In the previous section we have shown that the properties of real MRG thin films can be only partially explained by the properties of stoichiometric compounds. We are led to the conclusion that the films must be affected by a disorder of some sort. We have shown that the stoichiometric compounds lack both key features of a compensated ferrimagnetic half-metal (CFHM), namely the half-metallicity and the magnetization compensation are absent. The presence of disorder thus seems to be crucial for the emergence of these properties. Based on our results, disorder is expected to be stronger for compounds with a low Ru fraction, as the discrepancy between the experiment and the theory increases with the decreasing Ru concentration (cf. figure 3.4-b). For the compounds close to the Mn2RuGa composition we anticipate that the films have the Mn:Ru ratio different from 2:1, going towards 1:1. This is inferred from the neutron diffraction study of Hori et al. [177]. They have found that the stoichiometric Mn2RuGa has a magnetization of ≈1µBf.u.−1, in good agreement with our DFT results. We attribute the formation of a meta-stable, nearly cubic, structure of the nominal Mn2RuGa films to the substrate strain. We find that the properties of the films with a low Ru fraction can not be rationalized based on the DFT results presented so far. In this section we present the results of a high-throughput search for competing Mn-Ru-Ga phases having a (half-)Heusler crystal structure. We identify the energetically most favourable phases and then simulate the corresponding defects using supercells. The results allow us to qualitatively describe the effects of the disorder on the properties of realistic MRG alloys. Limited computational resources prevent an ab initio description of the defect forma- tion process in real materials. However, ab initio DFT calculations offer accurate total energies, which can in turn be used to compare the relative stability of compounds with different stoichiometry. We calculate the enthalpy of formation, ∆H, which is defined as a difference between the total energy of a compound and the sum of the energies of the most stable elemental, i.e. single element, phases of the constituents. For example, the enthalpy of formation of a hypothetical binary AmBn compound is calculated as
∆H= 1
N [E(AmBn)−m·E(A)−n·(B)], (3.3) where N = m+n is the number atoms in the formula unit for which the energy was calculated. The enthalpy of formation defined using the equation (3.3) measures the
# 4a 4c 4b 4d ∆H [eV] M [µB] c/a Vol [˚A3] 1 Mn Ru Ga Ru -0.28 2.18 1.0 56.33 2 Ga Mn Ga Ru -0.25 2.93 1.0 55.83 3 Mn Mn Ga Ru -0.24 0.07 1.2 55.88 4 Mn Mn Ga Ru -0.13 4.66 1.0 55.01 .. . ... ... ... ... ... ... ... ... 10 Ga Mn Ga – 0.01 3.14 1.0 52.01 12 Ru Mn Ga – 0.09 0.19 1.0 44.70 13 Ru Mn Ga Ru 0.10 4.44 1.0 59.20 14 – Mn Ga Ru 0.17 4.50 1.0 49.60 15 Mn Mn Ga – 0.18 0.47 1.0 46.54 .. . ... ... ... ... ... ... ... ...
Table 3.2: Calculated enthalpies of formation, ∆H, for the most stable competing Heusler phases of the 1221 structural and magnetic Mn-Ru-Ga cells investigated. The configurations investigated are limited to the primitive 4-atom Heusler cell (3-atom for
half-Heuslers).
