Magnetoelastic interactions and magnetic damping in Co2Fe0.4Mn0.6Si and Co2FeGa0.5Ge0.5 Heusler alloys thin films for spintronic applications

Co2Fe0.4Mn0.6Si (CFMS) and Co2FeGa0.5Ge0.5 (CFGG) Heusler alloys are among the most promising thin film materials for spintronic devices due to a high spin polarization, low magnetic damping and giant/tunneling magnetoresistance ratios. Despite numerous investigations of Heusler alloys magnetic properties performed up to now, magnetoelastic effects in these materials remain not fully understood; due to quite rare studies of correlations between magnetoelastic and other magnetic properties, such as magnetic dissipation or magnetic anisotropy. In this research we have investigated epitaxial CFMS and CFGG Heusler alloys thin films of thickness in the range of 15–50 nm. We have determined the magnetoelastic tensor components and magnetic damping parameters as a function of the magnetic layer thickness. Magnetic damping measurements revealed the existence of non-Gilbert dissipation related contributions, including two-magnon scattering and spin pumping phenomena. Magnetoelastic constant B11 values and the effective magnetic damping parameter αeff values were found to be in the range of − 6 to 30 × 106 erg/cm3 and between 1 and 12 × 10–3, respectively. The values of saturation magnetostriction λS for CFMS Heusler alloy thin films were also obtained using the strain modulated ferromagnetic resonance technique. The correlation between αeff and B11, depending on magnetic layer thickness was determined based on the performed investigations of the above mentioned magnetic properties.

Scientific Reports | (2021) 11:7608 | https://doi.org/10.1038/s41598-021-87205-y www.nature.com/scientificreports/ with an additional Ag buffer layer on the Cr one were grown with the aim of reducing of Cr atoms diffusion into a magnetic layer. Most of the films were additionally covered by a 5 nm Au or 3 nm Ta capping layer in order to prevent surface oxidation. Magnetic layers were deposited at room temperature (except for CFGG with Ag buffer samples which were grown at 250 °C) and were subsequently annealed at 500 °C for 20 min in order to promote the chemical ordering 35,83 . In situ reflection high-energy electron diffraction (RHEED) images after each layer deposition process were recorded and ex situ X-ray diffraction (XRD) measurements were performed to control the surface and structure quality of grown thin films. Series of the films with different magnetic layer thicknesses (from 15 to 50 nm) were grown in order to check how the thickness of the magnetic layer influence the quality of the epitaxial growth and structural ordering. Studying of the samples series with different magnetic layer thicknesses enables determination of the surface effects (surface anisotropy and surface magnetoelastic coupling) influence on magnetic, magnetoelastic and damping properties of the investigated samples.
To determine magnetoelastic constants of the magnetic thin films, Strain Modulated FMR technique was applied, which uses a standard X-band spectrometer equipped with an additional system for applying the controlled strain modulation (all details are given in Ref. 82 ). The observed FMR line shift due to the strain effect contains information on the magnetoelastic tensor B ijkl components. In this case magnetic free energy of the system includes also a magnetoelastic energy contribution.
where α i and α j are directional cosines of the magnetization vector, ε kl denotes the strain tensor. For an isotropic case the tensor B ijkl can be constructed using only one magnetoelastic constant, e.g. B 1111 ; in Voigt notation B 1111 = B 11 .
The thin film sample face was glued to the quartz rod, with square (3 × 3 mm 3 ) cross-section in which periodic strains were induced by a piezoelectric generator 82 . The strains induced in the sample plane were ε 11 and ε 22 , in the direction parallel and perpendicular to the rod, respectively. In our system no shear strain in the plane of the film was induced (i.e. ε 12 = 0). The crystallographic [110] axis of the epitaxially grown Heusler alloy layer was oriented parallel to the rod, and the external magnetic field in Strain Modulated FMR experiments was oriented perpendicular to the rod and parallel to the investigated film plane. From the shift of a FMR line induced by the strain the magnetoelastic constant B 11 was calculated assuming the magnetoelastic properties of the magnetic layer are isotropic.
For most of the studied samples Agilent Technologies Vector Network Analyzer FMR spectrometer with a micro-stripe line was used to record FMR curves either with a frequency swept at a fixed magnetic field strength or with magnetic field swept at a fixed frequency, using the measurement protocol given in 84 .
