstrate that the intrinsic OMM, originating from Berry curvature linearly corrected by electric field, is valley-contrasting which contributes to the nonlinear valley Hall current, and the shift OMM, originating from Fermi distribution function linearly corrected by electric field, is valley-independent which contributes to the nonlinear Berry-curvature-dipole (BCD) Hall current. Thus, we reveal that the nonlinear BCD Hall current and nonlinear valley current essentially have the same physics and the dependence of valley index of the orbital magnetic moment determines which nonlinear Hall effect emerges. Physically, we give an interpretation of two-step process: One electric field Ex induces an orbital magnetization and then the other electric field Ex generates the anomalous Hall effect under the orbital magnetization. These results establish a microscopic connection between orbital magnetization and nonlinear Hall responses. Furthermore, the strong dependence of OMM on strain provides a route to strain-engineered control of the nonlinear Hall transports.