2026/03/03 by Robin Silber, Maik Gaerner, Kamil Postava +2
Physics and Astronomy · Engineering · Materials Science · #Magnetic properties of thin films #Magneto-Optical Properties and Applications #Magnetism in coordination complexes
paper · pdf · doi:10.1103/c82k-w836
The existence of higher-order effects in magnetization in magneto-transport and magneto-optic phenomena is well known. These effects are of importance not only in research but also directly in applications. Anisotropic magneto-resistance, anisotropic magneto-thermopower, quadratic magneto-optic Kerr effect (QMOKE), Voigt effect, or x-ray linear magnetic dichroism (and birefringence)—all of these are effects of second order in magnetization. Recently, we have reported on systematic observations of the cubic-in-magnetization magneto-optic Kerr effect (CMOKE) in Ni(111) thin films. In this paper, we introduce the detailed theory of CMOKE by deriving the magneto-optic tensor of third order in magnetization, denoted as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="bold-italic">H</a:mi> </a:math> , and comparing the strength of CMOKE for different crystal orientations theoretically and experimentally. In crystals with cubic symmetry, the tensor <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi mathvariant="bold-italic">H</c:mi> </c:math> is described by two independent parameters <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>H</e:mi> <e:mn>123</e:mn> </e:msub> </e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msub> <f:mi>H</f:mi> <f:mn>125</f:mn> </f:msub> </f:math> . Together with the linear magneto-optic tensor <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi mathvariant="bold-italic">K</g:mi> </g:math> and quadratic magneto-optic tensor <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi mathvariant="bold-italic">G</i:mi> </i:math> , the permittivity tensor is described up to third order in magnetization. We analytically describe equations of the magneto-optic Kerr effect (MOKE), including the contribution of QMOKE and CMOKE themselves for (001)- and (111)-oriented cubic crystal structures. Those are compared with experimental measurements of two samples with an (001)- and (111)-oriented fcc Ni layer, respectively. The experimental data are measured using the so-called eight-directional method, which was developed to separate linear and quadratic MOKE contributions and to observe their individual anisotropic dependence on the crystal lattice direction but is applicable to CMOKE contributions as well. Furthermore, we use Yeh's <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mn>4</k:mn> <k:mo>×</k:mo> <k:mn>4</k:mn> </k:mrow> </k:math> transfer matrix calculus to simulate and describe the experimental measurements phenomenologically from the permittivity tensor developed up to third order in magnetization. This allows us to obtain the values of the magneto-optic parameters of the Ni layer for both crystal orientations. We find that the MOKE anisotropy that stems from the magneto-optic tensor <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi mathvariant="bold-italic">H</l:mi> </l:math> described as <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:mi mathvariant="normal">Δ</n:mi> <n:mi>H</n:mi> <n:mo>=</n:mo> <n:msub> <n:mi>H</n:mi> <n:mn>123</n:mn> </n:msub> <n:mo>−</n:mo> <n:mn>3</n:mn> <n:msub> <n:mi>H</n:mi> <n:mn>125</n:mn> </n:msub> </n:mrow> </n:math> , is much more pronounced for the (111)-oriented cubic crystal structure, for which it manifests as threefold in-plane angular dependencies of MOKE with longitudinal and also with transversal magnetization direction, respectively. For (001)-oriented cubic crystal structures, <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi mathvariant="normal">Δ</p:mi> <p:mi>H</p:mi> </p:mrow> </p:math> should manifest as a fourfold angular dependence of the MOKE with longitudinal and transversal magnetization directions, but its amplitude is predicted to be smaller than in the case of the (111)-oriented cubic crystal structure, which may explain why the CMOKE has not yet been experimentally identified in (001)-oriented cubic crystal structures.