A versatile method for exploring the magnetooptical properties of polar saturated and unsaturated ferromagnetic metallic thin films

Polar unsaturated ferromagnetic thin films are promising for low-power and high-speed nonvolatile resistive and optical memories. Here we measure the magnetooptical (MO) response of polar unsaturated Co90Fe10 and Co40Fe40B20 thin films in the spectral range from 400 nm to 1000 nm using vector MO generalized ellipsometry (VMOGE) in an out-of-plane applied magnetic field of ±0.4 T where magnetization of the ferromagnetic (FM) thin film is not saturated. Using magnetooptical simulation software (MagOpS®), we extract the complex MO coupling constant ( Q ) of the polar unsaturated FM thin films from difference spectra of VMOGE data recorded in a polar configuration at Hz = +0.4 T and Hz = −0.4 T. The presented approach opens a path to determine Q of both polar saturated and polar unsaturated FM thin films for simulating the MO properties of application-relevant optical memory multilayer structures.


Introduction
Ferromagnetic materials have been used in magnetooptical (MO) memories [1][2][3][4][5][6][7][8] and magnetic memories [tunnel magnetoresistance (TMR), giant tunnel magnetoresistance (GMR)] [ [9][10][11].There have been remarkable achievements in the fields of integrated optical memory and optical random access memory [12].Materials with polar (perpendicular) magnetic anisotropy, for example CoFeB [13], have been developed as perpendicular recording media for use in highdensity spin-transfer torque magnetic random access memories.The magnetic properties of ferromagnetic materials have been extensively studied for their dependence on crystallinity, chemical composition and alloy broadening using magnetization measurements.There are few reports in the literature on the MO properties of ferromagnetic materials and the design of multilayer structures with enhanced MO effects.In our previous work [14], we discussed that, compared with the MO response of multilayer structures with ferromagnetic metallic thin films having transverse and longitudinal magnetization, the MO response of multilayer samples with ferromagnetic thin films with polar magnetization is much larger.However, at the external magnetic fields available in MO experiments most ferromagnetic metallic thin films are only saturated for transverse and longitudinal configurations and are unsaturated in the polar configuration.Furthermore, magnetization of a ferromagnetic metallic thin film with a large coercive field is not expected to vary across the film.Nevertheless, substrateinduced anisotropic stress may play a critical role in the transition from single-domain to multi-domain magnetization [15].
In our previous work [14] we analyzed the MO response of a polar saturated ferromagnetic Ni 81 Fe 19 thin film.The relative dielectric permittivity tensor of a medium is defined as Here we assume an isotropic medium.Without a magnetic field or without magnetization of the medium the dielectric permittivity tensor ϵ can be expressed as The introduction of magnetization (M) with components M x , M y and M z causes the medium to become anisotropc.We call the MO dielectric permeability tensor ϵ MO .ϵ MO can be expressed as follows: According to Onsager, the diagonal elements are even functions of M and the off-diagonal elements are odd functions of M [16].Here we focus on the MO response of multilayer systems with ferromagnetic thin films in reflection mode and will analyze the off-diagonal elements ϵ xy (M), ϵ xz (M) and ϵ yz (M) which introduce the MO Kerr effect.We will neglect the Cotton-Mouton effect, which is induced by differences in the diagonal elements The values of ϵ xx are calculated from the refractive index n and from the extinction coefficient k (see equation (B1)).We introduce the complex MO coupling constant Q = (Q x (λ), Q y (λ), Q z (λ)) [17,18] and define the MO coupling constant Q, namely Q = Re(Q) − i Im(Q).The MO coupling constant quantifies the influence of magnetization on the MO response of a magnetizable material [19].It depends on the net spin polarization and electronic band structure of the magnetizable medium [17].
Next, we decompose the off-diagonal elements of the MO dielectric permittivity tensor into a product of the magnetization and the MO coupling constant as follows: ϵ xy (M) = +M z (H)Q z (λ), ϵ xz (M) = +M y (H)Q y (λ) and ϵ yz (M) = +M x (H)Q x (λ).The dielectric tensor (equation ( 3)) for polar saturated ferromagnetic materials with M = (0, 0, M s ), with M s being the saturation magnetization, can be written as The dielectric tensor (equation ( 1)) for polar unsaturated ferromagnetic materials with M = (M x , M y , M z ) can be written as (5) If we assume an isotropic MO coupling constant, i.e.Q x (λ) = Q y (λ) = Q z (λ) = Q(λ), the dielectric tensor (equation ( 1)) for polar unsaturated ferromagnetic materials reads as follows: For polar unsaturated ferromagnetic materials it is imperative to consider the scenario in which only the out-of-plane magnetization (M z ) is known and where the amplitude of the inplane magnetization, but not M x and M y , can be determined from the relation between saturation magnetization (M s ) and the out-of-plane magnetization (M z ) Equation (7) holds true for ferromagnetic metallic thin films with single-domain magnetization.Because of the saturated in-plane magnetization and finite out-of-plane coercivity (figure 3) and the isotropic underlying ZnO or SiO 2 /Si substrate [15], we assume that Co 90 Fe 10 and Co 40 Fe 40 B 20 thin films are single-domain magnetized.A detailed methodology for simulating the MO response of polar unsaturated ferromagnetic materials is discussed in section 4. Next, we introduce the Jones matrix and the Müller matrix, which are typically related to depolarization-insensitive complex Kerr angle measurements and depolarization-sensitive vector MO generalized ellipsometry (VMOGE) measurements, respectively.The Jones matrix in reflection is defined as follows: J = r pp r ps r sp r ss (8) where r ss and r pp are the amplitude reflection coefficients for s-polarized reflected/s-polarized incident and p-polarized reflected/p-polarized incident light respectively.r sp and r ps are the amplitude reflection coefficients for s-polarized reflected/p-polarized incident and p-polarized reflected/spolarized incident light, respectively.The Jones matrix contains depolarization, which is an intrinsic error, but the off-diagonal block of the Müller matrix contains purely anisotropy information for a