Open Submitted: 10 March 2021 Accepted: 18 April 2021 Published Online: 03 May 2021
APL Materials 9, 051104 (2021); https://doi.org/10.1063/5.0049891
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• Peter Swekis
• Anastasios Markou
• Jörg Sichelschmidt
• Claudia Felser
• Sebastian T. B. Goennenwein

The magnetic anisotropy determines the equilibrium orientation of the magnetization in a ferromagnet. In Mn-based inverse tetragonal Heusler compounds, a large uniaxial anisotropy makes these materials excellent candidates for both spin-transfer-torque and skyrmionic devices. Here, we present systematic investigations of the magnetocrystalline anisotropies in MnxPtSn films (x = 1.0–1.6). We find that the MnxPtSn films, grown by magnetron sputtering on MgO substrates, show a structural transition between x = 1.0 and 1.2 from cubic to tetragonal, where the tetragonal structure shows a twinned in-plane c-axis orientation. From ferromagnetic resonance measurements, we determine the out-of-plane and in-plane uniaxial anisotropies, taking into account the particular structural properties of the films. We find a strong dependence of the uniaxial anisotropies on the Mn concentration, as well as on structural distortions due to the lattice-matched growth. From temperature-dependent ferromagnetic resonance measurements, we infer the evolution of the in-plane uniaxial anisotropy and observe the presence of additional magnetic interactions and magnetization relaxation mechanisms around the spin reorientation transition.
The preferred orientation of magnetic moments, may it be in collinear magnets or materials hosting unique types of noncoplanar spin textures, is connected to the crystal symmetry via the magnetocrystalline anisotropy (MCA). Controlling the MCA is, therefore, an important goal in developing spintronic materials, e.g., for the design of spin-transfer-torque1,21. S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, “Current-induced magnetization reversal in nanopillars with perpendicular anisotropy,” Nat. Mater. 5, 210 (2006). https://doi.org/10.1038/nmat15952. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.201201879 or skyrmion-based devices.3–53. A. Fert, V. Cros, and J. Sampaio, “Skyrmions on the track,” Nat. Nanotechnol. 8, 152 (2013). https://doi.org/10.1038/nnano.2013.294. S. Parkin and S.-H. Yang, “Memory on the racetrack,” Nat. Nanotechnol. 10, 195 (2015). https://doi.org/10.1038/nnano.2015.415. C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert, M. Garst, T. Ma, S. Mankovsky, T. L. Monchesky, M. Mostovoy, N. Nagaosa, S. S. P. Parkin, C. Pfleiderer, N. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von Bergmann, and J. Zang, “The 2020 skyrmionics roadmap,” J. Phys. D: Appl. Phys. 53, 363001 (2020). https://doi.org/10.1088/1361-6463/ab8418 Materials that allow flexible tuning of the magnetic properties, especially the MCA, are highly desirable in this process. Particularly, the inverse tetragonal MnxYZ Heusler compounds (Y—transition metal and Z—main-group element) stand out as promising materials in this endeavor. Initially, they attracted much interest for the design of spin-transfer-torque applications, achieving compensated ferrimagnetism, high spin polarization, and a high perpendicular magnetic anisotropy.2,6–102. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.2012018796. A. K. Nayak, M. Nicklas, S. Chadov, P. Khuntia, C. Shekhar, A. Kalache, M. Baenitz, Y. Skourski, V. K. Guduru, A. Puri, U. Zeitler, J. M. D. Coey, and C. Felser, “Design of compensated ferrimagnetic Heusler alloys for giant tunable exchange bias,” Nat. Mater. 14, 679 (2015). https://doi.org/10.1038/nmat42487. L. Wollmann, S. Chadov, J. Kübler, and C. Felser, “Magnetism in tetragonal manganese-rich Heusler compounds,” Phys. Rev. B 92, 064417 (2015). https://doi.org/10.1103/physrevb.92.0644178. R. Sahoo, L. Wollmann, S. Selle, T. Höche, B. Ernst, A. Kalache, C. Shekhar, N. Kumar, S. Chadov, C. Felser, S. S. P. Parkin, and A. K. Nayak, “Compensated ferrimagnetic tetragonal Heusler thin films for antiferromagnetic spintronics,” Adv. Mater. 28, 8499 (2016). https://doi.org/10.1002/adma.2016029639. A. Kalache, A. Markou, S. Selle, T. Höche, R. Sahoo, G. H. Fecher, and C. Felser, “Heteroepitaxial growth of tetragonal Mn2.7−xFexGa1.3 (0 ≤ x ≤ 1.2) Heusler films with perpendicular magnetic anisotropy,” APL Mater. 5, 096102 (2017). https://doi.org/10.1063/1.499146810. A. Kalache, S. Selle, W. Schnelle, G. H. Fecher, T. Höche, C. Felser, and A. Markou, “Tunable magnetic properties in tetragonal Mn-Fe-Ga Heusler films with perpendicular anisotropy for spintronics applications,” Phys. Rev. Mater. 