Amorphous ferrimagnetic thin films composed of rare earth elements and transition metals attracted considerable attention because of their useful technological applications1,2,3,4. As one of their important representatives, GdxFe(100-x) has significant advantages, such as large magnetization and possibility to adjust its compensation temperature, coercive field and saturation magnetization by changing the composition5,6,7. These properties make GdxFe(100-x) substantial for modern micro- and nano-electronic research, where it is often used in domain wall junctions or magneto-optical (MO) memories1,3,4. Recently, a novel concept of high speed MO spatial light modulator for holographic displays based on GMR with GdxFe(100-x) as a free layer was proposed2.

Another valuable feature of GdxFe(100-x) is that it enables direct access to its spins through the electromagnetic interactions, which makes this material subject of importance for future magnetic recording (such as heat assisted magnetic recording) and information processing technologies. Recent numerical atomic scale modeling simulations of the spin dynamic in Heisenberg GdxFe(100-x) ferrimagnet demonstrated that the rapid transfer of energy into the spin system leads to switching of the magnetization within a few ps without the necessity of applied magnetic field. The experiment in GdFeCo alloys, which used linearly polarized fs laser pulse to produce the ultra-fast heating, confirmed this prediction8,9,10. Moreover, by using circularly polarized laser pulses, it is possible to take an advantage of the magnetic circular dichroism effect to record a magnetic domain in which the helicity of the laser pulse influences the final magnetization direction8,10,11,12. These mechanisms allow the GdxFe(100-x) magnetic domain light spin manipulation and hence coherent control of the magnetization precession at fluencies as low as 6 μJ/cm213 and in rates of ps10,11,14.

The main purpose of our investigation was the determination of the complete permittivity tensors of GdxFe(100-x) thin films with various compositions (x = 18.3, 20.0, 24.7, 26.7) since their spectra provide a deeper look at optical and MO properties of this material. Moreover, the knowledge of the complete tensor allows the theoretical prediction of complex physical properties in complicated multilayered nanostructures containing GdxFe(100-x) layer.

In this work, we used spectroscopic ellipsometry (SE) at energies from 1.5 to 6 eV and MO spectral measurements at energies from 1.5 to 5.5 eV. From SE data, we derived the diagonal permittivity tensor elements ε1r and ε1i spectra of GdxFe(100-x) thin films. We examined MO properties by polar MO Kerr effect (MOKE) rotation and ellipticity measurements. From these results we determined the spectral dependence of the off-diagonal GdxFe(100-x) permittivity tensor elements ε2r and ε2i. We also performed MOKE hysteresis loop measurements, which demonstrated changes in magnetization in dependence on GdxFe(100-x) composition.

GdxFe(100-x) is usually covered by a capping to avoid the oxidation process15. However, this fact complicates its analysis. Optical properties of capping materials (here Ru, SiO2) may slightly differ in dependence on material they are deposited on. The reason behind this behavior is usually the lattice mismatch between the film and substrate, which induces strains of various kinds16,17,18. In order to deal with this issue, we used two different capping materials which allowed more precise determination of GdxFe(100-x) permittivity tensors. Measurement of SE showed very similar optical properties of individual GdxFe(100-x) compositions for both cappings.

Results and Discussion

We parameterized spectral dependences of optical functions (diagonal elements of permittivity tensor) to ensure KK consistent results. We used the summation of two Lorentz oscillators and Drude term in the spectral range from 1.5 to 6 eV. The parameters of the dispersion functions were fitted together with thicknesses of individual layers of investigated samples. Figure 1 shows that this theoretical approach describes both Gd20Fe80/Ru and Gd20Fe80/SiO2 SE experimental spectra adequately. Sample parameters and optical properties of Si, SiO2 and Ru, used in the SE analysis are shown in the Methods section.

Figure 1
figure 1

Measured variable angle SE Psi and Delta spectra of (a) Gd20Fe/Ru and (b) Gd20Fe/SiO2 samples compared to the theoretical model.

Figure 2 shows derived spectral dependences of the real part of the diagonal permittivity tensor elements ε1r while Fig. 3 shows the spectra of the imaginary parts ε1i. The ε1r spectra are dominated by one global minimum at 2.9 eV while the ε1i amplitudes decrease at higher energies for all compositions. Obtained results show spectral behavior similar to previously published optical properties of Gd22Fe7819, Fe, Gd20,21 and GdxFe(100-x)6,22. The spectral dependence of ε1r between 1.5 and 3 eV, where ε1r decreases its amplitude with increasing energy, differs from typical Drude-like behavior (describing intra-band transitions) of metallic compounds, and is similar to the behavior of some transitions metals (including Cr, Gd, Ru, Ti20,21). This behavior is coming from the Lorentz contribution centered near 1.9 eV and most likely originates in the inter-band transition, which involves Fe 3d and Gd 5d states. The Fe 3d state lie around 1.5 eV below Fermi energy, while Gd 5d states are situated approx. 0.5 eV above the Fermi energy23,24. The second Lorentz oscillator centered near 2.5 eV does not significantly modify the spectral dependence of Drude behavior due to its small amplitude. This points on the origin of Gd d-d electron transition23,24, since this transition should be forbidden with small oscillator strength.

