First-principles study of the impact of grain boundary formation in the cathode material LiFePO4

A key challenge for developing efficient lithium ion batteries is to preserve homogeneous Li flows. Li inhomogeneity can appear at internal interphases within the lithium ion battery. Grain boundaries that result from heat treatments aimed to improve phase crystallinity of LiFeO4 cathodes can therefore hinder the homogeneity of the Li distribution. In fact, delithiation produces Li agglomeration at the grain boundaries. In this paper, we present first principles calculations rationalizing the impact of grain boundaries on the electrochemical performance of the LiFeO4 battery cathodes.


Introduction
LiFePO4 (LFP) or triphylite [1] is an environmental friendly material used to make cathodes of the lithium ion battery (LIB). It has and orthorhombic structure containing 28 atoms per unit cell, where the Li atoms can diffuse in and out while the LIB works. A major challenge for LIB is their ageing produced by several factors that have a complex hierarchical structure on various length and time scales. Due to these issues, in spite of several decades of research, the understanding of battery ageing remains challenging. Recent investigations [2][3][4][5][6] aimed at studying LFP have combined x-ray spectroscopy and theoretical modeling to monitor the evolution of the redox orbitals in nanoparticles and single-crystal LFP cathodes under different lithiation levels. These studies have provided advanced characterizations techniques for cathodes such as general methods for understanding the relation between lattice distortions and potential shifts [5]. Since LIB cathodes are made of powders, grain surfaces interfaces come to play. Grain boundaries (GB) (between the grains of the same material) are the simplest interfaces that can be studied using reliable Density Functional Theory (DFT) calculations. Recently Lachal et al. [7] have surmised that strong chemical delithiation kinetic degradation in LFP can be explained by grain boundaries that drastically hinder the lithium mobility. In this paper, we will present theoretical evidence based on DFT calculations supporting this hypothesis.

