Finite-size scaling as a cure for supercell approximation errors in calculations of neutral native defects inInP

Abstract

The relaxed and unrelaxed formation energies of neutral antisites and interstitial defects in InP are calculated using ab initio density functional theory and simple cubic supercells of up to 512 atoms. The finite-size errors in the formation energies of all the neutral defects arising from the supercell approximation are examined and corrected for using finite-size scaling methods, which are shown to be a very promising approach to the problem. Elastic errors scale linearly, while the errors arising from charge multipole interactions between the defect and its images in the periodic boundary conditions have a linear plus a higher order term, for which a cubic provides the best fit. These latter errors are shown to be significant even for neutral defects. Instances are also presented where even the 512 atom supercell is not sufficiently converged. Instead, physically relevant results can be obtained only by finite-size scaling the results of calculations in several supercells, up to and including the 512 atom cell and in extreme cases possibly even including the 1000 atom supercell.

Anisotropic charge screening and supercell size convergence of defect formation energies

“…Castleton and co-workers proposed an extrapolation procedure 9,35 for the study of defects in InP whereby the limit of a fit to a series of formation energies obtained from supercells of increasing size was used to determine the formation energy of the isolated defect. However, uniformly scaling all axes simultaneously rapidly increases the number of atoms in the supercell.…”

“…This problem is particulary acute in the case of charged defects as the Coulomb interaction decays slowly as a function of the separation between point charges 1 . A number of correction schemes have been devised to extract the formation energies in the desired dilute limit from simulations of relatively small supercells: these have been widely applied to systems such as silicon 2,3 , NaCl 2,4 , diamond 5,6 , GaAs 5-8 , InP 9 and Al 2 O 3 10, 11 . Inherent in all of these schemes is the assumption that the dielectric response of the material is isotropic and can be described by a single dielectric constant, ǫ.…”

Finite-size supercell correction schemes for charged defect calculations

“…In the present calculations, we ensure k-point sampling convergence of the formation energies, as the importance of this issue has been highlighted several times in the study of defect corrections. 2,34,35 The formation energies are plotted with respect L −1 , where L is the side of the supercell. In all figures, lines of the form aL −1 + bL −3 + c are fitted to the calculated values to obtain extrapolated values corresponding to the limit L → ∞.…”

“…In all figures, lines of the form aL −1 + bL −3 + c are fitted to the calculated values to obtain extrapolated values corresponding to the limit L → ∞. 2,3 We note that the MP and LZ correction schemes only have L −1 and L −3 terms. This implies that the MP-corrected and the LZ-corrected formation energies extrapolate by construction to the same values as the uncorrected formation energies, provided the potential alignment is carried out in the same way in the two schemes.…”

“…The straightforward approach is to perform calculations for supercells of varying size and to extrapolate to the limit of infinitely large supercell. [2][3][4][5] When this is excessively demanding from the computational point of view, one generally resorts to using small supercells and to adopting correction schemes. Apart from the local moment countercharge method which accounts for the multipole moments of the defect within the self-consistent electronic-structure cycle, 6 the most common approaches consist in correcting a posteriori the calculated formation energies.…”