Topological magneto-optics in the non-coplanar antiferromagnet Co_{1/3}NbS_2: Imaging and writing chiral magnetic domains

TL;DR

The study investigates topological magneto-optical effects in the fully compensated antiferromagnet Co$_{1/3}$NbS$_2$, where non-coplanar tetrahedral spin order yields a finite $\sigma_{xy}(\omega)$ and magnetic circular dichroism (MCD) despite negligible net magnetization. Broadband MCD spectroscopy across 400–1000 nm and high-resolution scanning MCD microscopy reveal spectral and micron-scale chiral AFM domains, and demonstrate optical writing of chiral domains via thermally assisted switching. First-principles calculations of $\sigma_{xx}(\omega)$ and $\sigma_{xy}(\omega)$ based on the non-coplanar order reproduce a characteristic spectral fingerprint (rise near $1.3$ eV, maximum near $1.5$ eV, sign change near $2.5$ eV) consistent with the Berry-curvature origin of the TMO response, independent of spin-orbit coupling. Collectively, these results establish optical methods as incisive probes of topological AFM order and show the feasibility of optically writing AFM domains, with implications for all-optical control and information storage in chiral antiferromagnets.

Abstract

Despite its tiny net magnetization, the antiferromagnetic (AFM) van der Waals material Co$_{1/3}$NbS$_2$ exhibits a large transverse Hall conductivity $σ_{xy}$ even at zero applied magnetic field, which arises, as recently shown, from the topological nature of its non-coplanar ``tetrahedral'' AFM order. This triple-Q magnetic order can be regarded as the short-lengthscale limit of a magnetic skyrmion lattice, and has an intrinsic spin chirality. Here we show, using optical wavelengths spanning the ultraviolet to infrared (400-1000 nm), that magnetic circular dichroism (MCD) provides an incisive optical probe of the topological AFM order in Co$_{1/3}$NbS$_2$. Measurements as a continuous function of photon energy are directly compared with first-principles calculations, revealing the influence of the underlying quantum geometry on optical conductivity. Leveraging the power and flexibility of optical methods, we use scanning MCD microscopy to directly image chiral AFM domains, and demonstrate writing of chiral AFM domains.

Figures (6)

• Figure 1: (a) Crystal structure and magnetic unit cell of Co$_{1/3}$NbS$_2$. Below its AFM ordering temperature $T_N$, the Co spins exhibit non-coplanar tetrahedral order. Time-reversed ($\mathcal{T}$) spin configurations exhibit opposite scalar spin chirality, $\chi_{ijk} = \mathbf{S}_i \cdot (\mathbf{S}_j \times \mathbf{S}_k)$, from which $\mathbf{b}_{\textrm{eff}}$, $\sigma_{xy}(\omega)$, and TMO effects arise. $\odot$ / $\otimes$ represent spins pointing out of /into the page, and spins with small/large arrowheads are canted into/out of the page. Grey spins depict Co on a neighboring (lower) 2D layer, so that the all-in/all-out tetrahedral spin configuration can be readily recognized. b) The MCD experiment: Wavelength-tunable light is linearly polarized (LP) and modulated between right- and left-circular polarization (RCP/LCP) by a photoelastic modulator (PEM), then reflected at near-normal incidence from the sample and detected by an avalanche photodiode (APD). c) MCD versus $B$ at different temperatures, showing the emergence of large and hysteretic MCD below $T_N \approx 28.2$ K, similar to the large and hysteretic Hall conductivity observed in transport studies (where $\omega \approx 0$) Ghimire:2018. d) MCD as Co$_{1/3}$NbS$_2$ is cooled in $B=\pm 0.5$ T and 0 T. The growth of the MCD below $T_N$ tracks the growth of the AFM order parameter $\chi_{ijk}$.
• Figure 2: a) MCD vs. $B$ at $T$=26 K, using different wavelengths (photon energies) of probe light. All curves are plotted on the same vertical scale (shown), but offset for clarity. The amplitude and sign of the hysteresis loops vary with wavelength, as does the slope of the linear background. b) Dependence of the amplitude of the hysteresis loop, $\Delta$MCD, on photon energy (i.e., the TMO response elicited solely by chiral AFM order). Dashed grey line shows the slope of the linear background (in units of MCD/T; smoothed). c) First-principles calculation of the energy-dependent MCD arising from non-coplanar tetrahedral AFM order in Co$_{1/3}$NbS$_2$.
• Figure 3: 50$\times$50 $\mu$m MCD images of Co$_{1/3}$NbS$_2$, at a) $T$=30K ($>T_N$), and after cooling to 9 K ($<T_N$) in $B$=+60 mT, 0 mT, and -60 mT respectively. Chiral AFM domains form spontaneously when cooled in $B$=0. Scale bars are 10 $\mu$m. b) Images of chiral AFM domains at $T$=7.5 K, in the same area, following three thermal cycles to 30 K at $B$=0.
• Figure 4: Writing AFM domains with opposite spin chirality in Co$_{1/3}$NbS$_2$ using light (see main text). The MCD images show examples of chiral AFM domain patterns comprising a) a triangular arrangement of local domains, b) vertical stripes, and c) the National High Magnetic Field Lab's logo, "$\vec{\textrm{M}}$". Scale bars are 25 $\mu$m.
• Figure S1: First-principles calculations: To calculate the energy-dependent optical conductivities and topological magneto-optical (TMO) response from the non-coplanar tetrahedral antiferromagnetic (AFM) spin order in Co$_{1/3}$NbS$_2$, we adopt the Vienna Ab-initio Simulation Package (VASP) Kresse:1996Kresse:1999 code for density functional theory (DFT) calculations using the experimental structure of Co$_{1/3}$NbS$_2$ measured from X-ray diffraction Anzenhofer:1970. The Perdew-Burke-Ernzerhof (PBE) Burke:1997 functional is used for the exchange-correlation functional with the plane-wave energy cut-off for 400 eV and a 14x14x4 k-point grid. For the magnetic band structure and the dynamical Hall conductivity calculations, we first construct the tight-binding Hamiltonian based on the non-magnetic primitive cell using both Co d and Nb d Wannier orbitals by adopting the Wannier90 code Marzari:2012. The magnetic unit cell is constructed as the 2x2x1 supercell from the primitive cell. Then, the spin-exchange interaction is incorporated into the Hamiltonian and treated using the Hartree-Fock approximation following previous approaches Park:2022. Left: The dynamical Hall conductivities $\sigma_{xx}(\omega)$ and $\sigma_{xy}(\omega)$ are computed based on the magnetic band structure by adopting the Wannier-Berri package Tsirkin:2021. For convergence, we used a 10x10x8 k-mesh with 10 recursive refinement iterations and the smoothing of the Fermi functions at temperature $T$=10 K. Right: The calculated Kerr ellipticity $\epsilon_K(\omega)$ (i.e., the imaginary part of the complex Kerr angle $\theta_K + i\epsilon_K = -\sigma_{xy}/[\sigma_{xx} \sqrt{1 + i(4\pi/\omega)\sigma_{xx}}~]$) Feng:2015Schilberth:2022. Shown below for comparison is the calculated MCD from the main text (estimated as $\sim$ Im[$\sigma_{xy}$]/Re[$\sigma_{xx}$]; the trends are similar except that the zero-crossing at low energy occurs at $\sim$1.5 eV instead of $\sim$1.3 eV.