Tuned Magnetic Properties of L10-MnGa/Co(001) Films by Epitaxial Strain

We demonstrate that the interface structure has a significant influence on the magnetic state of MnGa/Co films consisting of L10-MnGa on face-centered-cubic Co(001) surface. We reveal an antiferromagnetic to ferromagnetic magnetization reversal as a function of the lateral lattice constant. The magnetization reversal mainly originates from localized states and weak hybridization at interface due to charge redistribution between muffin-tin spheres and interstitial region. The magnetic anisotropy energy of Mn/Co interface system is enhanced with increasing in-plane lattice constant, which is ascribed to the interface interactions and the above magnetization reversal.

In this paper, we investigate the magnetic interaction between L1 0 -MnGa and face-centered cubic (fcc) Co films depending on interface structure and lateral lattice constant (a). The enhanced magneto-crystalline anisotropy (MCA) driven by lattice expansion is discussed with magnetization reversal and interface interaction.

Results and Discussion
In MnGa/Co(001) film calculations, two interfaces are considered due to polar surface of L1 0 -MnGa, Mn or Ga terminated surfaces. To find reasonable interface structure of Mn/Co or Ga/Co, we perform total energy calculation with four initial magnetic states. In Table 1, we show the calculated total energy differences according to the interface and initial magnetic configurations. As shown in Fig. 1, FM, AFM1, AFM2, and AFM3 correspond to uu/U, ud/U, du/U, and dd/U configuration, respectively. The magnetic states of Co layers are fixed to U, and u stands for the parallel spin direction of Mn atoms, and d for the anti-parallel spin configuration respect to Co layers. The first u is the spin configuration of Mn 3 (Mn 4 ) in Mn/Co (Ga/Co) interface and second one indicates spin configuration of Mn 1 (Mn 2 ), respectively. For instance, AFM3 (dd/U) means that the magnetization , where M refers to the magnetic configurations (FM, AFM1, AFM2, or AFM3), and "Int" to Mn/Co or Ga/Co. Accordingly, systems with positive energy difference in Table 1 are less stable than FM-Mn/Co structure.
It is found the stable interface is independent on a, and the Mn/Co interface is energetically more favorable than Ga/Co interface. Interestingly, the magnetic interactions between L1 0 -MnGa and Co are changed from AFM3 to FM depending on a in Mn/Co interface. In contrast, magnetization reversal is not observed in Ga/Co interface. This clearly indicates that the FM ordering between Mn and Co stems from the change of the Mn-Co hybridization, rather than the lattice expansion. Recently, the anti-parallel magnetic coupling between Mn-Ga alloys and Co has been observed at a = 2.880 Å, and they have assumed that Mn-Ga/Co films dominantly form the Ga/Co interface, due to similar composition dependent magnetic behaviors of Mn-Ga/Co films comparing with Mn-Ga alloys 2 . Our computational result of Ga/Co interface well describes the experiment. Indeed, if both interfaces are thermally stable, the interface could be adjusted by growing modes, such as Co on MnGa or MnGa on Co. Thus, the control of interface is very important tunning the magnetic properties of MnGa/Co films. In the following, we discuss the Mn/Co interfacial systems, since they are energetically stable and show interesting magnetic behavior.
To further confirm the magnetization reversal as a function of a, we calculate total energy difference between FM and AFM states, denoted as Δ E = E AFM − E FM in Fig. 2(a). The positive (negative) energy differences mean FM (AFM)ground state. One can see the AFM states at a = 2.507 and 2.553 Å, and the FM state appears from a = 2.618 Å. This result demonstrates that the epitaxial strain is an efficient way to tailor the magnetic interactions at the Mn/Co interfaces. We expect that in practice a can be adjusted by selection of substrates supporting the MnGa/Co films.
In Table 2, we display the calculated magnetic moment of Ga, Mn, and Co atoms within the muffin-tin (MT) sphere. For the AFM states, the magnetic moments of Co S and Mn 1 are reduced compared to those of Co S−1  and Mn 3 . For the FM states, however, the magnetic moments of Co S and Mn 1 are close to those of Co S−1 and Mn 3 , respectively. The suppressed magnetic moments can be understood by hybridization with neighboring layer. In AFM coupling, Mn 1 induces electrons in minority spin states for Co S because Mn S has negative magnetic moment. Inversely, Co S having positive magnetic moment provokes an increment of electrons in majority spin states for Mn 1 . Thus, the bilateral process between Mn 1 and Co S will decrease the spin asymmetry of the total number of electrons, which is also found in the spin polarized density of state spectra. In addition, one can see an enhancement of magnetic moments in Mn 1 and Co S depending on a (where Mn 1 and Co S stand for the atoms at the interface), and significant modifications are observed when the magnetic state changes from AFM to FM. It means that the spin asymmetry between majority and minority spin states is induced by epitaxial strain. The magnitude of magnetic moment is simply obtained by the difference of the electrons in the two spin parts, majority and minority. In Fig. 2(b-d), we present the electronic density of states (DOS) of Mn 1 and Co S atoms at a = 2.507 Å (AFM), 2.752 Å (FM) and 2.977 Å (FM) to monitor the magnetic behavior from electronic structure. In AFM state, the shape of DOS spectra of Co S and Mn 1 are broad. In addition, hybridization between Co S and Mn 1 is observed in large range of minority spin state. In FM states with a = 2.752 and 2.977 Å, marked differences are seen in the DOS compared to the AFM state. One can observe the DOS reversals of spin states. This phenomenon correlates well with the magnetization reversal. In addition, a weak hybridization is obtained with larger in-plane lattice constant. Interestingly, more peaks are observed in majority spin part when a is expanded. This indicates that Co S and Mn 1 become more localized with increasing a. Furthermore, the DOS of both Co S and Mn 1 near E F is decreased with increasing a. It indicates electron loss in the MT region and charge redistribution between MT and interstitial region. When a is expanded, the inter-atomic distance between Mn 1 and Co S increases corresponding to longer bond length. As a result, the charge redistribution is essential to maintain Co-Co, Mn-Mn and Co-Mn bonding.
