4.2 Point Defects
71
(a)
(b)
(c)
(d)
Fig. 4.2 (a) Schematic diamond lattice with vacancy, i.e. a missing Si atom without relaxation. (b) Si with neutral
vacancy (V 0 ), lattice relaxation and formation of two new bonds. (c) Schematic diagram showing the (inward) relaxation
of the neighbors around the neutral Si vacancy defect site (empty circle) calculated by an ab initio method. The distances
of the outer shell of atoms (red circles) from the vacant site is labeled (in nm). The bond lengths of the two new bonds
and the second-neighbor (blue circles) distance are also indicated. The bond length in bulk Si is 0.2352 nm, the secondneighbor distance 0.3840 nm. Adapted from [266]. (d) Si unit cell with positively charged vacancy (V + ). Parts (a, b, d)
reprinted with permission from [267]
Fig. 4.3 Silicon
tetrahedral interstitial Si T
i
and its next atoms in ideal
(white spheres) and relaxed
(black spheres) position.
Adapted from [175]
enthalpy of an isolated defect. This could be, e.g., the enthalpy of a vacancy, created by bringing an
atom from the (later) vacancy site to the surface, or an interstitial, created by bringing an atom from
the surface to the interstitial site. In the limit that the n defects do not interact with each other, i.e. their
concentration is sufficiently small, they can be considered independent and the enthalpy is given by
H = H 0 + n H
f
.
(4.2)
The increase of entropy due to increased disorder is split into the configurational disorder over the
possible sites, denoted as S
d , and the formation entropy S f due to localized vibrational modes. The
total change of the free energy is
71
(a)
(b)
(c)
(d)
Fig. 4.2 (a) Schematic diamond lattice with vacancy, i.e. a missing Si atom without relaxation. (b) Si with neutral
vacancy (V 0 ), lattice relaxation and formation of two new bonds. (c) Schematic diagram showing the (inward) relaxation
of the neighbors around the neutral Si vacancy defect site (empty circle) calculated by an ab initio method. The distances
of the outer shell of atoms (red circles) from the vacant site is labeled (in nm). The bond lengths of the two new bonds
and the second-neighbor (blue circles) distance are also indicated. The bond length in bulk Si is 0.2352 nm, the secondneighbor distance 0.3840 nm. Adapted from [266]. (d) Si unit cell with positively charged vacancy (V + ). Parts (a, b, d)
reprinted with permission from [267]
Fig. 4.3 Silicon
tetrahedral interstitial Si T
i
and its next atoms in ideal
(white spheres) and relaxed
(black spheres) position.
Adapted from [175]
enthalpy of an isolated defect. This could be, e.g., the enthalpy of a vacancy, created by bringing an
atom from the (later) vacancy site to the surface, or an interstitial, created by bringing an atom from
the surface to the interstitial site. In the limit that the n defects do not interact with each other, i.e. their
concentration is sufficiently small, they can be considered independent and the enthalpy is given by
H = H 0 + n H
f
.
(4.2)
The increase of entropy due to increased disorder is split into the configurational disorder over the
possible sites, denoted as S
d , and the formation entropy S f due to localized vibrational modes. The
total change of the free energy is