Light-enhanced nonlinear Hall effect

TL;DR

This work shows that optical Floquet driving can dramatically enhance the Berry curvature dipole and the nonlinear Hall response in a two-band tilted Dirac system by inducing a light-driven topological transition between CI and NI phases. Using high-frequency Floquet theory, the effective Hamiltonian H^(F) acquires light-dependent renormalizations that unlock a large D_{ac} near the transition, leading to strong second-harmonic Hall signals that deviate from quantized linear Hall values. The authors also propose Floquet-quench protocols with tunable durations to further control the BCD, enabling robust, tunable nonlinear electronic properties. The approach is argued to be broadly applicable across materials with broken inversion and time-reversal symmetry, offering practical paths to engineer nonlinear Hall effects in diverse media.

Abstract

It is well known that a nontrivial Chern number results in quantized Hall conductance. What is less known is that, generically, the Hall response can be dramatically different from its quantized value in materials with broken inversion symmetry. This stems from the leading Hall contribution beyond the linear order, known as the Berry curvature dipole (BCD). While the BCD is in principle always present, it is typically very small outside of a narrow window close to a topological transition and is thus experimentally elusive without careful tuning of external fields, temperature, or impurities. In this work, we transcend this challenge by devising optical driving and quench protocols that enable practical and direct access to large BCD and nonlinear Hall responses. Varying the amplitude of an incident circularly polarized laser drives a topological transition between normal and Chern insulator phases, and importantly allows the precise unlocking of nonlinear Hall currents comparable to or larger than the linear Hall contributions. This strong BCD engineering is even more versatile with our two-parameter quench protocol, as demonstrated in our experimental proposal. Our predictions are expected to hold qualitatively across a broad range of Hall materials, thereby paving the way for the controlled engineering of nonlinear electronic properties in diverse media.

Figures (9)

