Defects, Dopants and Lithium Mobility in Li9V3(P2O7)3(PO4)2

Layered Li9V3(P2O7)3(PO4)2 has attracted considerable interest as a novel cathode material for potential use in rechargeable lithium batteries. The defect chemistry, doping behavior and lithium diffusion paths in Li9V3(P2O7)3(PO4)2 are investigated using atomistic scale simulations. Here we show that the activation energy for Li migration via the vacancy mechanism is 0.72 eV along the c-axis. Additionally, the most favourable intrinsic defect type is Li Frenkel (0.44 eV/defect) ensuring the formation of Li vacancies that are required for Li diffusion via the vacancy mechanism. The only other intrinsic defect mechanism that is close in energy is the formation of anti-site defect, in which Li and V ions exchange their positions (1.02 eV/defect) and this can play a role at higher temperatures. Considering the solution of tetravalent dopants it is calculated that they require considerable solution energies, however, the solution of GeO2 will reduce the activation energy of migration to 0.66 eV.

assessment of the quality and efficacy of the classical pair potentials (refer to Table S1 in the supplementary information for the potentials parameters used and method section for the detailed description of the methodology) used in this study. The calculated equilibrium lattice constants (tabulated in Table 1) are in excellent agreement with experiment.

Intrinsic defect processes.
To understand the electrochemical behavior of an electrode material, intrinsic defect processes are crucial. A series of isolated point defect (vacancy and interstitial) energies were calculated, which were combined to determine the formation energies for Frenkel and Schottky-type defects in Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 .
The following equations represent the reactions involving these defects as written using Kröger-Vink notation 27 . Li The reaction energies for these intrinsic defect processes are reported in Fig. 2 and Table S2. The most favorable intrinsic disorder is Li Frenkel and the formation of other Frenkel and Schottky defects is unfavourable. The second most favorable defect process is calculated to be anti-site. This indicates that there will be a small percentage of Li on V sites ( ″ Li V ) and V on Li sites ( ⋅⋅ V ) Li particularly at higher temperatures. It should be noted that this defect has been observed in a variety of Li ion battery materials during cycling 8,28-32 . The formation enthalpy of  Lithium ion-diffusion. The intrinsic lithium ion diffusion of Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 material is of crucial importance when assessing its use as a possible high-rate cathode material in lithium batteries. Using static atomistic simulation it is possible to examine various possible diffusion paths responsible for lihium ion conduction, which are often difficult to explore on the atomic scale by experiment alone. For the Li vacancy migration, we identified three lower energy long range paths connecting local Li hops (A, B C and D as shown in Fig. 3). There are two long range paths exhibit a zig-zag pattern along ab plane including a local Li hop with lower activation energy of migration of 0.38 eV but with overall activation energy of 1.07 eV (refer to Table 2 and Fig. 4). The third long range migration path along the c axis has identical Li hops with the activation energy of 0.72 eV. Thus this long range Li diffusion channel will have the overall activation energy of 0.72 eV in good agreement with the value of 0.74 eV reported by Jain et al. 26 . The activation energy of migration calculated along the ab plane is 1.30 eV 26 , which is in agreement with our calculated value of 1.07 eV. The difference in activation energy is due to description of ions in different methodologies. Here the activation energy of migration is defined as the position of the highest potential energy along the migration path. This indicates that long range diffusion is likely slow.
Tetravalent doping. The Li Frenkel is calculated to be only 0.44 eV/defect; however, an increase in the concentration of Li will further increase the applicability of Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 as a cathode material for rechargeable lithium batteries. A way to increase the content of intrinsic defects in oxides is by the solution of aliovalent  dopants as it was previously demonstrated in CeO 2 (for example ref. 33 and references therein). Here we considered the solution of RO 2 (R = Ce, Zr, Ti, Si and Ge) via the following process (in Kröger-Vink notation): Figure 5 reports the solution energies of RO 2 and it can be observed that GeO 2 and ZrO 2 have the lowest ones 2.40 eV and 2.42 eV respectively. These solution energies are higher as compared to the Li Frenkel process nevertheless the solution of