Results

In the table below are listed the parameters used in the calculation and the results.

Parameter	Value	Source
a$_{cublic}$	5.387(10$^{-10}$)m	calculated
$\mu_{Si}$	-5.971 eV/atom	calculated
$\mu_{Zr\:(bulk)}$	-9.362 eV/atom	calculated
$H_{f\:ZrSi_{2}}$	-12.7 kcal/mol=-0.551eV/molicule	reference [20]
N $_{substitution}$	8/a$^{3}$=5.12(10$^{22}$)cm$^{-3}$
N $_{tetrahedral}$	4/a$^{3}$=2.56(10$^{22}$)cm$^{-3}$
N$_{octahedral}$	3/a$^{3}$=1.92(10$^{22}$)cm$^{-3}$
E$_{total\:sub}$	-382.684eV	calculated
E$_{total\:tet}$	-389.894eV	calculated
E$_{total\:oct}$	-389.896eV	calculated
E$_{f\:sub}$	-6.51-$\mu_{Zr}$
E$_{f\:tet}$	-7.75-$\mu_{Zr}$
E$_{f\:oct}$	-7.752-$\mu_{Zr}$

The energy to form a substitutional Zr point defects is approximately 2.8eV and the energy to form interstitial defects is 1.6eV. The error of energies calculated by VASP is on the order of $1(10^{-2})$. Ignoring the possible error introduced by the assumptions, one can approximate the error of these E$_{f}$ energies to be 0.6eV.

These results are of a believable magnitude although one may expect slightly higher E$_{f}$ values. Maroudas and Brown calculated, via EAM potential, the energy of formation for silicon vacancy, self-substitution, and self-interstitial$_{tet}$ defects to be 2.66eV, 3.09eV, and 4.84eV respectively. [21] Within the present context there is insufficient information to explain why the calculated energy of formation for Zr substitutional defects is lower than Maroudas's energy of formation for vacancies. A more detailed investigation should yield clues to determining if there is a failure in one of the assumptions from section 3 or if this discrepancy in results is indicative of the short comings of EAM methods.

From the results above the equilibrium concentrations of Zr point defects are plotted in figures 2. From the Arrhenius plot, it is apparent that in the Zr rich limit ( $\mu_{Zr}=\mu_{Zr\:(bulk)}$) the concentration of defects is slightly higher. At low temperatures the number of interstitial defects dominates the overall number of Zr defects, but this difference in concentration decreases with increasing temperature. A cross-over in concentration occurs at around 5000K, a temperature much higher than the melting point of silicon. It is therefore expected that for all temperatures the concentration of Zr interstitials will be greater than that of substitutional Zr.

Figure 2: The concentration of zirconium substitutional and interstitial defects in silicon as a function of temperature for Zr rich and Zr poor environments.