2, 084407 (2018). https://doi.org/10.1103/physrevmaterials.2.084407 More recently, they have also shown to host vortex-like spin textures, the skyrmions, in a broad temperature and magnetic field range11–1411. A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U. K. Rößler, C. Felser, and S. S. P. Parkin, “Magnetic antiskyrmions above room temperature in tetragonal Heusler materials,” Nature 548, 561 (2017). https://doi.org/10.1038/nature2346612. J. Jena, R. Stinshoff, R. Saha, A. K. Srivastava, T. Ma, H. Deniz, P. Werner, C. Felser, and S. S. P. Parkin, “Observation of magnetic antiskyrmions in the low magnetization ferrimagnet Mn2Rh0.95Ir0.05Sn,” Nano Lett. 20, 59 (2019). https://doi.org/10.1021/acs.nanolett.9b0297313. J. Jena, B. Göbel, T. Ma, V. Kumar, R. Saha, I. Mertig, C. Felser, and S. S. P. Parkin, “Elliptical Bloch skyrmion chiral twins in an antiskyrmion system,” Nat. Commun. 11, 1115 (2020). https://doi.org/10.1038/s41467-020-14925-614. L. Peng, R. Takagi, W. Koshibae, K. Shibata, K. Nakajima, T.-h. Arima, N. Nagaosa, S. Seki, X. Yu, and Y. Tokura, “Controlled transformation of skyrmions and antiskyrmions in a non-centrosymmetric magnet,” Nat. Nanotechnol. 15, 181 (2020). https://doi.org/10.1038/s41565-019-0616-6 connected to a reorientation of the magnetic moments of the Mn sublattices.1515. O. Meshcheriakova, S. Chadov, A. K. Nayak, U. K. Rößler, J. Kübler, G. André, A. A. Tsirlin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle, M. Nicklas, and C. Felser, “Large noncollinearity and spin reorientation in the novel Mn2RhSn Heusler magnet,” Phys. Rev. Lett. 113, 087203 (2014). https://doi.org/10.1103/physrevlett.113.087203 These spin textures originate from a competition of the magnetic interactions that, in turn, govern their type and stability. Effectively, all those properties can be attributed to the tetragonal multi-sublattice structure, where the non-centrosymmetric crystal symmetry (D2d) gives rise to an anisotropic Dzyaloshinskii–Moriya interaction16–1816. I. Dzyaloshinsky, “A thermodynamic theory of weak' ferromagnetism of antiferromagnetics,” J. Phys. Chem. Solids 4, 241 (1958). https://doi.org/10.1016/0022-3697(58)90076-317. T. Moriya, “Superexchange interaction and weak ferromagnetism,” Phys. Rev. 120, 91 (1960). https://doi.org/10.1103/physrev.120.9118. A. N. Bogdanov, U. K. Rößler, M. Wolf, and K. Müller, “Magnetic structures and reorientation transitions in noncentrosymmetric uniaxial antiferromagnets,” Phys. Rev. B 66, 214410 (2002). https://doi.org/10.1103/physrevb.66.214410 that is required to stabilize skyrmions.
Despite the growing interest in the tetragonal Heusler compounds, only minor efforts were taken toward experimentally quantifying the anisotropies of the skyrmion hosting MnxYSn (Y = Rh, Pd, Pt) systems,19–2119. K. G. Rana, O. Meshcheriakova, J. Kübler, B. Ernst, J. Karel, R. Hillebrand, E. Pippel, P. Werner, A. K. Nayak, C. Felser, and S. S. P. Parkin, “Observation of topological Hall effect in Mn2RhSn films,” New J. Phys. 18, 085007 (2016). https://doi.org/10.1088/1367-2630/18/8/08500720. Y. Huh, P. Kharel, A. Nelson, V. R. Shah, J. Pereiro, P. Manchanda, A. Kashyap, R. Skomski, and D. J. Sellmyer, “Effect of Co substitution on the magnetic and electron-transport properties of Mn2PtSn,” J. Phys.: Condens. Matter 27, 076002 (2015). https://doi.org/10.1088/0953-8984/27/7/07600221. Y. Jin, S. Valloppilly, P. Kharel, J. Waybright, P. Lukashev, X. Z. Li, and D. J. Sellmyer, “Structural, magnetic, and electron-transport properties of epitaxial Mn2PtSn films,” J. Appl. Phys. 124, 103903 (2018). https://doi.org/10.1063/1.5045667 following the prediction of a high MCA in Mn2PtSn (7.88 MJ/m3).22. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.201201879 Nevertheless, a systematic characterization of the magnetic anisotropies as a function of, e.g., composition and lattice strain is essential to address the usability of those compounds for spintronic applications as well as to gain a complete understanding of all interactions that govern the type and stability of their noncoplanar spin textures. In that context, the evolution of the MCA from the cubic to the tetragonal Heusler phase holds important information. Similarly, the magnetization dynamics, often distinct for different types of magnetic textures,2222. Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y. Tokura, “Observation of magnetic excitations of skyrmion crystal in a helimagnetic insulator Cu2OSeO3,” Phys. Rev. Lett. 109, 037603 (2012). https://doi.org/10.1103/physrevlett.109.037603 are yet unknown for these compounds. A better understanding of the spin reorientation between the collinear and noncoplanar spin textures could, therefore, be achieved by investigating the magnetic resonance conditions and magnetization relaxation around this transition.