sample.Hence, the 4 × 4 Müller matrix approach [20] is used to analyze the ellipsometry data of samples containing anisotropic thin films.The Müller matrix is defined as Note that similar symbols M and M are chosen conventionally for magnetization and the Müller matrix, respectively.When incident s-polarized light is used as the probe, the complex Kerr angle (Φ K ) can be calculated using [14] where ψ sp = arctan r sp r ss (0 and where θ K , η K , ψ sp and ∆ sp are the Kerr rotation, Kerr ellipticity, the amplitude ratio and the phase difference induced by reflection, respectively.Depolarization effects are not accounted for in the complex Kerr angle because it is derived from depolarization-independent elements of the Jones matrix (equation ( 10)).The M 14 and M 23 elements of the Müller matrix quantify the circular dichroism and circular birefringence of a depolarizing material, respectively.Therefore, the M 14 and M 23 elements of the Müller matrix directly relate to the optical anisotropy of ferromagnetic thin films in a polar configuration.Hence, the difference in off-diagonal Müller matrix elements M 14 and M 23 , i.e. ∆M 14 and ∆M 23 , defines the change in the optical anisotropy of polar unsaturated ferromagnetic thin films.The sign change in H z and M z translates into a sign change of M 14 and M 23 .So, to calculate the change in the anisotropy introduced just by the out-of-plane magnetic field (+H z , −H z ) and not by intrinsic depolarization we have calculated and fitted difference Müller matrix element spectra of the off-diagonal Müller matrix elements M 14 and M 23 , i.e. ∆M 14 and ∆M 23 , and compared them with difference Kerr angle spectra θ K and η K , i.e. ∆θ K and ∆η K .The difference Kerr angle spectra contain nonseparable information on MO anisotropy and intrinsic depolarization.The analysis scheme for the determination of the MO coupling constants (Q) of polar unsaturated ferromagnetic metallic thin films exploits the finding that the difference Müller matrix elements ∆M 14 and ∆M 23 depend on the amplitude and direction of out-of-plane magnetization and on the amplitude of inplane magnetization, but not on the direction of in-plane magnetization.This allows us to specify only the amplitude and direction of out-of-plane magnetization and the amplitude of in-plane magnetization, not of the M x and M y components, of polar unsaturated ferromagnetic thin films when modeling the MO response of multilayer structures with such polar unsaturated ferromagnetic thin films in external out-of-plane magnetic fields (+H z , −H z ).Using this analysis scheme we have calculated ∆M 14 and ∆M 23 of multilayer structures with polar unsaturated ferromagnetic thin films for a fixed amplitude of in-plane and out-of-plane magnetization and changing direction of out-of-plane magnetization (+M z , −M z ).We suggest that the use of polar unsaturated ferromagnets not only in stacks of magnetic memories but also in optical magnetic memories should be analyzed.This would push the boundaries of data transmission in electronic/photonic chips and enable the transmission of information at ultrahigh speed while generating minimal heat.Since the development of VMOGE, the MO properties of multilayer samples with magnetizable layers can also be studied without being limited by the depolarization effect [21] as long as the polar, transverse and longitudinal components are known.Oblak et al reported on a versatile method to enhance the sensitivity of MO measurement set-ups for multilayer samples with saturated longitudinal ferromagnetic metallic thin films [22] where it can be expected that the assumption M ∥ H is fulfilled.As reviewed by Kimel et al [23], conventional optical ellipsometry as well as ultrasensitive MO Kerr (MOKE) measurements were developed in the 1980s.VMOGE was developed as a depolarization-dependent method in the late 1990s [24,25].MOKE depends on the amplitude and direction of the inplane magnetization vector, i.e. on M x and M y , but VMOGE depends on the amplitude and not the direction of in-plane magnetization.Therefore, VMOGE can be used to analyze the MO response of polar unsaturated ferromagnetic thin films where M x and M y are not known.So far, reports are only available for MOKE [26,27] and VMOGE [14].However, there are very few analyses of the MO response of polar unsaturated ferromagnetic thin films with unknown M x and M y using MOKE or VMOGE.Non-destructive VMOGE is used to extract the MO coupling constant Q(λ) from off-diagonal elements of the 3 × 3 MO dielectric tensor and is typically carried out in the presence of a magnetic field in reflection or transmission mode and under the assumption that magnetization is parallel to the external magnetic field, i.e.M ∥ H.The MO coupling constant Q = Q(λ) defines the MO figure of merit of magnetizable materials in MO sensors or in MO memory devices [17].Analytical solutions can be used as an initial guess for the complex MO coupling constant Q [28].We have numerically calculated the off-diagonal dielectric tensor of permalloy (Ni 81 Fe 19 ) using the 4 × 4 Müller matrix and determined the complex MO coupling constants in the standard approach [14].The complex coupling constant of any arbitrarily anisotropic and depolarizing magnetic multilayer system can also be studied with VMOGE measurements [29][30][31][32].Following this, the wavelength-dependent, magnetic field independent complex MO coupling constant of a ferromagnetic metal layer can be extracted by fitting experimental VMOGE data obtained on multilayer samples with ferromagnetic metal layers whose magnetization is saturated parallel to the external magnetic field [21].Q(λ) is also of interest for the analysis of the spin-dependent electronic band structure of magnetizable materials.
Here, we study the MO properties of polar unsaturated Co 90 Fe 10 and Co 40 Fe 40 B 20 thin films with M ∦ H by analyzing selected Müller matrix elements to separate out-of-plane and in-plane magnetization.The work is structured as follows: after introducing the basic principles of how to extract the MO response coupling constants of polar unsaturated ferromagnetic thin films in polar unsaturated multilayer samples from the 16 measured Müller matrix elements, in section 2 we describe the MO analysis of multilayer samples with polar unsaturated ferromagnetic thin films by determining first its magnetization components M x , M y and M z and then by using M x , M y and M z for the MO dielectric constant of the polar unsaturated ferromagnetic thin film in the multilayer sample.Section

