Abstract:
We investigate the structural stability and magnetic properties of cubic, tetragonal and hexagonal phases of Mn3Z (Z=Ga, Sn and Ge) Heusler compounds using firstprinciples density-functional theory. We propose that the cubic phase plays an important role as an intermediate state in the phase transition from the hexagonal to the tetragonal phases. Consequently, Mn3Ga and Mn3Ge behave differently from Mn3Sn, because the relative energies of the cubic and hexagonal phases are different. This result agrees with experimental observations from these three compounds. The weak ferromagnetism of the hexagonal phase and the perpendicular magnetocrystalline anisotropy of the tetragonal phase obtained in our calculations are also consistent with experiment.
The Mn atoms in the hexagonal phase exhibit various possible magnetic configurations. On the one hand, in the plane of the triangle lattice, the magnetic moments of Mn atoms may point in-plane or out-of-plane. On the other hand, one finds that the Mn-Mn bonds between neighboring layers in of Mn3Z are a little shorter than the in-plane Mn-Mn bonds. Therefore the interlayer magnetic coupling is also important. As a summary, we display the most important configurations in Fig. 2. The AFM type of in-plane ordering in Fig. 2(d) is found to be the most stable. However, the directions of the magnetic moments are not equally separated by 120°, therefore they do not fully cancel each other, resulting in a weak ferromagnetic phase [30~32]. The calculated magnetic moments also agree with previous experiments for all three materials. For example, the magnetic moments are 0.03 µB (exp. 0.045 µB [1]) for Mn3Ga, 0.01 µB (exp. 0.009 µB [2]) for Mn3Sn and 0.01 µB (exp. 0.06 µB [3]) for Mn3Ge. In Mn3Ge, the variation between the experimental and calculated results originates mostly from the fact that the experimental compositions are off-stoichiometry [3, 22].
We propose that the hexagonal phase does not change into the tetragonal phase directly; rather it transitions through the cubic lattice, which is structurally intermediate. It is simple to see that distortion along the c direction of the cubic phase will lead to the tetragonal phase, while the transition between the cubic and hexagonal phases is not as straightforward. If the cubic lattice is projected along the diagonal direction, one obtains a trianglar lattice. In order to recover a hexagonal phase, four atomic layers including three Mn layers and one Z layer should be compressed into one layer. However, this cannot be realized by a simple projection of ABC sites, for the Z atom overlaps with 5 one Mn atom in this way (Figs. 1(d) and 1(e)). In order to host the additional Mn atoms inside a layer, the honeycomb lattice of Mn (Fig. 1(d)) should change into a Kagome lattice (Figs. 1(f) and 1(g)) to create more available sites for Mn atoms. On the one hand, the transition from the cubic to the hexagonal phase requires compressive pressure along the cubic diagonal direction in order to push Mn and Z atoms inside a plane. On the other hand, the transition from the hexagonal to the cubic phase also needs compressive strain in the ab plane of the hexagonal lattice.
We propose that the hexagonal phase does not change into the tetragonal phase directly; rather it transitions through the cubic lattice, which is structurally intermediate. It is simple to see that distortion along the c direction of the cubic phase will lead to the tetragonal phase, while the transition between the cubic and hexagonal phases is not as straightforward. If the cubic lattice is projected along the diagonal direction, one obtains a trianglar lattice. In order to recover a hexagonal phase, four atomic layers including three Mn layers and one Z layer should be compressed into one layer. However, this cannot be realized by a simple projection of ABC sites, for the Z atom overlaps with 5 one Mn atom in this way (Figs. 1(d) and 1(e)). In order to host the additional Mn atoms inside a layer, the honeycomb lattice of Mn (Fig. 1(d)) should change into a Kagome lattice (Figs. 1(f) and 1(g)) to create more available sites for Mn atoms. On the one hand, the transition from the cubic to the hexagonal phase requires compressive pressure along the cubic diagonal direction in order to push Mn and Z atoms inside a plane. On the other hand, the transition from the hexagonal to the cubic phase also needs compressive strain in the ab plane of the hexagonal lattice.
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