Figure 2
figure 2

Real parts of diagonal elements of the permittivity tensor of the GdxFe(100-x) thin films.

Figure 3
figure 3

Imaginary parts of diagonal elements of the permittivity tensor of the GdxFe(100-x) thin films. The inset displays the dependence of the imaginary parts on Gd concentration at E = 2.8 eV.

Finally, we discuss the Gd substitution effect. With increase of the Gd content the amplitude of the first Lorentz function around 1.9 eV decreased (see Table 2 in Methods). This is visible in Fig. 2 as a change of ε1r amplitude around 2.9 eV and in the inset of Fig. 3. Increasing Gd content decreases Fe density of states below Fermi energy, resulting in the suppression of Fe 3d to Gd 5d transition probability. In contrast to that, the increase of Gd content is increasing the amplitude of the second Lorentz function centered near 2.5 eV. Such behavior supports the assignment of the origin to Gd d-d transition.

Polar MOKE hysteresis loop measurements are shown in Fig. 4. Samples with higher Gd content exhibit square-like loops with sharp transition, indicating out-of-plane easy axis of the net magnetization. As the composition approaches the compensation point, the coercivity increases. Below the composition value of x ≈ 25, the Fe moments become more dominant forcing the sample accommodate the in-plane anisotropy. The higher net magnetization value results in stronger effects of dipolar interaction, inducing a complex multi-domain state in the sample with negligible coercivity. With lower Gd content, the hysteresis loop shape becomes more prolonged and together with the increase of the saturation field hints on tilted direction of the net magnetization to the OOP orientation. Such state is described by complex butterfly-like shaped hysteresis loops shown in Fig. 4 for samples Gd18.3Fe and Gd20Fe.

Figure 4
figure 4

Hysteresis loops of examined samples with (a) Ru and (b) SiO2 cappings at 2.38 eV.

Opposite direction of the magnetization as well as different coercivity for samples with the same Gd concentration x = 24.7, but different capping (Gd24.7Fe/Ru and Gd24.7Fe/SiO2) is noticeable in Fig. 4. Since this composition is extremely close to the compensation point, it would be reasonable to assume that the real composition of the sample might slight differ from the measured one (obtained integrally by EDX). Therefore, the sample with SiO2 capping might have lower concentration than x = 24.7 (situated below compensation point) and can be closer to the compensation point, since the coercivity is increasing when approaching the compensation. On the other hand, one cannot also exclude the influence of the capping layer itself, although its influence will be much smaller with respect to the layer/capping thickness ratio. Finally, the slight change in the temperature during the measurement can bring the sample above/below the compensation point.

Figure 5 shows polar MOKE rotation and ellipticity spectra. Firstly, both spectra are characteristic by increasing value of rotation and ellipticity towards smaller energies. Secondly, samples with SiO2 capping show notably higher MO signal than samples with Ru capping (also visible in hysteresis loops measurements in Fig. 4). Because both capping materials have different optical properties the total reflection coefficient rpp of the sample will differ (see Eq. 8 in Methods section). This influences the value of MOKE rotation, as well as ellipticity. Moreover, one can see that the substitution of Gd is slightly increasing amplitudes of the MOKE. Finally, as expected, amplitudes of MOKE rotation and ellipticity change their sign when Gd reaches the compensation concentration (x ≈ 25) and therefore when the net magnetization direction changes. All the data correspond to the hysteresis loops measurements.

Figure 5
figure 5

Polar MOKE (a) rotation and (b) ellipticity spectra of examined samples.

Combining the results from SE and MOKE measurements we calculated the off-diagonal elements of the GdxFe(100-x) permittivity tensors. In the calculation we used the thicknesses of the layers determined by SE. Figure 6 shows calculated spectrally dependent real parts of the off-diagonal elements ε2r and Fig. 7 imaginary parts of the off-diagonal elements ε2i. The spectra are consistent with the spectral behavior of diagonal elements ε1 and agree with previously reported results on Gd22Fe7819. From Figs 6 and 7 one can see a change in the sign when crossing the compensation point. This confirms the change of the direction of the net magnetic moment in the investigated samples. Apart from the sign change, the spectral behavior remains the same for all samples (it is clearly demonstrated by the same zero crossing near 1.8 eV in the Fig. 6). Magneto-optical Kerr effect mainly originates from electric-dipole transitions between particular spin-orbit split bands, which leads to the dispersion of the off-diagonal elements ε2r. Since the zero crossing in the Fig. 6 is situated near 1.8 eV for all samples with different composition, the electronic transitions have similar energy in all samples. This indicates only small changes in electronic structure of GdxFe(100-x) with composition around the compensation point. The slight change in the amplitude of the off-diagonal components of the permittivity tensor can be attributed to the change of the net magnetization near the compensation point. The results are consistent with hysteresis loop measurements in Fig. 4.