Magnetooptical analysis of multilayer samples
The capabilities of VMOGE in addition to MO spectroscopy and MO magnetometry are theoretically demonstrated by modeling the MO properties of application-relevant multilayer samples with ferromagnetic metallic thin films having parallel and antiparallel magnetization.
The first investigations focused on the interference [33] and depth-dependent [34] MO effects, as reviewed by Kuch et al [35].In the beginning, complex Kerr angles were been mostly used to study the MO response of magnetic multilayer samples [36][37][38][39].The Kerr angle approach is based on an analysis of the 2 × 2 Jones matrix and does not account for depolarization effects, for example due to geometric roughness on a scale at least one order of magnitude larger than the wavelength of the incident beam so that components of the light beam reflected from different regions add incoherently.Recently, the enhancement of MO effects has been studied in special multilayer samples, for example ultrathin magnetizable layers [40] or semi-infinite thin films, using the 2 × 2 Jones matrix approach or an extended finite element method, respectively.
Using the software package MagOpS ® [41], in this work we demonstrate the versatility of the 4 × 4 Müller matrix approach to model MO effects of multilayer samples with arbitrarily thick magnetizable layers of arbitrary magnetization and show how, without any approximation, thickness interference and depth-dependent effects are included.MO effects are strongest for a polar configuration, where the magnetization of magnetizable layers(s) in the multilayer sample is normal to the sample surface.Furthermore, for difference spectra of specific Müller matrix elements, the MO effect only depends on the amplitude of in-plane magnetization and not on its direction.We use these findings to develop a detection scheme for multilayer samples with polar unsaturated metallic thin films in mixed polar, longitudinal and transverse configurations.
The analyzed multilayer samples contain a single magnetizable layer, namely Co 90 Fe 10 and Co 40 Fe 40 B 20 in Ru/FM/Ta samples (FM = Co 90 Fe 10 and Co 40 Fe 40 B 20 ) on SiO 2 /Si and ZnO substrates.Pure metals and metal alloys have been widely investigated due to their notable and controllable MO response [42][43][44][45][46].The chosen FM materials, Co 90 Fe 10 and Co 40 Fe 40 B 20 , are commonly used in magneto-electronic devices, for example in TMR and in GMR structures [47,48].Superconducting quantum interference device (SQUID) measurements (not shown) in the polar [H = (0, 0, H z )] configuration have been used to determine the z-component of magnetization (M z ) of a polar unsaturated ferromagnetic thin film at µ 0 H z = 0.4 T. µ 0 H z = 0.4 T corresponds to the largest possible external magnetic field that can be applied during VMOGE measurements.Using the determined M z (table C.1), the in-plane magnetization components M x and M y were calculated for three different cases: where M s is the saturation magnetization of the ferromagnetic material.Cases I, II and III are used to distinguish between three different in-plane magnetizations M x and M y for a constant out-of-plane magnetization M z .Using the three different cases, we show that the difference spectra of the Müller matrix elements M 14 and M 23 for a fixed M s and experimentally given M z depend only on the amplitude of in-plane magnetization and not on the sign (direction) of M x and M y .Further simulation (not shown) reveals that in the event of the previous assumption of just M z changing sign, while M x and M y remain unchanged, not being satisfied, our findings will remain valid as long as the amplitude of the magnetization vector remains constant.This analysis scheme can be used to estimate the influence of in-plane magnetization on MO studies in polar geometry.For Cases I, II and III we have analyzed the difference spectra of Müller matrix elements M 14 and M 23 (i.e.∆M 14 and ∆M 23 ) for a fixed M s and have shown that difference Müller matrix element spectra ∆M 14 and ∆M 23 only depend on the amplitude of in-plane magnetization and not on the sign (direction) of in-plane magnetization (M x , M y ).If the magnetizable layer in the multilayer sample on the SiO 2 /Si substrate is thin enough, interference effects due to multiple reflections in the underlying SiO 2 layer can be observed.The two substrates are optically different in the spectral range from 400 nm to 1000 nm.The saturation magnetization (M s ) has been taken from the literature and agrees well with SQUID measurements.The non-zero M x , M y and M z magnetization components at µ 0 H =(0, 0, ±0.4 T) correspond to the polar unsaturated case.We have subsequently used the non-zero M x , M y and M z magnetization components to model MO dielectric tensor elements of ferromagnetic thin films and to determine the complex, wavelength-dependent and material-specific MO coupling constant Q(λ).It is found that MO coupling constants of the polycrystalline metal alloys Co 90 As an example, extracted MO coupling constants were used to model the complex Kerr angle of a multilayer sample in the spectral range from 400 nm to 1000 nm under varying detection conditions (presented in appendix C).

