Quadratic‐in‐magnetization permittivity and conductivity tensor in cubic crystals

Hamrlová, Jana; Hamrle, Jaroslav; Postava, Kamil; Pištora, Jaromı́r

doi:10.1002/pssb.201349031

Cited by 11 publications

(17 citation statements)

References 20 publications

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“…In order to express G-elements for crystals with cubic symmetry, the elements can be rearranged defining G s = G 11 −(G 12 +G 21 )/2 and ∆Γ = (G 12 −G 21 )/2. Note that alternative way used to express anisotropy of the permittivity is ∆G = G 11 − 2G 44 − (G 12 + G 21 )/2 [16,42]. In case of cubic crystals owning point O h symmetry, G 12 = G 21 and hence G s = G 11 − G 12 , ∆Γ = 0 [40].…”

Section: Permittivity Tensor In Cubic Crystalsmentioning

confidence: 99%

Principal spectra describing magnetooptic permittivity tensor in cubic crystals

Hamrlová

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Veis

et al. 2016

Journal of Magnetism and Magnetic Materials

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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...

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Section: Permittivity Tensor In Cubic Crystalsmentioning

confidence: 99%

Principal spectra describing magnetooptic permittivity tensor in cubic crystals

Hamrlová

Legut

Veis

et al. 2016

Journal of Magnetism and Magnetic Materials

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“…We conclude this appendix by explaining the relationship between terminology used in this paper (components of the effective permittivity tensor ε eff ) and the notation of "quadratic magneto-optic tensor components" [60,61] used elsewhere [62][63][64]. The basic conceptual difference between the two approaches is whether M is kept fixed and different polarizations of light are considered (the former approach) or vice versa (the latter approach).…”

Section: Consider An Electromagnetic Wave E( Rt) = E 0 E I(kz−ωt)mentioning

confidence: 99%

Systematic study of magnetic linear dichroism and birefringence in (Ga,Mn)As

et al. 2014

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Magnetic linear dichroism and birefringence in (Ga,Mn)As epitaxial layers is investigated by measuring the polarization plane rotation of reflected linearly polarized light when magnetization lies in the plane of the sample. We report on the spectral dependence of the rotation and ellipticity angles in a broad energy range of 0.12-2.7 eV for a series of optimized samples covering a wide range on Mn dopings and Curie temperatures and find a clear blueshift of the dominant peak at energy exceeding the host material band gap. These results are discussed within the framework of the k · p and mean-field kinetic-exchange model of the (Ga,Mn)As band structure. We infer that disorder-induced nondirect transitions significantly influence magneto-optical properties of (Ga,Mn)As.

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“…Namely, G s = ε − ε ⊥ for M 100 and 2G 44 = ε − ε ⊥ for M 110 , where the parallel ( ) and perpendicular (⊥) symbols denote the directions of linear light polarization (i.e. applied electric field) with respect to the magnetization direction, respectively [40].…”

Section: Theory Of Linear and Quadratic Mokementioning

confidence: 99%

“…Our separation process of different QMOKE contributions is stemming from 8-directional method [10], but we use a combination of just 4 directions and a sample rotation by 45 • as will be described later in the text. This approach allows us to isolate QMOKE spectra that stem mostly from individual MO parameters and thus subsequently determine spectral dependencies of G s = G 11 − G 12 and 2G 44 , denoting MO parameters quadratic in M [17,[38][39][40]. Therefore, we start our study on FM bcc Fe thin films grown on MgO(001) substrates to get a basic understanding of QMOKE spectroscopy for further studies of AFMs.…”

Section: Introductionmentioning

confidence: 99%

Quadratic magneto-optic Kerr effect spectroscopy of Fe epitaxial films on MgO(001) substrates

et al. 2019

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The magnetooptic 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 M . Nevertheless, a higher-order term being proportional to 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 8-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 Gs and 2G44 are yielded. The difference between those two parameters, known as ∆G, denotes the strength of the QMOKE anisotropy. Those QMOKE parameters give rise to the QMOKE tensor G, fully describing the perturbation of the permittivity tensor in the second order in 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 M are obtained, respectively. Finally, all those spectra are described by ab-initio calculations.

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Quadratic‐in‐magnetization permittivity and conductivity tensor in cubic crystals

Cited by 11 publications

References 20 publications

Principal spectra describing magnetooptic permittivity tensor in cubic crystals

Principal spectra describing magnetooptic permittivity tensor in cubic crystals

Systematic study of magnetic linear dichroism and birefringence in (Ga,Mn)As

Quadratic magneto-optic Kerr effect spectroscopy of Fe epitaxial films on MgO(001) substrates

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