Li intercalation in graphite: A van der Waals density-functional study

Modeling layered intercalation compounds from first principles poses a problem, as many of their properties are determined by a subtle balance between van der Waals interactions and chemical or Madelung terms, and a good description of van der Waals interactions is often lacking. Using van der Waals density functionals we study the structures, phonons and energetics of the archetype layered intercalation compound Li-graphite. Intercalation of Li in graphite leads to stable systems with calculated intercalation energies of $-0.2$ to $-0.3$ eV/Li atom, (referred to bulk graphite and Li metal). The fully loaded stage 1 and stage 2 compounds ${\mathrm{LiC}}_6$ and ${\mathrm{Li}}_{1/2}{\mathrm{C}}_6$ are stable, corresponding to two-dimensional $\sqrt{3}\times \sqrt{3}$ lattices of Li atoms intercalated between two graphene planes. Stage $N>2$ structures are unstable compared to dilute stage 2 compounds with the same concentration. At elevated temperatures dilute stage 2 compounds easily become disordered, but the structure of ${\mathrm{Li}}_{3/16}{\mathrm{C}}_6$ is relatively stable, corresponding to a $\sqrt{7}\times \sqrt{7}$ in-plane packing of Li atoms. First-principles calculations, along with a Bethe-Peierls model of finite temperature effects, allow for a microscopic description of the observed voltage profiles.

Figure 1

Graphite phonon dispersion calculated with the optB88-vdW functional starting from the optimized equilibrium structure [62]. On top left and right an enlargement of the low-frequency $\Gamma$-A region and the Brillouin zone with the high-symmetry points are shown, respectively.

Figure 2

From top to bottom: phonon density of states (PhDOS) of stage 1 compound ${\mathrm{LiC}}_6$, stage 2 compound ${\mathrm{LiC}}_{12}$, pure graphite, and bcc Li metal. The red line gives the total PhDOS, and the blue and black lines give the contributions of, respectively, the carbon and lithium atoms to the normal modes.

Figure 3

The intercalation enthalpy $H_{\mathrm{int}}$, see Eq. (2), entropy $S_{\mathrm{int}}$, and free energy $G_{\mathrm{int}}=H_{\mathrm{int}}-T{S}_{\mathrm{int}}$, of the stage 1 compound ${\mathrm{LiC}}_6$ and the stage 2 compound ${\mathrm{LiC}}_{12}$, including the phonon contributions.

Figure 4

Examples of stage 2 compounds with different in-plane Li ordering: (a) ${\mathrm{LiC}}_{12}$ with $\sqrt{3}\times \sqrt{3}$ Li ordering, (b) ${\mathrm{LiC}}_{16}$ with $2\times 2$ Li ordering, (c) ${\mathrm{LiC}}_{24}$ with $\sqrt{7}\times 2$ Li ordering, and (d) ${\mathrm{LiC}}_{32}$ with $\sqrt{7}\times \sqrt{7}$ Li ordering. For simplicity, only AA stackings are shown.

Figure 5

The intercalation energy $H_{\mathrm{int}}$, see Eq. (2) of the dilute stage 2 compounds ${\mathrm{Li}}_x{\mathrm{C}}_6$, calculated with the optB88-vdW (red) and PBE-PBE (blue) functionals.

Figure 6

(a) Intercalation energy (eV/Li) versus concentration $x$ in ${\mathrm{Li}}_x{\mathrm{C}}_6$ structures (b) Zero-temperature (free) energy $G_0\left(x\right)$ (eV) of ${\mathrm{Li}}_x{\mathrm{C}}_6$ structures versus Li concentration $x$; (c) Calculated zero-temperature voltage profile of ${\mathrm{Li}}_x{\mathrm{C}}_6$ structures versus Li concentration $x$. (d) Intercalation free energy (eV/Li) versus concentration $x$ in ${\mathrm{Li}}_x{\mathrm{C}}_6$ structures at room temperature ($T=300$ K) (e) Room-temperature (free) energy $G\left(x\right)$ (eV) of ${\mathrm{Li}}_x{\mathrm{C}}_6$ structures versus Li concentration $x$; (f) Calculated room-temperature voltage profile of ${\mathrm{Li}}_x{\mathrm{C}}_6$ structures versus Li concentration $x$.