Samples and experiment
We investigated the MO response in a Ru The deposition rate and thickness were calibrated with films of thickness larger than 200 nm using profilometry.An in-plane magnetic field of 5 mT was applied during sputtering to produce a well-defined deposition-induced anisotropy field [49].Optical measurements confirmed that the thickness homogeneity was better than 2% over the diameter of the 6 in Si/SiO 2 substrate (200 mm).
In general, MO effects depend on the optical contrast at the Ru/FM and FM/Ta interfaces (figure 1) and on the thickness and MO coupling constant of the FM layer.The MO response of each sample was measured using VMOGE with an RC2 rotating compensator ellipsometer from J. A. Woollam Co. Inc. and a magnetic field applied in the polar configuration H = (0, 0, ±H z ) (figure 2).A maximum 0.4 T magnetic field along the z-direction was applied with a home-designed octupole magnet [21].From SQUID magnetization measurements (figure 3) with an outof-plane magnetic field µ 0 H = (0, 0, +0.4 T) we know that the measured M z is not saturated.Using saturation   14) and ( 15), respectively.Difference Müller matrix spectra, i.e. ∆M 14 and ∆M 23 , were obtained from VMOGE measurements of Müller matrix elements M 14 andM 23 with applied magnetic field (a) (0, 0, +Hz) and (b) (0, 0, −Hz).magnetization from the literature and measured M z from the SQUID (equation ( 11)), we extracted the in-plane magnetization consisting of M x (longitudinal) and M y (transverse) (see table C.1).
Here we report on an analysis scheme which we have developed for the determination of the MO coupling constants of polar unsaturated ferromagnetic metallic thin films.We exploit the fact that the MO response of such films depends on the amplitude of both out-of-plane and in-plane magnetization and that it depends on the direction of out-of-plane magnetization but not on the direction of in-plane magnetization.Only the amplitude, not the direction, of in-plane magnetization components (longitudinal, transverse) influences MO response of thin films not saturated in the perpendicular direction.If our assumption that only M z changes sign while M x and M y remain constant is not met, our results will still hold true as long as the magnitude of the magnetization vector remains constant.Using the three different magnetization cases, we show that for a fixed M z and M s the difference spectra of Müller matrix elements M 14 and M 23 nearly depend on only the amplitude of in-plane magnetization and not on the single in-plane magnetization components M x (longitudinal) and M y (transverse).

Analysis of polar unsaturated ferromagnetic metallic thin films
We first calculate the off-diagonal elements (ϵ ij ) of the MO dielectric tensor (ϵ MO ) (see equation ( 6)).From this equation we can see that ϵ 12 and −ϵ 21 mainly depend on the magnetization along M z which changes sign when the direction of the applied field along the z-direction is reversed.Also, ϵ 23 (−ϵ 32 ) and ϵ 13 (−ϵ 31 ) depend on the magnetization along M x and M y , respectively.
We assume that the amplitude of the magnetization amounts to the saturation magnetization in the literature (M Lit s ) and take the magnetization vector components from table C.1 for three different cases: non-zero M x and M y = 0 (Case I), M x = 0 and non-zero M y (Case II) and M x = M y (Case III) by rearranging equation (11) Nonzero M x and M y components introduce the same offset in θ K and η K , M 14 and M 23 , which is removed in the difference spectra of VMOGE data ∆θ K , ∆η K , ∆M 14 and ∆M 23 .The calculated ∆M 14 and ∆M 23 for Cases I, II and III (table 1) for all samples are plotted in figure 4. We found that that difference spectra are the same for Cases I, II and III for all samples.This study suggests that, for a given magnetization value of a ferromagnetic thin film in a multilayer system, the strength of the MO effect depends on the direction of the magnetization vector along the z-direction and on the amplitude of in-plane magnetization.This opens a path to extract the complex MO coupling constant (Q) of polar unsaturated FM thin films from the difference spectra of VMOGE data recorded in the polar configuration even if the magnetization of the FM layer is not saturated along the z-direction.
For example, our VMOGE system allows us to apply a magnetic field of 0.4 T to multilayer systems with FM thin films whose saturation field amounts to 1.822 T and 1.571 T for Co 90 Fe 10 and Co 40 Fe 40 B 20 , respectively.Therefore, the magnetization along the z-direction (M z ) of the FM thin films measured with VMOGE in polar configuration at µ 0 H z = ±0.4T will not reach M Lit s and the magnetization M = (M x , M y , M z ) will also have non-zero in-plane components M x and M y (table 1).(11).These components are used to demonstrate that the contribution of non-zero Mx and My components to the MO response can be removed by analyzing difference spectra of specific Müller matrix elements (equation ( 16)).Note that the SI unit for magnetization is A m −1 and the unit conversion factor is 1 emu g −1 = (1/ρ) emu cm −3 = (10 3 /ρ) A m −1 where ρ is the density of FM as obtained from x-ray reflection (XRR) fitting (table 2).For example, magnetization of Co 90 Fe 10 (on SiO 2 /Si) along the z-direction corresponds to 318 emu cm −3 (= 318 × 10 3 A m −1 = 35.8emu g −1 ) and magnetization of Co 40 Fe 40 B 20 (on SiO 2 /Si) corresponds to 318 emu cm −3 (= 318 × 10 3 A m −1 = 41.1 emu g −1 ).

