X-ray magnetic circular dichroism evidence of intrinsic $d$-wave altermagnetism in rutile-structure NiF$_2$

Abstract

We present the x-ray magnetic circular dichroism (XMCD) at the Ni $L_{2,3}$-edge as an evidence of the $d$-wave altermagnetism in rutile-structure NiF$_2$. Sizable XMCD signal is observed in excellent agreement with theoretical simulations. Owing to a considerable net magnetization due to spin canting, the XMCD spectrum consists of an altermagnetic signal as well as a non-negligible ferromagnetic contribution. We verify experimentally that the XMCD spectrum can be written as a sum of contributions from altermagnetism and weak ferromagnetism. Two experimental methods to isolate the ferromagnetic contribution are shown to yield essentially the same result. These are dependence of XMCD on applied magnetic fields below the Néel temperature and the XMCD measured in applied field above the Néel temperature. Our results demonstrate the utility of XMCD as a probe for altermagnetic materials with the coexisting weak ferromagnetism induced by the relativistic spin-orbit coupling.

Figures (12)

• Figure 1: (a) Crystal and magnetic structures of rutile NiF$_2$. Red arrows indicate the spin directions of the two antiferromagnetic sublattices on the Ni sites. Here, x-, y-, and z- represent the crystal's a, b, and c axis, respectively. (b) Schematic picture of the experimental geometry, where the incident x-ray wave vector $\boldsymbol{k}_{\rm in}$ is parallel to the applied field $\boldsymbol{B}$ and the sample $y$-axis. In this configuration, the net magnetic moment $\boldsymbol{m}_{\rm FM}$ aligns along $y$-axis and the Néel vector $\mathbf{L}$ points along $x$-axis. The black dashed line represents the (110) surface of the sample. (c) Field dependence of the magnetization, measured by SQUID at 2 K with the magnetic field applied along the $y$-axis. The magnetization data point at 6 T and 220 K (green star) is shown which is extracted from the corresponding XMCD spectra. We also present the magnetization calculated within the present theory (blue dots).
• Figure 2: Ni $L_{2,3}$-edge XAS (top), XMCD (middle), and XLD (bottom) spectra measured at 2 K under an applied field of 0.1 T. Red solid lines show the experimental data, and blue dashed lines show the calculated spectra. The XMCD and XLD spectra are scaled by constant factors indicated in the figure. The theoretical XLD spectrum in the paramagnetic (PM) phase is also shown (green line). The calculated spectral intensities at the $L_3$ ($L_2$) edge are broadened by a Lorentzian of 0.20 eV (0.25 eV) and a Gaussian of 0.20 eV (0.35 eV) (HWHM).
• Figure 3: (a) Ni $L_{2,3}$-edge XMCD spectra measured at different magnetic fields up to 6 T (top). The corresponding simulated spectra are shown at the bottom. The XMCD is defined as $\Delta(\omega)=(\mu_{+}(\omega)-\mu_{-}(\omega))/(\mu_{+}(\omega)+\mu_{-}(\omega))_{\rm max}$. (b) XMCD spectra reconstructed as a linear combination of the extracted (c) $\Delta_{\rm ALT}(\omega)$ (blue, left axis) and $\Delta_{\rm FM}(\omega)$ (red, right axis) according to Eq. \ref{['eq:h']}. The inset in (b) shows the difference between the experimental XMCD spectra and the reconstructed spectra.
• Figure 4: (a) Ni $L_{2,3}$-edge $\mu_+$ (red) and $\mu_-$ (blue) spectra measured at 220 K under magnetic fields of 6.0 T. (b) XMCD spectra measured at 220 K (well above the Néel temperature) under magnetic fields of 6.0 T (black) and 0.1 T (orange). The red dashed line represents $\Delta_{\rm FM}(\omega)$ scaled by the magnetic moment $m_{\rm FM}(B=6~\rm {T},220~\rm {K})=0.036~\mu_{\rm B}$ extracted from the XMCD spectrum at 220 K under a 6 T magnetic field using the sum rule.
• Figure S1: X-ray Laue of NiF$_2$ sample with [110] surface normal.