Vacancies’ Formation Parameters 
for Carbon-Subgroup Crystals

M.N. Magomedov

Institute for Geothermal Research of Daghestan Scientific Center RAS,
Chamil str., 39a, Makhachkala, 367030, RUSSIA, E-mail: mahmag@dinet.ru

The parameters of formation of vacancies in the semiconductor crystals of elements of the carbon subgroup (C-diam, Si, Ge, -Sn) are calculated by method, whish described in work [1]. It is shown that consideration of atom delocalization brings about an increase in the values of enthalpy (hv) and entropy (sv) and the volume (vv) of the vacancy formation. At low temperatures the parameters of vacancy formation depend heavily on temperature, with the entropy of vacancy formation becoming negative: sv < 0. The reasons of negative value of sv were discussed in work [2]. The temperature dependences of vacancies parameters were studied at the isobaric (P = 0) heating from T = 0 to the melting temperature (Tm).
For diamond the next values are received at T = 300 K and 1200 K, respectively:
gv = 1.982 eV, hv = 0.395 eV, sv/kb = – 61.38, vv/va = 0.178;
gv = 3.544 eV, hv = 2.991 eV, sv/kb = – 5.351, vv/va = 0.717.
Here, gv is the thermodynamic potential (or free Gibbs energy) of the vacancy formation, kb is the Boltzmann constant, va(T) is the volume of an atom in a crystal.
For Si the next values are received at T = 300 K and Tm = 1685 K, respectively:
gv = 2.611 eV, hv = 1.892 eV, sv/kb = – 27.79, vv/va = 0.645;
gv = 3.135 eV, hv = 2.901 eV, sv/kb = – 1.607, vv/va = 0.896.
For Ge the next values are received at T = 300 K and Tm = 1212 K, respectively:
gv = 2.265 eV, hv = 2.027 eV, sv/kb = – 9.193, vv/va = 0.820;
gv = 2.422 eV, hv = 2.302 eV, sv/kb = – 1.345, vv/va = 0.895.
For -Sn the next values are received at T = 100 K and T = 240 K, respectively:
gv = 2.106 eV, hv = 1.434 eV, sv/kb = – 77.95, vv/va = 0.779;
gv = 2.412 eV, hv = 2.212 eV, sv/kb = – 9.671, vv/va = 1.076.
It is shown that, at low temperatures the use of the Arrhenius equation for determination of the energy parameters of the vacancy formation is not correct. 
This work was supported by RFBR (Project No. 09 – 08 – 96508), and by Programs of the Presidium of RAS (Project No. 12.1.19).

References

[1] M.N. Magomedov. Semiconductors. 42 (2008) 1133.
[2] M. N. Magomedov. Technical Physics Letters. 34 (2008) 414.