stability of a compound with respect to the decomposition into its constituents. In order to evaluate the most stable structure of a given stoichiometry one needs to minimize ∆H with respect to all possible decomposition paths. This makes the full stability analysis a formidable task. Here we are interested only in the relative stability, therefore it is sufficient to know ∆H for each compound considered. We have calculated the ∆H for 1221 structural and magnetic Mn-Ru-Ga Heusler cells [31]. The phases which are excluded by the experimental analysis, e.g. compounds having Ga-Ga and Ru-Ru nearest neighbors [32], were not included in the final stability analysis. We have also ignored the compositions where more than one atom per formula unit is replaced, e.g. we do not include the compounds like Ga2Ru in the stability analysis. This would correspond to formation of multiple defects, which we believe is less likely to occur. We will later show that this approximation is sufficient for the understanding of real MRG films. Our results are summarized in table 3.2. The most stable stoichiometry is Ru2MnGa, followed by Ga2MnRu and Mn2RuGa. These results are in agreement with our hypoth- esis on the formation of an antiferromagnetic Ru2MnGa phase in Ru rich MRG films. The formation of Ga2MnRu is not likely due to the difference in stoichiometry. We note that Ru2MnGa reported in table 3.2 is ferromagnetic. The apparent discrepancy comes from the fact that in the high-throughput calculations only primitive unit cells are considered. Thus Ru2MnGa is ferromagnetically ordered by construction. DFT yields a positive enthalpy of formation for all half-Heusler Mn-Ru-Ga phases, i.e it predicts them as unstable against decomposition. We therefore conclude that the corresponding MRG films should be meta-stable structures. We observe that the addition of Ru, in general, tends to increase the stability of the compounds. The most stable half-Heusler stoichiometry is Ga2Mn, followed by the RuMnGa. The Mn2Ga composition, which we
a||≈ 5.959Å a||≈ 5.959Å Δ E Ed Eg
V
DFTV
EXPE
[e
V
]
-20.2
-20.1
-20.0
-19.9
-19.8
-19.7
c/a
0.7
0.8
0.9
1
1.1
1.2
1.3
Figure 3.7: Isovolumetric energy profiles for stoichiometric Mn2Ga calculated at the
observed experimental and the relaxed DFT primitive unit cell volume, VEXP≈52.9 ˚A
and VDFT≈46.4 ˚A, respectively. The theoretically expectedc/aratios are denoted by
the vertical dashed lines. ∆E is the energy difference between the experimental and the relaxed DFT structure. Ed is the energy gain due to the formation of Ga defects.
The net energy gain by forming the defects and taking the experimental structure is denoted by Eg.
have considered as a representative of the nominal Mn2Ga films is predicted to be much higher in energy than the aforementioned, Ga and Ru rich, phases. It is therefore very likely that such defects occur in the real MRG films.
The high-throughput analysis of the relative phase stability of Mn-Ru-Ga compounds reveals the Mn-Ga and the Mn-Ru substitutions as the most likely defects to occur in the MRG films. In order to understand if the occurrence of these defects is more stable than the formation of the stoichiometric MRG phase, we construct supercells for the two MRG compositions, x=0 and 1/3, substitute a single Mn atom at the crystallographic positions 4a (or 4c) with Ga or Ru, and then compare the corresponding enthalpies of formation.
We find that the Mn-Ga substitution at the 4a site is energetically most favourable, for both compositions. For x = 0.0 we find ∆H = −0.26 eVf.u.−1 and for x ≈ 0.3 the ∆H = −0.15 eVf.u.−1. In both cases the most favorable Ru substitution does not improve the stability of the compound with respect to the stoichiometric composition. Increasing Ru content in the supercell promotes the Mn-Ru defect stability, which makes the Mn-Ga substitution favourable only for the compounds with a low Ru fraction. In figure 3.7 we illustrate the results obtained for the Mn2Ga supercell. In the absence of
Ga defects the lowest energy structure has the theoretically predicted volume, VDFT ≈
46.4 ˚A, and c/a ≈ 0.87. The in-plane lattice constant of the MRG films is pinned by the MgO substrate to 5.959 ˚A, which leads to the tetragonal distortion. Even after taking into account the substrate strain, the theoretical structure is more stable than the experimentally observed one, by ∆E ≈60 meVf.u.−1. Substituting 1/3 of the Mn 4a sites by Ga and taking the experimental structure results in a net energy gain of 200 meVf.u.−1, compared to the theoretically predicted unit cell. The formation of defects is therefore energetically more favourable than having a stoichiometric MRG phase.