Method

As proof of principle a few sample calculations will be performed to approximate the concentration of Zr point defects in Si.

The total energies are computed using the ultrasoft pseudopotential formalism embodied in the Vienna Ab-Initio Simulation Package (VASP). [16,17] The atoms will be allowed to relax into a minimum energy configuration using a conjugate gradient method. The changes discussed in reference [15] to allow for the calculation of charged defects are non-trivial considering the time limitations for this project. Only neutral defects will be examined.

It will be assumed that the ultrasoft pseudopotentials [18] provided by VASP are sufficiently accurate for the atomic configurations in the system. Periodic boundary conditions are employed and a basis of 64 Si atoms will used in the calculation. It will be assumed that a 64 atom basis will be sufficient in size to isolate the point defects through the periodic boundaries. [15] Sampling of k-space will be performed by a Monkhorst-Pack scheme [19] and it will be assumed that a mesh of 2x2x2 k-points will be sufficient to accurately determine the energy of the system. [15] For a more definitive calculation, these conditions can not be assumed but must be determined by convergence tests.

The constitutive relation used for Zr neutral substitutional and interstitial defects are given in 8 and 9 and the range for $\mu_{Zr}$ is give in 10

$$E_{form\:sub}=E_{total}-63\mu_{Si}-1\mu_{Zr}$$ (8)

$$E_{form\:inter}=E_{total}-64\mu_{Si}-1\mu_{Zr}$$ (9)

$$(\mu_{Zr\:(bulk)}+H_{f\:ZrSi_{2}})<\mu_{Zr}<\mu_{Zr\:(bulk)}$$ (10)