In all Vector Network Analyzer FMR experiments the external magnetic field was applied parallel to the film plane, and parallel to [110] crystallographic axis of the epitaxially grown magnetic layer. For this reason, the dragging effect, which occurs when the sample magnetization is not parallel to the external magnetic field,   85 . Bruker EMX X-band spectrometer with a resonant cavity was used for the basic FMR measurements. A SQUID magnetometer was used to determine the saturation magnetization of all samples. In order to check the crystallographic quality of the layer and possible deformation of its crystal lattice, a high-resolution X-ray diffractometer with a 4-reflection Ge (220) monochromator and X-ray mirror, radiation -Cu Kα1 and an analyzer in front of a proportional detector was used for the sample with the thickest CFMS layer (50 nm) grown on the Ag buffer.

Results and discussion
The RHEED patterns obtained for the magnetic Heusler layers revealed a low roughness surface and the epitaxial growth with relationships of crystallographic directions: MgO (001) || Cr (001) || magnetic layer (001) or MgO (001) || Cr (001) || Ag (001) || magnetic layer (001) and for all the samples superlattice (200) streaks were clearly observed. In accordance with the RHEED patterns, the (200) superlattice diffractions as well as the (400) fundamental diffractions of the Heusler layers were clearly observed in XRD patterns for all the samples (see Fig. 2). The well-defined (400) diffraction peak indicates that the samples are well crystallized, having cubic symmetry. The intense (200) superlattice peak indicates that all magnetic layers have at least the B2 ordered structure. Magnetoelastic interactions in the studied samples are relatively weak (part of the data was previously published 80 ). Experimentally determined magnetoelastic constants B 11 have values in the range of − (6-30) × 10 6 erg/cm 3 (see Table 1), which are close to the values reported for Co 2 YSi (with Y = Fe or Mn) 70 and for Co 2 FeAl 66 . Figure 3 illustrates the magnetoelastic constants dependence on the film thickness. The saturation magnetostriction λ S can be described as: where c 11 88 , the concept of the so-called surface magnetoelastic coupling was introduced 89,90 . In this case the magnetoelastic constant can be, formally, divided into two parts: where d is a magnetic layer thickness, B 11,V and B 11,S are the volume and surface components of the magnetoelastic constant, respectively. Using the experimental data of the thickness dependence of magnetoelastic constants B 11 , shown in Fig. 3, and assuming a linear fit (see dotted lines in Fig. 3) the B 11,V and B 11,S parameters values were determined for the samples series, containing 3 samples: for the CFMS thin films without the additional Ag buffer layer: B 11,V (CFMS) = − 17.0 × 10 6 erg/cm 3 , B 11,S (CFMS) = 7.5 erg/cm 2 , and for the Ag-buffered CFGG thin films: B 11,V (CFGG) = − 31.6 × 10 6 erg/cm 3 , B 11,S (CFGG) = 7.8 erg/cm 2 .
Very little experimental work was up to now devoted to the study of surface magnetoelastic coupling. The estimated by us surface components of the magnetoelastic constants for CFMS and CFGG films are very close to those found in epitaxial iron films deposited on GsAs 91,92 . Although the volume components of the magnetoelastic constants in these iron films were found to depend on both sample preparation procedure and the structure of the films (using additional overlayers 91 ), the surface components of all the investigated films were very similar: b 1 S ~ 11 erg/cm 3 . For an isotropic sample b 1 S = (3/2)B 11,S , which gives a good consistency with our results. For all investigated in this study series of samples the magnetoelastic constant absolute values decrease with decreasing magnetic layer thickness (see Fig. 3). Hence, the magnetoelastic effect in such thin layers is weaker than that expected in bulk materials.
The obtained for CFMS samples saturation magnetostriction value λ S = 1.44 × 10 -5 , is in accordance with the experimental value reported for Co 2 Cr 0.6 Fe 0.4 Al Heusler alloy bulk sample 93 and the recent theoretical calculations for Co 2 XAl Heusler alloys 94 , where X = V, Ti, Cr, Mn, Fe.
The concept of surface magnetoelastic coupling 89,90 can explain the incerase of the magnetoelastic constants absolute values with increasing thickness for the studied magnetic layers. Existing theoretical papers predict several mechanisms responsible for surface magnetoelastic coupling (see e.g. review paper 95 ). Among them the most important are spin-orbit interactions (single-ion model) and dipole-dipole interactions. Other mechanisms observed in both single layers and multilayers are non-linear contributions to bulk magnetoelastic coefficients due to surface strains and surface roughness effects. Also the presence of interdiffusion layers which are formed at the interface should be taken into account.