Figure 6
figure 6

Real parts of the off-diagonal elements of the permittivity tensors of GdxFe(100-x).

Figure 7
figure 7

Imaginary parts of the off-diagonal elements of the permittivity tensors of GdxFe(100-x).

The combination of SE and MOKE spectroscopy allowed us to successfully determine the spectral dependence of full permittivity tensor of GdxFe(100-x). The analysis of optical properties revealed contribution of intra- and inter-band transitions described by Drude and Lorentz approaches. The inter-band transitions were assigned to Fe-Gd and Gd-Gd 3d bands. MOKE hysteresis loops confirmed the changes in magnetic ordering when the composition crosses the compensation point. Complex shape of hysteresis loops suggests multidomain structure after crossing the compensation point. MOKE spectroscopy and subsequent calculation of off-diagonal permittivity elements demonstrated, in agreement with SE, only small changes in electronic structure in the vicinity of the compensation point.



It is a common approach to describe the response of a material to electromagnetic wave in the presence of external magnetic field by permittivity and permeability tensors. At optical frequencies, we can assume the permeability to be unit scalar25. Since the magnetic field acts as a small perturbation of the system, the permittivity tensor, as a function of magnetization, can be expanded in the series to the second order as

$${\varepsilon }_{ij}={\varepsilon }_{ij}+{K}_{ijk}{M}_{k}+{G}_{ijkl}{M}_{k}{M}_{l},$$

where εij is the permittivity independent on magnetization, Mk,Ml are the components of magnetization vector and Kijk and Gijkl are linear and quadratic magneto-optical tensors, respectively. In the present work, we restrict ourselves only to the linear magneto-optical effect and omit the third term in Eq. (1).

If the magnetization is parallel to the z-axis of the Cartesian coordinate system (the magnetic film-ambient interface is normal to the z-axis, light is propagating along the z-axis) and if we restrict ourselves to linear MO effects, the permittivity tensor simplifies to the form26

$$(\begin{array}{ccc}{\varepsilon }_{1} & -i\cdot {\varepsilon }_{2} & 0\\ i\cdot {\varepsilon }_{2} & {\varepsilon }_{1} & 0\\ 0 & 0 & {\varepsilon }_{1}\end{array}).$$

Generally, the tensor elements are complex having real and imaginary parts:

$$\begin{array}{ccc}{\varepsilon }_{1} & = & {\varepsilon }_{1r}-i\cdot {\varepsilon }_{1i}\\ {\varepsilon }_{2} & = & {\varepsilon }_{2r}-i\cdot {\varepsilon }_{2i}\end{array}.$$

In the base of s and p polarized incident light, we can describe the optical response of a sample upon light reflection within the framework of the Jones matrix of reflection25

$${J}_{sp}^{R}=(\begin{array}{cc}{r}_{ss} & {r}_{sp}\\ {r}_{ps} & {r}_{pp}\end{array}){J}_{sp}^{I},$$

where its elements are amplitude reflection coefficients.

It is possible to derive the diagonal elements of the permittivity tensor analyzing the experimental data from SE. The change in the polarization state of the reflected light can be expressed the by the SE parameters Psi (ψ) and Delta (Δ), which are defined as

$$\tan \,{\rm{\psi }}\cdot {e}^{i\Delta }=\rho =\frac{{r}_{pp}}{{r}_{ss}}.$$

In this equation, tan ψ is the magnitude of the reflectivity ratio and Δ is the phase change between s and p polarized light. The important part of SE analysis is the proper parametrization of the dispersion relations of investigated material. In this work, we used Kramers-Kronig (KK) consistent Lorentz and Drude approaches. The phenomenological Lorentz approach is commonly used to describe the dispersion of inter-band transitions in matter as

$${\varepsilon }_{{1}_{Lorentz}}=\frac{A\Gamma {E}_{0}}{{E}_{0}^{2}-{E}^{2}-i\cdot E\Gamma }.$$

Here, the parameters E0, A, Γ denote the transition energy, amplitude and the broadening parameter respectively27 (p. 343)28. The Drude approach was used to model the free carrier contribution to the optical properties using the following equation

$${\varepsilon }_{{1}_{Drude}}=\frac{-{\hslash }^{2}}{{\varepsilon }_{0}\rho (\tau {E}^{2}+i\hslash E)}.$$

Parameters ρ and τ denote the resistivity and mean scattering time respectively.

