Point defect formation energies in graphene from diffusion quantum Monte Carlo and density functional theory

Density functional theory (DFT) is widely used to study defects in monolayer graphene with a view to applications ranging from water filtration to electronics to investigations of radiation damage in graphite moderators. To assess the accuracy of DFT in such applications, we report diffusion quantum Monte Carlo (DMC) calculations of the formation energies of some common and important point defects in monolayer graphene: monovacancies, Stone-Wales defects, and silicon substitutions. We find that standard DFT methods underestimate monovacancy formation energies by around 1 eV. The disagreement between DFT and DMC is somewhat smaller for Stone-Wales defects and silicon substitutions. We examine vibrational contributions to the free energies of formation for these defects, finding that vibrational effects are non-negligible. Finally, we compare the DMC atomization energies of monolayer graphene, monolayer silicene, and bulk silicon, finding that bulk silicon is significantly more stable than monolayer silicene by 0.7522(5) eV per atom.

Figure 1

TB standard error in the TA-DMC pure MV formation energy in a $3\times 3$ supercell against the number of blocks into which the 24 original twists are divided. Within each block, Eq. (4) is used to find the TA pure formation energy. The standard error in the single-block case is obtained by Gaussian propagation of the Monte Carlo random errors, with no account being taken of residual twist-sampling errors.

Figure 2

DFT pure formation energies of (a) MV, (b) SiS, and (c) SW defects in graphene against the reciprocal of the supercell size $N$. Fine $\mathbf{k}$-point grids were used in each supercell. Ultrasoft pseudopotentials were used. The dashed lines show fits of Eq. (5) to the data.

Figure 3

(a) Top-down and (b) in-plane views of the DFT-PBE-relaxed MV structure in a $5\times 5$ supercell. The undercoordinated carbon atom is shown in red.

Figure 4

(a) Top-down and (b) in-plane views of the DFT-PBE-relaxed SiS structure in a $5\times 5$ supercell. The silicon atom is shown in blue.

Figure 5

(a) Top-down and (b) in-plane views of the DFT-PBE-relaxed “sinelike” SW defect structure in a $5\times 5$ supercell.

Figure 6

DFT and DMC formation energies against reciprocal of supercell size $N$, using different methods for dealing with momentum quantization errors, for (a) MV, (b) SiS, and (c) SW defects. The red dashed lines show an unweighted least-squares fit of Eq. (5) to the TA-DMC data. Both the DFT and DMC calculations used Dirac-Fock pseudopotentials. The carbon and silicon chemical potentials were taken to be the energy per atom of monolayer graphene and bulk silicon, extrapolated to infinite system size; hence the $N$-dependence shown in this figure only arises from the finite-concentration and finite-size effects in the pure formation energy.

Figure 7

TA-DMC static-nucleus atomization energies of (a) graphene, (b) bulk silicon, and (c) silicene against $N^{-5/4}$ for graphene and silicene, and $N^{-1}$ for bulk silicon, where $N$ is the number of primitive cells in the supercell. The atomization energies are defined with respect to the DMC spin-polarized ${}^3{\text{P}}_0$ ground states of isolated carbon and silicon atoms.

Figure 8

DMC total energies per supercell of (a) MV, (b) SiS, and (c) SW defects in a $3\times 3$ supercell of graphene against DMC time step $\tau$ at a single, randomly chosen twist ${\mathbf{k}}_{\text{s}}$. The dashed lines show quadratic fits to the energy as a function of time step.

Figure 9

DMC pure formation energies of (a) MV, (b) SiS, and (c) SW defects in a $3\times 3$ supercell of graphene against DMC time step $\tau$ at the twist ${\mathbf{k}}_{\text{s}}$ used in Fig. 8. The dashed lines show linear fits to the pure formation energy as a function of time step.