Energy landscape and diffusion kinetics of lithiated silicon: A kinetic activation-relaxation technique study

With large specific and volumetric capacity, lithiated silicon is an excellent anode for lithium-ion batteries. Its application is challenged today, however, by the formation of an amorphous $a-{\text{Li}}_x\text{Si}$ phase associated with a large volume change that occurs at relatively low Li concentration and remains only very partly understood at the microscopic level. In this paper, we characterize the full energy landscape associated with the onset of Li insertion in crystalline Si as a first step for understanding the lithiation process. We identify the diffusion mechanisms and migration energies for one to ten Li atoms in a Si crystal as well as the average lifetime of small lithium aggregates, using the kinetic activation-relaxation technique (kART), an off-lattice kinetic Monte-Carlo method with on-the-fly catalog building capabilities coupled to a newly developed force field (ReaxFF) used as potential based on ab initio results. We show that the short lifetimes of the bound states (from meV to ten meV) mean that Li atoms move in the interstitial sublattice with little interactions, explaining how high Li concentration in Si can be reached.

Figure 1

Diffusion mechanism for ${\text{1-Li}}_{Td}$ in c-Si. (a) Initial state: ${\text{1-Li}}_{Td}$ states. $E_f=-0.973\mathrm{eV}$. (b) Saddle point state: hexagonal position. $E_a=0.52\mathrm{eV}$. (c) Final state: ${\text{1-Li}}_{Td}$ states. $E_f=-0.973\mathrm{eV}$.

Figure 2

2 IV pairs formation with ${\text{1-Li}}_{\left(1\right)}$. Beige: Si atoms; purple: Li; blue: empty Si sites. (a) Initial state: ${\text{1-Li}}_{Td}$ states. $E_f=-0.973\mathrm{eV}$. (b) Saddle point state. $E_a=2.00\mathrm{eV}$. (c) Final state: 2 IV pairs+${\text{1-Li}}_{\left(1\right)}, E_f=0.777\mathrm{eV}$.

Figure 3

Representation of the dominant states for 2-Li in c-Si. Bond lengths are displayed according to the scale on the right of each panel. Beige: Si atoms; purple: Li. (a) State dimer $E_f=-2.004\mathrm{eV}$; $E_b=-0.057\mathrm{eV}$. (b) State ${\text{2-Li}}_{\left(2\right)} E_f=-1.978\mathrm{eV}; E_b=-0.031\mathrm{eV}$. (c) State ${\text{2-Li}}_{\left(3\right)} E_f=-1.959\mathrm{eV}$; $E_b=-0.012\mathrm{eV}$. (d) State ${\text{2-Li}}_{\left(4\right)} E_f=-1.956\mathrm{eV}$; $E_b=-0.009\mathrm{eV}$.

Figure 4

Evolution of the energy landscape associated with Li atoms only as a function of KMC steps for the two-Li-atom system. Left axis: activation energy for the available (green symbols) and selected (red symbols) events. Right axis: number of available activation energies per KMC steps (blue symbols). Grayed regions indicate the presence of a di-Li bound state.

Figure 5

Evolution of the di-Li system in a bound state (KMC steps 500 to 700). Top panel: energy histogram for the initial states with $\delta E$ measured from the ground state set at 0.0 eV. Middle panel: histogram of the corresponding activation energies. Bottom panel: energy histogram of the final states. Initials states are color coded: ${\text{2-Li}}_{\left(1\right)}$ in blue, ${\text{2-Li}}_{\left(2\right)}$ in red, ${\text{2-Li}}_{\left(3\right)}$ in green, ${\text{2-Li}}_{\left(4\right)}$ in yellow, and ${\text{2-Li}}_{\left(5\right)}$ or higher in cyan and purple. Colors are propagated throughout the activation energy and the final state histograms.

Figure 6

Representation of the ${\text{3-Li}}_{(2,2,2)}$ (left) trigonal and (right) ring states. Li occupied and empty sites are displayed in purple and red, respectively. Si atoms are represented in beige.

Figure 7

Evolution of the energy landscape associated with Li atoms only as a function of KMC steps for the three-Li-atom system. Left axis: activation energy for the available (green symbols) and selected (red symbols) events. Right axis: number of available activation energies per KMC steps (blue symbols). The gray and blue region indicated the presence of tri-Li and di-Li bound compounds, respectively.

Figure 8

Evolution of the di-Li system in a bound state (first 150 KMC steps). Top panel: energy histogram of the initial states with $\delta E$ measured from the ground state set at 0.0 eV. Middle panel: histogram of the corresponding activation energies. Bottom panel: energy histogram of the final states. Initials states are color coded. Blue: trimer ${\text{3-Li}}_{(1,1,2)}$; red: ${\text{3-Li}}_{(1,2,5)}$; green: ${\text{3-Li}}_{(2,2,2)}$ ring; first yellow bin at $0.013\mathrm{eV}$: ${\text{3-Li}}_{(1,2,3)}$; cyan: dimer+monomer; purple at $0.047\mathrm{eV}$: ${\text{3-Li}}_{(2,2,2)}$ trigonal; orange: di-Li bound compound. Colors are propagated throughout the activation energy and the final state histograms.

Figure 9

Representation of the (left) translational and (right) rotational mechanism for the 3-Li system. Li and Si atoms are displayed in purple and beige. The purple arrow points to the displacement of Li atoms throughout the entire mechanism.

Figure 10

Evolution of distance between 5 Li pairs out of 45 for the 10-Li system as a function of simulated 10. The horizontal dashed black corresponds to the $5\text{th}$ NN distance $\left(5.92\AA \right)$.

Figure 11

Evolution of the 10-Li system during the 993 KMC steps. Top panel: energy histogram for the initial states with $\delta E$ measured from the ground state set at 0.0 eV. Middle panel: histogram of the corresponding activation energies. Bottom panel: energy histogram of the final states. Initial states are color coded with colors assigned based on initial energy. Colors are propagated throughout the activation energy and the final state histograms.