FM layer
Mx@µ 0 Hz = ±0.4T My@µ 0 Hz = ±0.4T Mz@µ 0 Hz = ±0.4T (emu cm −3 ) (emu cm −3 ) (emu cm  The extracted real and imaginary parts of the off-diagonal elements are shown in figures 5 and 6 for Cases I, II and III (table 1) on Si/SiO 2 and ZnO substrates, respectively.Figures 5 and 6 show the values of ϵ 12 = −ϵ 21 (polar configuration), ϵ 13 = −ϵ 31 (transverse configuration) and ϵ 23 = −ϵ 32 (longitudinal configuration) for the two samples on SiO 2 /Si and ZnO substrates, respectively.The large values of the offdiagonal elements of the MO dielectric tensor (equation ( 1)) can be neglected when modeling difference spectra of the complex Kerr angle (∆θ K , ∆η K ) and of the Müller matrix elements (∆M 14 , ∆M 23 ).By dividing the off-diagonal element by M z (as calculated using equation ( 11)) we obtain the MO coupling constant Q for all cases.Detailed discussions can be found in section 5.

Magnetooptical coupling constant of Co 90 Fe 10 and Co 40 Fe 40 B 20
The permittivity tensor (ϵ MO ) and MO conductivity tensor (σ MO ) of a medium depend not only on the electronic band structure but also on the spin polarization.ϵ MO and σ MO are a measure of the dependence of the optical anisotropy of a medium on its magnetization and on the wavelength of the incident light.Corresponding off-diagonal elements of ϵ MO are calculated from the product of the magnetization M and MO coupling constant Q.Applying an out-of-plane magnetic field of 0.4 T, i.e. the maximum possible magnetic field of the VMOGE setup used, magnetization of the investigated multilayer samples cannot be saturated in the polar configuration.In this work we show that this does not impact the analysis of the VMOGE data for ∆M 14 and ∆M 23 .We took the saturation magnetization (table C.1) from the literature and determined the out-of-plane saturation magnetic field (H x + M 2 y ) (see equation ( 11)) at µ 0 H = (0, 0, ±0.4 T).VMOGE data for the multilayer sample were analyzed by using the material-dependent dielectric tensor of every non-magnetic layer and the MO dielectric tensor (equation ( 1)) of the ferromagnetic thin films.In equation ( 1), we used an isotropic MO coupling constant (Q(λ)).As demonstrated in appendix C, only the amplitude of the in-plane magnetization M 2 x + M 2 y and not the single components M x and M y of in-plane magnetization of polar unsaturated thin films have to be included when analyzing Müller matrix difference spectra to extract the MO coupling constant.The complex MO coupling constant Q(λ) depends only on the wavelength λ and not on the applied magnetic field H, whereas magnetization M(H) depends on the applied magnetic field H. Spectroscopic ellipsometry data were modeled with a 2 × 2 Jones matrix approach using CompleteEASE software to determine the on-diagonal elements of the dielectric tensor (equation ( 1)) of each layer in the multilayer system (also of the FM layer in each sample).The 4 × 4 Müller matrix difference spectra were analyzed using the software package MagOpS ® .For ϵ MO  ii of the FM layer we used a combination of a Drude oscillator (describing free electrons) and three Tauc-Lorentz oscillators (describing interband transitions) to model the on-diagonal elements of the dielectric tensor.Modeled optical constants of all the other layers in the multilayer system (Ru, Ta, SiO 2 , Si, ZnO) agree well with the corresponding optical constants from the CompleteEASE software database.Plasma frequency was calculated from the Drude contribution for all samples (see table B.1 in the appendix) and is found to agree with the literature data [51,52].The on-diagonal MO conductivity elements of the FM layers on both substrates were calculated from the on-diagonal elements of the dielectric tensor (for the on-diagonal dielectric constant see figure B.4 in the appendix) and are shown in figure 7. Differences in dependence on the underlying substrate are clearly visible.This is attributed to the possible effect of the substrate on the structure and electrical properties of the polycrystalline metal alloy Co 90   [38] and fcc Co (−△−) [38].
Table 2. Layer thickness, density of FM, surface roughness (air/Ru) and interface roughness (Ru/FM, FM/Ta, Ta/substrate) of each sample extracted from XRR modeling using the following densities: Ru = 12.37 g cm −3 , Ta = 16.65 g cm −3 , Si = 2.33 g cm −3 , ZnO = 5.61 g cm −3 .d FM denotes the thickness of the FM layer.Crystallite sizes of all samples are calculated from respective experimental x-ray diffraction (XRD) data using the Scherrer equation.Experimental XRD data and XRR data are shown in figures A. 1 1)) and the MO conductivity tensor σ MO are related as follows [14]: Equations ( 12) and ( 13) were used to extract on-diagonal and off-diagonal elements of the conductivity tensor, respectively.Difference Kerr angle spectra (∆θ K and ∆η K ) are shown in figure 8 and difference Müller matrix difference spectra in figure 9.The Kerr angle was determined from measured ellipsometric parameters ψ sp and ∆ sp after converting them into Jones matrix elements (equation ( 10)).The Müller matrix elements were directly measured.We removed a possible offset in the experimental MO data due to non-zero in-plane magnetization by fitting difference spectra [14].The difference spectra of Kerr rotation and Kerr ellipticity are defined as follows : In this work, we model the MO response of polar unsaturated ferromagnetic metal thin films in mixed polar, longitudinal and transverse configuration (figure 2) with the mag- and modeled difference spectra of the Kerr rotation (∆θ k ) and difference spectra of Kerr ellipticity (∆η k ) are shown in figure 8.Note that the difference in the lineshape of difference Kerr spectra (figure 8) from the reported values of Sharma et al [45] and Hoffmann et al [42] is due to different film thicknesses of the CoFeB sample.Also, the opposite sign in the spectra can be attributed to different sign conventions.
For the modeling we used the Jones matrix method [50].The best fitting result, i.e. the closest agreement between experimental and modeled data, was obtained for Co 90 Fe 10 samples on SiO 2 /Si (figures 8(a)-(f)) and on ZnO (figures 8(g)-(l)).
The observed mismatch between the experimental and simulated results is attributed to the presence of depolarization within the 2 × 2 Jones matrix which is not accounted for in the modeling approach.Due to depolarization effects, in the next step we used a 4 × 4 Müller matrix modeling approach in which difference where ρ is the density of the FM obtained from XRR fitting (table 2).spectra of Müller matrix elements (∆M ij ) were considered for fitting, Among the Müller matrix elements M 14 and M 23 show the strongest dependence under reversal of the applied magnetic field along the z-direction.We obtained the MO coupling constant by fitting difference spectra of Müller matrix elements ∆M 14 and ∆M 23 .Again, we explicitly modeled the polar unsaturated configuration (figure 2) with the magnetiza- The fitted ∆M 14 and ∆M 23 data agree well with the experimental data for all six samples and the SiO 2 thickness fringes (figures 9(a)-(f)) also match with experimental data.The modeled real and imaginary parts of the off-diagonal elements of the dielectric tensor are shown in figure B.5.In the following, we discuss the MO coupling constant, which was determined from extracted off-diagonal elements of the complex dielectric tensor by fitting ∆M 14 and ∆M 23 (figure 9).Extracted real and imaginary parts of the MO coupling constants of the two different ferromagnetic thin film materials are shown in figure 10.As expected, the magnetic fieldindependent and thickness-independent complex MO coupling constants depend on the wavelength, i.e.Q = Q(λ).10).This influences the zero-crossing point where the real part of Q changes sign.The epsilon-near-zero behavior [53] or zero-crossing position [51] for all samples was calculated and is summarized According to the zero-crossing position can be directly related to the width of the d-band [51].that the definition of ϵ ij used in our report (equation ( 1)) is different by a factor i from the definition used by Tikuišis et al [51].Reports can be found in which the zero-crossing position is determined from the imaginary part of the off-diagonal dielectric constant [54,55], which in turn is directly proportional to the real part of the MO coupling constant Q.The zero-crossing position was calculated and is summarized in table B.1.Alloy broadening refers to the phenomenon where the electronic states of the alloy are a mixture of the electronic states of the pure elements.Alloy broadening can lead to a shift and broadening of the energy levels of the electronic states of the alloy.As a result, 3d energy levels of 3d alloys show alloy broadening.We observe the effects of alloy broadening in the extracted MO coupling constants.In agreement with theory, alloy broadening causes a shift in the zero-crossing position from the real part [54,55] (optics convention) or the imaginary part [51] (physics convention) of the off-diagonal dielectric element.Note that we use the real part of the MO coupling constant Q (Re(Q)) in our analysis.Negative values of Re(Q) (i.e.positive values of Im(ϵ ij )) correspond to the transition of electrons in the 3d ↓ sub-band and vice versa.Hence, this position corresponds to the energy difference between the energy of unfilled 3d ↑ sub-bands and the energy of filled 3d ↓ energy bands.A smaller width of the unfilled part of the 3d ↑ sub-band causes a reduction in the contribution of the 3d ↑ transition [56].Lattice spacing and atomic coordination number also strongly influence the shift in the zero-crossing position [51].The reported value of the zero-crossing position for Fe is 6.6 eV [56] and for Co it is 6.0 eV [56].For the Co 90 Fe 10 and Co 40 Fe 40 B 20 alloys the zero-crossing position is red-shifted (table B.1).This can be interpreted as an increased width of the d-bands in the 3d alloys compared with the pure 3d metals.For Co 90 Fe 10 and Co 40 Fe 40 B 20 on a SiO 2 /Si substrate, the zero-crossing positions are found to have very similar values.However, the distinctive magnetic properties (table B.1) of Co 90 Fe 10 and Co 40 Fe 40 B 20 on a ZnO substrate responsible for the notable red-shift in the zero-crossing position from that of the respective material on a SiO 2 /Si substrate.
The figure of merit is defined by value of the mean square error (MSE) [57]  Off-diagonal elements of the MO conductivity tensor (for the off-diagonal dielectric constant see figure B.5) were calculated using the obtained MO coupling constant (figure 10) and are shown in figure 11.The 2p absorption edges of individual Fe and Co are visible around 1.5 eV [39] in the calculated off-diagonal MO conductivity tensor of these ferromagnetic materials, although spin-orbit splitting of the 2p absorption edge is hardly visible in the plot.Apart from that, orbital moments which depend on the spin moment, bandfilling effects, and short-range order also influence the crystal field splitting [58].
Using the obtained MO coupling constant, it is possible to simulate the MO response of a multilayer sample with a ferromagnetic layer for a given magnetic field, angle of incidence and wavelength range.The operating wavelength for the  maximum MO response of MO multilayer samples with FM metal layers can be determined.Knowledge of this operating wavelength is important for tuning MO sensors and hence for designing MO sensors.Because Q(λ) and M(H) can be separated and because the MO response can be detected for dynamic changes of magnetization, magnetic coherence and damping of such ferromagnetic metals of can be investigated by magneto-dynamic measurements [59].

