Hall Shift Current and Nonlinear Anomalous Hall Effect in Gapped Dirac Fermion Systems

Technical Summary: Hall Shift Current and Nonlinear Anomalous Hall Effect in Gapped Dirac Fermion Systems

Problem Statement
The nonlinear anomalous Hall effect (NLAHE) is a topological transport phenomenon where a Hall current proportional to the square of the electric field arises in nonmagnetic crystals with time-reversal symmetry (TRS) at zero magnetic field. Conventionally, this effect is attributed to an intraband mechanism driven by the Berry curvature dipole (BCD), which results from the modification of the electron distribution function under an electric field. However, the organic Dirac fermion system $\alpha$-(ET)$_2$I$_3$ presents a unique case: it exhibits NLAHE in a weak charge-ordered (CO) state characterized by an extremely small energy gap ($\sim$0.2 meV) and low carrier density. In this regime, preliminary measurements indicate the presence of nonlinear longitudinal conduction, suggesting that the standard intraband BCD mechanism may not fully account for the observed transport phenomena. Specifically, the coexistence of nonlinear Hall and longitudinal responses hints at a contribution from interband tunneling processes, which are typically suppressed in gapped systems but may become significant when the gap is minute.

Methodology
The authors propose and analyze a new interband mechanism for NLAHE based on Landau-Zener (LZ) tunneling across the mass gap in tilted two-dimensional gapped Dirac fermion systems. The theoretical framework is built upon the following steps:

1. Model System: The study utilizes a 2D Dirac fermion Hamiltonian describing tilted cones with a mass gap ($\Delta$). The system includes two valleys (right-handed and left-handed) related by time-reversal symmetry.
2. LZ Tunneling and Shift Vector: The authors apply the theory of nonreciprocal transport in LZ tunneling systems. When an electric field drives electrons in the valence band, they undergo LZ transitions to the conduction band. Crucially, in systems lacking space-inversion symmetry, this tunneling is accompanied by a "shift vector" ($R_{cv}$), representing a real-space displacement of the wave packet center perpendicular to the driving direction.
3. Hall Shift Current Calculation: The authors derive the Hall shift current density ($j_y^{(shift)}$) as the summation of these side-jump displacements weighted by the transition rate. They calculate the transition probability $P(k)$ and the local transition rate $W(k)$ using the time-dependent Schrödinger equation. To model dissipative steady-state transport, they employ a phenomenological non-oscillatory probability $P'(k)$ to account for decoherence.
4. Valley Summation and Distribution Effects: The total current is calculated by integrating over the Brillouin zone, considering the Fermi surface (FS) anisotropy of the tilted cones. The authors explicitly analyze the contributions from both time-reversal valleys and the effect of the nonequilibrium distribution shift ($\delta f$).

Key Contributions and Results

• Identification of an Interband Mechanism: The paper establishes that in lightly doped, tilted gapped Dirac systems with a small gap, an interband mechanism driven by LZ tunneling contributes to the NLAHE. This mechanism generates a "Hall shift current" originating from the side-jump displacement associated with the tunneling process.
• Dependence on Tilt and Gap: The Hall shift current is finite only when the Dirac cone is tilted, which breaks the inversion symmetry of the Fermi surface. The current arises from the summation of side jumps over the Fermi surface, where the shift vector texture and the off-centered elliptical FS align to produce a net transverse current.
• Field Dependence: The calculated Hall shift current ($j_y^{(shift)}$) exhibits a distinct field dependence. At low electric fields, it scales as $|E_x|^2$, consistent with the conventional NLAHE. However, in the large-field limit, it crosses over to a linear dependence on $|E_x|$, reflecting the saturation of the LZ transition probability.
• Valley Cancellation and Additivity: The authors demonstrate that while the shift vector texture reverses between the two time-reversal valleys, the Fermi surface off-centering also reverses. Consequently, the Hall shift currents from both valleys add constructively. In contrast, contributions arising from the field-induced shift of the Fermi surface ($\delta k$) cancel out between the valleys, justifying the use of the equilibrium distribution function in the integral.
• Comparison with BCD Mechanism: The study compares the magnitude of the Hall shift current with the conventional BCD-driven current. For the specific parameters of $\alpha$-(ET)$_2$I$_3$ (gap $\sim$0.17 meV, carrier density $n_{2D} \sim 1.8 \times 10^{10}$ cm$^{-2}$), the authors estimate that the Hall shift current is comparable in magnitude to the BCD contribution, particularly when the electric field is not strong enough to induce a large distribution shift.

Significance
The paper claims that this interband mechanism provides a plausible explanation for the nonlinear anomalous Hall effect observed in $\alpha$-(ET)$_2$I$_3$, specifically addressing the regime where nonlinear longitudinal transport is also present. The authors argue that the standard intraband BCD mechanism alone may be insufficient to explain the experimental observations in this weakly gapped, lightly doped system. By introducing the Hall shift current associated with LZ tunneling, the work offers a theoretical framework that unifies the observation of nonlinear Hall and longitudinal responses in organic Dirac materials with extremely small gaps. The authors note that while domain cancellation in real crystals reduces the observable signal, the mechanism remains a significant component of the transport physics in these systems.