We present diagonal and off‐diagonal elements of the permittivity and conductivity tensor up to the second order in magnetization for cubic crystals. We express all tensor elements as a function of a general sample orientation, for arbitrary magnetization direction and for (001), (011), and (111) surface orientations. Finally, we discuss, how to extract values of quadratic elements G of the second‐order permittivity tensors for different sample surface orientations from both experiment and ab initio calculations.
Magneto-optic (MO) ellipsometry of ferromagnetic materials is extremely sensitive to ultra-thin films, multilayers, and nanostructures. It gives a possibility to measure all components of the magnetization vector in the frame of the magneto-optic vector magnetometry and enable us to separate magnetic contributions from different depths and materials in nanostructures, which is reviewed in this article. The method is based on ellipsometric separation using the selective MO Kerr effect. The figure of merit used to quantify the ellipsometric selectivity to magnetic nanostructures is defined on the basis of linear matrix algebra. We show that the method can be also used to separate MO contributions from areas of the same ferromagnetic materials deposited on different buffer layers. The method is demonstrated using both: (i) modeling of the MO ellipsometry response and (ii) MO measurement of ultra-thin Co islands epitaxially grown on self-organized gold islands on Mo/Al 2 O 3 buffer layer prepared using the molecular beam epitaxy at elevated temperatures. The system is studied using longitudinal (in-plane) and polar (perpendicular) MO Kerr effects.
Recently, we have shown that the approach of depth sensitivity of magneto-optic ellipsometry can be generalized to selectivity from different materials in nanostructures. We use the condition number as the figure of merit to quantify the magneto-optic selectivity to two different magnetic contributions in magnetic nanostructure. The method is demonstrated on nanostructures containing magnetically hard Fe particles in surface layer of soft FeNbB amorphous ribbon. We separated both magnetic contributions from measurement of hysteresis loops using magneto-optic Kerr effect in longitudinal configuration. Magneto-optic selectivity is discussed and theoretical model on the basis of effective medium is compared with experimental data of longitudinal magneto-optic Kerr effect depending on angle of incidence.
We provide unified phenomenological description of magnetooptic effects being linear and quadratic in magnetization. The description is based on few principal spectra, describing elements of permittivity tensor up to the second order in magnetization. Each permittivity tensor element for any magnetization direction and any sample surface orientation is simply determined by weighted summation of the principal spectra, where weights are given by crystallographic and magnetization orientations. The number of principal spectra depends on the symmetry of the crystal. In cubic crystals owning point symmetry we need only four principal spectra. Here, the principal spectra are expressed by ab-initio calculations for bcc Fe, fcc Co and fcc Ni in optical range as well as in hard and soft x-ray energy range, i.e. at the 2p-and 3p-edges. We also express principal spectra analytically using modified Kubo formula. PACS numbers: 42.50.Ct, 78.20.Ls, 78.70.Dm, 78.40.Kc There is a vast number of physical phenomena proportional to quadratic form of magnetization. In case of dc transport phenomena, the most well-known examples are anisotropy magnetoresistance (AMR) [1,2] or longitudinal Hall effect [3]. Within the magnetooptic community, the field of magnetooptic effects quadratic in magnetization is complicated by incredible number of nomenclature, being called Cotton-Mouton effect, Voigt effect, quadratic magnetooptic Kerr effect (QMOKE) [4], magnetic linear birefringence, X-ray magnetic linear dichroism (XMLD) [5,6], magnetic double refraction, magnetooptic orientation effect, magnetooptic anisotropy, Hubert-Schäfer effect or magnetorefractive effect [7,8]. The nomenclature is not strictly defined, however it refers either to type of samples (liquid, gas, solid state) or it refers to experimental configurations of the setup (namely detecting change of light intensity or detecting change of polarization state upon variation of magnetization direction). Although those effects are usually not considered as single phenomena, they originate from equal parts of permittivity tensors (i.e. from equal symmetry breaking). Notice, that recently new types of quadratic-in-magnetization effects arose, for example anisotropic magneto-thermopower [9, 10] in spin-caloritronics. Within some generalization, one can expect that any magneto-transport linear in magnetization will have its quadratic-in-magnetization counterpart.The first observation of magnetooptic effects quadratic in magnetization dates back more than century ago in works of Kerr [11], Majorana [12] and Cotton and Moutton [13], where magnetic birefringence was observed in liquids and colloids. Later, those quadratic magnetooptic response was observed in gases, solid state materials and obviously also in ferromagnetic materials. See large reviews of Smolenskii et al [14] and Ferré, Gehring [15] from 80's. The anisotropy of magnetooptic effects (i.e. dependence of QMOKE on crystal and field orientations) was investigated for various systems. However, the investigation were mostly d...
Magneto-optic (MO) methods are used for characterisation of magnetic ultrathin films and nanostructures. The paper deals with the sensitivity of the complex MO effect to films at different depths and to different materials in periodic multilayers and self assembled nanostructures. It is shown, that the basic principle and separation methods are the same for depth sensitive selectivity and the material sensitivity in nanostructures. Method of MO signal separation from different films and phases and the figure of merit used to quantify the MO ellipsometric selectivity of magnetic nanostructures is defined on the basis of linear matrix algebra. We review applicability of the method to separate MO contributions from (Au/Co) N , N = 2, 3; (Co/Au/NiFe/Au)10 multilayers; BiFeO3-CoFe2O4 self-assembled nanostructure; cobalt film on self-organised Au islands; and magnetically hard Fe nanoparticles near the surface of soft FeNbB amorphous ribbons.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.