Magnetooptical response of multilayer samples with Co 90 Fe 10 and Co 40 Fe 40 B 20
The MO response of different multilayers was simulated in dependence on the angle of incidence (figure 12), on the thickness of the FM layer (figure 13), on the magnetization direction of FM layers in a multilayer sample with two FM layers (figure C.1) and on the thickness of the Ta buffer layer (figure C.2) in the spectral range from 400 nm to 1000 nm.Strong focus was put on understanding interference effects and depth sensitivity in multilayer samples with FM layers.The motivation for the above choice of simulations is the possible application of such multilayer samples in MO Kerr microscopy where the wavelength-dependent MO response needs to be maximized in order to visualize magnetic domains [35].Extraction of Q(λ) and simulation of difference spectra of Kerr angle (∆θ K and ∆η K ) and of Müller matrix elements (∆M 14 and ∆M 23 ) was carried out in the MagOpS ® software package [41] in a 4 × 4 matrix algorithm.Input parameters were the thickness and dielectric tensor of each layer in the multilayer sample (figure 1) and the off-diagonal elements of the MO dielectric tensor (equation ( 1)) of each magnetizable layer.The off-diagonal elements are described as the product of the material-specific MO coupling constant and the magnetization of the given magnetizable layer.Figure 12 shows the variation of MO response with wavelength for three different angles of incidence (30 • , 45 • and 60 • ) for all samples.A phase shift in SiO 2 is visible in the thickness fringes of Kerr rotation (∆θ K ) and Kerr ellipticity (∆η K ) of all samples on a SiO 2 /Si substrate.For samples on a ZnO substrate, no thickness fringes were observed.Another simulation (figure 13) shows the MO response of all samples in dependence on FM layer thickness.We used three different ferromagnetic layer thicknesses, namely (d FM − 10) nm, d FM nm and (d FM + 10) nm where d FM nm is the thickness of the corresponding FM layer as obtained from the XRR result (table 2).The amplitudes of the Kerr rotation (∆θ K ) and Kerr ellipticity (∆η K ) for samples on a SiO 2 /Si substrate vary in dependence on the thickness of the FM layer.For samples on a ZnO substrate, the MO response shifts in dependence on FM layer thickness.Two additional simulations were done (see appendix C) to demonstrate the effect of magnetization in two FM layers and the effect of Ta layer thickness on the MO response.
With these simulations one can not only determine the MO response of multilayer samples relevant for MO applications but also simulate the maximized MO response before fabrication of such multilayer samples with optimized thickness and chemical composition of each layer in the multilayer sample.

Summary and outlook
The 16 Müller matrix elements of multilayer samples with a single ferromagnetic thin film, namely Co 90 In order to extract the MO coupling constants of such polar unsaturated ferromagnetic metallic thin films we developed a novel analysis scheme.For this scheme we exploit the fact that the MO response of polar unsaturated ferromagnetic metallic thin films depends on the amplitude of both out-of-plane and inplane magnetization and on the direction of out-of-plane magnetization but not on the direction of in-plane magnetization.Using this analysis scheme we extracted the MO coupling constant of Co 90 With this work we extend the applicability of our VMOGE approach for the analysis of the MO response of multilayers with polar saturated ferromagnetic thin films to multilayers with polar unsaturated ferromagnetic thin films.As an example, we have chosen application-relevant TMR and GMR multilayer structures with two polar unsaturated ferromagnetic thin films and analyzed the wavelength-dependent MO response in an external out-of-plane magnetic field.The thickness and roughness of each layer were extracted from XRR data.As shown in figure A.2, the experimental (symbols in respective color) and simulated (solid lines in respective color) results are in good agreement.The difference in surface energies between the different materials mainly influences the crystallinity of the multilayer sample, for example interface roughness, homogeneity.We used fixed densities for the different materials within the multilayer sample (see to table 2).The SiO 2 /Si interface cannot be resolved due to the large thickness of the SiO 2 layer (1000 nm) on top of Si.The roughness of the Ru/FM and FM/Ta interfaces (table 2) is largest for the FM layer Co 90 Fe 10 with the largest crystallite size.

Appendix B. Magnetooptical dielectric tensor of ferromagnetic metal layers
We observed that the refractive index (n) of Co 90 Fe 10 and Co 40 Fe 40 B 20 on SiO 2 /Si is larger than the refractive index of   and extinction coefficient k.N is related to the diagonal elements of the dielectric permittivity tensor as follows: We have modeled the refractive index n and extinction coefficient k of the FM layers Co 90 Fe 10 and Co 40 Fe 40 B 20 .We assume isotropic on-diagonal elements of the MO dielectric tensor We also assume isotropic, off-diagonal elements of the MO dielectric tensor  For a given wavelength, the absolute value of the MO response ∆θ K of ptpb and ntnb and of ptnb and ntpb is the same.For a given wavelength, the value of the MO response ∆η K of ptpb The saturation field of corresponding thin films along the z-direction (polar configuration) (H Sat z ) was measured with a SQUID (figure 3).H Sat z is much larger than the maximum magnetic field (0.4 T) of the 0.4 T octupole magnet of the VMOGE setup.Therefore, experimentally, the magnetization along the z-direction (Mz) of the FM thin films measured with VMOGE in a polar configuration at µ 0 Hz = ±0.4T will not reach the saturation magnetization (Ms), and hence the magnetization M = (Mx, My, Mz) will also have non-zero in-plane components Mx and My.The magnetization at µ 0 Hz = +0.4T along the z-direction is calculated from Mz = (H Exp z /H Sat z )Ms.The non-zero in-plane components Mx and My were calculated for three different cases: My = 0 (Case I), Mx = 0 (Case II) and Mx = My (Case III), using M 2 x + M 2 y = (Ms) 2 − M 2 z .For all three cases the amplitudes of the in-plane components are the same.Obtained Mx, My and Mz values were used to fit experimental VMOGE data to calculate Q(λ) of polar unsaturated Co 90 Fe 10 and Co 40 Fe 40 B 20 with an external magnetic field along the z-direction (µ 0 Hz = 0.4 T).Here we exploit the fact that the difference spectra of the Müller matrix elements M 14 [∆M 14 ] (figure 9) and M 23 [∆M 23 ] (figure 9) depend on the amplitude √ M 2 x + M 2 y and not on the magnetization direction of Mx and My.Note that the SI unit for magnetization is A m −1 and the unit conversion factor is 1 emu g −1 = (1/ρ) emu cm −3 = (10 3 /ρ) A m −1 where ρ is the density of FM as obtained from XRR fitting (table 2).For example, magnetization of Co 90 Fe 10 (on SiO 2 /Si) along the z-direction corresponds to 318 emu cm  and ntnb and of ptnb and ntpb is the same.The MO response in ∆θ K and ∆η K is stronger if the magnetization of both FM thin films shows in the same direction (ptpb and ntnb).The sign of ∆θ K is reversed if one compares ptpb and ntnb and ptnb and ntpb.The sign of ∆η K is not reversed of one compares ptpb and ntnb and ptnb and ntpb.Furthermore, we have calculated the MO response for a 45 • angle of incidence of a multilayer sample shown in figure 1 for three different thicknesses, d Ta , (d Ta + 10) and (d Ta + 20), where d Ta is the thickness of the Ta buffer layer in the respective samples tabulated in table 2. The difference spectra of the complex Kerr angle for the multilayer sample with d Ta is comparable to the difference spectra of the complex Kerr angle shown in figure 8.It clearly visible that the SiO 2 thickness fringes in the MO response of the multilayers on the SiO 2 /Si substrate reduce with increasing thickness of the Ta buffer layer (figures C.2(a)-(f)).For the multilayers on the ZnO substrate it is clearly visible that the dependence of MO response on the wavelength is not changed, i.e. for both the Ta thicknesses ∆θ K increases with increasing wavelength and ∆η K slightly decreases with increasing wavelength for Co 90
Fe 10 and Co 40 Fe 40 B 20 are smeared out compared with corresponding d-d * interband transitions of Co and Fe due to alloy broadening in the electronic band structure of Co 90 Fe 10 and Co 40 Fe 40 B 20 .
(3 nm)/FM (20 nm)/Ta (3 nm) multilayer sample with FM = Co 90 Fe 10 (blue line) and FM = Co 40 Fe 40 B 20 (green line).Both ferromagnetic thin films are polar unsaturated under a magnetic field of 0.4 T, which is the maximum field of the VMOGE setup.FM thin films were deposited on a 6 in thermally oxidized Si/SiO 2 (1000 nm) wafer (figure 1(a)) and cut into 1 cm × 1 cm large sample pieces after deposition.To understand the effect of the underlying substrate on MO response, we deposited the same multilayer sample on 1 cm × 1 cm ZnO substrates (figure 1(b)) in the same run.The ZnO substrate was placed in the center of the 6 in Si/SiO 2 substrates during magnetron sputtering.The 3 nm thick Ta buffer layer serves as an adhesion layer.The 3 nm thick Ru layer on top of the thin film sample prevents oxidation of the underlying Co 90 Fe 10 and Co 40 Fe 40 B 20 layers.The base pressure of the ultra-high vacuum-compatible deposition chamber was kept below 2 × 10 −8 mbar and the deposition was carried out at a pressure of 5 × 10 −3 mbar in a 99.9999% pure argon gas atmosphere.

