Magnetoelastic anisotropy in Heusler-type Mn_2-δCoGa_1+δ films

Surface morphologies were observed using AFM and the results are shown in figures 1(a)–1(d) for the series A and B with $t$ of 5 and 30 nm. For the 5-nm-thick samples, several islands with a height of several nanometers are observed in series A (Fig. 1(a)), while relatively flat surface is observed in series B (Fig. 1(c)). Regarding the 30-nm-thick samples, the surfaces are relatively flat for both series A and B.

The thickness dependence of the average surface roughness ($R_{\mathrm{a}}$) and peak-to-valley height (P-V) are summarized in figures. 2(a) and 2(b), respectively. The values of $R_{\mathrm{a}}$ and P-V drastically drop at $t=10$ nm for series A, while, those values show no dependence on $t$ and relatively small values for series B. These results suggest a difference in the growth mode of the MCG layers depending on the buffer material: For the series A using Ag buffer layer, the Volmer-Weber (VW) mode, which describes a three-dimensional growth, is considered for the initial growth of the MCG layer. It then possibly changes to the Frank-van-der-Merwe (FM) mode for a layer-by-layer growth mode with increasing $t$. For the series B using Cr buffer layer, the film was grown possibly in the FM mode throughout the whole thicknesses. The lattice spacings of both buffer materials are smaller than that of a bulk Mn₂CoGa [28], and the lattice mismatches are approximately 1.6% and 1.4% to the Cr and Ag buffer layers, respectively, which are of the same order. A reason for the difference in the growth modes between the two sample series can be qualitatively understood by considering surface energies for the buffer$/$MCG layer system in the following way: According to the quasi-equilibrium description, a sum of surface energies ($\sigma_{\mathrm{sum}}$) can be written as $\sigma_{\mathrm{sum}}=\sigma_{\mathrm{f}}-\sigma_{\mathrm{b}}+\sigma_{\mathrm{i}}$, where $\sigma_{\mathrm{f}}$, $\sigma_{\mathrm{b}}$, and $\sigma_{\mathrm{i}}$ are surface energies of a film, a buffer layer, and the interface energy between the film and the buffer layer, respectively [42, 43]. Here, strain energies for the film and the buffer layer are neglected because the lattice strain is small ($\sim$ a few %) in the present samples. We may also neglect the difference between $\sigma_{\mathrm{i}}$ values for the cases of series A (Ag$/$MCG interface) and series B (Cr$/$MCG interface), because $\sigma_{\mathrm{i}}$ depends on the lattice misfits between the layers which are comparable to each other for the present sample series. In addition, contribution of $\sigma_{\mathrm{i}}$ to the value of $\sigma_{\mathrm{sum}}$ could be very small because the potential energy for the interface energy exhibits exponential decrease when the misfit is close to zero [44, 45, 46]. Based on the points stated above, $\sigma_{\mathrm{f}}$ and $\sigma_{\mathrm{b}}$ can be dominant parameters affecting the sign of $\sigma_{\mathrm{sum}}\sim\sigma_{\mathrm{f}}-\sigma_{\mathrm{b}}$, i.e. VW (FM) mode is expected for $\sigma_{\mathrm{sum}}>0$ ($<0$) for which $\sigma_{\mathrm{f}}$ is larger (smaller) than $\sigma_{\mathrm{b}}$.

Reported surface energies for poly-crystalline and (001) surface of the cubic systems are summarized in Table 1 [47, 48, 49]. Although the surface energy of MCG is unknown here, we may qualitatively understand the different growth modes considering the surface energies of Cr and Ag as $\sigma_{\mathrm{b}}$, i.e., the surface energies of Cr are larger than those of Ag for all the cases shown in Table 1. Thus, the $\sigma_{\mathrm{sum}}$ can be relatively small for the Cr buffer sample compared to that using the Ag buffer sample, which possibly resulted in better coverage of the MCG layer deposited on the Cr buffer than that on the Ag buffer. In addition to the possibility of the different growth modes explained above, different situations for the surface reconfiguration caused by the post-annealing can be another factor: i.e., the MCG layer forms islands (the layer structure) on the Ag (Cr) buffer after the annealing because of the relatively small (large) $\sigma_{\mathrm{b}}$, especially for the relatively thin 5-nm-thick samples. For $t\geq$ 10 nm, the good coverage was likely realized because of the thick (relatively large volume) of the MCG layer for the as-deposited stage and surface reconfiguration caused by the post-annealing.

