Abstract
The nonlinear Hall effect has opened the door towards deeper understanding of topological states of matter. Disorder plays indispensable roles in various linear Hall effects, such as the localization in the quantized Hall effects and the extrinsic mechanisms of the anomalous, spin, and valley Hall effects. Unlike in the linear Hall effects, disorder enters the nonlinear Hall effect even in the leading order. Here, we derive the formulas of the nonlinear Hall conductivity in the presence of disorder scattering. We apply the formulas to calculate the nonlinear Hall response of the tilted 2D Dirac model, which is the symmetry-allowed minimal model for the nonlinear Hall effect and can serve as a building block in realistic band structures. More importantly, we construct the general scaling law of the nonlinear Hall effect, which may help in experiments to distinguish disorder-induced contributions to the nonlinear Hall effect in the future.
Highlights
The nonlinear Hall effect has opened the door towards deeper understanding of topological states of matter
All previous Hall effects are in the linear-response regime, that is, the transverse voltage is linearly proportional to the driving current, and a measurable Hall voltage requires that time-reversal symmetry is broken by magnetic fields or magnetism[1,2,3,4]
The debate on the origin of the anomalous Hall effect lasted for one century, until recently the mechanisms are summarized in terms of intrinsic and extrinsic contributions[3]
Summary
If the electric field is along the y direction, there is no such a Hall signal because χxyy 1⁄4 0, as required by the ydirection mirror reflection symmetry This indicates that the dc nonlinear Hall signal σNxy has one-fold angular dependence[11,12]. To measure the nonzero χyxx, the driving electric current is applied along the x direction and the nonlinear Hall voltage is measured along the y direction An advantage of this variable is that the intrinsic and side-jump parts become disorder independent. Parameters are chosen as t 1⁄4 0:1 eV Á Å, v 1⁄4 1 eV Á Å, m 1⁄4 0:1 eV, niV02 1⁄4 102 eV2 Á Å2 and niV13 1⁄4 104 eV3 Á Å4
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