In this work, we employ magnetic field orientation-dependent ferromagnetic resonance (FMR) experiments at a constant microwave frequency to determine the evolution of the magnetic anisotropies in MnxPtSn thin films (x = 1.0–1.6), transitioning from the cubic to the tetragonal structure. Furthermore, we investigate the temperature dependence of the in-plane (IP) uniaxial anisotropy as well as the out-of-plane (OOP) resonance fields and linewidths in a Mn1.6PtSn film around the spin reorientation transition.
Epitaxial thin films of MnxPtSn were grown in a BESTEC UHV magnetron sputtering system on single-crystal MgO(001) substrates and capped with 3 nm Si to prevent oxidation. The details of the growth are provided in Refs. 2323. P. Swekis, A. Markou, D. Kriegner, J. Gayles, R. Schlitz, W. Schnelle, S. T. B. Gönnenwein, and C. Felser, “Topological Hall effect in thin films of Mn1.5PtSn,” Phys. Rev. Mater. 3, 013001 (2019). https://doi.org/10.1103/physrevmaterials.3.013001 and 2424. P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, “Role of magnetic exchange interactions in chiral-type Hall effects of epitaxial MnxPtSn films,” ACS Appl. Electron. Mater. 3, 1323 (2021). https://doi.org/10.1021/acsaelm.0c01104. The stoichiometries of the films were confirmed by energy-dispersive x-ray spectroscopy (EDXS), with an experimental uncertainty of less than 5 at. %. X-ray diffraction (XRD) and x-ray reflectivity (XRR) measurements were conducted using a PANalytical X’Pert3 MRD diffractometer employing Cu-Kα1 radiation (λ = 1.5406 Å).
Magnetization measurements were performed on a superconducting quantum interference device (SQUID) vibrating sample magnetometer (MPMS 3, Quantum Design) to infer the magnetic field-dependent magnetization of the films. For this, the diamagnetic substrate contribution was subtracted from the raw data.
FMR measurements as a function of the external magnetic field orientation were performed on a continuous-wave Elexsys E500 spectrometer by Bruker. The measurements were conducted at 9.4 GHz (X-band) and 34.0 GHz (Q-band) in cylindrical cavities operating in the TE011 and TE012 mode, respectively. The resonance signal was recorded in the field-derivative $dPdH$ of the absorbed microwave power (P) using a lock-in technique that modulated the external magnetic field (H) at a frequency of 100 kHz. The sample orientation was manipulated with a goniometer, rotating perpendicular to H. The line shapes of the obtained spectra were fitted with at least one first-derivative Lorentzian to obtain the resonance field Hres and resonance linewidth ΔH. The temperature at the sample was controlled by a continuous flow of either nitrogen (X-band) or helium gas (Q-band) and a heating element.
A. Structural characterization
In our previous work, we described the unique crystallographic orientation of MnxPtSn films,2323. P. Swekis, A. Markou, D. Kriegner, J. Gayles, R. Schlitz, W. Schnelle, S. T. B. Gönnenwein, and C. Felser, “Topological Hall effect in thin films of Mn1.5PtSn,” Phys. Rev. Mater. 3, 013001 (2019). https://doi.org/10.1103/physrevmaterials.3.013001 visualized in Fig. 1(a). The films grow with two distinct orientations of the c-axis [001] in the film plane, i.e., aligning either along the [110] or the [1$1̄$0] directions of the MgO substrate as MnxPtSn(100)[001]∥MgO(001)[110] and MnxPtSn(100)[001]∥MgO(001)[1$1̄$0]. Furthermore, we determined that a lattice mismatch (2.5% and 6.5%) with the substrate results in an in-plane compressive strain, breaking the equivalence between the tetragonal a and b lattice constants. This effectively results in an orthorhombically distorted structure (I212121)2323. P. Swekis, A. Markou, D. Kriegner, J. Gayles, R. Schlitz, W. Schnelle, S. T. B. Gönnenwein, and C. Felser, “Topological Hall effect in thin films of Mn1.5PtSn,” Phys. Rev. Mater. 3, 013001 (2019). https://doi.org/10.1103/physrevmaterials.3.013001 as compared to the tetragonal bulk superstructure ($I4̄2d$).2525. P. Vir, N. Kumar, H. Borrmann, B. Jamijansuren, G. Kreiner, C. Shekhar, and C. Felser, “Tetragonal superstructure of the antiskyrmion hosting Heusler compounds Mn1.4PtSn,” Chem. Mater. 31, 5876 (2019). https://doi.org/10.1021/acs.chemmater.9b02013 For simplicity, we shall still refer to the films as tetragonal in the following discussion.
