Colossal Gilbert Damping Anisotropy in Heusler‐Alloy Thin Films

Manipulating Gilbert damping is of critical importance in spintronic devices. Here, this work reports the damping anisotropy of Heusler‐alloy Co3−xFexAl single crystalline thin films using inductive ferromagnetic resonance. A colossal Gilbert damping anisotropy ratio of up to 1400%, nearly three times higher than the highest value reported among known metallic ferromagnets, is observed in the optimized sample Co1.4Fe1.6Al. The Gilbert damping anisotropy with strong fourfold symmetry is attributed to the variation of the spin–orbit coupling, which is further corroborated by the current‐orientation‐dependent anisotropic magnetoresistance ratio in Co1.4Fe1.6Al thin film. These results provide a practical approach to adjust Gilbert damping continuously and show great promise for future spintronic applications.


Introduction
The energy dissipation rate of magnetization dynamics is quantified by a dimensionless Gilbert damping constant (α) in the Landau-Lifshitz-Gilbert (LLG) equation, [1] which plays a pivotal role in the spintronic devices. [2][3][4] Smaller α leads to a lower critical current density of magnetization switching, [5] while larger α results in a higher speed to operate magnetic memory devices. [6] www.advelectronicmat. de have been demonstrated in our previous work. [35] The anisotropic damping constants were measured using coplanar waveguide-based differential FMR. As shown in Figure 1a, the typical (derivative) FMR absorption spectra were obtained at a frequency of 14 GHz for all the samples with the magnetic field along [110] and [100] orientations and can be well-fitted by a sum of symmetric and antisymmetric Lorentzian derivatives [36] where H r and ΔH represent the resonance field and the linewidth, respectively. L and D are the symmetric and antisymmetric fitting parameters, respectively. Analyzing the frequency f dependent on the linewidth ΔH is a well-known approach to extracting the damping constant α [37] Figure 2b summarizes the anisotropic damping ratio [39] [(α max − α min )/α min × 100%] for all the samples and shows the comparison for the other ferromagnetic films. Surprisingly, a breakthrough has been made by the optimized Co 1.4 Fe 1.6 Al thin film with an anisotropic damping ratio of 1400%, which, to the best of our knowledge, is the highest value among ferromagnetic materials. Anisotropic damping generally arises from the extrinsic two-magnon scattering (TMS) [40][41] or intrinsic properties of the material. [31,32] Hence,  anisotropic damping ratio for ferromagnetic films at room temperature. "*" denotes effective damping, and the others are intrinsic damping. Data taken from the literature include Fe, [38] FeSi, [61] FeGa, [62] CoFeB, [39] CoFe, [54] FeCo, [34] Co 2 FeSi, [30] Co 2 MnSi, [31] and Co 2 FeAl. [32,40] www.advelectronicmat.de it is significant to further unveil the contribution mechanism of damping anisotropy in this series of Heusler alloys. We chose the optimized sample Co 1.4 Fe 1.6 Al to answer this question because of its colossal anisotropic damping ratio. Figure 3a shows the geometry of differential FMR measurement and defines the orientation of the external magnetic field ϕ H and magnetization ϕ M with respect to the MgO [100] orientation (or Co 1.4 Fe 1.6 Al [110]). The Co 3−x Fe x Al lattice structure rotates by 45° relative to the MgO substrate. From the previous work, [35] the easy axis is parallel to the 〈110〉 directions while the hard axis lies in 〈100〉 directions. Figure 3b illustrates the angular-dependent FMR spectra of the Co 1.4 Fe 1.6 Al film for ϕ H from 0° to 90° at a frequency of 18 GHz. The resonance signals can be well fitted by Equation (1). Both resonance fields H r and linewidths ΔH increase markedly from the easy axis to the hard axis. In addition, for easy axis [110] direction, it is notable that an additional resonance mode, i.e., perpendicular standing spin wave (PSSW) mode, [42][43][44] was excited at a lower resonance field, [42,45] which may be attributed to the surface partial spin pinning affected by anisotropy field. [45][46][47] Figure 3c,d shows the frequency and angular dependence of the resonance field H r . To quantify the magnetocrystalline anisotropy, H r is fitted using the Kittel equation [48,49] f HH a

