Nonlinear Hall Effect with Time‐Reversal Symmetry: Theory and Material Realizations

Abstract

The appearance of a Hall conductance necessarily requires breaking of time‐reversal symmetry, either by an external magnetic field or by the internal magnetization of a material. As a second response, however, Hall dissipationless transverse currents can appear even in time‐reversal symmetric conditions provided the material is non‐centrosymmetric. This non‐linear Hall effect has a quantum origin: it is related to the geometric properties of the electronic wavefunctions and encoded in the dipole moment of the Berry curvature. Here, the general theory underpinning this effect is reviewed and various material platforms where non‐linear Hall transverse responses have been found are discussed. On the theoretical front, the link between the non‐linear Hall effect and the Berry curvature dipole is discussed using Boltzmann transport theory. On the material front, different platforms, including topological crystalline insulators, transition metal dichalcogenides, graphene, and Weyl semimetals are reviewed.

On the separation of Hall and Ohmic nonlinear responses

“…Let us now revisit the prescriptions proposed in Eqs. ( 8) and ( 9) for defining σ H αβγ , which consist in first symmetrizing the full σ αβγ in the last two indices, and then antisymmetrizing the first index with either the second or the third [15,19]. When applied to a concrete example in Sec.…”

“…Building on seminal works from the 1970s [1][2][3], there is at present renewed interest in nonlinear effects in solids arising from broken symmetries [4]. The nonlinear transport effects that are being actively investigated include unidirectional magnetoresistance (both induced by a magnetic field [5][6][7] and spontaneous [8][9][10]), and various nonlinear Hall effects [11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Nonlinear anomalous Hall effects probe topological phase-transitions in twisted double bilayer graphene

“…Having explored the electronic band structure of TDBG, we now focus on the electric, thermoelectric and thermal NLA Hall effects 38,39 . It has been recently discovered 36 that SIS broken systems possess a secondorder nonlinear Hall charge current, which is finite even in the presence of TRS.…”

“…This is because the presence of time reversal symmetry (TRS) suppresses the Berry curvature related linear response phenomena, such as the anomalous Hall effect 34,35 , in these materials. However, the nonlinear anomalous (NLA) Hall effect induced by the Berry curvature dipole (BCD) [36][37][38][39][40][41] can probe the Berry curvature in systems with TRS [Fig. 1(a)].…”

Tunable topology and berry curvature dipole in transition metal dichalcogenide Janus monolayers

“…Following the prediction of Sodemann and Fu, the first moment of the Berry curvature, termed Berry curvature dipole (BCD), has been identified as an important quantity responsible for non-linear Hall effects [34]. Such a non-linear Hall effect can also be generated in systems preserving time reversal (TR) symmetry, in contrast to the linear Hall effect [34][35][36]. Diverse set of materials have been predicted to show finite BCD and the resulting second order Hall effect, including two-dimensional materials [37][38][39][40][41], and Dirac and Weyl semimetals [42][43][44][45][46], strained bilayer and monolayer graphene [47] and Rashba systems [48].…”