It is possible to derive the off-diagonal elements of the permittivity tensor from spectroscopic MOKE data analysis. Here, we used the Yeh matrix formalism25,26,29 for theoretical calculations in studied multilayers. This formalism solves the wave equation in optically anisotropic layers, finds the eigenstates of propagation and subsequently uses the boundary conditions to relate the propagation across all layers. As a result, the Jones reflection and transmission matrices, containing Fresnel reflection coefficients, are obtained. In the polar MOKE experiment, we can express the change in the polarization state of the reflected beam by the complex MO Kerr angle ΦK, which is for incident p-polarization and small angles defined as follows

$${\Phi }_{K}={\theta }_{K}-i\cdot {e}_{K}=\frac{{r}_{sp}}{{r}_{pp}}.$$

In this equation, ϴk is the Kerr rotation, ek is the Kerr ellipticity. To derive the off-diagonal elements of permittivity tensor, we considered a model of semi-infinite Si substrate and three additional layers. Each layer was characterized by the complex permittivity tensor and the thickness. Combining the experimental values of MOKE, Eq. (8) and Yeh formalism, we were able to fit the off-diagonal elements ε2 for each energy of incident light.

Samples and measurement details

Thin films of GdxFe(100-x), x = [18.3–26.7], were deposited by DC sputtering on thermally oxidized silicon substrates (100) in Kr gas (8.7 × 10−2 Pa). The growth rate was 3.6 nm/min. Ru capping was grown by the same technique. SiO2 capping was deposited by ion beam sputtering technique with rf ion source. Table 1 shows the model structures and nominal thicknesses (determined by XRF) used for the theoretical analysis of SE and MOKE experimental data. The thicknesses of capping layers were chosen to be smaller compared to the penetration depth of the incident light in the whole spectral range. Model structures contained Si substrate, 300 nm thick layer of SiO2, 100 nm thick layer of GdxFe(100-x), and a 3 nm thick capping Ru layer or 20 nm thick SiO2 layer on top. A surface roughness was modelled as a separate layer using a Bruggeman Effective Medium Approximation formula30 of the mixture of 50% of hosting material (ε) and 50% of void. The roughness was deduced as a thickness of this layer.

Table 1 Structure compositions and nominal thicknesses of examined samples.

A Mueller matrix ellipsometer Woollam RC2 was employed to perform SE measurements of ellipsometric parameters Psi and Delta in the spectral range from 1.5 to 6 eV for incident angles 55°, 60° and 65°. Obtained experimental data were analyzed using the CompleteEase software. In the case of samples with both cappings (Gd18.3Fe81.7, Gd24.7Fe75.3) a “Multi Sample Analysis” was used. This allowed to simultaneously fit the optical properties of GdFe from both samples together while keeping optical properties of Ru and SiO2 constant27.

The spectral dependences of optical functions (diagonal elements of permittivity tensor) were parametrized with the summation of two Lorentz oscillators and Drude term in the spectral range from 1.5 to 6 eV to ensure KK consistent results. The parameters of the dispersion functions were fitted together with thicknesses of individual layers of investigated samples. Used parameters are listed in Table 2. Optical properties of Si, SiO2 and Ru, used in the SE analysis, were determined from SE measurements on individual samples. Resulting thicknesses of individual layers of investigated samples are summarized in Table 3. To verify the thicknesses of GdxFe(100-x) layers obtained from SE, XRF measurements were performed as well, giving similar results, as follows from Table 3.

Table 2 Fitted parameters of Lorentz oscillators and Drude term used to parameterize optical functions of GdxFe(100-x) optical functions in the spectral range from 1.5 to 6 eV.
Table 3 Fitted thicknesses used for the model of GdxFe(100-x) in SE and X-ray in the spectral range from 1.5 to 6 eV.

Similarly to our previous work19, MOKE rotation and ellipticity spectra in the polar configuration were obtained using a method of generalized MO ellipsometry with rotating analyzer, which allowed the determination of the rotation angles with high accuracy. All the spectra were measured at the room temperature and nearly normal light incidence. Applied magnetic field was 1.2 T, which was enough for magnetic saturation of all the samples. Incident light was p-polarized. The data were recorded in the photon energy range from 1.5 to 5.5 eV.

Differential intensity detection method was employed to acquire MOKE rotation hysteresis loops at 2.38 eV. We performed all measurements in the polar geometry and at the room temperature. Applied magnetic field was ranging from −1.8 T up to 1.8 T, which was beyond saturation point.