Figure 1 .
Figure 1.Schematic representation of the a Ru/FM/Ta multilayer on (a) SiO 2 /Si and (b) ZnO substrates with FM = Co 90 Fe 10 and FM = Co 40 Fe 40 B 20 .The sample is placed inside the 0.4 T octupole magnet of the VMOGE measurement setup (not shown here) with the plane of incidence spanned by the x-axis and z-axis [50].The external applied out-of-plane magnetic field is pointing in the +Hz and −Hz directions.In this illustration, the +Hz magnetic field (H = (0, 0, +Hz)) is pointing from the sample surface towards the inside of the sample.The Kerr rotation θ K and the Kerr ellipticity η K depend on the product of the magnetic field-dependent magnetization M(H) and the wavelength-dependent MO coupling constant, Q(λ).Once Q(λ) and M(H) of a given FM layer are known, the MO properties of the multilayer samples containing such a FM layer can be modeled in dependence on the angle of incidence, polarization state of incident light, external magnetic field H, wavelength and thickness of the FM layer.The thickness of the SiO 2 layer amounts to 998 nm and 984 nm for the Ru/FM/Ta multilayer with FM = Co 90 Fe 10 and FM = Co 40 Fe 40 B 20 , respectively.The SiO 2 layer causes thickness fringes.

Figure 3 .
Figure 3. (a), (b) Out-of-plane and (c), (d) in-plane magnetization measured by a SQUID magnetometer.Magnetization is normalized to the sample volume for polar unsaturated Co 90 Fe 10 (blue line) and Co 40 Fe 40 B 20 (green line) thin films on (a), (c) SiO 2 /Si and (b), (d) ZnO substrates.

Figure 4 .
Figure 4. Modeled difference spectra of Müller matrix elements ∆M 14 and ∆M 23 of Co 90 Fe 10 (blue) and of Co 40 Fe 40 B 20 (green) not completely saturated in the out-of-plane direction on (a)-(d) SiO 2 /Si and (e)-(h) ZnO substrates for Cases I, II and III (table1).
Fe 10 and Co 40 Fe 40 B 20 films.The refractive index of Co 90 Fe 10 and Co 40 Fe 40 B 20 on SiO 2 /Si is larger than the refractive index of Co 90 Fe 10 and Co 40 Fe 40 B 20 on ZnO (figure B.3(a)).A possible reason for this is that the Drude term depends on the crystalline

Figure 9 .
Figure 9. Experimental (light solid lines) and fitted data (dark solid lines) for (a), (b) Müller matrix element difference spectra ∆M 14 and (c), (d) Müller matrix element difference spectra ∆M 23 for Co 90 Fe 10 (blue lines) and Co 40 Fe 40 B 20 (green lines) on a SiO 2 /Si substrate.Experimental (light solid lines) and fitted data (dark solid lines) for (e), (f) Müller matrix element difference spectra ∆M 14 and (g), (h) Müller matrix element difference spectra ∆M 23 for Co 90 Fe 10 (blue lines) and Co 40 Fe 40 B 20 (green lines) on a ZnO substrate.

Figure 10 .
Figure 10.(a), (c) Real and (b), (d) imaginary part of the MO coupling constant as extracted by fitting Müller matrix elements difference spectra ∆M 14 and ∆M 23 (see figure 9) for Co 90 Fe 10 (blue lines) and Co 40 Fe 40 B 20 (green lines) on (a), (b) SiO 2 /Si, and (c), (d) ZnO substrates.Grey dashed lines in (a) and (c) define zero y-axis lines.The zero-crossing point is the point where the real part of Q(λ) changes sign.Note that the unit conversion factor is 1 g emu−1 = ρ cm 3 emu −1 = ρ×10 −3 m A −1 ,where ρ is the density of the FM obtained from XRR fitting (table2).

Figure 9
shows the measured and modeled difference spectra of Müller matrix elements ∆M 14 and ∆M 23 of Co 90 Fe 10 and Co 40 Fe 40 B 20 on SiO 2 /Si and ZnO substrates.
A slightly different Q(λ) was obtained for Co 90 Fe 10 and Co 40 Fe 40 B 20 on SiO 2 /Si and ZnO substrates.In the main, the real part of the MO coupling constant in the longer-wavelength range of Co 90 Fe 10 on ZnO and of Co 40 Fe 40 B 20 on ZnO is larger than that of Co 90 Fe 10 on SiO 2 /Si and of Co 40 Fe 40 B 20 on SiO 2 /Si (figure

Figure 13 .
Figure 13.Simulated (a), (b) and (e), (f) difference spectra of Kerr rotation (∆θ K ) and (c), (d) and (g), (h) difference spectra of Kerr ellipticity (∆η K ) of multilayer samples with Co 90 Fe 10 (blue lines) and Co 40 Fe 40 B 20 (green lines) single thin films on (a)-(d) SiO 2 /Si and (e)-(h) ZnO substrates in dependence on the ferromagnetic layer thickness of (d FM − 10) nm, d FM nm and (d FM + 10) nm (where d FM is the thickness of the corresponding FM layer determined by XRR).The thicknesses of Ru, FM and Ta single layers are given in table 2. The angle of incidence kept constant at 45 • .
Fe 10 and Co 40 Fe 40 B 20 in Ru/FM/Ta stacks (FM = Co 90 Fe 10 and Co 40 Fe 40 B 20 ) on SiO 2 /Si and ZnO substrates, were measured by VMOGE in the spectral range from 400 nm to 1000 nm and a polar external field of 0.4 T and analyzed using the 4 × 4 Müller matrix approach.At an external magnetic field of 0.4 T, the investigated ferromagnetic Co 90 Fe 10 and Co 40 Fe 40 B 20 thin films are polar unsaturated.
Fe 10 and Co 40 Fe 40 B 20 metal alloys.Furthermore, the MO coupling constants of Co 90 Fe 10 and Co 40 Fe 40 B 20 obtained from VMOGE experiments were converted into offdiagonal MO conductivity tensors and compared with offdiagonal elements of the MO conductivity tensor from spindependent band structure calculations.As expected, compared with the d-d * interband transitions in pure and Fe metals, the d-d * interband transitions in the metal alloys Co 90 Fe 10 and Co 40 Fe 40 B 20 are smeared out due to alloy broadening in the electronic band structure of Co 90 Fe 10 and Co 40 Fe 40 B 20 metal alloys.