Figure 1(b) shows the symmetric radial (ω − 2θ) XRD scans of the 38 nm thick MnxPtSn films (x = 1.0–1.6). For MnPtSn, the presence of the (h00) Bragg reflections, indexed based on the bulk structure ($F4̄3m$),2626. H. Masumoto and K. Watanabe, “An intermetallic fluorite-type compound PtMnSn in the Pt-Mn-Sn system and its magnetic properties,” Trans. Jpn. Inst. Met. 14, 408 (1973). https://doi.org/10.2320/matertrans1960.14.408 confirms the growth of the cubic half-Heusler phase. Increasing the Mn concentration leads to a splitting of the cubic 400 reflection around x = 1.2. By considering the tetragonal bulk structure of Mn1.5PtSn,2525. P. Vir, N. Kumar, H. Borrmann, B. Jamijansuren, G. Kreiner, C. Shekhar, and C. Felser, “Tetragonal superstructure of the antiskyrmion hosting Heusler compounds Mn1.4PtSn,” Chem. Mater. 31, 5876 (2019). https://doi.org/10.1021/acs.chemmater.9b02013 the split peaks can be indexed as the tetragonal 400 and 008 reflections. Therefore, we identify the splitting as a structural transition of the films from the cubic half-Heusler phase (x = 1.0) into the inverse tetragonal phase (x ≥ 1.2), where the formation and ordering of vacancies, up to x = 1.6, leads to the preferred stabilization of a tetragonal superstructure as opposed to the typical inverse tetragonal structure.2525. P. Vir, N. Kumar, H. Borrmann, B. Jamijansuren, G. Kreiner, C. Shekhar, and C. Felser, “Tetragonal superstructure of the antiskyrmion hosting Heusler compounds Mn1.4PtSn,” Chem. Mater. 31, 5876 (2019). https://doi.org/10.1021/acs.chemmater.9b02013 Since for all films with x ≤ 1.4, additional unidentified reflections are observed (40°–41°), phase pure epitaxial growth can only be achieved in Mn1.6PtSn.2424. P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, “Role of magnetic exchange interactions in chiral-type Hall effects of epitaxial MnxPtSn films,” ACS Appl. Electron. Mater. 3, 1323 (2021). https://doi.org/10.1021/acsaelm.0c01104 The corresponding out-of-plane a and in-plane c lattice constants (indexing the 512 and 536 Bragg reflections; not shown) are summarized in Table I, increasing and decreasing with x, respectively.
TABLE I. Structural and magnetic parameters as well as FMR parameters, determined from the simulation of the H orientation-dependent Hres, of MnxPtSn films.
Mn concentration (x)a (Å)c (Å)TC (K)Ms (kA/m)gKu,[001] (kJ/m3)Ku,[100] (kJ/m3)Ku,eff (kJ/m3)
1.66.37712.183743802.0598.9−91.3182
1.46.31712.323985042.1043.0−22.7182
1.26.27812.443894652.1517.7−9.3145
1.06.2573412651.98011.333
B. Magnetization measurements
Figure 2 shows the magnetization hysteresis loops of the MnxPtSn films at 300 K, recorded with the magnetic field applied in-plane as well as out-of-plane. The OOP loops are reminiscent of hard-axis behavior with small coercive fields, whereas the IP loops indicate that the magnetic easy-axis lies within the film plane. This is in agreement with the orientation of the tetragonal c-axis in the film plane, known to induce an easy axis of the magnetization M along its direction.2,7,102. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.2012018797. L. Wollmann, S. Chadov, J. Kübler, and C. Felser, “Magnetism in tetragonal manganese-rich Heusler compounds,” Phys. Rev. B 92, 064417 (2015). https://doi.org/10.1103/physrevb.92.06441710. A. Kalache, S. Selle, W. Schnelle, G. H. Fecher, T. Höche, C. Felser, and A. Markou, “Tunable magnetic properties in tetragonal Mn-Fe-Ga Heusler films with perpendicular anisotropy for spintronics applications,” Phys. Rev. Mater. 2, 084407 (2018). https://doi.org/10.1103/physrevmaterials.2.084407 Table I summarizes the saturation magnetizations (Ms) obtained from the magnetometry experiments, which increase up to x = 1.4 before decreasing toward x = 1.6. The same evolution is observed in the Curie temperatures (TC) determined from field cooled (0.1 T) magnetization measurements (not shown). As described in Ref. 2424. P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, “Role of magnetic exchange interactions in chiral-type Hall effects of epitaxial MnxPtSn films,” ACS Appl. Electron. Mater. 3, 1323 (2021). https://doi.org/10.1021/acsaelm.0c01104, this behavior is related to the sublattice occupation of the Mn atoms, with their magnetic moments aligning antiferromagnetically to the net moment above x = 1.5, thereby decreasing the total moment. Such a tuning of Ms and TC was also reported in similar Mn-based tetragonal Heusler compounds,2,272. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.20120187927. V. Kumar, N. Kumar, M. Reehuis, J. Gayles, A. S. Sukhanov, A. Hoser, F. Damay, C. Shekhar, P. Adler, and C. Felser, “Detection of antiskyrmions by topological Hall effect in Heusler compounds,” Phys. Rev. B 101, 014424 (2020). https://doi.org/10.1103/physrevb.101.014424 accessing the responsible magnetic interaction through chemical substitution.
