Broadband nonlinear Hall response and multiple wave mixing in a room temperature altermagnet

TL;DR

This work shows that altermagnetic CrSb hosts a Berry curvature quadrupole (BCQ) in its non-relativistic spin-split electronic structure, enabling a broadband third-order nonlinear anomalous Hall effect at room temperature. Symmetry analysis and DFT identify a single BCQ component $\mathcal{Q}_{xxxz}$ in the $xz$-plane, which accounts for the observed third-order transverse voltage $V_{zxxx}^{3\omega}$ and its anisotropy. The authors verify the BCQ origin experimentally via angle-resolved photoelectron spectroscopy and scaling laws, and exploit the nonlinear susceptibility $\chi^{3\omega}$ to realize a room-temperature multiple wave mixing device, with outputs at combinations of input frequencies that hint at THz-range applications. Overall, the results establish that the nonlinear electric properties of altermagnets are governed by their magnetic crystalline order, expanding potential applications from spintronics to high-frequency electronics and energy harvesting.

Abstract

Crystalline symmetries determine the linear and nonlinear response of materials to external stimuli such as mechanical pressure and electromagnetic fields, governing phenomena such as piezoelectricity, optical activity, and multiple wave mixing with wide ranging technological applications. Altermagnets present a new class of materials with magnetic crystalline order where specific crystal symmetry operations connect antiferromagnetic sublattices, leading to non-relativistic spin-splitting of the electronic band structure. Hence, the electric material properties of altermagnets should uniquely mirror these underlying symmetry properties, potentially giving rise to novel phenomena in response to external driving fields. Here, we report the discovery of a broadband third-order nonlinear anomalous Hall effect in altermagnetic CrSb at room temperature. The comparison of our observations with symmetry analyses and model calculations shows that this nonlinear Hall response is induced by the nonlinear electric susceptibility of a Berry curvature quadrupole, which exists within the spin-split band structure of CrSb and is characterized by the underlying crystalline and magnetic symmetries. We then utilize this third-order nonlinear electric susceptibility of CrSb to realize a multiple wave mixing device with pronounced four wave mixing output, which could, in principle, be extended to THz frequencies. Our study discovers that the crystalline magnetic order of altermagnets determines their nonlinear electric material properties, which could facilitate applications in high-frequency electronics, THz generation, communication networks, and energy harvesting.

Figures (6)

