Crystal Structure Influences Migration along Li and Mg Surfaces

Dendrite formation on Li metal anodes hinders commercialization of more energy-dense rechargeable batteries. Here, we use the migration energy barrier (MEB) for surface transport as a descriptor for dendrite nucleation and compare Li to Mg. Density functional theory calculations show that the MEB for the hexagonal close-packed structure is 40 and 270 meV lower than that of the body-centered cubic structure for Li and Mg, respectively. This is suggested as a reason why Mg surfaces are less prone to form dendrites than Li. We show that the close-packed facets exhibit lower MEBs because of smaller changes in atomic coordination during migration and thereby less surface distortion.


Simulation details
The density functional theory (DFT) calculations were performed with the Vienna ab initio Simulation Package (VASP) S1,S2 and the Li sv and Mg projector augmented wave (PAW) S3 pseudopotentials supplied with VASP.5.4. Plane waves were expanded up to cutoff energies of 500 eV for Li and 350 eV for Mg and the PBEsol functional was used. S4 The bcc, fcc and hcp bulk structures of Li and Mg are relaxed to within 1 meV/Å. The obtained lattice constants are given in Tab. S1. direction. This corresponds to a 4x4 or 5x5 geometry in the xy-plane, and more than 8 layers in the z-direction, as shown in Fig. S1 for Li bcc (001). Five images are used in the NEB calculations with a spring constant of 5 eV/Å between them, and the max force in each image has converged to less then 10 meV/Å.
S-2 Figure S1: The supercell of Li bcc (001), indicating the vacuum, and the free and fixed layers.

Convergence testing
Convergence testing on the supercells were performed to ensure a "correct" behaviour of the surface. The atoms in the four uppermost layers of the supercells were allowed to move freely, while the rest of the atoms were fixed in space, mimicking the bulk in a real crystal.
The convergence of the atomic positions of the free layers are shown in Fig. S2 as a function of total number of layers in the supercell.
In addition to the number of layers in the supercell, the extent of the xy-plane is important for realistic MEBs. If the surface is too small, the migrating atom will feel itself across the periodic boundaries, and thus affect the MEBs. Figure S3 shows the dependence of the MEB on the xy-plane area for the minimum energy path on Li bcc (001).

Surface Energy Calculations and Adaptive Common Neighbour Analysis
The surface energies of the different facets were calculated using: where E slab is the energy of the slab, E bulk is the energy of the bulk per atom, n is the number of atoms in the slab and A is the surface area. The obtained surface energies are displayed in Tab. S2. The adaptive common neighbour analysis (cna) was performed using Ovito. S16 Tab. S3 shows the number of atoms in a supercell that has changed its structure from the stable position to the saddle point, normalized over the number of atoms that are free to move.
A positive number means that the atoms in the saddle point structure has changed their structure to something different than the bulk. Oppositely, a negative result signifies that the atoms of the stable position was recognized in a different structure, and changed to the bulk structure at the saddle point.

Density of States Calulations
The electronic density of states (DOS) was investigated for the stable and saddle positions for every migration path using VASP. The resulting DOS of the stable and saddle positions of the minimum energy path on the bcc (001) surface for Li and Mg is shown in Fig. S5.
Based on the DOSes, the difference between the stable and saddle position was calculated with: where dos E,i is the density of states at energy E, normalized with respect to the Fermi energy, of position i (i.e. the stable or saddle position). The ∆DOS was calculated for two different energy ranges; one summed over the valence band, termed 'vDOS', up to the Fermi energy, and one over the s-and p-bands, termed 'all'.

S-7
(a) (b) (c) (d) Figure S5: The DOS of a) and c) the stable, and b) and d) saddle position of the minimum energy path on the Li (a) and b)) and Mg (c) and d)) bcc (001) surface