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Condens. Matter 2019, 4, 80 4 of 12 Condens. Matter 2019, 4 4 added or removed in the present GB configuration, which was constructed to be charge neutral (the charge neutral (the same holds for C1). The result of this transformation (including the relaxation of same holds for C1). The result of this transformation (including the relaxation of atoms and the last cell atoms and the last cell dimension) is shown in Figure 1b, and it is called configuration 2 (or C2). dimension) is shown in Figure 1b, and it is called configuration 2 (or C2). From the viewpoint of the From the viewpoint of the magnetic structure, we keep the magnetic moments opposite for Fe atoms magnetic structure, we keep the magnetic moments opposite for Fe atoms with y = 1/4 and 3/4, having with y = 1⁄4 and 3⁄4, having effectively an AF structure. In the relaxed structure C2, the coordination effectively an AF structure. In the relaxed structure C2, the coordination number of Fe atoms close to number of Fe atoms close to the GB reaches 4–6, whereas the C1 structure yields 4–5 only. This the GB reaches 4–6, whereas the C1 structure yields 4–5 only. This means that some Fe atoms close means that some Fe atoms close to the interfaces in C2 exhibit the same (octahedral) coordination as to the interfaces in C2 exhibit the same (octahedral) coordination as in the bulk regions. The oxygen in the bulk regions. The oxygen coordination is also modified at the GB region and increased slightly coordination is also modified at the GB region and increased slightly upon the shift performed to upon the shift performed to obtain configuration C2. The coordination of P ions is not affected at the obtain configuration C2. The coordination of P ions is not affected at the GB regions. Regardless of GB regions. Regardless of configuration, Li atoms are fivefold coordinated at all the GB interfaces. configuration, Li atoms are fivefold coordinated at all the GB interfaces. These findings indicate that These findings indicate that the presence of cations at the studied GBs is important in order to the presence of cations at the studied GBs is important in order to maintain cohesion at these interfaces. maintain cohesion at these interfaces. Figure 1. Σ 3 (101)/[010] Grain boundaries (GB) for lithium iron phosphate (LFP): (a) Configuration 1 Figure 1. Σ 3 (101)/[010] Grain boundaries (GB) for lithium iron phosphate (LFP): (a) Configuration 1 (C1); (b) configuration 2 (C2). Color scheme is as follows: Li—yellow, Fe—blue, P—green, O—red; the (C1); (b) configuration 2 (C2). Color scheme is as follows: Li—yellow, Fe—blue, P—green, O—red; LFP unit cell is also shown as a dashed green cuboid. The GBs in the middle of the boxes are indicated the LFP unit cell is also shown as a dashed green cuboid. The GBs in the middle of the boxes are by tilted rectangles. The supercell translation vectors are also outlined. The grayish objects identify the indicated by tilted rectangles. The supercell translation vectors are also outlined. The grayish objects regions with the highest positron density (see the text for explanations). identify the regions with the highest positron density (see the text for explanations). The final C1 and C2 structures (shown in Figure 1a,b) were both relaxed with respect to atomic The final C1 and C2 structures (shown in Figure 1a,b) were both relaxed with respect to atomic coordinates as well as the supercell last dimension. This procedure is now explained in more detail as coordinates as well as the supercell last dimension. This procedure is now explained in more detail follows. The ‘bulk’ supercell (not shown here), in which the upper part/grain is not rotated, is relaxed as follows. The ‘bulk’ supercell (not shown here), in which the upper part/grain is not rotated, is with respect to all its cell dimensions (including atomic positions). The corresponding lattice vectors are relaxed with respect to all its cell dimensions (including atomic positions). The corresponding lattice ar,br,andcr anditholdsapproximatelythatar ≈a−c,br ≈b,andcr ≈6c. Inthisway,weobtainthe vectors are ar, br, and cr and it holds approximately that ar ≈ a–c, br ≈ b, and cr ≈ 6c. In this way, we supercell containing no GB/interface, and this supercell is used as a reference for GB energy calculations obtain the supercell containing no GB/interface, and this supercell is used as a reference for GB (the corresponding total energy is denoted as Etot(bulk)). First, two box dimensions along cell vectors energy calculations (the corresponding total energy is denoted as Etot(bulk)). First, two box ar and br are taken to be the corresponding dimensions of supercells for configurations C1 and C2. dimensions along cell vectors ar and br are taken to be the corresponding dimensions of supercells The last cell dimension, called further zc, was taken initially to be cr, and its size was then optimized to for configurations C1 and C2. The last cell dimension, called further zc, was taken initially to be cr, obtain the lowest energy, Etot(GB). The final/equilibrium value of zc is larger than cr, since the cells and its size was then optimized to obtain the lowest energy, Etot(GB). The final/equilibrium value of zc is larger than cr, since the cells expand along the last cell vector (the adaptation of the structure at the grain interfaces takes normally larger volume compared to bulk). Whereas dimensions parallel toPDF Image | First-Principles Grain Boundary Formation in the Cathode Material LiFePO4
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