• Figure 1: CrSb: a g-wave altermagnet. (a) Top left: Shown is the lattice structure of a $\mathcal{P}\mathcal{T}$-symmetric antiferromagnet, where the atomic sublattices are connected via inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetry. Up and down spin orientations are color-coded in red and blue color, respectively. Top right: Shown is the lattice structure of a t$\mathcal{T}$ antiferromagnet, where the inversion is broken due to the presence of an extra grey atom out of the plane. Center: Shown is the schematics of the spin-degenerate electronic band structure of both $\mathcal{P}\mathcal{T}$ and t$\mathcal{T}$ antiferromagnets within the Brillouin zone. Bottom: The table illustrates which Berry curvature contributions are allowed (green check) and forbidden (red cross) to exist in antiferromagnets by symmetry arguments. BCM, BCD, and BCQ denote the Berry curvature monopole, dipole, and quadrupole, respectively. (b) Top: Shown is the illustration of the lattice structure of an altermagnet, schematically indicating the presence of $C$-symmetry operation to connect the two spin sublattices. Center: Shown is a schematic illutstration of the spin-split electronic band structure of an altermagnet within the Brillouin zone. Bottom: The table illustrates which Berry curvature multipoles are allowed to exist in altermagnets by symmetry arguments. (c) Shown is the crystallographic and magnetic structure of CrSb. The Chromium (Cr) atoms with up-spin and down-spin are shown in red and blue color, respectively and antimony (Sb) atoms are shown as grey spheres. (d) Top: Shown are the anisotropic spin sublattices of CrSb connected via the combined symmetry rotation operation $[C_{2}^s {\parallel} C_{6z}t]$. Bottom: Shown are the anisotropic spin sublattices of CrSb connected via the mirror operation $[C_{2}^s {\parallel} M_{z}]$. (e) Shown is the momentum $k_x$ and energy $E$-resoled angle-resolved photo-electron spectroscopy spectrum recorded on the surface of a cleaved CrSb crystal at $k_y= 3.3\,Å^{-1}$ and $k_z=1.2\,Å^{-1}$ along a direction parallel to $\Gamma-K$ (see Methods section for details).
• Figure 1: Reproduction of the third-order NLAHE on device $'XZ2'$. We have fabricated another Hall bar device $'XZ2'$ from a CrSb lamella cut along the crystallographic $xz-$direction (see Methods section). (a) Shown is the third-order transverse voltage $V_{zxxx}^{3\omega}$ (circular symbols) recorded as a function of the cubed longitudinal bias current $I_x$ at room temperature $T=300\,$K. The measurement geometry and a false colored scanning electric microscopy image of the device are shown as an inset. The linear fit (dashed line) to the data establishes the third-order nature of the detected signal. The amplitude of $V_{zxxx}^{3\omega}$ measured in device $'XZ2'$ quantitatively reproduces the amplitude of $V_{zxxx}^{3\omega}$ measured in device $'XZ'$. (b) Shown is the magnetic field ($B$) dependence of $V_{zxxx}^{3\omega}$ at room temperature. Consistent with our observation on device $'XZ'$, $V_{zxxx}^{3\omega}$ is independent of the magnetic field amplitude over the examined field range. (c) Scaling law analysis of the third-order NLAHE. Shown is $E_{zxxx}^{3\omega}/(\sigma_{xx}^{1\omega}(E_{xx}^{1\omega})^3)$ (circular symbols) and linear fit to the data (dashed line) plotted as a function of the squared longitudinal conductivity $\left( \sigma_{\rm xx}^{1\omega} \right)^2$.From the fit, we extract a finite vertical intercept $\beta=(-106\pm3)\times10^3\,\Omega\mu\text{m}^3V^{-2}$ and linear slope $\alpha=(2381\pm16)\times10^{-6}\Omega^3\mu\text{m}^5V^{-2}$.
• Figure 2: Discovery of the third-order nonlinear anomalous Hall effect in altermagnetic CrSb at room temperature. (a) Shown is a summary of theoretically possible contributions--Drude scattering, quantum metric quadrupole (QMQ), and Berry curvature quadrupole (QMQ)--to the longitudinal ($\sigma_{\rm L}^{n\omega}$) and transverse ($\sigma_{\rm T}^{n\omega}$) linear ($1\omega$) and nonlinear ($2\omega,\,3\omega$) conductivities, respectively obtained from symmetry analysis within different crystallographic planes of CrSb. Red crosses indicate that contributions are prohibited by symmetry arguments. (b) Shown is a schematic illustration of the prism-shaped CrSb bulk crystal (light purple) as well as the spatial orientation of the lamellae ($'XZ'$ red, $'YZ'$ blue, and $'ZX'$ grey) with respect to the bulk crystal. (c) Shown is a false-colored scanning electron microscopy image of the finished Hall bar device $'XZ'$ made from a crystalline lamella cut within the crystallographic $xz-$plane. The lamella, FIB-deposited Pt contacts, and prepatterned gold electrodes are highlighted by pink, blue, and yellow color, respectively. The SiO$_{2}$/Si substrate appears as black background. (d)-(f) Shown are schematic illustrations of the measurement geometry of the $'XZ'$, $'YZ'$, and $'ZX'$ devices, respectively. The bias current $I_{\alpha}$ and the longitudinal $V_{\alpha\alpha\alpha\alpha}$ and transverse $V_{\beta\alpha\alpha\alpha}$ voltage responses are denoted within the respective coordinate frame of each device. (g)-(i) Shown are the third-order nonlinear transverse voltage $V_{\beta\alpha\alpha\alpha}^{3\omega}$ as a function of the cubed bias current $I_{\rm \alpha}^3$ for the $'XZ'$ (red), $'YZ'$ (blue), and $'ZX'$ (grey) devices, respectively.
• Figure 2: Finite Berry curvature quadrupole at Fermi energy from DFT calculations. Shown is the momentum-space integrated Berry curvature quadrupole $BCQ_{xxxz}$ plotted as a function of the chemical potential $\mu$. The actual chemical potential of CrSb $\mu_0=8.8972 eV$ is indicated by a vertical dashed line. The underlying Berry curvature distribution $\mathcal{Q}_{xxxz}(k_x,\,k_z)$ in the $xz$-plane of the Brillouin zone is shown in Fig. \ref{['fig:fig3']}(e). Details of the model calculation are presented in the Methods section.
• Figure 3: Characterization of the third-order NLAHE of CrSb. (a) Shown is the third-order nonlinear transverse voltage $V_{zxxx}^{3\omega}$ as a function of an external magnetic field (${\it B}$) applied along the $y-$direction. (b) Shown are the first-order longitudinal conductivity $\sigma_{xx}^{1\omega}$ (blue symbols) and the third-order transverse conductivity $\sigma_{zxxx}^{3\omega}$ (red symbols) as a function of temperature $T$. (c) Shown is $E_{zxxx}^{3\omega}/(\sigma_{xx}^{1\omega}(E_{xx}^{1\omega})^3)$ and fit to the data (dashed line) plotted as a function of the squared longitudinal conductivity $\left( \sigma_{\rm xx}^{1\omega} \right)^2$. (d) Shown is the electronic band structure of CrSb, as obtained from DFT calculations (see Methods section), overlaid with the amplitude of the Berry curvature $\Omega_{xz}$. The band structure is plotted along the $k_z$ direction at $k_y=0$ and $k_z=0.4\,Å^{-1}$. (e) Shown is the momentum space distribution of $Q_{xxxz}$ within the $xz-$plane of the Brillouin zone at Fermi energy.