Angular Dependence of Gilbert Damping in Co 1.4 Fe 1.6 Al Film
Here, M eff is effective saturation magnetization, γ (=gμ B /ℏ) is the gyromagnetic ratio, g is the Landé factor, μ B is the Bohr magneton, and ℏ is the reduced Planck constant. H 2 and H 4 are uniaxial and fourfold magnetic anisotropy fields, respectively. The angular dependence of the resonance field shows a clear fourfold symmetry (Figure 3d), indicative of a dominant in-plane cubic magnetocrystalline anisotropy [50] which originates from the face-center-cubic lattice texture of the Heusler-alloy Co 1.4 Fe 1.6 Al. By fitting, we obtain g = 2.09 (γ = 0.184 GHz mT −1 ), H 2 = 0.3 mT, H 4 = 11.45 mT, M eff = 1.34 T, where the parameters are close to the fitting of H r versus frequency f dispersion curves in Figure 3c. The H 2 value is about two orders of magnitude lower than H 4 , which reveals that uniaxial anisotropy, [23] on the other hand, is negligible. Such a weak uniaxial anisotropy field may be related to the symmetry breaking [51] at the Co 1.4 Fe 1.6 Al/MgO interface. Figure 3e shows the angular dependence of the linewidths ΔH taken at f = 18 GHz. Again, similar anisotropic behavior is observed, and the extrema of linewidths ΔH are commensurate with those of the resonance fields H r and the magnetocrystalline anisotropy energy. [35] Figure 4a shows the frequency dependent linewidth in the Co 1.4 Fe 1. 6 Al film, which can be fitted by [38] H Im where ΔH 0 is the inhomogeneous broadening, and Δ[Im(χ)] is the linewidth of the imaginary part of the dynamic magnetic susceptibility Im(χ) which is obtained by solving the LLG equation [1] Im