As the magnetoelastic constant can be changed by a strain, their changes with the thickness of the magnetic layers may be also a result of structural relaxation. In our previous studies 80 , we found the magnitude of the perpendicular magnetocrystalline anisotropy of investigated samples to be decreasing with increasing magnetic layer thickness. Such a behavior could suggest an occurrence of structural relaxation. For the strained Heusler alloys their cubic symmetry will be lowered and in order to check such a possibility additional X-ray studies were performed.
For the sample with the thickest CFMS layer (50 nm) grown on the Ag buffer additional detailed XRD studies were performed (see Figs. 4 and 5). From the rocking curve measurements of the (004) and (224) reflections, we obtained the half-widths of the ω-scans equal to 0.75° and 0.65°, respectively. This indicates that, in addition to the undoubted widening associated with a small layer thickness, we are probably observing the influence of mosaic-type defects. In order to calculate the lattice deformation, 2-dimensional X-ray diffusion scattering maps around the symmetric (004) (Fig. 4a) and asymmetric (224) (Fig. 5a) reflexes were obtained. Projections of the map nodes on the 2θ axis and fitting Gaussian curves to the obtained measurement points allowed to determine the interplanar distances for (004) and (224) planes (see Figs. 4b and 5b, respectively). Hence, we obtained lattice www.nature.com/scientificreports/ unit parameters: a ⊥ = 5.675 ± 0.001 Å and a || = 5.643 ± 0.005 Å indicating a tetragonal deformation of about 0.5%, which results in the appearance of the strain induced anisotropy. Errors in determining lattice unit parameters were assessed from the ambiguity of fitting Gaussian curves to projections of maps and measurement method.
As the tetragonal distortion of the investigated sample was found to be very small, it seems that not the structural relaxation but rather the surface magnetoelastic coupling is responsible for the observed magnetoelastic constants absolute values increase with increasing thickness of the magnetic layer. It is possible that the observed changes of the magnetoelastic constants may be also connected with the thickness dependent structural ordering occurring for thicker layers as it was reported for Co 2 FeSi layers 96 .
Using the formula for magnetoelastic energy (1) for an isotropic sample case and assuming the tetragonal distortion (ε 11 = ε 22 ) the formula for the strain induced contribution to the uniaxial anisotropy constant K SI can be written as: where the strain component ε 33 = − 2(c 12 /c 11 )ε 11 .
Taking the data from our experiments: ε 11 = − 2.44 × 10 -3 , B 11 = − 17.50 × 10 6 and elastic constants 87 , we have obtained the value of K SI = (1.39 ± 0.23) × 10 5 erg/cm 3 . It has the opposite sign and a small value in comparison to the overall magnetocrystalline anisotropy of relatively low and negative value |K|< 1.5 × 10 6 erg/cm 3 which was determined from FMR/SQUID studies 80 . Strain causes an increase of the anisotropy constant and reduces its absolute value. The minimal tetragonal distortion value, which is necessary to switch the magnetic layer anisotropy from an easy-plane to an easy axis type, was estimated to be at least ε 11-min ≈ − 0.07; but such a large strain is not likely to be present in our samples.  www.nature.com/scientificreports/ Such a large estimated critical value of the distortion results from the fact that a large demagnetizing energy in a thin film (proportional to the magnetization) must be overcome by the strain induced anisotropy, which is proportional to a (relatively small) magnetoelastic constant.
Magnetic damping is proportional to the FMR linewidth. However there are many mechanisms that may be responsible for the dissipation of magnetic energy in thin magnetic films, thus the full width at half maximum in registered FMR spectra can include the following contributions 85 : ΔH = ΔH 0 + ΔH TMS + ΔH G+sp , where ΔH 0 denotes the frequency-independent sample inhomogeneity contribution, ΔH TMS denotes the two-magnon scattering contribution (TMS), while the last term ΔH G+sp contains the spin pumping (SP) and Gilbert damping contributions and is proportional to α eff × f, where α eff is an effective magnetic damping parameter and f is the FMR frequency 85 . Considering the TMS mechanism is especially important for the samples characterized by low Gilbert damping when the external magnetic field is parallel to the film 97 . Figure 6 shows the experimental frequency dependence of half-maximum width ∆H for the 30 nm CFMS sample without additional silver buffer layer. It is clearly seen that at frequencies below 8 GHz this dependence is nonlinear and a slope of the curve decreases with increasing frequency but above 8 GHz, almost linear dependence was observed. Such a dependence of ∆H is typical if the TMS contributes to the damping processes 98,99 . In the low frequency range the ∆H TMS vs. frequency is nonlinear, while it saturates at high frequencies. For this reason in the high frequency range, the slope of the ∆H can be assumed to be proportional to effective magnetic damping parameter α eff . The estimated values of α eff for all samples studied in our experiments are presented in Tab.2. After subtraction of ΔH G+sp and ΔH 0 the ΔH TMS can be evaluated, which is also presented in Fig. 6. It can be seen that the ΔH TMS contribution to the total ∆H is relatively strong, which is typical for in-plane broad band FMR experiments if the effective magnetic damping parameter is relatively small 97 .