GeO 2 or ZrO 2 during synthesis should be examined experimentally as they can increase the Li vacancy concentration (via relation (8)). Figure 6 depicts the local coordination (including bond lengths and angles) with oxygen of the dopants occupying the V site and for comparison the octahedral VO 6 unit in the relaxed structure of undoped Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 . The ionic radius of V 3+ in octahedral coordination is 0.64 Å. The ionic radius of Si 4+ is 0.38 Å smaller that that of V 3+ . In the SiO 6 unit, there are two shorter bonds present compared to the other four Si-O bonds. This indicates that Si prefers SiO 4 unit as observed in most silicates and this is reflected in the solution energy. The lowest solution energy is calculated for Ge. There are six Ge-O bonds present with approximately equal bond distances. The bond distances are ~0.1 Å shorter than the V-O bond lengths. Though Ge forms tetrahedral coordination in most of the complexes, the exact reason for the lowest solution energy should be due to other factors. The solution energy of Ti is ~0.40 eV higher than that of Ge. The second lowest solution energy is found for Zr. Zirconium normally forms octahedral six-coordinate complexes in their crystal structures and its ionic radius is closer to the ionic radius of V 3+ . This is reflected in the solution energy. In the relaxed structure of CeO 6 unit, Ce-O bond lengths are approximately the same but ~0.20 Å longer than V-O bond lengths present in VO 6 unit. Furthermore, the ionic radius of Ce 4+ is 0.26 Å longer than V 3+ . Thus the solution energy   is slightly high. However, the current solution energy values are still large and positive indicating that they are highly unfavourable. Introducing dopants in a lattice can also have an impact on the activation energies of migration. We present in Fig. 7 the energy profile diagrams for Li vacancy hoping closer to the Ge and Zr substitutionals as these are the lowest solution enthalpy dopants. The presence of the Ge and Zr substitutionals will increase the migration energy barriers of Li in the ab plane, but will reduce them in the c-axis mechanism where it matters as it is the lowest energy mechanism (refer to Figs 4 and 7). The activation energy of Li migration in the vicinity of Ge substitutionals is 0.66 eV that is 0.08 eV lower than in undoped Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 .
Summary. In the present study, the atomistic simulation techniques have been used to provide detailed insights into intrinsic defects, lithium ion mobility and tetravalent doping, which are relevant to the general electrochemical behavior of layered Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 as lithium battery cathodes. An advantage of this material is its low energy Li Frenkel (0.44 eV/defect). This will ensure that there will be considerable number of Li vacancies that are necessary as they act as vehicles for Li diffusion. We have considered the solution energies of RO 2 (R = Ce, Zr, Ti, Si and Ge) and calculated that GeO 2 and ZrO 2 have the lowest solution energies. These are far higher than the Li Frenkel process. At any rate if Li 9 V 3 (P 2 O 7 ) 3 (PO 4 ) 2 doped with GeO 2 is synthesized it will have a lower  Methods. In order to calculate the energetics for the formation of intrinsic defects and possible Li ion diffusion pathways, the classical pair potential method as implemented in the GULP package 34 was employed. This method is based on the classical Born model description of an ionic crystal lattice. All systems were treated as crystalline solids with interactions between ions consisting of the long-range attractions and short-range repulsive forces representing electron-electron repulsion and van der Waals interactions. The short range interactions  Table S1). Simulation boxes and the corresponding atom positions were relaxed using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm 35 . The Mott-Littleton method 36 was used to investigate the lattice relaxation about point defects and the migrating ions. It divides the crystal lattice into two concentric spherical regions, where the ions within the inner spherical region (on the order of >700 ions) immediately surrounding the defect relaxed explicitly. Li ion diffusion was calculated considering two adjacent vacancy sites as initial and final configurations. Seven interstitial Li ions were considered in a direct linear route and they were fixed while all other ions were free to relax. The local maximum energy along this diffusion path is calculated and reported as activation energy of migration. As the present model assumes a full charge ionic model with the calculations corresponding to the dilute limit the defect enthalpies will be overestimated, however, relative energies and trends will be consistent.