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Calculation of electronic structures
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Catalysis-tunable Heusler alloys in selective hydrogenation of alkynes: A new potential for old materials

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The electronic structures were calculated using the full-potential linearized augmented plane wave method (36). The generalized gradient approximation developed by Perdew et al. (37) was used for the exchange-correlation potential. The plane wave cutoff was RKmax = 7.0, where R is the smallest atomic sphere radius and Kmax is the magnitude of the largest K vector. For the atomic sphere radius, we used 2.22 atomic unit (a.u.) for the 3d transition metals and 2.09 a.u. for Ga and Ge. To calculate the electronic structure of each Heusler alloy listed in table S2, we used the model structure with the space group given in the table. The optimized lattice constants are also listed in the table. The lattice vectors for Co2Mn0.5Fe0.5Ge and Co2FeGe0.5Ga0.5 are a′ = (a + b)/2, b′ = (−a + b)/2, and c′ = c, where a, b, and c are lattice vectors for the cubic lattice. We used a 20 by 20 by 20 K-mesh for the Brillouin zone integration for ternary alloys such as Co2FeGe. For quaternary alloys, the K-meshes were chosen so that the size of the K-mesh corresponds to that for the ternary alloys.

The most stable surface of Heusler alloys seems to be a close-packed (110) surface, which has been indicated by calculation for seven types of surfaces, as shown in table S3. To simulate a thin (110) film of Co2FeGe and Co2FeGa used for Fig. 5B, we considered a slab in a supercell specified by three primitive vectors: a′ = (−a + b)/2 = a(−0.5,0.5,0), b′ = c = a(0,0,1), and c′ = 7(a + b) = a(7,7,0), where a denotes the length of the side of the conventional unit cell of a cubic lattice. The film used was constructed on 11 layers with a total thickness of 5| c′|/14 and an empty (vacuum) region with a thickness of 9| c′|/14. We used a = 5.741 and 5.720 Å, which gave a minimum total energy for the bulk ferromagnetic Co2FeGe and Co2FeGa, respectively (see table S2). For both films, we estimated the equilibrium positions of the atoms in the direction of c′. The K-mesh adopted was 14 by 10 by 2. See (38) for more information.

The d-band center (εd) is defined as (20) (8)where D(ε) is the PDOS of the d-band at an energy, ε. We estimated the εd values from the sum of D(ε) for the transition metal components with the integration range from −13.000 to 6.457 eV, which was enough for a relative comparison of εd. The εd values were estimated from the bulk electronic structures because the calculation of the surface electronic structures for Co2Mn0.75Fe0.25Ge, Co2Mn0.25Fe0.75Ge, Co2FeGa0.25Ge0.75, and Co2FeGa0.75Ge0.25 made it difficult to assume appropriate supercells and slabs and would have very high calculation costs. The difference in DOS between the surface and the bulk originates from symmetry breaking at the surface, which indicates that materials with the same crystal structure and similar component elements make similar changes in the DOS from the bulk to the surface (39). Thus, the bulk εd values could be used for relative comparisons among our samples.

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