• Magnetocrystalline anisotropies in Mn_xPtSn thin films

In our previous work, we described the unique crystallographic orientation of Mn_xPtSn 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 Mn_1.5PtSn,” 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$\bar{1}$0] directions of the MgO substrate as Mn_xPtSn(100)[001]∥MgO(001)[110] and Mn_xPtSn(100)[001]∥MgO(001)[1$\bar{1}$0]. Furthermore, we determined that a lattice mismatch (2.5% and 6.5%) with the substrate results in an in-plane compressive strain, breaking the equivalence between the tetragonal a and b lattice constants. This effectively results in an orthorhombically distorted structure (I2₁2₁2₁)²³23. P. Swekis, A. Markou, D. Kriegner, J. Gayles, R. Schlitz, W. Schnelle, S. T. B. Gönnenwein, and C. Felser, “Topological Hall effect in thin films of Mn_1.5PtSn,” Phys. Rev. Mater. 3, 013001 (2019). https://doi.org/10.1103/physrevmaterials.3.013001 as compared to the tetragonal bulk superstructure ($I\bar{4}2d$).²⁵25. P. Vir, N. Kumar, H. Borrmann, B. Jamijansuren, G. Kreiner, C. Shekhar, and C. Felser, “Tetragonal superstructure of the antiskyrmion hosting Heusler compounds Mn_1.4PtSn,” 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 Mn_xPtSn films (x = 1.0–1.6). For MnPtSn, the presence of the (h00) Bragg reflections, indexed based on the bulk structure ($F\bar{4}3m$),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 Mn_1.5PtSn,²⁵25. P. Vir, N. Kumar, H. Borrmann, B. Jamijansuren, G. Kreiner, C. Shekhar, and C. Felser, “Tetragonal superstructure of the antiskyrmion hosting Heusler compounds Mn_1.4PtSn,” 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 Mn_1.4PtSn,” 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 Mn_1.6PtSn.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 Mn_xPtSn 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.

Figure 2 shows the magnetization hysteresis loops of the Mn_xPtSn 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 (M_s) 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 (T_C) 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 Mn_xPtSn 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 M_s and T_C 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.

The magnetic anisotropies of the Mn_xPtSn 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 Mn_xPtSn 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 Mn_xPtSn structure.

Based on the crystallographic orientations in the described coordinate systems [Figs. 3(a)–3(c)], the total free energy density (F_tot), required to determine the anisotropies of the Mn_xPtSn films, can be defined. Here, F_tot includes the Zeeman energy (F_z), the shape anisotropy (F_d), a cubic anisotropy term (F_c; only MnPtSn), and two different uniaxial anisotropy terms, one perpendicular to the film plane (F_u,[100]) as well as one along each of the two crystallographic c-axes ($F_{\mathrm{u},[\mathrm{001}]}^{\pm }$),

$$\begin{array}{ccc}\hfill F_{\text{tot}}& =\hfill & F_{\mathrm{z}}+F_{\mathrm{d}}+F_{\mathrm{u},[\mathrm{100}]}+F_{\mathrm{u},[\mathrm{001}]}^{+}+F_{\mathrm{u},[\mathrm{001}]}^{-}+F_{\mathrm{c}}\hfill \\ \hfill & =\hfill & -{\mu }_0{M}_{\mathrm{s}}H(\sin \mathrm{\Theta }\sin \mathrm{\Phi }\sin \theta \sin \varphi +\hfill \\ \hfill & \hfill & \cos \mathrm{\Theta }\cos \theta +\sin \mathrm{\Theta }\cos \mathrm{\Phi }\sin \theta \cos \varphi )\hfill \\ \hfill & \hfill & +\frac{{\mu }_0}{2}{M}_{\mathrm{s}}^2{\left(\sin \mathrm{\Theta }\cos \mathrm{\Phi }\right)}^2-K_{\mathrm{u},[\mathrm{100}]}{\left(\sin \mathrm{\Theta }\cos \mathrm{\Phi }\right)}^2\hfill \\ \hfill & \hfill & +K_{\mathrm{u},[\mathrm{001}]}\frac{1}{2}{\left(\sin \mathrm{\Theta }\sin \mathrm{\Phi }+\cos \mathrm{\Theta }\right)}^2\hfill \\ \hfill & \hfill & +K_{\mathrm{u},[\mathrm{001}]}\frac{1}{2}{\left(\sin \mathrm{\Theta }\sin \mathrm{\Phi }-\cos \mathrm{\Theta }\right)}^2\hfill \\ \hfill & \hfill & +K_{\mathrm{c}}\frac{1}{4}\left({\sin }^2(2\mathrm{\Theta })+{\sin }^4\mathrm{\Theta }{\sin }^2(2\mathrm{\Theta })\right).\hfill \end{array}$$