C. Magnetic anisotropies
The magnetic anisotropies of the MnxPtSn films can be inferred from the dependence of the magnetic resonance field on the direction of the applied magnetic field. These orientation dependencies can be interpreted by considering the crystallographic directions of the films with respect to the measurement coordinate system shown in Fig. 3(a). This coordinate system allows us to define the orientation of H and M with respect to the film surface and crystal structure through two polar angles. We use lower case θ and ϕ for H and upper case Θ and Φ for M. Figure 3(b) shows the orientation of the cubic MnPtSn film in the coordinate system, growing 45° rotated relative to the MgO substrate such that MnPtSn(100)[001]∥MgO(001)[110]. Figure 3(c) shows the orientation of the tetragonal MnxPtSn films (x = 1.2–1.6) in the coordinate system, growing with two equivalent c-axes in the film plane [Fig. 1(a)] such that the c-axes point along θ = 45° with ϕ = 90° as well as θ = 135° with ϕ = 90° [black arrows in Fig. 3(c)], respectively. In the following, the given crystallographic directions always refer to the MnxPtSn structure.
Based on the crystallographic orientations in the described coordinate systems [Figs. 3(a)3(c)], the total free energy density (Ftot), required to determine the anisotropies of the MnxPtSn films, can be defined. Here, Ftot includes the Zeeman energy (Fz), the shape anisotropy (Fd), a cubic anisotropy term (Fc; only MnPtSn), and two different uniaxial anisotropy terms, one perpendicular to the film plane (Fu,[100]) as well as one along each of the two crystallographic c-axes ($Fu,[001]±$),
 $Ftot=Fz+Fd+Fu,[100]+Fu,[001]++Fu,[001]−+Fc=−μ0MsH(sinΘ⁡sinΦ⁡sin⁡θ⁡sin⁡ϕ+cosΘ⁡cos⁡θ+sinΘ⁡cosΦ⁡sin⁡θ⁡cos⁡ϕ)+μ02Ms2sinΘ⁡cosΦ2−Ku,[100]sinΘ⁡cosΦ2+Ku,[001]12sinΘ⁡sinΦ+cosΘ2+Ku,[001]12sinΘ⁡sinΦ−cosΘ2+Kc14sin2(2Θ)+sin4Θ⁡sin2(2Θ).$ (1)
The shape anisotropy and Ku,[100] are then summarized in the effective perpendicular uniaxial anisotropy $Ku,eff=12μ0Ms2−Ku,[100]$. Note that since the tetragonal MnxPtSn films (x = 1.2–1.6) feature crystallites with two independent crystalline orientations [Fig. 3(c)], we simulated two independent Hres orientation dependencies. For that purpose, two different Fu,[001] were defined, where MnxPtSn(100)[001]∥MgO(001)[110] was accounted for by $Fu,[001]+$, while MnxPtSn(100)[001]∥MgO(001)[1$1̄$0] was accounted for by $Fu,[001]−$. The two independent simulations were, in turn, performed by setting either $Fu,[001]+$ or $Fu,[001]−$ to zero. For the simulation of the cubic film, both $Fu,[001]+$ and $Fu,[001]−$ were set to zero.
The orientation dependencies of Hres on H of the MnxPtSn films for a rotation from OOP to IP [(010) or (001) plane; purple line in Fig. 3(c)] and a rotation in the film plane [(100) plane; yellow line Fig. 3(b)] are shown in Figs. 3(d)3(g) and 3(h)3(k), respectively. Qualitatively, the OOP to IP orientation dependencies of Hres for all films agree well with the magnetization measurements (cf. Fig. 2). The magnetic hard axis perpendicular to the film plane, inferred from the magnetization data, shows up as a maximum in Hres for the respective field orientation in the FMR data. Moreover, Hres along the OOP direction ([100], ϕ = 0°) increases with increasing Mn concentration, and a second Hres appears for the Mn1.2PtSn film. Notably, a second Hres is not present for x = 1.4 and 1.6. The IP orientation dependencies of Hres show a distinct evolution from the cubic film to the tetragonal films. The MnPtSn film shows a minor fourfold angular dependence of Hres related to the cubic structure, whereas the tetragonal films show two twofold angular dependencies, increasing with increasing Mn concentration. These observations can be interpreted in view of the crystallographic orientation of the films, shown in Figs. 3(b) and 3(c). The IP growth directions of the two tetragonal c-axes are reflected in the two twofold orientation dependencies of Hres [Figs. 3(i)3(k)], each accordingly shifted by 90° relative to each other. As mentioned before, the easy-axes lie along the respective crystallographic c-axes, θ = 45° and θ = 135°. Notably, in order to fulfill the resonance condition along the c-axes, the IP rotation of the Mn1.6PtSn and Mn1.4PtSn films had to be measured at Q-band frequency, whereas all the other measurements were performed at X-band frequency. This shift of the resonance condition stems from the increasingly significant contribution of the accompanying in-plane uniaxial anisotropy along the c-axes [Eq. (1)]. For the same reason, only the Hres related to the rotation in the (001) plane are observed in the OOP to IP rotations in Figs. 3(f) and 3(g), while Fig. 3(e) also includes the Hres of the rotation in the (010) plane. In contrast, the single Hres orientation dependencies of the cubic MnPtSn film [Figs. 3(d) and 3(h)] only stem from the effective uniaxial anisotropy perpendicular to the film plane.
