Local-density-functional calculations of the vacancy-oxygen center in Ge

We carry out a comprehensive density-functional study of the vacancy-oxygen (VO) center in germanium using large H-terminated Ge clusters. The importance of a nonlinear core correction to account for the involvement of the $3d$ electrons in Ge-O bonds is discussed. We calculate the electrical levels and the vibrational modes of ${\mathrm{VO}}^0$, ${\mathrm{VO}}^{-}$, and ${\mathrm{VO}}^{=}$ finding close agreement with experiment. We also explore the reorientation, migration, and dissociation mechanisms of neutral and negatively charged VO and compare the calculated energy barriers with experimental data. We conclude that the defect is likely to anneal through both mechanisms.

Figure 1

(Color online) Atomic energy levels of the O, N, Ga, Ge, and Si elements. The density-functional-theory–local-density-approximation orbital energies are obtained from an all-electron relativistic calculation performed with the ADF code (Ref. 31). The Hartree-Fock (HF) values from Ref. 32 for the $\mathrm{Ge}3d$ and $4s$ orbitals and the $\mathrm{O}2s$ orbital are also shown for comparison. Despite the discrepancy between the absolute electron energies for the free atoms found in different references, the close proximity of the $\mathrm{Ge}3d$ and $\mathrm{O}2s$ orbitals suggests a significant hybridization.

Figure 2

The VO center. Germanium and oxygen atoms are represented in gray and white, respectively. The dark bond between atoms ${\mathrm{Ge}}_3$ and ${\mathrm{Ge}}_4$ represents a reconstructed bond.

Figure 3

(Color online) Representation of the electron wave function of the (a) highest occupied level of ${\mathrm{VO}}^0$, (b) and (c) highest occupied level of ${\mathrm{VO}}^{-}$ (two different projections), and (d) lowest unoccupied level of ${\mathrm{VO}}^{-}$. The oxygen atom is represented as a small sphere and the germanium atoms are represented as large spheres.

Figure 4

Reorientation of ${\mathrm{VO}}^0$: (a) initial position, (b) saddle point, and (c) final position. Germanium and oxygen atoms are represented by gray and white spheres, respectively. In structure (b), two O-Ge bonds are represented in a darker shade to highlight that they are longer than the third Ge-O bond.

Figure 5

Diffusion paths: [(a)–(c)], [(d)–(f)], and [(g)–(i)] are three possible mechanisms of transformation of VO into a metastable structure ${\mathrm{VO}}^{*}$. Germanium and oxygen atoms are represented by gray and white spheres, respectively. Crosses indicating the vacancy site and dashed arrows suggesting the movement of the atoms were added to help in the visualization.

Figure 6

(Color online) Proposed configuration-coordinate diagram for the VO defect including the relevant transformations involved in the lowest-energy migration and dissociation mechanisms. VO and ${\mathrm{VO}}^{\prime }$ represent equivalent vacancy-oxygen defects centered in different lattice sites. The corresponding energies for the single negatively charged defect are given in parentheses.

Figure 7

Second step of the dissociation path of VO from (a) ${\mathrm{VO}}^{*}$ to (c) $\mathrm{V}+\mathrm{O}$. Crosses indicate the vacancy site.