(1)

The shape anisotropy and K_u,[100] are then summarized in the effective perpendicular uniaxial anisotropy $K_{\mathrm{u},\mathrm{e}\mathrm{ff}}=\frac{1}{2}{\mu }_0{M}_{\mathrm{s}}^2-K_{\mathrm{u},[\mathrm{100}]}$. Note that since the tetragonal Mn_xPtSn films (x = 1.2–1.6) feature crystallites with two independent crystalline orientations [Fig. 3(c)], we simulated two independent H_res orientation dependencies. For that purpose, two different F_u,[001] were defined, where Mn_xPtSn(100)[001]∥MgO(001)[110] was accounted for by $F_{\mathrm{u},[\mathrm{001}]}^{+}$, while Mn_xPtSn(100)[001]∥MgO(001)[1$\bar{1}$0] was accounted for by $F_{\mathrm{u},[\mathrm{001}]}^{-}$. The two independent simulations were, in turn, performed by setting either $F_{\mathrm{u},[\mathrm{001}]}^{+}$ or $F_{\mathrm{u},[\mathrm{001}]}^{-}$ to zero. For the simulation of the cubic film, both $F_{\mathrm{u},[\mathrm{001}]}^{+}$ and $F_{\mathrm{u},[\mathrm{001}]}^{-}$ were set to zero.

The orientation dependencies of H_res on H of the Mn_xPtSn 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 H_res 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 H_res for the respective field orientation in the FMR data. Moreover, H_res along the OOP direction ([100], ϕ = 0°) increases with increasing Mn concentration, and a second H_res appears for the Mn_1.2PtSn film. Notably, a second H_res is not present for x = 1.4 and 1.6. The IP orientation dependencies of H_res show a distinct evolution from the cubic film to the tetragonal films. The MnPtSn film shows a minor fourfold angular dependence of H_res 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 H_res [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 Mn_1.6PtSn and Mn_1.4PtSn 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 H_res 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 H_res of the rotation in the (010) plane. In contrast, the single H_res 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 Mn_xPtSn films [Figs. 3(d)–3(k)] can be derived from F_tot [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).

$${\left(\frac{\omega }{\gamma }\right)}^2=\frac{1}{M_{\mathrm{s}}^2\sin \mathrm{\Theta }}\left(\frac{{\partial }^2{F}_{\text{tot}}}{\partial {\mathrm{\Phi }}^2}\frac{{\partial }^2{F}_{\text{tot}}}{\partial {\mathrm{\Theta }}^2}-{\left(\frac{{\partial }^2{F}_{\text{tot}}}{\partial \mathrm{\Theta }\partial \mathrm{\Phi }}\right)}^2\right),$$

(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 $\gamma =\frac{g{\mu }_B}{\hslash }$) of the Mn_xPtSn films were derived, using M_s from Table I. The simulations, included in Fig. 3(d)–3(k) as solid lines, well reproduce the orientation dependencies of H_res, and hence, the obtained anisotropy constants are summarized in Table I, except for K_c = 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 ($F_{\mathrm{u},[\mathrm{001}]}^{+}=0$), whereas Fig. 3(e) also includes the simulation of the (010) plane ($F_{\mathrm{u},[\mathrm{001}]}^{-}=0$). As introduced before, this absence is related to the resonance condition, not fulfilled along the [001] direction at X-band frequency for the Mn_1.4PtSn and Mn_1.6PtSn films. We verified this noncompliance by solving an analytical special case of the resonance condition [Eqs. (1) and (2)] along the [001] direction,

$$\begin{array}{ccc}\hfill {\left(\frac{\omega }{\gamma }\right)}^2& =\hfill & \left({\mu }_0{H}_{\text{res,}\left[001\right]}+2\frac{K_{\mathrm{u},[\mathrm{001}]}}{M_{\mathrm{s}}}\right)\hfill \\ \hfill & \hfill & \times \left({\mu }_0{H}_{\text{res,}\left[001\right]}+2\frac{K_{\mathrm{u},\mathrm{e}\mathrm{ff}}}{M_{\mathrm{s}}}\right),\hfill \end{array}$$

(3)

employing the anisotropies and g determined from the in-plane and (001) plane simulations.