The resonance conditions for the MnxPtSn films [Figs. 3(d)3(k)] can be derived from Ftot [Eq. (1)] for arbitrary orientations of the external magnetic field with respect to the sample as2828. S. V. Vonsovskii, Ferromagnetic Resonance, 1st ed. (Pergamon, 1966).
 $ωγ2=1Ms2⁡sinΘ∂2Ftot∂Φ2∂2Ftot∂Θ2−∂2Ftot∂Θ∂Φ2,$ (2)
with the derivatives evaluated for the equilibrium direction of the magnetization. In addition, for the simulation of the resonance condition in the presented coordinate system, ϕ was fixed at 90° for the IP rotation, whereas the OOP to IP rotations are accounted for by setting θ = 90° + 45°(sin ϕ).
From the simulations, using Eqs. (1) and (2), the anisotropy constants and g-values (entering via $γ=gμBℏ$) of the MnxPtSn films were derived, using Ms from Table I. The simulations, included in Fig. 3(d)3(k) as solid lines, well reproduce the orientation dependencies of Hres, and hence, the obtained anisotropy constants are summarized in Table I, except for Kc = 0.3 kJ/m3 (only MnPtSn). Note that Figs. 3(f) and 3(g) only include one simulation, namely, for the rotation in the (001) plane ($Fu,[001]+=0$), whereas Fig. 3(e) also includes the simulation of the (010) plane ($Fu,[001]−=0$). As introduced before, this absence is related to the resonance condition, not fulfilled along the [001] direction at X-band frequency for the Mn1.4PtSn and Mn1.6PtSn films. We verified this noncompliance by solving an analytical special case of the resonance condition [Eqs. (1) and (2)] along the [001] direction,
 $ωγ2=μ0Hres,[001]+2Ku,[001]Ms×μ0Hres,[001]+2Ku,effMs,$ (3)
employing the anisotropies and g determined from the in-plane and (001) plane simulations.
1. Stoichiometry dependence
Figure 4 shows the dependence of Ku,[100] and Ku,[001] on the Mn concentration x. Ku,[100] (Ku,[001]) gradually decreases (increases) with increasing x, going from the cubic to the tetragonal structure. This evolution underlines the structural transition followed by the gradual occupation of the Mn sub-lattice, directly affecting the size of the MCA via spin–orbit coupling.22. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.201201879 Here, the sign of Ku,[001] is in agreement with the c-axis orientation IP, while the transition from the cubic to the tetragonal structure is reflected in a sign change of Ku,[100]. Notably, the negative Ku,[100] for the tetragonal films cannot be accounted for by the lattice distortion from a mismatch between the MnxPtSn films and the MgO substrate alone. This distortion, resulting in an IP compression and OOP elongation of the lattice constants,2424. P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, “Role of magnetic exchange interactions in chiral-type Hall effects of epitaxial MnxPtSn films,” ACS Appl. Electron. Mater. 3, 1323 (2021). https://doi.org/10.1021/acsaelm.0c01104 is expected to give positive values as for the cubic film. Therefore, additional uniaxial terms with the same angular dependence play a role after the transition, where the similar trend and size of Ku,[100] and Ku,[001] imply that the additional term might stem from the MCA. This is further supported by the absolute values of Ku,[100] and Ku,[001], differing by 8–20 kJ/m3. In the context of potential device-based applications, requiring a high MCA, the Mn1.6PtSn film stands out as the most suitable candidate. Simultaneously, for controlling and stabilizing noncoplanar spin textures, the MCA trend highlights the anisotropies as tunable parameters, accessible through lattice-matched thin film growth as well as through the Mn sub-lattice occupation.
