A study on density functional theory of the effect of pressure on the formation and migration enthalpies of intrinsic point defects in growing single crystal Si

摘要

In 1982, Voronkov presented a model describing point defect behavior during the growth of single crystal Si from a melt and derived an expression to predict if the crystal was vacancy- or self-interstitial-rich. Recently, Vanhellemont claimed that one should take into account the impact of compressive stress introduced by the thermal gradient at the melt/solid interface by considering the hydrostatic pressure dependence of the formation enthalpy of the intrinsic point defects. To evaluate the impact of thermal stress more correctly, the pressure dependence of both the formation enthalpy (H_f) and the migration enthalpy (H_m) of the intrinsic point defects should be taken into account. Furthermore, growing single crystal Si is not under hydrostatic pressure but almost free of external pressure (generally in Ar gas under reduced pressure). In the present paper, the dependence of H_f and H_m on the pressure P, or in other words, the pressure dependence of the formation energy (E_f) and the relaxation volume (v_f), is quantified by density functional theory calculations. Although a large number of ab initio calculations of the properties of intrinsic point defects have been published during the last years, calculations for Si crystals under pressure are rather scarce. For vacancies V, the reported pressure dependences of H_f~V are inconsistent. In the present study, by using 216-atom supercells with a sufficient cut-off energy and mesh of k-points, the neutral I and V are found to have nearly constant formation energies E_f~I and E_f~V for pressures up to 1 GPa. For the relaxation volume, v_f~I is almost constant while v/ decreases linearly with increasing pressure P. In case of the hydrostatic pressure P_h, the calculated formation enthalpy H_f~I and migration enthalpy H~m~I at the [110] dumbbell site are given by H_f~I = 3.425 - 0.057 × P_h (eV) and H_m~I = 0.981 - 0.039 ×P_h (eV), respectively, with P_h given in GPa. The calculated H_f~V and H_m~V dependencies on P_h given by H_f~V= 3.543 - 0.021 × P_h~2 - 0.019 × P_h (eV) and H_m~V = 0.249 + 0.018 × P_h~2 - 0.037 × P_h (eV), respectively. These results indicate that, when assuming that the pre-factors in the Arrhenius equation are not influenced, hydrostatic pressure up to 1 GPa leads to a slight increase of the thermal equilibrium concentration and diffusion of vacancies but this increase is much smaller than that of self-interstitials. The thermal stress in growing Si crystal is compressive, and thus the point defects are under internal pressure. Taking into account the differences in the enthalpies of point defects between hydrostatic pressure and internal pressure, Si crystal shifts to being V-rich with an increase in thermal stress during crystal growth.