Berry Curvature Dipole and Nonlinear Hall Effect in Type-II Semi-Dirac Systems

The Berry curvature dipole (BCD) and the resulting nonlinear Hall effect have been investigated in various time-reversal (TR) invariant but inversion-breaking materials, where the primary mechanisms are typically attributed to low-energy Dirac models with tilt, Fermi surface (FS) warping, or semi-Dirac dispersion with quadratic momentum dependence in one direction. This study proposes that a nonzero BCD arises in a special type-II semi-Dirac model formed by the merging of three conventional Dirac points in the absence of TR symmetry. The BCD in this model initially increases, then decreases, as the chemical potential varies or as the Dirac cones merge. This non-monotonic behavior is strongly linked to the evolution of the FS and the distribution of the BCD density across different parameter regimes. Detailed comparison with other models further reveals distinct characteristics of this model. Additionally, it is shown that a pair of TR counterparts can exhibit a net BCD, suggesting potential applications in TR-invariant materials. As a concrete example, a pronounced BCD is demonstrated in bilayer graphene, where type-II semi-Dirac dispersion is achieved by fine-tuning interlayer sliding. These findings provide insights into BCD behavior in semi-Dirac materials and establish a foundation for exploring the nonlinear Hall effect in related systems.