# Replacement of Cobalt in Lithium-Rich Layered Oxides by n-Doping: A DFT Study
## 3 Results
### Crystal Structures of Cobalt Doped LRLO
We considered the following equation:  
\[\sigma^{2} = \frac{\sum_{i=1}^{N}[(x_{i}-x^{\prime}_{i})]^{2}+[(y_{i}-y^{ \prime}_{i})]^{2}+[(z_{i}-z^{\prime}_{i})]^{2}}{N} \tag{1}\]  
where \((x_{i}\,y_{i}\,z_{i})\) are the fractional coordinates of each atomic species in the site \(i\) within the supercells of the LNMC12, LNMC08, LNMC04, and LNM structures; \((x^{\prime}_{i}\,y^{\prime}_{i}\,z^{\prime}_{i})\) are the fractional coordinates of the same atomic site \(i\) in the ordered LiCoO\({}_{2}\) supercell; and \(N\) is the total number of atoms with the same atomic identity. The displacement factor is an evaluation of the mean buckling of the layers, with respect to the perfectly planar arrangement in the LICoO\({}_{2}\) lattice. We have calculated mean values of \(\sigma^{2}\) for Li, LiTM (lithium ions in the TM layer), Ni, Mn, Co, and O for each composition, as shown in Figure 2.  
All atomic species in LRLO supercells show large and similar displacements compared to LiCoO\({}_{2}\), ranging between 0.0029 and 0.00023. In particular, whereas the oxygen displacement is almost constant and insensitive with changes in the cobalt content, differences are observed for all other metals. The reduction of the cobalt content from LNMC12 to LNM increases the displacement of the Li\({}^{+}\) ions in the lithium layers of approximately 600%, and in the TM layers (+770%). Similar increases are observed also for Ni (+594%) and Mn (+450%). On passing, it is interesting to observe that passing from LNMC4 to LNM the bucking of all atomic species seems to slightly decrease. Therefore, the LNMC4 composition shows the mostly buckled layers among all supercells. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Supercells**} \\ \cline{2-5}  & **LNMC12** & **LNMC08** & **LNMC04** & **LNM** \\ \hline a (Å) & 2.893 & 2.902 (2.874) & 2.907 (2.893) \\ c (Å) & 14.430 & 14.477 (14.28) & 14.500 (14.292) \\ V (Å\({}^{3}\)/f.u.) & 34.87 & 35.21 (34.24) & 35.38 (34.72) \\ \(\alpha=\beta\) (\({}^{\circ}\)) & 90 & 90 & 90 & 90 \\ \(\gamma\) (\({}^{\circ}\)) & 120 & 120 & 120 & 120 \\ a/c & 0.200 & 0.200 & 0.200 & 0.200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hexagonal hR12-reduced cell parameters of the optimized supercells. Available experimental literature values are reported in parentheses [54; 56; 58]. In summary, the removal of cobalt and the simultaneous p-doping of the LRLO lattice induce larger cell volumes, originating from expanded distances among stacked planes, and a parallel increase of the layer buckling. Turning to the interatomic bonds, we analyzed the distances between transition metals and the first neighbor's oxygen atoms d\({}_{\text{M-O}}\): all transition metals are surrounded by six oxygen atoms, forming distorted octahedra. The distribution of the M-O (M = Ni, Mn, and Co) bond distances is shown in Figure 3 in the form of Pair Distribution Functions (PDFs). Qualitatively PDFs are similar for all compositions, being that MnO\({}_{6}\) octahedra very close in dimensions to the CoO\({}_{6}\), whereas NiO\({}_{6}\) octahedra are larger. The reduction of the cobalt content from LNMC12 to LNM leads to a slight monotonic increase in the Mn-O bond distances, from 1.949 A to 1.951 A, as well as a slight decrease for Ni-O, from 2.058 A to 2.048 A. It is remarkable to observe that the reduction of cobalt in the structure occurs in parallel with the sparse elongation/compression of few Ni-O and Mn-O bonds. This  
Figure 3: Pair Distribution Functions (PDFs) of LNMC12 (upper panel), LNMC08 (middle panel), LNMC04 (middle panel), and LNM (bottom panel) for Mn-O, Ni-O, and Co-O bond distances. Figure 2: Mean (a-dimensional) displacements factors \(\sigma^{2}\) calculated on fractional coordinates for each atomic species in LNMC12, LNMC08, LNMC04, and LNM supercells. may be a clue of the appearance/increase of Jahn-Teller distortions in the LRDO lattices induced by the substitution of Co with a blend of Mn/Ni (see below for more details and discussions in the electronic structure section) [60].