Energetic and vibrational analysis of hydrogenated silicon $m$ vacancies above saturation

We present a systematic study on hydrogenated silicon $m$ vacancies above saturation. For each system a global geometry optimization search for low-lying local minima is performed using a newly developed SiH tight binding model. Subsequently a large number of low-energy structures are examined by density functional calculations using a minimal basis set. Finally the energetically favorable structures are reexamined using a systematically extendable basis set with local, semilocal, and hybrid exchange-correlation functionals. Particular attention is paid to the divacancy to which the Raman peak at 3822 ${\mathrm{cm}}^{-1}$ associated with the ${\mathrm{H}}_2$ molecule had previously been assigned. Both the energetics and vibrational analysis of divacancy-related stable configurations suggest a revision of the above conclusion.

Figure 1

The three configurations of the 1 vacancy listed in Table 2: (a) $A_1^{1,6}$, (b) $A_2^{1,6}$, (c) $A_3^{1,6}$. Si atoms (large spheres) are shown in orange, H atoms (small spheres) in yellow, and vacancies in blue. All other figures follow the same scheme.

Figure 2

(a) The structure $S_1^{2,8}$. The monovacancies are second nearest neighbors and are connected by ${\mathrm{SiH}}_2$. (b) The structure $A_1^{2,8}$. ${\mathrm{H}}_2$ is at the T${}_d$ site dislodged towards the divacancy between two neighboring (001) planes. The ${\mathrm{H}}_2$ is along $\left[010\right]$ with bond length of $0.747$ Å.

Figure 3

(a) The free energy, which includes the potential energy, the ZPE, as well as vibrational and configurational contributions to the entropy, is illustrated for the two lowest energy configurations in the case of the 2 vacancy (${\mathrm{V}}_2$${\mathrm{H}}_8$). The free energies are plotted relative to that of configuration $S_1^{2,8}$ at each temperature. The free energies of configurations $S_1^{2,8}$ and $A_1^{2,8}$ are shown by squares (red) and triangles (green), respectively. The solid line (blue) is the potential energy of configuration $A_1^{2,8}$ relative to that of $S_1^{2,8}$. (b) The normalized Boltzmann occupations are illustrated as a function of temperature for the configurations $S_1^{2,8}$ and $A_1^{2,8}$ by squares (red) and triangles (green), respectively.

Figure 4

The three configurations of the 3 vacancies listed in Table 4: (a) $A_1^{3,10}$, (b) $A_2^{3,10}$, (c) $S_1^{3,10}$.

Figure 5

The five configurations of the 4 vacancies listed in Table 5: (a) $A_1^{4,12}$, (b) $A_2^{4,12}$, (c) $S_1^{4,12}$, (d) $S_2^{4,12}$, (e) $S_3^{4,12}$.

Figure 6

The five configurations of the 5 vacancies listed in Table 6: (a) $S_1^{5,14}$, (b) $A_1^{5,14}$, (c) $A_2^{5,14}$, (d) $A_3^{5,14}$, (e) $A_4^{5,14}$.

Figure 7

The 10 configurations of the 6 vacancies listed in Table 7 are illustrated: (a) ${A^{\prime }}_1^{6,16}$, (b) ${A^{\prime }}_2^{6,16}$, (c) $S_1^{6,16}$, (d) $S_2^{6,16}$, (e) $A_1^{6,16}$, (f) $S_3^{6,16}$, (g) $A_2^{6,16}$, (h) $A_3^{6,16}$, (i) $A_4^{6,16}$, (j) ${A^{\prime }}_3^{6,16}$.

Figure 8

The five configurations of the 6 vacancies listed in Table 8 are illustrated: (a) $A_1^{6,18}$, (b) $A_2^{6,18}$, (c) $S_1^{6,18}$, (d) $A_3^{6,18}$, (e) ${A^{\prime }}_1^{6,18}$.