First-principles calculations

In order to confirm the presence of the particular diffusion pathway along which the diffusion of Li ions is favored, the valence orbital of c-Si (diamond cubic) was calculated using first-principles calculations. The calculation cell was constructed to a 2 × 2 × 2 supercell structure prepared from the unit cell of a Si crystal. Periodic boundary conditions were enforced along the x-, y-, and z-directions. The calculations were performed with the plane-wave basis set and projector augmented wave (PBE-PAW) pseudopotentials^31,32, as implemented in Quantum Espresso³³. The k-point mesh was set to 8 × 8 × 8 using the Monkhorst–Pack scheme for the Brillouin zone sampling. The energy cutoff was 750 eV and convergence in the energy was achieved down to 10⁻⁴ eV.

To understand the mechanism of the Li diffusion and its orientation dependency, we calculated the energy barriers required for the Li diffusion using the nudged elastic band (NEB) method³⁴ by considering the changes in the free energy of the Li ion during diffusion into c-Si with different crystallographic orientations. The NEB method adjusts the intentionally selected initially diffusion pathway of the Li ion so that the Li ion can travel along the pathway that costs the least energy. In order to calculate the energy barriers for the Li diffusion, it is therefore necessary to prepare the initial diffusion pathway of the Li ion and the c-Si through which the Li ion migrates. For this purpose, we first constructed two different c-Si slabs with their respective surface normal vectors parallel to <110> and <111> (hereinafter denoted as Si<110> and Si<111> slabs). The Si<110> and Si<111> slabs were modeled as the eight-layered (110) and (111) planes of c-Si, and a 30-Å-thick vacuum layer was added above the free surface. We then placed a Li ion in the c-Si slabs such that the initial diffusion pathway of the Li ion for NEB calculations was consistent with the trajectories of the Li pathways evaluated using the MD simulations. The Li diffusion pathways were optimized until the force on each atom was smaller than 0.05 eV/Å. The bottom two layers of the c-Si slabs were held fixed for all calculations.