According to Heilter and London (HL) model for magnetic ordering of H 2 , weaker hybridization and more localized electron prefer the FM order resulting in a gain in the magnetic energy 17,18 . As suggested above, the weaker hybridization and localized effect originates from charge redistribution. In Fig. 3(a), we display the relative number of electrons (Δ n e ) as a function of a, with respect to those at a = 2.507 Å system. Indeed, it is clearly observed that a loss of electrons from the MT sphere of Mn 1 and Co S , and a gain of electrons in the interstitial region with increasing a. The loss of electrons mainly occurs in specific spin part to increase magnitude of magnetic moment of each atoms (not shown here), and this is confirmed from enhancement of magnetic moment with increasing a in Table 2. Therefore, the increased spin asymmetry is mainly originated from the charge redistribution which can be simply parameterized by charge difference in interstitial region ( ∆ ) n e i . The increased spin asymmetry between Mn 1 and Co S may affect the magnetic ordering between them. The magnetic ordering can be determined by competition between Coulomb and kinetic energies. According to Stoner model, total energy variation due to electron transfer is expressed with kinetic and magnetic energies 31 . The magnetic energy is proportional to square of charge and spin asymmetry, and kinetic energy is linear function of charge and spin asymmetry. The charge redistribution and the modified spin asymmetry can induce changes in the magnetic order. Therefore, we should be able to represent the magnetization reversal of MnGa/Co films in terms of ∆n e i , which is a parameter including the charge redistribution and the change of the spin asymmetry.
In Fig. 3(b), we plot the energy difference between FM and AFM3 as a function of ∆n e i . The Δ E is fitted with a quadric function where the coefficients may be interpreted as the Coulomb (Δ E M ) and kinetic (Δ E K ) energies, viz, where C is a positive constant. We notice that whereas the data in Fig. 3(b) weakly dependence on the muffin-tin radius as going from 2.15/2.25 to 2.25/2.35 a.u. for the 3d transition metals/Ga atoms (shown are results only for radii 2.20/2.30 a.u.), the conclusions below are not affected by the actual MT radius. According to Fig. 3(b), the magnetic energy difference can be understood as the competition between the Coulomb and the kinetic energies. For AFM states which have negative Δ E and small charge redistribution, the kinetic energy terms should have larger contribution. On the other hand, the FM states corresponding to positive energy difference dominantly have Coulomb interaction terms. It is concluded that from the combination of the HL and Stoner model the weak hybridization and localized states due to charge redistribution may induce modification of magnetic interactions and this well describes the predicted magnetization reversal as a function of a.
Next we discuss the MCA energies as a function of a. As shown above, interface and in-plane lattice constant are essential factors for tailoring the magnetic structure of MnGa/Co films. Previously, we found that the MCA energy (E MCA ) of L1 0 MnGa alloy can be tuned by epitaxial strain 22 . Here, we calculate the E MCA (in μeV/atom, including Co, Mn and Ga atoms) depending on a using the torque method 32 . In film structure, the E MCA , arising from spin-orbit coupling (SOC), is written as , where E and ⊥ E correspond total energies with in-plane and perpendicular magnetization to film surface, respectively. Therefore, positive MCA energies are associated with PMA, and negative ones correspond to in-plane magnetization. In Fig. 4(a), we display the calculated E MCA . All MnGa/Co films show large PMA, and E MCA is increased with a. It seems like that the enhanced MCA energy results mainly from the epitaxial strain effect on MnGa alloys. However, the interface effects cannot be ignored due to the a-dependent hybridization between Mn 1 and Co S layers. To check the interface effect on MCA energy, we also calculate E MCA for the Ga/Co interface with a = 2.507 Å and 2.752 Å. The so obtained MCA energies, 59.46 μeV/atom and 28.69 μeV/atom, respectively, show opposite trend compare to the Mn/Co interface. This means that the interaction at the interface is important to understand the a-dependent MCA energy.
To reveal the origin of the PMA and enhancement of the MCA energy with increasing a, we explore the distribution of the E MCA over two-dimensional (2D) Brillouin Zone (BZ) as shown in Fig. 4(b,c). The circles are contributions of spin-orbit interaction between the occupied and unoccupied state at given k-points. Red (blue) circles mean perpendicular (in-plane) magnetization. The magnitude of E MCA is proportional to the size of circles. Thus, total E MCA is determined by sum of E MCA at given k-points over 2D-BZ. One can see that there is no dominant PMA contributions of single point nor any particular directions. Furthermore, the changes of E MCA and direction of magnetization occur around not only zone center (Γ ) but also around corners (M). Actually, the modification of MAE with increasing a is observed in the whole k-space. We think that no simple picture can explain the PMA and magnetic anisotropy behavior as a function of a.
According to perturbation theory 28 , E MCA is defined by the SOC interaction between occupied and unoccupied states with magnetic quantum number (m) through the l z and l x operators, as where o (u) and ε , o s 2 ε ( ) , u s 1 represent eigenstates and eigenvalues of occupied (unoccupied), respectively. s 1 (s 2 ) is spin state of occupied (unoccupied) states, majority(↑ ) or minority(↓ ) spin, and ξ means the SOC strength. From Eq. (2), the MCA can be analyzed by decomposing E MCA into spin channels, namely ↓↓ E MCA , ↑↑ E MCA , and ↑↓ E MCA . For the same spin channel interaction, the positive contribution to E MCA comes from the SOC between occupied and unoccupied state with the same m through the l z operator. On the other hand, the SOC with the different m through the l z operator has positive contribution for spin-flip channel interaction 15,23,24 .