• Figure 1: Light-induced topological band inversion and its large Berry curvature dipole (BCD) contribution. Top row (a1-e1): The $k_y=0$ slice of the Floquet band structure [Eq. \ref{['eq:energy']}] for our nonlinear Hall medium [Eq. \ref{['eq:H_F']}], which exhibits a band inversion at laser amplitude $A_0\to A_{0c}\approx1.0541~\text{nm}^{-1}$. Second row (a2)-(e2): Berry curvature $\Omega^{(-)}_{\bold{k},z}$ [Eq. \ref{['eq:BC']}] of the lower band, which integrates to $+1$ for $A_0<A_{0c}$ and $0$ for $A_0>A_{0c}$. Third row (a3)-(e3): The resultant linear Hall conductance [Eq. \ref{['eq:Hall_0']}] as a function of the Fermi energy $E_F$, which is quantized at $+e^2/h$ in the gap (pale yellow) for $A_0<A_{0c}$ and $0$ for $A_0>A_{0c}$, corresponding to the Chern insulator (CI) and normal insulator (NI) phases, respectively. Fourth row (a4)-(e4): The nonlinear BCD $D_{xz}$ [Eq. \ref{['eq:BCD_0']}], which vanishes when $E_F$ is in the gap, but which peaks when $E_F$ is near the band edge. Near the transition at $A_0\approx A_{0c}$, the peak becomes drastically higher and narrower. The other parameters are $\varphi=\pi/2$ (right-handed circularly polarized light), $\hbar\tilde{\omega}=1$ eV, $t_{0}=0.05$ eV$\cdot$nm, $v=0.1$ eV$\cdot$nm, $\alpha=0.1$ eV$\cdot$nm$^2$, $m=0.1$ eV, $\eta=-1$, and $k_{B}T=0.003$ eV, i.e., $T\approx34.8136$ K, which are of the same order as those in Refs. ma2019observationmuechler2016topologicalliu2021short.
• Figure 2: Light enhanced Berry curvature dipole (BCD) and significant departure from linear Hall response. (a) BCD [Eq. \ref{['eq:BCD_0']}] peak max$(|D_{xz}|)$ (black) and band gap $\Delta={\rm min}[\epsilon_{\bf k}^{(+)}]-{\rm max}[\epsilon_{\bf k}^{(-)}]$ (blue) as a function of light amplitude $A_0$. While the gap $\Delta$ vanishes as $A_0\to A_{0c}\approx1.0541$ nm$^{-1}$, leading to a transition between the Chern insulator (CI) and normal insulator (NI) phases, the BCD is dramatically enhanced. The BCD peak values correspond to the maximum of $D_{xz}$ over the broad window $E_F\in[-0.2,0.2]$ eV. (b) The root-mean-square Hall current density $\sqrt{\langle J_y^2\rangle}$ [Eq. \ref{['eq:J_total']}] as a function of $A_{0}$, from metallic ($E_F=0.1$ eV) to purely insulating ($E_F\to 0$) cases. While $\sqrt{\langle J_y^2\rangle}$ is understandably not quantized along the dashed line when the system is not insulating (purple, yellow), it is still not completely quantized even when $E_F$ is well within the gap (red). Instead, it exhibits a pronounced spike when $A_0$ is very close to $A_{0c}$ due to the large BCD contribution. Despite its narrowness, this spike is experimentally accessible due to the experimental ease of tuning $A_0$. Other than the electric field amplitude $\mathcal{E}_{x}$=0.1 V/m, $\tau\approx4.12434\times10^{-14}$ s ma2019observation, a.c. frequency $\omega=17.777$ Hz ma2019observation, and the other parameters used are identical to those in Fig. \ref{['fig:E_BC_C_BCD_phi05pi_together']}.
• Figure 3: Proposed experimental setup for Floquet quench. In our nonlinear Hall material (taking the left figure for example), a lock-in amplifier is used to measure the nonlinear Hall voltage $V_{y}^{2\omega}$ resulting from a longitudinal electrical field $E_{x}^{\omega}={\rm Re}(\mathcal{E}_{x}e^{i\omega t})$ induced by an a.c. $I^\omega_x$ with ultralow frequency $\omega$, which is much lower than the optical driving frequency $\tilde{\omega}$. Left figure: In the duration time $T_{1}$, there is no illumination on the nonlinear Hall material. Right figure: In the duration time $T_{2}$, there is a high-frequency irradiated light (purple) with amplitude $A_{0}=\sqrt{2}A_{0c}$ and high frequency $\tilde{\omega}$ illuminating the nonlinear Hall material. The total periodic time is $T=T_{1}+T_{2}$.
• Figure 4: Quench-induced topological band inversion and Berry curvature dipole (BCD) peaks. (a1-c1) Floquet energy bands [Eq. \ref{['eq:Heff']}] for the $k_{y}=0$ slice. In (a1) where $T_{1}=0.1T_{2}$ and (c1) where $T_2=0.1T_1$, the ${\cal H}_{2}({\bf k})$ with light amplitude $A_0=\sqrt{2}A_{0c}$ and ${\cal H}_{1}({\bf k})$ with $A_0=0$ are respectively dominant. They respectively correspond to the normal insulator (NI) and Chern insulator (CI) phases and are both gapped (pale yellow). But in (b1) where $T_{1}=T_{2}$, nontrivial contributions from ${\cal H}_{1}({\bf k})$ and ${\cal H}_{2}({\bf k})$ cancel each other out, leaving a gapless band structure. (a2-c2) Berry curvature [Eq. \ref{['eq:BC']}] of the lower band, whose integral over the $k_x$-$k_y$ plane interpolates between the quantized values of $0$ and $+1$ as $T_1/T_2$ increases. (a3-c3) The corresponding Hall conductance [Eq. \ref{['eq:Hall_0']}], which exhibits these quantized values when $E_{F}$ is within the band gap (pale yellow). (a4-c4) The BCD ($D_{xz}$) [Eq. \ref{['eq:BCD_0']}], which vanishes for $E_F$ within the band gap but which peaks at the band edges, particularly when $T_1=T_2$. The other parameters are identical to those in Fig. \ref{['fig:E_BC_C_BCD_phi05pi_together']}. Here, $T_{2}$ is fixed at $0.1\hbar/$eV$\approx6.58212\times10^{-17}$ s.
• Figure 5: Tunable band gap and Berry curvature dipole (BCD) peaks from Floquet quench. Our quenching protocol offers versatility in the tuning of the nonlinear Hall response through the two independent quench durations $T_1$ and $T_2$. (a) The band gap $\Delta$ [Eq. \ref{['eq:Heff']}] in the $(T_2,T_1)$ parameter space. (b) The quench-enhanced BCD peak [Eq. \ref{['eq:BCD_0']}], which saliently peaks around the $T_{1}=T_{2}$ line where the gap closes. Here NI denotes the normal insulator phase and CI denotes the Chern insulator phase. The other parameters are identical to those in Fig. \ref{['fig:E_BC_C_BCD_phi05pi_together']}.