Second Harmonic Hall Response in Insulators: Inter-band Quantum Geometry and Breakdown of Kleinman's Conjecture

Abstract

The nonlinear Hall effect has recently garnered significant attention as a powerful probe of Fermi surface quantum geometry in metals. While current studies mainly focus on the nonlinear Hall response driven by quasi-static electric fields of low frequencies, the extension of the response to higher frequencies is another promising frontier, which introduces quantum geometry into inter-band transitions. Here, we demonstrate that a specific nonlinear Hall response, namely the second harmonic Hall (SHH) response, can arise from inter-band transitions. We establish the quantum geometric origin of the SHH response and show that inter-band quantum geometry dominates the SHH response when driven near inter-band resonance. Crucially, we find that the inter-band SHH response in insulators exhibits strong frequecy dispersion, manifesting the breakdown of Kleinman's conjecture in nonlinear optics. This connects the SHH response to the breakdown of Kleinman's conjecture and reveals that frequency dispersive insulators generally allow the SHH response. Furthermore, we predict a giant SHH susceptibility in gated strained bilayer graphene and propose that one can apply the polarization resolved second harmonic microscopy to detect the SHH response there.

Figures (3)

• Figure 1: Illustration of the SHH response involving inter-band processes. (a) The SHG transverse to the applied electric field serves as a distinctive signature of SHH response. (b) The BCD induced transverse SHG in a metal. The redistribution of electrons near the Fermi surface imbalances the Berry curvature and induces the SHH current response. In (c) and (d), the valence bands get fully occupied and the system is insulating. In the resonant regime in (c), electrons in the occupied bands are pumped onto the empty conduction bands, while in the slightly off-resonant regime in (d), electrons go through a virtual inter-band transition. In the inter-band processes, frequency dependent quantum geometric quantities in the occupied bands can give rise to a finite $\bm{J}_\perp\left(2\omega\right)$ in (c) and a finite $\bm{P}_\perp\left(2\omega\right)$ in (d) respectively. Both $\bm{J}_\perp\left(2\omega\right)$ and $\bm{P}_\perp\left(2\omega\right)$ manifest the SHH response.
• Figure 2: The SHH response in an insulating strained Bernal bilayer graphene of C$_{1v}$ symmetry. (a) The lattice structure of Bernal bilayer graphene with uniaxial strain applied along the zigzag direction ($x$-direction). (b) The band dispersions near the $K$ valley ($\delta k_x=k_x-K_x$). A gap of $\Delta=84$ meV is introduced by a $z$-directional electric field. The color variations denote $\gamma_{xz}\left(\omega,\bm{k}\right)$, with $\hbar\omega=42$ meV being the resonant frequency. Here $\gamma_{xz}\left(\omega,\bm{k}\right)$ is in unit of mA$\cdot$nm$^3$/V$^2$. The strain amplitude is $\epsilon=0.03$. (c) The SHH conductivity $\sigma_{yxx}^\perp\left(2\omega;\omega,\omega\right)$ and susceptibility $\chi_{yxx}^\perp\left(2\omega;\omega,\omega\right)$ as functions of the driving frequency. The susceptibility $\chi_{yxx}^\perp\left(2\omega;\omega,\omega\right)$ reaches a local maxima at the resonant frequency, while the local maxima of $\sigma^\perp_{xxy}\left(2\omega;\omega,\omega\right)$ is achieved at a higher frequency. The relative shift of the peaks between $\sigma^\perp_{yxx}\left(2\omega;\omega,\omega\right)$ and $\chi^\perp_{yxx}\left(2\omega;\omega,\omega\right)$ stems from the frequency dependence in the definition: $\sigma^\perp_{yxx}\left(2\omega;\omega,\omega\right)=-i2\omega\epsilon_0\chi^\perp_{yxx}\left(2\omega;\omega,\omega\right)$. The finite $\sigma^\perp_{yxx}\left(2\omega;\omega,\omega\right)$ in the off-resonance regime $\left(2\hbar\omega<\Delta\right)$ arises from the finite relaxation time we introduced in the calculation.
• Figure 3: Detection of the SHH response. (a) The schematic plot of the SHI measurement. The upper and lower panels correspond to the SHI measured in the $y$ and $x$ directions respectively. Here $I_x$ and $I_y$ denotes the SHI measured in the $x$ and $y$ directions, respectively. (b) The SHH susceptibility as a function of $\omega$. In the far off-resonant regime, $\chi_{yxx}$ and $\chi_{xyy}$ are approximately the same value. As the driving frequency approaches inter-band resonance ($2\hbar\omega\rightarrow\Delta$), $\chi_{yxx}$ and $\chi_{xxy}$ start to differ significantly, which manifests the SHH response.