The magnetic damping parameters in the studied samples are evaluated to be of the order of 10 -3 ; but the application of the additional Ag buffer layer in case of the CFGG samples resulted in a significant reduction of the α eff value.
It should be noted that contrary to other samples, the Ag-buffered CFGG thin films have the cover layer of Ta, which has a strong spin-orbit coupling and as a consequence, the value of the frequency proportional contribution ΔH G+sp ~ α eff × f is increased due to the presence of the spin pumping phenomenon 100 and only for these samples α eff is decreasing with increasing magnetic layer thickness (see column 4 in Table 2). In the case of other investigated samples which were without Ta -but having Au cover layer-it was found that the effective magnetic damping parameter α eff increases with the magnetic layer thickness (see Fig. 7).

Figure 6.
Resonance line full width at half maximum for the 30 nm CFMS sample without additional silver buffer layer as a function of frequency at room temperature. Solid line presents a linear fit for the frequencies above 8 GHz, which slope is proportional to α eff . Empty symbols present the extracted as described above, two magnon scattering contribution (TMS). www.nature.com/scientificreports/ It should be emphasized that for all studied samples, the increase of magnetic layer thickness is accompanied by the increase of the magnetoelastic constant B 11 absolute value. Therefore, for the samples, where effective magnetic damping parameter α eff may be considered as Gilbert damping parameter α (without spin pumping contribution due to a presence of Ta), the increase of the absolute value of magnetoelastic constant is accompanied by the increase of the damping parameter (see also Fig. 8).
It is generally assumed 77,79,101,102 that the amplification of the magnetoelastic effect should lead to an increase of magnetic damping. In the case of thin magnetic films, this hypothesis was verified by a number of experimental studies conducted mainly for Ni-Fe films. For example the correlation between magnetic damping and magnetostriction in permalloy films of different compositions was reported by several authors 77,79 . The variation of the magnetostriction constant from − 0.7 × 10 -5 to 1.5 × 10 -5 was caused by changing of a Ni-to-Fe ratio where λ = 0 for Ni 80 Fe 20 composition. Nevertheless, it is difficult to find formulas which correlate the magnetoelastic constants and damping parameters quantitatively. Bonin et al. 79 assumed that the correlation of magnetic damping with strain fluctuations was induced by the magnetoelastic effect. To describe the fluctuation processes that affect damping, originating during the magnetization reorientation process, the use of nonequilibrium statistical mechanics is required. In Ref. 79 , the self-organized criticality 103 was used to perform the experimental data analysis. A model of correlation between the magnetostriction and relaxation mechanism was also presented in Ref. 101 , where the coupling between the magnetic motion and lattice was based purely on continuum arguments concerning magnetostriction. In Ref. 102 , dynamics of magnetization coupled to a thermal bath of elastic modes was analyzed. Both models, in Refs. 101 and 102 , predict a damping factor to be proportional to the square of the  www.nature.com/scientificreports/ magnetoelastic constants. Most importantly in Ref. 102 , an increase of the damping factor with the increase of the magnetic layer thickness was anticipated what is confirmed by our results. Figure 8 shows the correlation between the absolute values of the magnetoelastic constants B 11 , and the effective magnetic damping parameter α eff for the three series of the samples studied in our experiments, for which the influence of the spin pumping on the magnetic damping can be neglected (no Ta capping layer). Experimental data of all three series were collected in one plot. Although the scatter of experimental data is large, one can see a general tendency of the effective damping increasing with an increase of B 11 magnitude. In Fig. 8 a fit using the quadratic function is also shown. As discussed above, such dependence was predicted by some of theoretical models describing correlations between magnetoelastic properties and damping 101,102 .