Conclusion
In summary, we have investigated magnetic properties of L1 0 -MnGa on fcc Co (001) film depending on interface structure and in-plane lattice constant. We have obtained magnetization reversal from AFM to FM coupling between L1 0 -MnGa and fcc Co (001) layers as a function of a. In Mn/Co interface structures, the reason for the a-dependent magnetization reversal is found to be the weak hybridization and more localized electrons due to charge re-distribution between MT and interstitial region. Furthermore, all MnGa/Co(001) films show perpendicular magnetic energy, and the magnetocrystalline anisotropy energy is enhanced with increasing a. The behavior of magnetic anisotropy energy can be explained by interface interaction and magnetization reversal. Finally, we have realized that the magnetic properties of MnGa/Co film can be tailored by controlling of the interface interaction, and the change of the in-plane lattice constant is one of the most effective methods.

Methods
We have employed the thin film version of all-electron full potential linearized augmented plane (FLAPW) method. Therefore, no shape approximation is assumed in charge, potential, and wave-function expansions [33][34][35] . We treat the core electrons fully relativistically, and the spin orbit interaction among valence electrons are dealt with second variationally 36 . The generalized gradient approximation (GGA) exchange-correlation potentials is used to describe exchange and correlation interaction 37 . Spherical harmonics with l max = 8 are used to expand the charge, potential, and wave-functions in the muffin tin region. Energy cut-offs of 225 Ry and 13.7 Ry are implemented for the plane wave star function and basis expansions in the interstitial region. We use 21 × 21 k-points with the Monkhorst-Pack method 38 . The muffin-tin radius is considered as 2.2 a.u. for 3d transition metals and 2.3 a.u. for Ga atom. The muffin-tin radii for all atoms are kept constant upon lateral lattice constant change. We assume four layers of L1 0 -MnGa film, consisting of two Mn and two Ga atoms grown pesudomorphically on fcc Co p(1 × 1) sublayer. The Co sublayer is simulated by seven fcc Co(001) layers. To apply epitaxial strain, we change a of the film. Here, we select a values corresponding to certain substrates, such as Co, Cu, Pd, Pt, Al, MgO and InAs. The vertical distance of films with various a is fully relaxed with force and total energy minimization procedure. The convergence for all physical quantities investigated in the present work has been carefully checked. The Co atom at the interface between MnGa and fcc Co (001) surface is denoted by Co S and the subsurface layers by Co s−i . Furthermore, Mn i means the i-th ad-layer counted from interface (Fig. 1).