The out-of-plane and in-plane XRD measurements were carried out to investigate the crystal structure. The XRD profiles of 5-nm-thick samples are shown in figure 3 for series A (Figs. 3(c) and 3(d)) and B (Figs. 3(e) and 3(f)). From the MCG layer, a series of $h~{}0~{}0$ ($h=$ 2 or 4) diffractions are observed for the diffraction profile with $\vec{Q}||$MgO[110], and only 2 2 0 diffractions are observed for $\vec{Q}||$MgO[100] where $\vec{Q}$ is a diffraction vector. Features of these diffraction profiles were similar for all other samples indicating the epitaxial growth with a relationship of MgO(001)[100] $|$ MCG(001)[110].

Figures 4(a), 4(b), and 4(c) show the $t$ dependence of the out-of-plane lattice constant ($c$), the in-plane lattice constants ($a$), and the $c/a$ ratio, respectively. For the series A, the $c/a$ ratio shows nearly no dependence on $t$, on the other hand for the series B, although the $c/a$ ratio is 1.04 at $t=5$ nm and relaxes with increasing $t$, the tetragonal distortion is maintained at a relatively thick $t$ of 30 nm for which the $c/a$ ratio is 1.02. The different lattice strain possibly originates from the difference in the growth modes of the MCG layers: For the series A, as it was found in the observation of morphology using the AFM images, the strain inside the lattice could be relaxed because of the three-dimensional growth mode in the initial stage. For the series B, on the other hand, the layer-by-layer growth in the initial stage resulted in the restriction of the MCG lattice to the in-plane direction, which remained unrelaxed even at the relatively large thickness.

Figures 5 and 6 show magnetization curves of all samples for the series A and B, respectively. In-plane magnetization is observed for the series A, and perpendicular magnetization is observed for the series B. The perpendicular magnetization in series B is consistent with a previous study on a Mn_1.8Co_1.2Ga_1.0 film [33]. These features for the easy magnetization directions were the same for all other layer thicknesses down to 5 nm in each sample series.

Figure 7(a) shows the $t$ dependence of saturation magnetic moments ($\mu_{\mathrm{s}}$) which were calculated using $M_{\mathrm{s}}$ values obtained from the magnetization curves (Figs. 5 and 6) and the lattice constants (Fig. 4). The $\mu_{\mathrm{s}}$ shows nearly no dependence on $t$, which suggests a similar degree of chemical order among the samples even for the relatively low thickness region. The magnitudes of $\mu_{\mathrm{s}}$ range from 1.1 to 1.4 $\mu_{\mathrm{B}}$ per a formula unit (f.u.), which are relatively small compared to other ferromagnetic Heusler alloys showing the moment of the order of 4 – 6 $\mu_{\mathrm{B}}$ because of the ferrimagnetic alignment of the atomic moments. Although the experimental values are smaller than theoretical values of approximately 2 $\mu_{\mathrm{B}}$ for Mn₂CoGa in literature [50] as well as in our calculation (see Sec. IV), those are of the same order of other calculated values of about 1 $\mu_{B}$ in which the off-stoichiometry and the disorder effects were considered ⁴⁴4The calculation details for the magnetic moments with the offstoichiometry and the disorder are described in Supplementary Material.. Comparing the two sample series, the samples in series A exhibit slightly larger $\mu_{\mathrm{s}}$ than those in series B. The difference is possibly because of a small difference of chemical order for each sample series[52]. As another possibility, the difference in composition distribution probably caused by a difference in a small amount of interdiffusion around the Ag (or Cr) / MCG interface may be considered[53]. Figure 7(b) shows the $t$ dependence of saturation field for hard magnetization directions which are the out-of-plane field ($H_{\mathrm{k}}^{\perp}$) and the in-plane field ($H_{\mathrm{k}}^{||}$) for series A and B, respectively. The $H_{\mathrm{k}}^{\perp}$ monotonically decreases as $t$ increases for the series A, while $H_{\mathrm{k}}^{||}$ shows a drop around the $t$ of greater than 10 nm for the series B.