However, it must be kept in mind that beside the magnetoelastic effect also other parameters influence the magnetic damping, and if these parameters change with the thickness of the magnetic layer this can also result in the changes of magnetic damping. In particular, half-metallicity and Gilbert damping factor in Co 2 Fe x Mn 1-x Si Heusler alloys depending on the film composition were investigated in Ref. 57 and Gilbert damping parameter in Co 2 Fe x Mn 1-x Si Heusler alloys was shown to be correlated with their band structure The minimum of the damping parameter (α ≈ 0.003) was found for the composition with x = 0.4. The results were discussed assuming the Gilbert damping constant to be proportional to the square of the spin-orbit coupling parameter and total density of states of the d-band at Fermi energy.
In the experimental studies performed earlier by other researches 77,79 , the variation of magnetoelastic properties was induced by the change of film composition. In the case of the results obtained and discussed in this paper, the samples have a constant composition in one series of samples and an increase of magnetic damping is correlated with an increase of the absolute value of magnetoelastic constants, which increases with increasing magnetic layer thickness.
Such an effect has not been observed so far. The observation and study of such correlations are important also from the point of view of applications in spin-mechanical devices, for which thin film materials possessing low magnetic damping with a sizable magnetoelastic effect are sought 76 .
Although the correlation between the magnetoelastic properties and damping factor in thin films is not fully understood yet, most models developed up to date predict an increase of the damping factor with the increase of the magnetoelastic constants magnitude and/or with the thickness of a magnetic layer. This is consistent with our experimental observations. Because of the lack of several important parameters included in the existing theoretical models like the elastic relaxation time or the exchange stiffness constant, quantitative comparison of our results with these models is difficult. It should also be remembered that the magnetoelastic mechanism may not be the only one responsible for the changes of the damping coefficient values observed in our experiments, e.g. the conductivity of the samples was neglected in our considerations. The changes of the structural ordering with the thickness of magnetic layer may also influence the band structure, and thus the Gilbert damping.

Conclusions
Magnetoelastic properties and magnetic damping for several series of quaternary Co 2 Fe 0.4 Mn 0.6 Si and Co 2 FeGa 0.5 Ge 0.5 Heusler alloy thin magnetic films were determined by the Strain Modulated FMR and the Vector Network Analyzer FMR methods, respectively. The magnetoelastic constants were found to have relatively small and negative values while saturation magnetostriction for all the studied samples was positive. It was shown that two-magnon scattering and spin pumping phenomena play an important role in the magnetic damping of microwaves in the studied magnetic thin films.
The frequency dependent FMR linewidths were found to be strongly inhomogeneous; the resonance line FWHM nonlinear dependence evidences that the inhomogeneous broadening is correlated mainly with two magnon scattering processes. It was revealed that the magnetic damping in the investigated samples is strongly influenced by the type of metal used for a buffer or cover layer; where specifically the use of Ta cover layer leads to a strong spin pumping phenomenon appearance. Linear dependence of the resonance line at high frequencies enabled determination of the magnetic Gilbert damping parameter for the studied samples, assuming that the TMS contribution at these frequencies is saturated.
The saturation magnetostriction (λ ≈ 10 -5 ) and low magnetic damping parameter (α eff ≈ 10 -3 ) values in the investigated Heusler alloys thin films are similar to the results of the pioneering work 73 , where a new class of materials promising for spin-mechanical devices was reported. Calculated strain, which is necessary to switch the magnetic anisotropy from the easy-plane to the magnetization easy axis type, is too large (ε 11-min ≈ − 0.07) to be achieved in the epitaxially grown magnetic layers.
In the samples for which it was possible to neglect the spin pumping phenomenon (i.e. without the Ta cover layer), the correlation between the Gilbert damping parameter and magnetoelastic constants was obtained. Based on the fact that both the magnetic damping parameter and the absolute value of the magnetoelastic constant increase with magnetic layer thickness, it was concluded that the enhanced magnetoelastic effects are accompanied by a stronger magnetic damping. An increase of the absolute values of magnetoelastic constants with increasing thickness of the magnetic layer can be explained as a result of surface magnetoelastic coupling and/or the thickness dependent structural ordering. An increase of Gilbert damping can be correlated with increasing magnetoelastic constants or changes in the band structure caused by the changes of structural ordering.

Data availability
All data reported in this manuscript is available from the corresponding author on a reasonable request.