Results
There can be many types of GBs for a given material. We will consider just planar GB with periodic boundary conditions that can be handled with ab initio electronic structure codes (classical molecular dynamics could in principle be performed too if appropriate interatomic potentials are available). A geometrical method called 'coincidence site lattice' (CSL) allows the construction of various GB boxes suitable for ab initio calculations [8]. Thus, CSL yields elemental GBs consisting in tilt and twist GBs. Next, we will consider a tilt GB in LFP. CSL works well for cubic systems where the coincidence (in CSL sense) can be made exact. However, this is not the case of lower symmetry structures (like orthorhombic LFP). For such a case, the coincidence can be made approximate only. is at 8d positions. The GB construction starts by selecting the tilt axis. One can take [010] (perpendicular to the structure mirrors at y = ¼ and ¾). Then by trying various rotation angles, a reasonably good coincidence occurs at ~48 degrees, which corresponds to the (101) as the GB plane. This correspond to a Σ3 GB since the CSL unit cell volume is 3 times that of LFP unit cell volume. The total volume of the GB box is then 6 times the LFP unit cell volume (the lower part is un-rotated and the upper part is rotated by ~48 degrees along the [010] axis. In this way, each box contains two GB boundaries (one is in the middle and the other one at the upper and lower faces). The result called configuration 1 is shown in Fig 1(a). The GB box (unit cell) is not anymore orthogonal and contains 168 atoms. We have used the following color scheme: Li-gray, Fe-blue, P-red, O-yellow; the LFP unit cell is also shown in green. Since, during the GB structure construction, the crystal is cut along some atomic plane and joined with other part of the crystal at different orientation (though the atomic plane is the same), atoms at the interface might have problems to form bonds with surrounding atoms (for metals this is not an important issue). Therefore, in addition to atomic relaxations it might be necessary to shift grains along the GB plane by some amount in a given direction to find an optimal interface. A closer inspection of the GB suggests that a shift by b/2 along [010] direction could be appropriate to avoid that some Fe atoms are too close. The result of this transformation is shown in Fig. 1(b) and it is called configuration 2. Interestingly, no atoms are added or removed in the present GB. Once the crystal is cut, then one part is rotated and then reconnected to the other part. Magnetic structures should be taken into account since the non-magnetic LFP unit cell relaxations lead to an equilibrium cell volume which is about 9% smaller than the experimental one whereas magnetic calculations (ferromagnetic and antiferromagnetic) lead to a volume which is just 2% larger than experiment. The antiferromagnetic arrangement gives the lowest energy for the bulk materials. After sorting out questions concerning the bulk LFP magnetic structure, one can continue with the GB relaxations. Even if the lattice parameters a and b do not change much when going from LFP to FP by removing the Li atoms, the c/a ratio change more significantly, which results in a Σ4 (101)/[010] GB rather than Σ3 we have for LFP. For this reason, there are more atoms in the boxes than before even if Li is not present. The boxes now contain 192 atoms (32x Fe, 32x P, 128x O). Nevertheless, a comparison of such GBs in LFP and FP makes still sense since the GB planes (and tilt axes) are the same and thereby the type of bonding at the GB is also the same (or should be). Figure 2 shows the relaxed GBs configuration 1 and configuration 2 of FP.
The magnetic calculations for all GBs are quite challenging. The AF order in the bulk becomes ferrimagnetic since the GB disturbs the magnetic order. In fact, the self-consistent cycle implemented in VASP can bring far the solution from the initial wave functions. The final point in the GB relaxation consist in manually relaxing the box/cell height. The minimum along this relaxation path corresponds to the equilibrium volume as shown in Fig. 3.
When the GB is relaxed, one needs a bulk supercell as a reference to calculate the GB (or interface) energy. Then, the interface energy is given by relation γ = (Etot(GB) -Etot(bulk)) / 2A, where A is the area of the GB in the supercell/box. Using the dimensions of the small relaxed bulk A can be calculated as b(a 2 +c 2 ) 1/2 . The factor 2 comes from the fact that there are two GBs in the box/supercell. Our results on GB interface energy are given in Table 1.    Table 1 confirms that configuration 2 has lower interface energy than configuration 1 which is expected according to our previous discussion. Nevertheless, this difference is only 2 meV/Å 3 for LFP and 3 meV/Å 2 . The table expresses the interface energy values in J/m 2 as well since these units are more frequently used in the literature (1.0 J/m 2 corresponds to 62.4 meV/ Å 2 ). Typical GB interface energies varies from 0.3 to 2.0 J/m 2 (i.e. from 20 to 125 meV/ Å 2 ). Therefore, LFP is closer to the lower end of the interface energy spectrum. One should keep in mind that twin GBs and stacking faults have even lower interface energies. In configuration 1 the LFP GB is 7 meV/ Å 2 lower in surface energy than the FP GB while the same difference is 8 meV/Å 2 . This numbers clear demonstrate that Li ions have a significant affinity for the GBs. Therefore, GB can trap Li leading to lithium mobility degradation and to impedance enhancement at the battery cathode [7].
Finally, we expect that positron annihilation lifetime spectroscopy (PALS) can facilitate the understanding of the Li atom transport. In fact, a positron in bulk FP occupies one-dimensional channels where Li is located in LFP [11,12]. Moreover, the positron lifetime changes dramatically with lithiation passing from 170 ps in LFP to 207 ps in FP. PALS can also provide useful information regarding GB formation since positrons like Li ions have positive affinities to GB and their lifetime consequently varies in presence of GBs.

Methods
The GB energies were calculated using the Vienna ab-initio simulation package (VASP) with the projector-augmented wave (PAW) method and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional within the spin-polarized generalized gradient approximation (GGA). PAW potentials with valence electron configurations 2s 1 for lithium, 3p 6 3d 7 4s 1 for iron, 3s 2 3p 3 for phosphorus and 2s 2 2p 4 for oxygen were used. The energy cutoff was set to 400eV in calculations involving cell shape relaxations, whereas 300eV was used in cases where the cell size and shape remained fixed. A 3×3×2 Monkhorst-Pack k-point grid and Gaussian smearing with width 0.1eV was used to sample the Brillouin zone. The criterion for energy convergence was 0.1meV and in geometry optimizations the forces were converged within 0.01eV/Å. The geometry of a bulk cell with the same number of atoms as the GB cell was first optimized to get the lengths of two-unit cell vectors, which were kept fixed in the GB cells. In the GB cells, the third vector length was varied to find the cell size with minimum energy, which corresponds to equilibrium GB energy. The cell shape and size was kept fixed, and the atomic positions were allowed to relax in the GB calculations.

Conclusions
Our DFT calculations show that LFP GBs can affect the Li ions transport by blocking the ion propagation. Moreover, this Li-ion accumulation increases the stress at the GB interphase and can lead to cracks in the battery cathode. Our DFT results provide a solid foundation to understand GB formation and the impact of this effect on the impedance and state of health of the battery. Therefore, this preliminary study motivates future research focusing on important GB issues affecting battery performance and ageing.