Figure A. 1 .
Figure A.1.XRD patterns of Co 90 Fe 10 and Co 40 Fe 40 B 20 films with a 3 nm thick Ru protection layer and a 3 nm thick Ta adhesion layer (a) on a 1000 nm SiO 2 /Si substrate and (b) on a ZnO substrate.Reflections marked with * are caused by half of the primary Cu Kα wavelength. ).

Figure A. 2 .
Figure A.2. XRR patterns of Co 90 Fe 10 and Co 40 Fe 40 B 20 thin films with a 3 nm thick Ru protection layer and a 3 nm thick Ta adhesion layer (a) on a 1000 nm SiO 2 /Si substrate and (b) on a ZnO substrate.Solid lines in respective color show the fit curves of corresponding data.

Figure B. 2 .Figure B. 3 .
Figure B.2. Real part (dashed lines) and imaginary part (solid lines) of on-diagonal element ϵxx of the dielectric permittivity tensor of FM = Co 90 Fe 10 (blue lines) and FM = Co 40 Fe 40 B 20 (green lines) on (a) SiO 2 /Si and (b) ZnO substrates.

Figure C. 2 .
Figure C.2. Simulated (a), (b) and (e), (f) difference spectra of Kerr rotation (∆θ K ) and (c), (d) and (g), (h) difference spectra of Kerr ellipticity (∆η K ) of multilayered samples with Co 90 Fe 10 (blue lines) and Co 40 Fe 40 B 20 (green lines) single thin films on (a)-(d) SiO 2 /Si and (e)-(h) ZnO substrates in dependence on the Ta buffer layer thickness d Ta , (d Ta + 10) and (d Ta + 20).Thickness values of Ru, FM and Ta single layers are taken from table 2. The angle of incidence is kept constant at 45 • .
3 describes the preparation of multilayer samples with a polar unsaturated ferromagnetic thin film, namely Co 90 Fe 10 and Co 40 Fe 40 B 20 in Ru/FM/Ta stacks (FM = Co 90 Fe 10 and Co 40 Fe 40 B 20 ) on SiO 2 /Si and ZnO substrates, and the VMOGE measurement techniques used in this work.Section 4 describes the extraction of the MO coupling constant of polar unsaturated ferromagnetic thin films from VMOGE data.Section 5 discusses the determined wavelength dependence of the complex MO coupling constant of Co 90 Fe 10 and Co 40 Fe 40 B 20 .Section 6 summarizes the work and gives an outlook on how to use the presented approach, for example for analyzing GMR and TMR multilayer structures with polar unsaturated ferromagnetic thin films.Structural properties and the MO dielectric tensors of Co 90 Fe 10 and Co 40 Fe 40 B 20 thin films are presented in appendices A and B, respectively.The MO response of application-relevant multilayer samples in the geometry of TMR and GMR memories with two polar unsaturated Co 90 Fe 10 and Co 40 Fe 40 B 20 thin films is presented for different detection conditions, for example wavelength and angle of incidence, in appendix C.

Table 1 .
Magnetization components (Mx, My, Mz) calculated from equation Sat z ).H Sat z amounts to 1.52 T for Co 90 Fe 10 and 1.25 T for Co 40 Fe 40 B 20 (table C.1). Applying linear interpolation we calculated the out-of-plane magnetization below the saturation field H Sat Sat z )M s , where M s is the literature value of saturation magnetization (table C.1).For µ 0 H z = 0.4 T, the outof-plane magnetization of Co 90 Fe 10 and Co 40 Fe 40 B 20 is given in table C.1.Finally, we calculated the amplitude of the inplane magnetization (M 2 and A.2.
The modeled on-diagonal MO conductivity of Co 40 Fe 40 B 20 , but not of Co 90 Fe 10 , shows the feature of bcc Fe at 2.75 eV.This would be expected due to the small Fe content in Co 90 Fe 10 .The modeled on-diagonal MO conductivities of Co 90 Fe 10 and of Co 40 Fe 40 B 20 show the feature of fcc Co at 1.75 eV.
(combined weight of ∆M 14 and ∆M 23 ), which amounts to 0.508 and 0.619 for Co 90 Fe 10 and Co 40 Fe 40 B 20 on a SiO 2 /Si substrate and to 0.508 and 0.512 for Co 90 Fe 10 and Co 40 Fe 40 B 20 on a ZnO substrate.From this, one can conclude that agreement between experiment and model is best for Co 90 Fe 10 on SiO 2 /Si and Co 40 Fe 40 B 20 on ZnO.
• , 45 • and 60 • .values of Ru, FM and Ta single layers are taken from table 2.

Table B . 1 .
Extracted zero-crossing position of Re(Q) (from figure 10) and the lattice constant of FM (from figure A.1) for Co 90 Fe 10 and Co 40 Fe 40 B 20 on SiO 2 /Si and ZnO substrates.

Magnetooptical response of multilayer samples with Co 90 Fe 10 and Co 40 Fe 40 B 20
Co 90 Fe 10 and Co 40 Fe 40 B 20 .The strength of the MO response can be estimated by comparing only the magnitude of complex Kerr angle data that amounts to approximately 0.1 • (figure8).Thickness values of Ru, FM and Ta single layers are taken from table The angle of incidence is kept constant at • .We have simulated the MO response at a 45 • angle of incidence for multilayer samples with two FM thin films of thickness d FM /2 (where d FM is the thickness of the FM layer (see table2)).Such TMR structures exhibit a greater change in resistivity than of the anisotropic magnetoresistance and the GMR structures and can be developed into magnetoresistance sensors for industrial applications.In TMR structures the tunnel barrier layer is typically made of SiO 2 or Al 2 O 3. The three main layers in a magnetic tunnel junction element are, from bottom to top, a pinned FM layer, a tunnel barrier layer and a free ferromagnetic layer.We modeled the MO response

Table C . 1 .
Literature values of saturation magnetization Ms in units of emu cm −3 of the ferromagnetic bulk materials Co 90 Fe 10 and Co 40 Fe 40 B 20 . )