Probing quantum geometric nonlinear magnetization via second-harmonic magneto-optical Kerr effect

Abstract

Quantum geometry provides an intrinsic framework for characterizing the geometric structure of quantum states. It highlights its relevance to various aspects of fundamental physics. However, its direct implications for magnetic phenomena remain largely unexplored. Here, we report the observation of electric-field-induced nonlinear magnetization in the nonmagnetic semimetal WTe$_2$ by using a second-harmonic magneto-optical Kerr effect (SMOKE) spectroscopy. We observe a robust nonlinear SMOKE signal that scales quadratically with current and persists up to 200 K. Theoretical modeling and scaling analysis indicate that this nonlinear magnetization is dominated by the orbital contribution and is intrinsically linked to the quantum Christoffel symbol. Just as the Christoffel symbol is a fundamental quantity encoding spacetime geometry in Einstein's general relativity, our work establishes a direct link between quantum geometry and nonlinear magnetization, and provides a geometric perspective for designing future orbitronic devices.

Figures (4)

• Figure 1: SMOKE detection and quantum geometric nonlinear magnetization. (a) Demonstration of second-harmonic nonlinear magnetization $\mathbf{M}^{2\omega}$ at frequency $2\omega$ induced by an alternating electric field $\mathbf{E}^{\omega}$ at frequency $\omega$, i.e., ($\mathbf{M}^{2\omega}\propto\mathbf{E}^\omega\mathbf{E}^\omega$). (b) Schematic of SMOKE detection to the nonlinear magnetization, the red downward (upward) arrow represents incident (reflected) light, the green arrow represents induced magnetization with frequency $2\omega$, and purple arrow represents applied alternating electric field with frequency $\omega$. (c) Schematic of nonlinear magnetization origin. Orbital magnetization arises from the self-rotation of Bloch wavepacket (circular arrows) Xiao10rmpGaoY14prl. An applied electric field modifies the wave packets (solid to dashed curves) and induces a nonequilibrium distribution. This prevents the mutual cancellation of time-reversal counterparts (red and blue), and leads to a quadratic electric-field dependence.
• Figure 2: Crystal structure and electrical transport properties of WTe$_2$. (a) The crystal structure of WTe$_2$. The ideal crystal structure of Td-phase WTe$_2$ only preserves a mirror symmetry $\mathcal{M}_a$ (with the mirror plane indicated by the dashed line). (b) Angle-dependent polarized Raman spectral intensity of a WTe$_2$ flake used in the device. The measurements were obtained by rotating the laser polarization with respect to the a axes. (c) Representative angle-dependent intensities of the Raman peak at 212 cm$^{-1}$. (d) First-harmonic longitudinal response $V^\omega$ with the alternating current applied along the a and b axes. (e) Second-harmonic transverse response $V_{\perp}^{2\omega}$ with the applied alternating current along the a and b axes (the horizontal coordinate has been converted to the voltage). The dots represent the experimental data, and the solid lines represent the fitted quadratic curves. The inset in (e) shows the measurement setup. Data in (d) and (e) were collected at $T = 8$ K.
• Figure 3: Observation of SMOKE and temperature-dependence. (a) Optical path for polar Kerr measurements. (b) Optical image of the WTe$_2$ device. The red arrow indicates the light spot. (c) The MOKE signal $\theta_{\text{K}}^{\omega}$ as a function of current $I_a^{\omega}$, and there is no signal within the measurement accuracy. (d) The SMOKE signal $\theta_{\text{K}}^{2\omega}$ as a function of current $I_{a}^{\omega}$, the solid line represents a quadratic fit to the data. (e) Temperature dependence of the SMOKE from 8 K to 200 K.
• Figure 4: Theoretical results and scaling analysis. (a) The evolution of Fermi surfaces at Fermi level $\varepsilon_F=0.25$ eV by breaking two-fold rotational symmetries ($\mathcal{C}_{a,b}^2$) and mirror symmetries ($\mathcal{M}_{a,b}$) successively, the colorbar depicts the strength of $\Lambda_n^{\text{O},caa}$. (b) Calculated nonlinear magnetization coefficient $\alpha_{caa}$ as a function of the Fermi energy $\varepsilon_F$. The green, red, and blue represent spin ($\times$10 for visibility), orbital, and total contributions, respectively. The inset depicts the energy dispersion. The model parameters are $v=1$ eV$\cdot$nm, $t/v=0.3$, $m=0.1$ eV, and $b=1$ eV$\cdot$nm$^2$Lu18prl. (c) Scaling behavior between the observed SMOKE signal and linear longitudinal conductivity for different temperatures from $T=8$ K to $T=200$ K. (d) Scaling behavior between the observed SMOKE signal and nonlinear Hall signal. The two data points lie on the vertical axis because the nonlinear Hall signal becomes undetectable at $T=150$ K and 200 K.