Prior to this work, the Ku,[001] of Mn2PtSn ($I4̄m2$) as bulk crystals, thin films, and from theoretical calculations were reported as 110 kJ/m3 (300 K), 550 kJ/m3 (300 K), and 7.88 MJ/m3, respectively.2,20,212. J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi, K. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. Kübler, G.-D. Liu, L. Gao, S. S. P. Parkin, and C. Felser, “Design scheme of new tetragonal Heusler compounds for spin-transfer torque applications and its experimental realization,” Adv. Mater. 24, 6283 (2012). https://doi.org/10.1002/adma.20120187920. Y. Huh, P. Kharel, A. Nelson, V. R. Shah, J. Pereiro, P. Manchanda, A. Kashyap, R. Skomski, and D. J. Sellmyer, “Effect of Co substitution on the magnetic and electron-transport properties of Mn2PtSn,” J. Phys.: Condens. Matter 27, 076002 (2015). https://doi.org/10.1088/0953-8984/27/7/07600221. Y. Jin, S. Valloppilly, P. Kharel, J. Waybright, P. Lukashev, X. Z. Li, and D. J. Sellmyer, “Structural, magnetic, and electron-transport properties of epitaxial Mn2PtSn films,” J. Appl. Phys. 124, 103903 (2018). https://doi.org/10.1063/1.5045667 The experimental values were determined from magnetization hysteresis loops. While the bulk values well agree with the presented results of our Mn1.6PtSn films, the Mn2PtSn film values are significantly higher. Considering the trend in Ku,[001] presented in Fig. 4(b), an increase from x = 1.6 to x = 2.0 appears reasonable. As reported in Ref. 2424. P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, “Role of magnetic exchange interactions in chiral-type Hall effects of epitaxial MnxPtSn films,” ACS Appl. Electron. Mater. 3, 1323 (2021). https://doi.org/10.1021/acsaelm.0c01104, however, neither the lattice constants nor the structure change in that regime. This implies a qualitative difference between the presented Mn1.6PtSn film and the Mn2PtSn film. Similarly, the significantly higher theoretical values indicate that the calculations based on Mn2PtSn ($I4̄m2$) are qualitatively different and are not an accurate representation of our films based on Mn1.5PtSn in the tetragonal superstructure ($I4̄2d$).2525. P. Vir, N. Kumar, H. Borrmann, B. Jamijansuren, G. Kreiner, C. Shekhar, and C. Felser, “Tetragonal superstructure of the antiskyrmion hosting Heusler compounds Mn1.4PtSn,” Chem. Mater. 31, 5876 (2019). https://doi.org/10.1021/acs.chemmater.9b02013 Notably, Mn2RhSn films ($I4̄m2$), with the c-axis growing along the OOP direction, have a perpendicular magnetic anisotropy with a significantly smaller Ku,[001] of 51.5 kJ/m3 at 200 K (measurements not shown).
2. Temperature dependence
Figures 5(a) and 5(b) show the temperature dependence of the OOP Ms and Hres (μ0H∥[100]; X-band) for the Mn1.6PtSn film, respectively. The Curie temperature (TC = 377 K) and the spin reorientation transition temperature (Ts = 190 K) from a collinear to a noncoplanar phase2323. P. Swekis, A. Markou, D. Kriegner, J. Gayles, R. Schlitz, W. Schnelle, S. T. B. Gönnenwein, and C. Felser, “Topological Hall effect in thin films of Mn1.5PtSn,” Phys. Rev. Mater. 3, 013001 (2019). https://doi.org/10.1103/physrevmaterials.3.013001 are highlighted by dashed lines. The gradual increase in both Ms and Hres with a decrease in T for T < TC can be attributed to additional internal fields in the ferromagnetic phase that contribute to the effective magnetic field. Since Hres along the OOP direction is mainly governed by Ms as well as Ku,[100] [cf. Eq. (1)], an increase in Hres proportional to Ms is observed until close to Ts. Around Ts, Hres abruptly starts to decrease, which we attribute to the phase transition, i.e., the appearance of a noncoplanar spin texture. Note that no additional FMR mode was observed below Ts at the X-band excitation frequency. In contrast, Ms(T) does not show clear features around Ts, which is attributed to the field polarization of the noncoplanar spin texture.1515. O. Meshcheriakova, S. Chadov, A. K. Nayak, U. K. Rößler, J. Kübler, G. André, A. A. Tsirlin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle, M. Nicklas, and C. Felser, “Large noncollinearity and spin reorientation in the novel Mn2RhSn Heusler magnet,” Phys. Rev. Lett. 113, 087203 (2014). https://doi.org/10.1103/physrevlett.113.087203 Below Ts, the interaction between the ferromagnetic and noncoplanar domains leads to an additional effective field, resulting in a reduction of Hres. Around 170 K, Hres disappears. We interpret this as evidence that the full transition into the noncoplanar phase is complete. In this spirit, the fact that Hres does not directly disappear at or immediately below Ts is evidence for the coexistence of the collinear ferromagnetic phase and the noncoplanar spin texture. Previous neutron diffraction studies showed that the canting in this regime is smaller than that in the low-temperature regime,2929. P. Vir, J. Gayles, A. S. Sukhanov, N. Kumar, F. Damay, Y. Sun, J. Kübler, C. Shekhar, and C. Felser, “Anisotropic topological Hall effect with real and momentum space Berry curvature in the antiskrymion-hosting Heusler compound Mn1.4PtSn,” Phys. Rev. B 99, 140406 (2019). https://doi.org/10.1103/physrevb.99.140406 corroborating this interpretation. A similar conclusion can be drawn from topological Hall effect measurements, which show a maximum in the topological Hall response more than 30 K below Ts.2424. P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, “Role of magnetic exchange interactions in chiral-type Hall effects of epitaxial MnxPtSn films,” ACS Appl. Electron. Mater. 3, 1323 (2021). https://doi.org/10.1021/acsaelm.0c01104
