Unified Semiclassical Theory of Nonlinear Hall Effect:Bridging Ballistic and Diffusive Transport Regime

Abstract: The nonlinear Hall effect has attracted considerable attention and undergone extensive investigation in recent years. However, theoretical studies addressing size-dependent effects remain largely unexplored. In this work, we establish a unified semiclassical framework based on the Boltzmann transport equation, incorporating generalized boundary conditions to bridge the ballistic and diffusive transport regimes. Our analysis reveals that the nonlinear Hall effect arises from the combined action of two distinct mechanisms: the Berry curvature dipole and the Fermi-surface integral of Berry curvature. Furthermore, we investigate the Hall effect in topological crystalline insulators (TCIs), elucidating that the size dependence originates from competition between the two transport mechanisms. By connecting the two distinct regimes, our theoretical framework provides a comprehensive understanding of the nonlinear Hall effect in finite-sized systems, offering both fundamental insights and a useful analytical tool for more size-dependent investigations.