Cubic-in-magnetization contributions to the magneto-optic Kerr effect investigated for Ni(001) and Ni(111) thin films

Abstract

*The abstract of this article is too long to be included in the arXiv metadata; please see the paper for the full abstract.* ...In this paper, we introduce the detailed theory of cubic-in-magnetization magneto-optic Kerr effect (CMOKE) by deriving the magneto-optic tensor of third order in magnetization, denoted as $\bm{H}$, and comparing the strength of CMOKE for different crystal orientations theoretically and experimentally. In crystals with cubic symmetry, the tensor $\bm{H}$ is described by two independent parameters $H_{123}$ and $H_{125}$. Together with the linear magneto-optic tensor $\bm{K}$ and quadratic magento-optic tensor $\bm{G}$, the permittivity tensor is described up to third order in magnetization. We analytically describe equations of the MOKE including the contribution of QMOKE and CMOKE itself for (001)- and (111)-oriented cubic crystal structures. Those are compared to experimental measurements of two samples with an (001)- and (111)-oriented fcc Ni layer, respectively. Further, we use Yeh's 4$\times$4 transfer matrix calculus to simulate and describe the experimental measurements phenomenologically from the permittivity tensor up to third order in $\bm{M}$. We find that the MOKE anisotropy that stems from the magneto-optic tensor $\bm{H}$ described as $ΔH = H_{123}-3H_{125}$, is much more pronounced for the (111)-oriented cubic crystal structure, for which it manifests as three-fold in-plane angular dependencies of MOKE with longitudinal and also with transversal magnetization direction, respectively.

Figures (9)

• Figure 1: Eight-directional method measurement of sample Ni(111) with $p$-polarized light and at an AoI of 45$^\circ$. Graphs present the dependence on the in-plane sample orientation $\alpha$ of the (a) Kerr rotation and (b) Kerr ellipticity at a wavelength of 406 nm as well as (c) Kerr rotation and (d) Kerr ellipticity at a wavelength of 635 nm. Experimental data points are plotted as "x" marks, whereas solid lines represent the fit of the numerical model based on Yeh's 4$\times$4 transfer matrix formalism to the experimental data.
• Figure 2: Eight-directional method measurement of sample Ni(111) at normal AoI and at wavelength 635 nm. The solid lines represent the prediction of Yeh's 4$\times$4 transfer matrix numerical model using the MO parameters from Tab. \ref{['tab:RS111_fit_result']}.
• Figure 3: Eight-directional method measurement of sample Ni(001) with $p$-polarized light and at an AoI of 45$^\circ$. Graphs present the dependence on the in-plane sample orientation $\alpha$ of the (a) Kerr rotation and (b) Kerr ellipticity at a wavelength of 406 nm as well as (c) Kerr rotation and (d) Kerr ellipticity at a wavelength of 635 nm. Experimental data points are plotted as "x" marks, whereas solid lines represent the fit of the numerical model based on Yeh's 4$\times$4 transfer matrix formalism to the experimental data.
• Figure 4: (a),(c) Kerr rotation and (b),(d) Kerr ellipticity of the amplitude of (a),(b) the four-fold angular dependence for (001)-oriented cubic crystal structure and (c),(d) the three-fold angular dependence for (111)-oriented cubic crystal structure depending on the AoI. The data were simulated using the numerical model based on Yeh's $4\times 4$ matrix formalism, for which structural and optical parameters of sample Ni(111) were used, but MO parameters were chosen to be $K=0+0i$, $\Delta G=0.01+0.01i$ (with $G_s=0+0i$ and $2G_{44}=0.01+0.01i$) and $\Delta H=0.01+0.01i$ (with $H_{123}=0.01+0.01i$ and $H_{125}=0+0i$). Vicinal structural parameter $\varepsilon_S=0$ here. The simulation is done for a wavelength of 635 nm.
• Figure A.1: (a) The plane of incidence and the surface of the sample define the Cartesian coordinate system $\hat{x}$, $\hat{y}$, $\hat{z}$, where $\hat{x}$ is perpendicular to the plane of incidence and parallel to the surface of the sample. (b) Definition of the $\hat{s}$, $\hat{p}$, $\hat{k}$ Cartesian system for the incident and reflected beam and definition of the directions of the normalized magnetization components $M_T \parallel \hat{x}$, $M_L\parallel \hat{y}$ and $M_P\parallel \hat{z}$. (c) Visualisation of the positive in-plane rotation of the sample and the magnetization within the $\hat{x}$, $\hat{y}$, $\hat{z}$ coordinate system, described by the angle $\alpha$ and $\mu$, respectively. The figure is taken from Ref. Silber19 and slightly modified.