M H H H H H H H H H H H H H H
The experimental data can be well fitted using Equation (3) with the values of H 2 , H 4 , and M eff determined above.
Apart from the intrinsic Gilbert damping, the extrinsic effects, including TMS, inhomogeneous, and mosaicity contribution, [52] can also contribute to the FMR linewidth. Regarding the easy and hard axes, firstly, the inhomogeneous broadening ΔH 0 values are almost equal. Secondly, the mosaicity broadening can be excluded because it usually vanishes when the external field is along the easy or hard axis. [53] Lastly, the TMS contribution can also be ruled out. It is reported [25,41] that the frequency dependence of linewidth has a kink near 10 GHz if the TMS is present in the CoFe-based Heusler alloys. There is no kink in the entire frequency range, indicating that the extrinsic contribution induced by TMS is negligible. The absence of TMS may be caused by the high quality [38] of our single-crystalline films grown by MBE, as demonstrated in Figure S1 (Supporting Information). To sum up, the linewidth difference should be attributed to the intrinsic Gilbert damping contribution.
Generally, the magnetic dragging effect, [38,54,55] stemming from the magnetization deviation from the external field direction ϕ H , cannot be neglected at other axes except for the easy and hard axes, as shown in Figure S4 (Supporting Information). Hence, we considered this magnetic dragging effect in calculating intrinsic Gilbert damping constants. Figure 4b summarizes the calculated intrinsic Gilbert damping constants, which shows a significant fourfold symmetry, with the lowest Gilbert damping (α = 0.001) for the [110] direction and the largest (α = 0.0154) for [100] direction. The Gilbert damping variation yields an ultrahigh anisotropy ratio of 1400%, a much higher than ever reported [28,34,54] value of 420% in CoFe films. Such a colossal intrinsic damping anisotropy implies ≈15 times smaller critical current density to switch magnetization [54] simply by rotating the magnetization orientation from the hard axis [100] to the easy axis [110] direction. In addition, it is noteworthy that the ultralow damping constant is obtained as 0.001 for the easy axis [110] direction, which is comparable with the lowest value of 0.001 observed in Heusler alloy Co 2 FeAl [25] and 0.0013-0.002 in CoFe alloy systems. [56] Such ultralow damping may be determined by the half-metallicity of Co 1.4 Fe 1.6 Al film discussed later, for which the spin-minority channel has no contribution to the damping. [5] Furthermore, to gain insights into the origin of such colossal Gilbert damping anisotropy, a practical criterion of intrinsic damping is described by Kambersky's torque-correlation model [43] for transition metal ferromagnetic material, i.e., , where N(E F ) is the density of state (DOS) at Fermi level and ξ is the SOC parameter. In order to check the role of N(E F ), first-principles calculations were done for the ordered B2 phase supercell, as shown in Figure S6 (Supporting Information). However, the anisotropy of N(E F ) (less than 0.1%) is negligible for different magnetization orientations, and even this slight variation trend of N(E F ) is opposite to that of anisotropic intrinsic damping. Our result differs substantially from the study on Gilbert damping anisotropy reported in ultrathin Fe films, which is manifested by the uniaxial anisotropy of N(E F ) at the Fe/GaAs interface [38] and is the interfacial property.
Therefore, we infer that the colossal Gilbert damping anisotropy of 1400% should be attributed to the variation of the SOC at different crystalline orientations. Depending on the magnetization orientation, the twofold AMR is derived from the asymmetric s-d electron scatterings, [57] where the conduction electrons on s orbitals are scattered into localized d orbitals by nonmagnetic impurities due to SOC. [58] Hence, the anisotropy of the SOC can be reflected by the AMR ratio with the current along different crystalline directions. [54] For AMR measurement, the Co 1.4 Fe 1.6 Al film was fabricated into arcuate Hall bars by standard photolithography and ion milling so that the current orientation varies continuously, as schematized in Figure 5a.
The longitudinal magnetoresistances of the different current directions were obtained via rotating the in-plane field orientation, as shown in Figure 5b. All the data show remarkable twofold symmetry and can be fitted by cos 2 (ϕ H − θ I ) in Co 1.4 Fe 1. 6 Al, similar to the traditional polycrystalline system. [59] The extrema of magnetoresistance appear at the same (ϕ H − θ I ) directions for all the Hall bars, indicating strong dependence on the www.advelectronicmat.de current orientation. In addition, recent reports [34,57] found fourfold symmetry in AMR measurement with the current along the specific direction, which is attributed to the relaxation time anisotropy. However, we did not observe the fourfold symmetry AMR behavior in Co 1.4 Fe 1.6 Al film, suggesting that the Gilbert damping anisotropy is unlikely associated with the relaxation time anisotropy.
The AMR ratio can be calculated by [59] R R AMR ratio 1 100% where R ∥ (θ I ) and R ⊥ (θ I ) are the longitudinal resistances for the external field parallel and perpendicular to the current orientation, respectively. Figure 5c quantitatively compares the renormalized α(ϕ H )/α max with AMR(θ I ) ratios for this sample, where the AMR ratios were calculated using Equation (5) with different θ I . The negative sign of the AMR ratio indicates the half-metallicity [58,60] of Co 1.4 Fe 1.6 Al film. The intrinsic Gilbert damping would be reduced by enhancing the negative AMR effect and vice versa. [60] The AMR ratio is minimal (maximal) for the current along the easy (hard) axis, with a substantial anisotropy by a factor of 9. It is clear that the AMR ratio anisotropy and its fourfold symmetry with the current orientation highly coincide with the damping anisotropy, concluding that the colossal Gilbert damping anisotropy stems mainly from strong SOC anisotropy in the Co 1.4 Fe 1.6 Al sample.

