Orbital magnetization as the origin of the nonlinear Hall effect

Abstract

The nonlinear Hall effect is a new type of Hall effect that has recently attracted significant attention. For the physical origin of the nonlinear Hall effect, while orbital magnetization has long been hypothesized to underpin the nonlinear Hall effect, a general relation between the two quantities remains elusive. Here, we resolve the problem by deriving the first explicit formula connecting the electric field induced orbital magnetization to the second order Hall conductivity. Our theory reveals that the applied electric field plays dual roles in generating the nonlinear Hall effect: it first generates nonequlibrium orbital magnetization associated with an edge current, and subsequently perturbs the circulating edge states to produce transverse Hall voltage. For the experimental verification, we propose to apply a combination of direct and alternative currents to identify the circulating edge current in the nonlinear Hall effect. Based on the orbital magnetization origin, we point out that in isotropic chiral metals of T and O point groups, the crystalline symmetry suppresses the nonlinear Hall response for a monochromatic linear polarized electric field, but a non-collinear bichromatic electric field can generate a finite nonlinear Hall current that manifests the chiral correlation of the field. This discovery finally enables us to incorporate both the nonlinear Hall effect and circular photo-galvanic effect into the framework of orbital magnetization.

Figures (4)

• Figure 1: The Hall effect and the associated orbital magnetization. (a) In the linear Hall effect, a $z$-directional orbital magnetization $M_z$ arises from the circulating edge current (the red loop). An $x$-directonal electric field $E_x$ creates chemical potential difference at sample edges, yielding a Hall voltage in the $y$ direction. (b) The second order nonlinear Hall effect in metals with finite Berry curvature (BC) dipole and quantum metric (QM) dipole on Fermi surfaces. The first order effect of $E_x$ is to induce an orbital magnetization $M_z=\chi_{zx}E_x$, and the induced $M_z$ also has a corresponding edge current (the purple dashed loop). The second order effect of $E_x$ differs the chemical potentials at sample edges, generating the second order Hall voltage in the $y$ direction of the current loop.
• Figure 2: The transport experiment to verify orbital magnetization as the origin of the nonlinear Hall effect. (a) The experimental setup that combines both DC and AC currents. The DC current $I^{\textrm{DC}}$ is to induce the nonlinear Hall effect and the associated orbital magnetization. The AC current $I^\omega\ll I^{\textrm{DC}}$ serves as a perturbation to detect the orbital magnetization. (b) The Hall voltages scale quadratically with the $x$-directional voltage, which confirms the nonlinear Hall effect. (c) The first harmonic Hall voltage scales linearly with the $x$-directional AC voltage, which verifies the orbital magnetization and the affliated edge current.
• Figure 3: Schematic showing of the nonlinear Hall effect driven by a non-collinear bichromatic electric field. (a) In a chiral metal, the field components $E_x\left(\omega_1\right)$ and $E_y\left(\omega_2\right)$ induce orbital magnetizations along the $x$ and $y$ directions, respectively. The affliated edge currents (the orange and purple loops) are further perturbed by the electric field components, generating $z$-directional Hall voltages $V_{H_1}\left(\omega_1\right)$ and $-V_{H_2}\left(\omega_2\right)$, respectively. The total Hall voltage thus becomes $V_H=V_{H_1}\left(\omega_1\right)-V_{H_2}\left(\omega_2\right)$. (b) The Lissajous curve of a non-collinear bichromatic electric field. The total electric field is represented by red arrows, and its time evolution gives the Lissajous curve colored in blue. The precession of the non-collinear bichromatic electric field generates the dynamical chirality, which drives a fintie nonlinear Hall response in isotropic chiral metals. (c) In isotropic chiral metals, both the circular photo-galvanic effect and the nonlinear Hall effect originate from orbital magnetization.
• Figure 4: The nonlinear Hall current density driven by a Gaussian pulse field $\bm{E}\left(t\right)=\left[E_1\left(t-\frac{\tau_0}{2}\right),E_2\left(t+\frac{\tau_0}{2}\right),0\right]$ with $E_i\left(t\right)=2E_0\sin\left(\omega_it+\varphi_i\right)\exp\left(-\frac{t^2}{4\sigma_i^2}\right)$, $i=1,2$. (a) The $x$ and $y$ components of the Gaussian pulse field $\bm{E}\left(t\right)$ in the time domain. Here in the simulation, we have set $\omega_1=8~\textrm{THz},\omega_2=24~\textrm{THz},\sigma_1=1.5~\textrm{ps},\sigma_2=1.0~\textrm{ps},\varphi_1=\frac{\pi}{4},\varphi_2=\frac{\pi}{5},\tau_0=2.0~\textrm{ps}$, so the Gaussian pulses in the two perpendicular directions are not synchronized. (b) The resulting Hall current pulse $\bm J_H(t)$ in the $z$ direction. Here, $J_0=\frac{e^3\gamma E_0^2}{\hbar^2}$.