The magneto-optic Kerr effect (MOKE) is a well known and handy tool to characterize ferro-, ferri-, and antiferromagnetic materials. Many of the MOKE techniques employ effects solely linear in magnetization $\mathit{M}$. Nevertheless, a higher-order term being proportional to ${\mathit{M}}^2$ and called quadratic MOKE (QMOKE) can additionally contribute to the experimental data. Here, we present detailed QMOKE spectroscopy measurements in the range of 0.8–5.5 eV based on a modified eight-directional method applied on ferromagnetic bcc Fe thin films grown on MgO substrates. From the measured QMOKE spectra, two further complex spectra of the QMOKE parameters $G_s$ and $2G_{44}$ are yielded. The difference between those two parameters, known as $\mathrm{\Delta }G$, denotes the strength of the QMOKE anisotropy. Those QMOKE parameters give rise to the QMOKE tensor $\mathit{G}$, fully describing the perturbation of the permittivity tensor in the second order in $\mathit{M}$ for cubic crystal structures. We further present experimental measurements of ellipsometry and linear MOKE spectra, wherefrom permittivity in the zeroth and the first order in $\mathit{M}$ are obtained, respectively. Finally, all those spectra are described by ab initio calculations.

Spectral ellipsometry is a useful technique allowing fast, nondestructive, and contactless characterization of thin films and constituent materials. In this Letter, we show that both the linear magneto-optic Kerr effect (LinMOKE, proportional to the magnetization M) and the quadratic MOKE (QMOKE, proportional to ${\mathit{M}}^2$ ) can be a useful extension of spectral ellipsometry and are able to sense the crystallographic ordering of Heusler compounds. This is demonstrated for the Heusler compound Co₂MnSi, which has a crystallographic transition from a B2 to an L2₁ structure with increasing annealing temperature T_a. We investigated a set of Co₂MnSi thin films deposited on MgO(001) substrates and annealed from 300 to 500 °C. The amplitude of LinMOKE and QMOKE spectra, detected in the extended visible spectral range of 0.8–5.5 eV, scales linearly with T_a, and this effect is pronounced at the resonant peaks of the QMOKE spectra below 2.0 eV. Furthermore, the spectra of the magneto-optic (MO) parameters, which fully describe the MO response of Co₂MnSi up to the second order in M, are obtained depending on T_a. Finally, the spectra are compared with ab initio calculations of a purely L2₁-ordered Co₂MnSi Heusler compound.

In most studies utilizing the magneto-optic Kerr effect (MOKE), the detected change of polarized light upon reflection from a magnetized sample is supposed to be proportional to the magnetization $\mathbf{M}$. However, MOKE signatures quadratic in $\mathbf{M}$ have also been identified and utilized, e.g., to sense the structural order in Heusler compounds, to detect spin-orbit torque or to image antiferromagnetic domains. In our study, we observe a strong anisotropic MOKE contribution of third order in $\mathbf{M}$ in $\mathrm{Ni}$(111) thin films, attributed to a cubic magneto-optic tensor $\propto {\mathbf{M}}^3$. We further show that the angular dependence of cubic MOKE (CMOKE) is affected by the amount of structural domain twinning in the sample. Our detailed study on CMOKE will open up opportunities for CMOKE applications with sensitivity to twinning properties of thin films, e.g., CMOKE spectroscopy and microscopy or time-resolved CMOKE. Furthermore, the in-plane magnetization orientation can be detected with CMOKE when perpendicular incidence light is used, which is suitable for MOKE setups that do not provide oblique incidence.

The Heusler compound ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ is an interesting material for spintronic applications due to its predicted half-metal ferromagnetic character with a large band gap in the minority states. In this work, we present the optical and magneto-optical properties of a ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ bulk sample at room temperature. The spectral permittivity is found to be dominated by an absorption peak at 1.4 eV, originating mostly from absorptions by minority electrons. The best agreement between the experimentally obtained spectral permittivity tensor and the ab initio calculated one is for the local density approximation potential, closely followed by the generalized gradient approximation potential. However, the half-metallic ferromagnet character with $100\%$ spin polarization is preserved only in calculations employing the Coulomb interaction correction (+U). This suggests that ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ is probably not a half metal as a single minority $d$-band dips below the Fermi level.

The magneto-optic Kerr effect (MOKE) is often applied as a tool for the magnetic characterization of thin films. Here, the change in polarization upon reflection from the magnetized sample is mainly regarded as being linearly proportional to the magnetization M (LinMOKE). Recently, we reported on third-order MOKE contributions, named cubic MOKE (CMOKE) being ∝ M3 in Ni(111) thin films. This CMOKE manifests itself as an anisotropic contribution to the MOKE signal (with regard to the crystallographic orientation) measured in longitudinal or transversal configuration in full magnetic saturation. While LinMOKE (odd in M) and quadratic MOKE (QMOKE) being ∝M2 (even in M) can easily be separated by methods based on magnetization parity, this no longer holds true for LinMOKE and CMOKE (odd in M). Here, we report on the observation of CMOKE in thin-film heterostructures with structurally twinned and untwinned Co(111) layers, demonstrating that a large CMOKE is not only present in Ni thin films. Additionally, we show that the observed anisotropic contributions cannot stem from LinMOKE by analyzing their dependence on the angle of incidence (AoI) of light. While the QMOKE is almost vanishing in Co(111) using light with wavelengths of 635 and 406 nm, the CMOKE contributions reach up to about 30% of the LinMOKE contribution at an AoI of 45 degrees and become even more dominant toward normal AoI, which emphasizes the importance of higher-order MOKE effects in magneto-optic experiments.