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

TL;DR

This paper develops a unified semiclassical theory to explain the nonlinear Hall effect across ballistic and diffusive regimes, emphasizing size-dependent effects and mechanisms involving Berry curvature in finite systems.

Contribution

It introduces a comprehensive semiclassical framework based on the Boltzmann equation that bridges different transport regimes and explains size-dependent nonlinear Hall phenomena.

Findings

Nonlinear Hall effect arises from Berry curvature dipole and Fermi-surface Berry curvature.

Size dependence results from the competition between two transport mechanisms.

The framework applies to topological crystalline insulators and finite-sized systems.

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.