Discussion
Using first-principles calculation, we found that the anisotropy of N(E F ) is negligible for all the samples, as shown in Figure S7 (Supporting Information). According to Kambersky's torquecorrelation model, we speculate that the iron composition dependence of Gilbert damping anisotropy should be attributed to the SOC anisotropy.
Based on the prior studies, we discuss the microscopic mechanism of SOC variation. First, recent work [54] demonstrated that local tetragonal distortions cause the change of the SOC in CoFe thin films. However, the postannealing treatment in our samples has relaxed all the strains in our films and minimized the local distortions. Second, Chen et al. [38] found that the Gilbert damping anisotropy is due to the interfacial SOC in Fe/GaAs, which vanishes as the film gets thicker. In comparison, our samples are much thicker than theirs, showing that Gilbert damping anisotropy is a bulk property. Third, the global next-nearest-neighbor Co-Fe interaction is likely to be responsible for the SOC anisotropy, [28] which can be influenced by the Co(4c)-Fe(4d) disorder. This disorder is higher in the intermediate composition of this series, [35] which can understand the composition-dependent Gilbert damping anisotropy. However, further theoretical analysis is greatly desirable.

Conclusion
In summary, the Gilbert damping constants of the easy and hard axes have been studied in single-crystalline Co 3−x Fe x Al thin films grown on the MgO (001) substrates. The anisotropic damping ratio first rises then descends with the increasing x, and the sample Co 1.4 Fe 1.6 Al shows a record value of up to 1400%, which is almost three times higher than the highest value of 420% ever reported in CoFe alloy. [34] The intrinsic Gilbert damping exhibits a strong fourfold symmetry, and the minimal (maximal) damping value is observed along the easy (hard) axis. Within Kambersky's torque-correlation model, we www.advelectronicmat.de demonstrate that such colossal Gilbert damping anisotropy is attributed to the variation of spin-orbit coupling, which is substantiated by comparing the Gilbert damping with the currentorientation-dependent AMR ratio. The finding opens an avenue for manipulating the Gilbert damping of the Heusler alloy within the same devices, which helps design advanced spintronic applications on different demands.

Experimental Section
Thin Films Growth: The high-quality 20 nm thick Co 3−x Fe x Al films with x = 1-2 were epitaxially grown on MgO (001) substrates in an MBE system with a base pressure blow 5 × 10 −10 mbar. The composition ratio was determined by the atomic deposition rates measured by a quartz microbalance. After postannealing at 723 K for half an hour, a 2 nm thick Al film was deposited as a capping layer to avoid oxidation. The microstructural characterization, including surface morphology and the crystal structure, has been studied in the previous work. [35] FMR Measurements: The dynamic properties are investigated by a homebuilt FMR setup with a coplanar waveguide (CPW) for microwave field excitation. The sample is placed on the CPW with the film surface down. The microwave field H rf was supplied through the central signal line (S) connecting with the signal generator, and the FMR absorption signal was detected using the lock-in amplifier. All the measurements were performed by scanning the magnetic field while fixing the microwave frequency. The external field was applied parallel to the sample plane. The dependence of FMR spectra on the magnetic field orientation was performed by rotating H or the sample in the film plane. The resonance frequency can be adjusted in the range of 1-20 GHz.
AMR Measurements: The Co 1.4 Fe 1.6 Al film was patterned into many arcuate Hall bars so that the current orientation can vary continuously, as shown in Figure 4a. The Hall bars have an identical size of 100 µm × 200 µm. The AMR measurements were performed in an Oxford instruments cryogenic system (TeslatronPT, Oxford) with a magnetic field of up to 2T at room temperature. Since a high magnetic field makes the magnetic field saturated, the external field angle ϕ H and magnetization orientation ϕ M are equal.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.