The uniaxial magnetocrystalline anisotropy constant ($K_{\mathrm{u}}$) of MCG films was evaluated from the magnetization curves using the following formula; $K_{\mathrm{u}}=K_{\mathrm{u}}^{\mathrm{eff}}+(\mu_{0}/2)M_{\mathrm{s}}^{2}$, where $(\mu_{0}/2)M_{\mathrm{s}}^{2}$ is the shape anisotropy energy, and $K_{\mathrm{u}}^{\mathrm{eff}}$ was derived from the area enclosed by the out-of-plane and in-plane magnetization curves. A positive value of $K_{\mathrm{u}}^{\mathrm{eff}}$ represents the perpendicular magnetization in this study. The values of $K_{\mathrm{u}}$ are summarized as a function of the $c/a$ ratios in figure 8. Here, the $K_{\mathrm{u}}$ exhibits positive correlation with the $c/a$ ratio indicating that it is caused by the magnetoelastic effect. The maximum value of $K_{\mathrm{u}}$ was 1.6 $\times$ 10⁵ J/m³ at the $c/a$ of 1.04. The $c/a$ dependent $K_{\mathrm{u}}$ is theoretically discussed in Sec. IV. Although the strain could be maximum around the interface region and be relaxed around the surface region in the real samples, the following discussion is based on the uniform strain for the simplicity.

The values of $K_{\mathrm{u}}$ exhibit clear dependence on $t$, especially in the samples of series B, for which the difference in magnetic domain structures was also observed using Kerr microscopy [54, 55]. Figure 9 shows hysteresis loops for the samples of series B with $t=$ 10 (Fig. 9(a)), 20 (Fig. 9(b)) and 30 nm (Fig. 9(c)) measured by MOKE in polar mode. The square like shapes of the hysteresis indicate that the magnetization reversal occurs via domain nucleation and domain wall motion for all the samples [54, 56]. To clarify the change of domain pattern, five points were chosen for MOKE microscopy observation: (i) a saturation point at a positive field ($+H_{\mathrm{sat}}$), (ii) a point around nucleation at a negative field ($-H_{\mathrm{nucl}}$), (iii) a point around coercivity at a negative field ($-H_{\mathrm{c}}$), (iv) a point near saturation in a negative field ($\sim-H_{\mathrm{sat}}$), and (v) a saturation point for the negative field ($-H_{\mathrm{sat}}$). The numbers, i – v are also annotated in each panel of Fig. 9, and the respective field values are shown in each panel of MOKE images in Fig. 10.

The domain images observed by MOKE microscopy in polar mode of three samples are shown in Fig. 10. Note that the domain image was not studied for $t$ = 5 nm because of large $H_{\mathrm{c}}$ ($\mu_{0}H_{\mathrm{c}}\sim$ 210 mT) which was out of the range of the MOKE setup. For the 10-nm-thick sample, relatively large size of domains appears, which contain a 180° domain wall and are visible in Figs. 10(b) to 10(d). The domain pattern changes for $t$ = 20 nm, in which large patch-like domains are observed (Figs. 10(g) – 10(i)). With increasing $t$ to 30 nm, the domain size becomes small (Figs. 10(l) – 10(n)). It is observed here that the domain size is observed to be comparatively large for the sample with large anisotropy energy. The surface roughness likely causes the magnetic domain pinning[57].

Element specific magnetic moment of the 30-nm-thick sample showing perpendicular magnetization (series B) was characterized using synchrotron radiation soft x-ray at BL23SU of SPring-8 [41]. The XAS ($=\mu^{-}+\mu^{+}$) and XMCD ($=\mu^{-}-\mu^{+}$) spectra are shown in figure 11 for Mn $L_{2,3}$ and Co $L_{2,3}$ absorption edges measured at $\theta$ of 0° and 54.7°, where $\mu^{+(-)}$ represents the soft x-ray absorption for the positive (negative) helicity. The XAS and XMCD spectra of previously studied bulk Mn₂VAl [38] and Co₂MnGa [19] samples are also shown in Fig. 11 to discuss the origin of structures observed in the spectra for the series B sample, which is to be described in the latter part of this section. Element specific hysteresis loops measured at the $L_{3}$ absorption edges of Mn and Co atoms are shown in figure 12. MCD signals for Mn and Co atoms simultaneously switch around the coercive field. For $\theta=0$, the coercive field is consistent with that observed in VSM measurements shown in Figs. 5 and  6. These results indicate strong ferromagnetic coupling between the Co moments and the net moment for Mn atoms. To quantitatively discuss the angular dependence of XMCD, orbital- ($m_{\mathrm{orb}}^{\theta}$) and effective spin-moments ($m_{\mathrm{spin}}^{\mathrm{eff}}$) per an atom were evaluated using magneto-optical sum rules [58, 59, 60, 61, 62, 63] which state the moments using the following equations;

