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Condens. Matter 2019, 4, 80 7 of 12 the Bohr magneton; only magnitudes are given). The net magnetic moment of the supercell is zero, as required for an AF order. The magnetic moment of Fe is close to the anticipated value of 4 μB (spin-only contribution). The magnetic moments of oxygen ions are nearly negligible. Concerning the GB configurations, both C1 and C2 have average Fe and O magnetic moments that are nearly the same as those in bulk within 0.02 μB. In the case of C1, Fe atoms close to the interface have magnetic moments that are about 0.12 μB smaller than those farther from the interfaces (magnitude is considered). This is related to the effect of diminished charge at such Fe ions. When C2 is considered, Fe atoms with sixfold coordination at the GB interfaces have their magnetic moment with the same value as that in the bulk regions, but those with a lower coordination number have their moment lowered by about 0.02 μB compared to the average value. This indicates that a better coherency of C2 is also detectable via the magnetic moments. The net magnetic moment of the whole supercell is negligible (~0.01 μB) for both configurations, confirming an AF magnetic state. 3.2. Delithiated System In the case of a delithiated system, i.e., FPO4 or FP, we proceed in a similar way to that employed for LFP. Even if the lattice parameters a, b, and c do not change much when going from LFP to FP by removing Li atoms, the c/a ratio important for CSL construction varies more significantly (increases by ~7%). This change affects the tilt angle, becoming now ~51.4◦, and increases the Σ parameter to 4, since Σ 3 yields poor coincidence (certainly, the tilt angle and Σ are correlated). The result is a near-CSL symmetrical tilt Σ4 (101)/[010] GB for FP [32]. For this reason, there are more atoms in the boxes/supercells than before, even if Li is not present. The supercells now contain 192 atoms (32× Fe, 32× P, 128× O). Nevertheless, a comparison of constructed GBs in LFP and FP still makes sense, since the GB planes (and tilt axes) are the same, and thereby, the geometrical relationship of the lower and upper blocks (expressed by the tilt angle) is very similar in both cases. However, we have no atoms/cations lying on the GB planes—in contrast to the LFP case, which may pose some problems related to the GB cohesion. In any case, this GB model for FP may be ideal to study Li GB diffusion at a later stage. As in the case of LFP, we construct a GB configuration shifted along [010] by b/2 (configuration 2) in order to improve the GB coherence compared to the configuration 1 obtained just by rotation along the tilt axis. The perfect bulk box was constructed and relaxed as well, and its ar, br, and cr translation vectors were used to make initial configurations of GBs similar to those for LFP. The magnetic order was again considered to be antiferromagnetic, and the directions of magnetic moments of Fe atoms were arranged in the same way as in LFP. While checking the atomic neighborhood of interfaces, one can notice that the rebonding of atoms at the bottom/top interface might be impeded by the matching plane of the rotated part being shifted by ar/2, which does not happen for the middle interface. This is an effect of having a Σ4 GB for FP, and it does not happen for the Σ 3 GB studied in LFP. Figure 3 displays en face the relaxed GB configuration 1 (C1) and configuration 2 (C2) of FP without performing box dimension optimization (it has length cr, as in the bulk box). The reason is that we need to examine first the GB structure before proceeding further, given the difficulty with the rebonding process mentioned above. Configuration 1 (Figure 3a) clearly has no bonds across the interfaces, showing also large interstitial space at these regions. In contrast, configuration 2 exhibits Fe–O bonding across the middle GB, whereas bonding across the bottom/top GB appears weak. This might indicate an insufficient GB cohesion in FP, at least for the GB type studied. Preliminary calculations show that increasing the last box dimension leads to upper and lower grain separation at both interfaces for C1. In the case of C2, only the bottom/top interface splits. Concerning the coordination of Fe atoms close to the interfaces, C1 shows fourfold or fivefold coordination at both interfaces, whereas in C2, Fe is fourfold or sixfold coordinated in the middle, and fourfold to sixfold coordinated at the bottom/top, which is similar to that observed in LFP. GB energies are calculated formally according to Equation (1). The results are reported in Table 1, and provide the first idea regarding interface energies for the delithiated system. These values correspond to boxes with the last dimension unrelaxed. We can see that C1 has a larger interface energy compared to C2, which is again an expected trend. Both γ’s—toPDF Image | First-Principles Grain Boundary Formation in the Cathode Material LiFePO4
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