Determination of Formation and Ionization Energies of Charged Defects in Two-Dimensional Materials

We present a simple and efficient approach to evaluate the formation energy and, in particular, the ionization energy (IE) of charged defects in two-dimensional (2D) systems using the supercell approximation. So far, first-principles results for such systems can scatter widely due to the divergence of the Coulomb energy with vacuum dimension, denoted here as $L_z$. Numerous attempts have been made in the past to fix the problem under various approximations. Here, we show that the problem can be resolved without any such assumption, and a converged IE can be obtained by an extrapolation of the asymptotic IE expression at large $L_z$ (with a fixed lateral area $S$) back to the value at $L_z=0$. Application to defects in monolayer boron nitride reveal that defects in 2D systems can be unexpectedly deep, much deeper than the bulk.

Figure 1

Calculated formation energy of charged $C_N^{-}$ and $C_B^{+}$ as a function of (a) supercell size $L_z$ (with a fixed lateral dimension: $S$ for $L_x\times L_y=6\times 6$) in 2D BN and (b) cube root of bulk volume in 3D BN. Insets show the local structures around the defects. Blue, pink, and gray balls are N, B, and C atoms, respectively. For bulk cubic BN, a $3\times 3\times 3$ Monkhorst-Pack mesh grid was used, and the dimension of the supercell is varied from 8, 16, 32, 48, 64, 96, 144, to 216 atoms.

Figure 2

Calculated IEs of (a) donors: $C_B$ and $V_N$, and (b) acceptors: $C_N$ and $V_B$, as a function of $L_z$ (with a fixed lateral dimension: $S$ for $6\times 6$). (c) and (d): schematic illustration of the corresponding electrostatic potentials. (e) and (f): acceptor state in $C_N^{-}$ at different $L_z=20$ and 70 Å, respectively. The shade areas at the top and bottom of panel (f) are the calculated defect states unphysically delocalized into the vacuum. The isosurface of the electron density is $1\times {10}^{-4}/{\alpha }_0^3$, where ${\alpha }_0$ is the Bohr radius.

Figure 3

Calculated IEs at different lateral dimensions ($S$ for $6\times 6$, $7\times 7$, $8\times 8$, and $9\times 9$) as a function of $L_z$. The insets show the value of $\overline{\mathrm{IE}(S,L_z)}=\mathrm{IE}(S,L_z)-\alpha /\sqrt{S}$ as a function of $L_z$. The converged ${\mathrm{IE}}_0\mathrm{s}$ (red-circle highlight) are deduced from the insets at $L_z=0$.

Figure 4

Ionization energies, calculated at finite lateral dimension $S$ with different ${\alpha }_i$, are shown as the error bars. Symbols are the averaged results. Dashed lines are the converged IEs for comparison.