where $m_{\mathrm{spin}}$ and $m_{T}^{\theta}$ are the spin-moment and magnetic dipole moment, respectively. The quantities, $p$, $q$, and $r$ are the XMCD integrations for the $L_{3}$-edge, $L_{3}+L_{2}$-edges, and the integral of XAS which are defined as the following equations: $p=\int_{L_{3}}(\mu^{-}-\mu^{+})dE$, $q=\int_{L_{3}+L_{2}}(\mu^{-}-\mu^{+})dE$, and $r=\int_{L_{3}+L_{2}}(\mu^{-}+\mu^{+}-\mathrm{b.g})dE$ (b.g is a background spectrum showing a step-function shape). A correction factor ($C$) is included in equation (2) to take into account the so-called jj mixing effect [64, 65, 66]. For $m_{\mathrm{spin}}^{\mathrm{eff}}$ of Mn and Co, the $C$ of 1.5 and 1.1 were assumed, respectively [64, 65, 38]. The numbers of 3d holes ($n_{\mathrm{h}}$) were assumed to be 4.44 $\pm$ 0.08 and 2.29 $\pm$ 0.02 for the Mn and Co atoms, respectively, based on the theoretical values for Mn₂CoGa obtained from the first principles calculations in section IV. Here, the values of $n_{h}$ are averaged over four cases of Mn₂CoGa with X_A and L2_1b phases and $c/a$ ratios of 1.00 and 1.04. The values of error are the standard deviation of all cases including possible inequivalent atomic sites for Mn atoms which are described in detail later. In a strict manner, the inequivalent sites should be deconvoluted from the XAS$/$XMCD spectra, and the equations (1) and (2) are applied to each site, however, the present $n_{\mathrm{h}}$ for the Mn sites showing similar values enable us to evaluate average values using the equations (1) and (2) as an approximation [67]. The values of $m_{\mathrm{spin}}^{\mathrm{eff}}$, $m_{\mathrm{orb}}^{\theta}$ and $m_{\mathrm{orb}}/m_{\mathrm{spin}}^{\mathrm{eff}}$ are summarized in figure 13. Corresponding total magnetic moment was calculated to be $1.18\pm 0.03~{}\mu_{\mathrm{B}}/\mathrm{f.u.}$ using $m_{\mathrm{spin}}^{\mathrm{eff}}$ and $m_{\mathrm{orb}}^{\theta}$ at $\theta=$ 54.7°, and the film composition. The total magnetic moment by the sum rule is of the same order as the $\mu_{\mathrm{s}}$ of 1.27 $\mu_{\mathrm{B}}/\mathrm{f.u.}$ measured by VSM. Concerning the angular dependence, no $\theta$ dependence was found either for $m_{\mathrm{spin}}^{\mathrm{eff}}$ or $m_{\mathrm{orb}}^{\theta}$. Here, the $\theta$ of 54.7° is a magic angle for which $m_{T}^{\theta}=0$ by applying the approximate symmetry relation of $2m_{T}^{90\degree}+m_{T}^{0\degree}=0$, where $m_{T}^{\theta}=m_{T}^{90\degree}\sin^{2}\theta+m_{T}^{0\degree}\cos^{2}\theta$ for 3d metals [60, 68]. Thus, the angular independent $m_{\mathrm{spin}}^{\mathrm{eff}}$ values suggest that the anisotropy of the magnetic dipole moment ($\Delta m_{\mathrm{T}}=m_{T}^{0\degree}-m_{T}^{90\degree}=\frac{3}{2}m_{T}^{0\degree}$) is less than 0.036 $\mu_{\mathrm{B}}$/atom which is in an error-bar range in the present sample. The angular independent $m_{\mathrm{orb}}^{\theta}$ values provide information for the discussion of origin of PMA in series B: The magnetocrystalline anisotropy energy ($E_{\mathrm{MCA}}$) has been theoretically evaluated using expressions based on the second-order perturbation theory in terms of spin-orbit interactions [69, 70, 71]. Note that $E_{\mathrm{MCA}}$ is defined as the energy difference when the magnetization points to the perpendicular and in-plane direction: $E_{\mathrm{MCA}}=E^{\theta=90}-E^{\theta=0}$, i.e., the positive sign of $E_{\mathrm{MCA}}$ indicates PMA in this definition. Within a formulation $E_{\mathrm{MCA}}$ can be expressed as follows [69, 70]:

