Thermodynamic and kinetic properties of the Li-graphite system from first-principles calculations

We present an ab initio study of the thermodynamics and kinetics of ${\text{Li}}_{\text{x}}{\text{C}}_6$, relevant for anode Li intercalation in rechargeable Li batteries. In graphite, the interlayer interactions are dominated by Van der Waals forces, which are not captured with standard density-functional theory (DFT). By calculating the voltage profile for Li intercalation into graphite and comparing it to experimental results, we find that only by correcting for vdW interactions between the graphene planes is it possible to reproduce the experimentally observed sequence of phases, as a function of Li content. At higher Li content the interlayer binding forces are increasingly due to Li-C interactions, which are well characterized by DFT. Using the calculated energies, corrected for the vdW interactions, we derive an ab initio lattice model, based on the cluster-expansion formalism, that accounts for interactions among Li ions in ${\text{Li}}_{\text{x}}{\text{C}}_6$ having a stage I and stage II structure. We find that the resulting cluster expansions are dominated by Li-Li repulsive interactions. The phase diagram, obtained from Monte Carlo simulations, agrees well with experiments except at low Li concentrations as we exclude stage III and stage IV compounds. Furthermore, we calculate Li migration barriers for stage I and stage II compounds and identify limiting factors for Li mobility in the in-plane dilute as well as in the high Li concentration range. The Li diffusivity, obtained through kinetic Monte Carlo simulations, slowly decreases as a function of Li content, consistent with increasing Li-Li repulsions. However, overall we find very fast Li diffusion in bulk graphite, which may have important implications for Li battery anode optimizations.

Figure 1

(Color online) (a) 001 Li-C stacking in stage II (left) and (b) stage I (right). Note that in stage II, because of the AABB alternating graphite plane sequence, the Li atoms are not quite aligned in the 001-direction.

Figure 2

(Color) First-principles calculated ground states from combined stage I (red circles) and stage II (blue squares) compounds. Stage II compound energies are shifted down by the vdW correction (20 meV/C) for every empty layer. The convex hull including both stage I and stage II compounds is indicated by the black line. It should be noted that the $x=0.25$ compound is very close to but not actually on the hull.

Figure 3

(Color online) (a) In-plane Li ordering in fully lithiated stage I and stage II (left) and (b) $2\times 2$ Li ordering (found in stage ${\text{II}}^{\prime }$) (right) seen from above on the carbon honeycomb lattice.

Figure 4

(Color online) Pairwise ECI for the stage I CE illustrated on the honeycomb carbon lattice. Filled circles represent in-plane interactions while open circles represent next-plane interactions. The numbers denote the magnitude of the ECIs in meV.

Figure 5

(Color online) Pairwise Li-Li ECI for the stage II CE illustrated on the honeycomb carbon lattice. Filled circles represent in-plane interactions while open circles represent next-plane interactions. The numbers denote the magnitude of the ECIs in meV.

Figure 6

(Color) Voltage profile for the Li-graphite system obtained by standard DFT data (red line), DFT data corrected for vdW interactions (blue line), compared to experiments (black curve) (Ref. 2).

Figure 7

First-principles phase diagram obtained from Monte Carlo simulations based on stage I and stage II separate CEs. The phase regions are denoted in the following way: G (graphite), II (stage II), IID (disordered stage II), I (stage I), ID (disordered stage I), and ${\text{II}}^{\prime }$ (stage II with $2\times 2$ Li ordering—see Fig. 3b. Different symbols refer to cooling and heating runs, respectively.

Figure 8

(Color) [(a) and (b)] The picture to the left (a) schematically illustrates an in-plane Li hop from one site to an adjacent one and (b) shows the corresponding energy barriers for this path in dilute stage II and stage IV calculated from first principles for different $c$ lattice parameters (equilibrium, 10% expanded and 10% contracted interlayer distance).

Figure 9

(Color) [(a) and (b)] The pictures schematically illustrate how the Li moves in-plane from one site, through an adjacent nearest-neighbor site, to a vacancy—the left-hand “vacancy” hop in (a)—or a NN site—the right-hand NN hop in (b)—under high Li concentration conditions. The migrating Li are shown as highlighted blue circles and the trajectories are indicated by arrows and dashed circles. The NN hop is energetically extremely unfavorable, compared to the vacancy hop.

Figure 10

(Color) Energy barriers calculated from first principles in stage II (blue squares) and stage I (red circles), in the high Li content limit. The result clearly shows that the energy barrier in the high Li concentration limit is the same for both stage I, and stage II as well as independent of increased interlayer distance.

Figure 11

(Color) (a) Thermodynamic factor for stage I (red circles) and stage II (blue squares) compounds, calculated with Monte Carlo simulations. (b) The self-diffusivity for stage I (red circles) and stage II (blue squares) calculated with kinetic Monte Carlo simulations. Both stage I and stage II are calculated using the barriers for the high Li concentration limit ($E_{\text{h}}=283\text{meV}$ for stage I and $E_{\text{h}}=297\text{meV}$ for stage II). (c) The chemical diffusivity for stage I (red circles) and stage II (blue squares) obtained from the tracer diffusivity and the thermodynamic factor. Both stage I and stage II are calculated using the barrier for the high Li concentration limit ($E_{\text{h}}=283\text{meV}$ in stage I and $E_{\text{h}}=297\text{meV}$ in stage II). We emphasize that the pure ordered phases are only observed at their stoichiometric compositions and that the stable phase for $0.5<x<1$ is the two-phase mixture.