Electronic surface error in the Si interstitial formation energy

The results for Si interstitial formation energies differ substantially if calculated with quantum Monte Carlo (QMC) or density functional theory (DFT) techniques. In fact, not even DFT results using different exchange-correlation functionals agree well for these energies. We carefully quantify the differences between the DFT results by accurate calculations with large supercells. A similar discrepancy for vacancy formation energies in metals has previously been resolved by introducing the concept of an “electronic surface error,” and this view is adopted and shown relevant also for the present DFT results for interstitials in semiconductors. The origin of the surface error for the Si interstitial is explained by careful examination of the electron density. A postcorrection for the surface error brings all the results obtained with the tested functionals close to the results of the AM05 functional. However, it remains an important puzzle that while the surface error correction aligns the DFT results, they are still in large disagreement with QMC results.

Figure 1

(Color) (a) A perfect Si crystal cell. The red atoms are on the cell sides in an fcc pattern and the blue atoms are inside the cell. [(b)–(d)] Top views of density contours in the three planes shown in (a), revealing the “holes” in the density at the interstitial hexagonal $\left(H\right)$ and tetragonal $\left(T\right)$ sites. The contours are spaced $0.0051/{\text{bohr}}^3$ and the red contours show a density of $0.041/{\text{bohr}}^3$, about half the maximum density. The holes at the atomic sites are due to the omission of the core density in our pseudopotential treatment. In (d), only the charge density inside the cell is shown. The density shown is calculated with AM05, but does not differ substantially from that calculated with other functionals.

Figure 2

(Color) The $⟨110⟩$-split interstitial. The two yellow five-coordinated atoms are equivalent to the atoms in the atomic positions at the bottom and left side of Fig. 1c. The interstitial atom (blue) pairs up with an existing lattice atom and they both relax upward toward the tetragonal sites. This configuration is obtained with AM05 but other functionals give very similar results.

Figure 3

(Color) Investigating the bonds of the $⟨110⟩$-split interstitial. (a) The density on a sphere with a radius of half the perfect crystal interatomic distance around an $⟨110⟩$-split interstitial atom (upper blue in Fig. 2). This interstitial atom is four coordinated but only has one strong, high density, bond of the type seen for the four-coordinated atoms in a bulk lattice shown in (b). (c) The density around a five-coordinated atom in the $⟨110⟩$-split interstitial (yellow in Fig. 2) shows three bulk type bonds and two weak, smeared out, bonds to the two interstitial atoms (blue). $\theta$ is the angle from the positive $z$ axes, while $\phi$ is the angle in the $x\text{−}y$ plane measured counterclockwise from the $x$ axes. The density shown is calculated with AM05, but does not differ substantially from densities calculated with other functionals. The colored dots show the direction to the equally colored atoms in Fig. 2.

Figure 4

(Color) The density in a plane cutting through the four bonds connecting the five-coordinated and the interstitial atoms (yellow and blue, respectively, in Fig. 2). The density between the atoms composing the interstitial defect is substantially more homogeneous than the density in the bulk regions. Notice that even though the two interstitial atoms are four coordinated, they do not have the clear bond picture of the four-coordinated atoms in the bulk [see Fig. 3b]. The contours are the same as in Fig. 1.