where $\xi$, $E_{\mathrm{ex}}$, and $\Delta m_{\mathrm{orb}}$ are the spin–orbit coupling constant, exchange splitting of 3d bands, and the anisotropy of orbital magnetic moment defined as $\Delta m_{\mathrm{orb}}=m_{\mathrm{orb}}^{\theta=0}-m_{\mathrm{orb}}^{\theta=90\degree}$, respectively, for which the first term on the right side corresponds to the contribution to the energy difference from the spin-conservation terms in virtual excitations, and the second term corresponds to that from the spin-flip term. The experimentally observed angular independent $m_{\mathrm{orb}}^{\theta}$ implies small contribution of the spin-conservation terms to $K_{\mathrm{u}}$, and the other contribution of the spin-flip terms discussed in refs. 70, 71 is more important. Theoretical values of $K_{\mathrm{u}}$ and $m_{\mathrm{orb}}$ as well as the origin of PMA with respect to the spin-flip term are to be discussed further in section IV.

The sub-peaks and shoulder structures in XAS and XMCD spectra provide additional information on the chemical order of the atoms in Heusler structures [17, 18, 52, 29, 39, 40, 72, 38, 19]. Note that a possibility of oxidization is excluded, because the shapes of XAS and XMCD spectra are completely different from those reported for materials containing Mn- and/or Co-oxides [73, 74]. In addition, shoulders and sub-peaks are clearly observed at 300 K for MCD spectra shown in Figs. 11(c) and 11(d), for which the measurement temperature is higher than the magnetic transition temperatures for most of Mn-/Co-oxide materials.

Figure 14 shows a schematic illustration of a crystal structure for Heusler alloys and the notations of the atomic sites for the inverse-Heusler structure (X_A phase), the full-Heusler structure (L2₁ phase), and an L2_1b phase. In addition to these three phases, fully-disordered D0₃ phase was considered as possible chemical phases in previous studies on bulk samples [75, 50]. According to total energy calculations in ref. 50, the X_A phase was the most stable at the ground state, and relatively small energy difference of 0.136 eV/f.u. was reported for the L2_1b phase. The energy differences with respect to the X_A phase were much higher for the L2₁ and D0₃ phases than that for the L2_1b phase. Concerning the magnitudes of magnetic moment for each atom, the values were similar in X_A and L2_1b phases to each other and ferrimagnetic coupling was reported as schematically shown in Fig. 14, which resulted in a total moment of about 2.0 $\mu_{\mathrm{B}}$/f.u. for both. On the other hand, those were relatively large (small) for the L2₁ (D0₃) phase showing ferromagnetic (ferrimagnetic) coupling with a total moment of about 7.7 (1.0) $\mu_{\mathrm{B}}$/f.u. As it was discussed on Fig. 7 in Sec. III.2, the total magnetic moments of the present samples are of the order of the reported values for the X_A and L2_1b phases, and considering the calculated energy difference mentioned above, the L2₁ and D0₃ phases are excluded in the following discussion. Note that the L2_1b phase was experimentally proposed for the previous bulk Mn₂CoGa samples based on high-angle annular dark field scanning electron transmission microscope images [75] and neutron diffraction experiments [50]. On the other hand, the X_A phase was proposed in another previous study on an epitaxially grown Mn₂CoGa film based on XAS and XMCD results [39]. The shapes of XAS and XMCD spectra of Mn₂CoGa films were previously discussed based on the first-principles calculation for the X_A phase. In the previous studies [39, 40], several shoulders, sub-peaks, and dip structures were mentioned. In the present results, similar structures are observed, i.e., broad shoulders in the XAS, and sub-peaks and dip structures in the XMCD spectra are observed, which are marked by solid and broken lines in Fig. 11. The marked structures in the XMCD spectra can also be understood as a sum of two spectra [29, 76, 39, 40, 72] which are, e.g., MVA [38] and CMG [19] for the markers of broken and solid lines, respectively. Here, MVA is a typical example of a material with Mn atom(s) at D(C)-site, and CMG is another example with a Mn atom at B-site in Fig. 14. The broad feature is mainly from the Mn atom(s) D(C)-site showing relatively itinerant nature of electrons [38], on the other hand, the Mn atom at the B-site exhibits relatively localized nature showing remarkable shoulders in XAS [17, 18, 19]. Concerning the XMCD spectra for Co, shoulders are also observed around the post-absorption edges with the solid markers in Fig. 11(d) which also includes a reference spectrum of CMG. The origin of the shoulders is attributed to the 2p $\rightarrow$ 3d $e_{g}$ (or $e_{u}$) transition [19].