The temperature dependence of Ku,[001] for the Mn1.6PtSn film is depicted in Fig. 5(c). A decrease in temperature, from 360 to 225 K, is accompanied by a linear eightfold increase of Ku,[001]. In comparison, Ms shows only a non-linear 2.5-fold increase. The relation between those temperature dependencies can be captured by the Callen–Callen power law,3030. H. B. Callen and E. Callen, “The present status of the temperature dependence of magnetocrystalline anisotropy, and the l(l+1)2 power law,” J. Phys. Chem. Solids 27, 1271 (1966). https://doi.org/10.1016/0022-3697(66)90012-6 Ku,[001](T) ∝ M(T)Γ, where the unusual exponent of Γ = 2.25 could potentially originate from an effective anisotropic exchange mediated by an induced Pt moment.3131. O. N. Mryasov, U. Nowak, K. Y. Guslienko, and R. W. Chantrell, “Temperature-dependent magnetic properties of FePt: Effective spin Hamiltonian model,” Europhys. Lett. 69, 805 (2005). https://doi.org/10.1209/epl/i2004-10404-2 In the context of the stability of noncoplanar spin textures in MnxPtSn, the increase in the MCA toward Ts suggests that the MCA might be indeed the parameter that tips the scales toward the transition into the noncoplanar phase.15,3215. O. Meshcheriakova, S. Chadov, A. K. Nayak, U. K. Rößler, J. Kübler, G. André, A. A. Tsirlin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle, M. Nicklas, and C. Felser, “Large noncollinearity and spin reorientation in the novel Mn2RhSn Heusler magnet,” Phys. Rev. Lett. 113, 087203 (2014). https://doi.org/10.1103/physrevlett.113.08720332. A. Bogdanov and A. Hubert, “Thermodynamically stable magnetic vortex states in magnetic crystals,” J. Magn. Magn. Mater. 138, 255 (1994). https://doi.org/10.1016/0304-8853(94)90046-9 Therefore, the accurate quantification of the presented evolution of Ku,[001] should allow to evaluate the exact contribution of the MCA in future investigations. Note that due to experimental limitations related to the resonance condition, Ku,[001] could not be determined closer to Ts.
In order to get a glimpse into the magnetization relaxation mechanisms of the Mn1.6PtSn film around Ts, we also show the temperature dependence of the FMR linewidths ΔH [Fig. 5(d)] that correspond to the Hres in Fig. 5(b). Note that the OOP configuration ensured a minimization of the two-magnon scattering contribution to the extrinsic broadening of ΔH.3333. K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. von Hörsten, H. Wende, W. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and Z. Frait, “Spin dynamics in ferromagnets: Gilbert damping and two-magnon scattering,” Phys. Rev. B 76, 104416 (2007). https://doi.org/10.1103/physrevb.76.104416 First, ΔH decreases from the paramagnetic state to the ferromagnetic state. From TC down to Ts, ΔH shows a minor linear increase. The magnetization relaxation mechanisms thus appear to be mostly independent of T in this regime. Approaching Ts, ΔH drastically increases (diverges). In this transition region, additional relaxation mechanisms thus emerge. A clear allocation of the mechanisms responsible for this divergence, however, is not possible at present. Nevertheless, since this transition region is dominated by the coexistence of the collinear ferromagnetic phase and the noncoplanar phase, the most intuitive interpretation of the ΔH-divergence is a strong local variation of the spin configurations, resulting in very efficient spin relaxation. Interestingly, the FMR does not re-emerge in the fully noncoplanar phase at low T, which might indicate that the magnetization relaxation is very efficient in this phase. Similar observations were made in FeRh around the ferromagnetic–antiferromagnetic phase transition, where a dramatic enhancement of ΔH was observed.3434. Y. Wang, M. M. Decker, T. N. G. Meier, X. Chen, C. Song, T. Grünbaum, W. Zhao, J. Zhang, L. Chen, and C. H. Back, “Spin pumping during the antiferromagnetic–ferromagnetic phase transition of iron–rhodium,” Nat. Commun. 11, 275 (2020). https://doi.org/10.1038/s41467-019-14061-w
In this work, we systematically determine the magnetic anisotropies in MnxPtSn films (x = 1.0–1.6) and correlate their evolution to the cubic-to-tetragonal transition of the film structure. We show the temperature dependence of the uniaxial anisotropies and their accurate control through the variation of the Mn concentration or lattice-matched film growth. Furthermore, our temperature-dependent FMR measurements reveal a distinct change in the resonance condition and the magnetization relaxation mechanisms in the proximity of the spin reorientation transition, which we attribute to the coexistence of the collinear and noncoplanar magnetic phases. The increasing interest in the skyrmion hosting Mn-based inverse tetragonal Heusler compounds corroborates the importance of the present study and motivates more detailed investigations of the involved magnetic interactions beyond the MCA, as well as the magnetic relaxation mechanisms, ideally by means of broadband FMR.
The authors acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under SPP 2137 (Project No. 403502666) and SFB 1143 (Project No. C08) as well as by the European Union’s Horizon 2020 research and innovation program under the FET-Proactive Grant Agreement No. 824123 (SKYTOP).
The data that support the findings of this study are available within the article or from the corresponding author upon reasonable request.
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