Germanium diffusion mechanisms in silicon from first principles

We present an extensive numerical study of the basic mechanisms that describe germanium diffusion in silicon mediated by point defects. This diffusion can be created by vacancies, interstitial atoms, or fourfold coordinated defects. All energies and elementary barriers have been precisely determined by ab initio calculations. The results for vacancies are compared with recently published values. The complex interstitial landscape is systematized and the key role of the hexagonal location is stressed as a halfway stable state between two, more stable, dumbbell [110] states. Finally, the mechanism of a concerted exchange linking two fourfold coordinated defects is fully calculated. Its activation energy is higher than for interstitial or vacancy mediated movements.

Figure 1

(Color online) Representation of interstitial sites in the silicon lattice. Spheres labeled with letters stand for interstitial site, whereas other spheres are substitutional sites. $d$-type sites stand for dumbbell along [110] axis and are labeled with a “d” letter; $h$-type sites represent hexagonal interstitials and are labeled with an “h.” Spheres labeled with prime letters do not belong to the represented ten-atom shape but are drawn, since they are used by [110] dumbbells. The pyramid is constituted of the four big blue (mat gray) spheres and all big spheres stand for the ten-atom shape.

Figure 2

(Color online) Relative energy variations while one silicon and one germanium move from different interstitial defects. The A (B) sites are occupied either by Si (Ge) or Ge (Si). The energy of the mixed $D^{110}$ defect is considered as reference energy. Delimited area corresponds to the ten-atom shape and the gray eight-atom shape areas symbolize the energy basins of the displaced hexagonal defects. On (a), paths to link $D^{110}$ defects to hexagonal defects are depicted. On (b), jumps between hexagonal defects of the same pyramid are depicted, and (c) corresponds to movements around the hexagonal position.

Figure 3

(Color online) Energy profile during an exchange between a germanium and a vacancy in silicon, followed by a separation movement. All reported energies have been calculated in 216-site $2\times 2\times 2$ systems. The main curve is an interpolation between these points. The dashed curve is taken from the work of Ramanarayanan et al. (Ref. 7), resulting from calculations in 64-site $2\times 2\times 2$ systems.

Figure 4

(Color online) Evolution of the total energy during a switch process between one substitutional silicon and one substitutional germanium. The diamonds represent fully relaxed ab initio calculations in boxes of 216 atoms (the line is an interpolation between the calculated energies).