Complex Defect: defect atoms and charged defects

The addition of defect atoms can complicate the constitutive equation. Mention of the defect atom must be included in the reference state. For example, to place a single defect atom, say Zr, on a silicon lattice site, equation 2 must be modified to include a term $-1\mu_{Zr}$. The allowed range for $\mu_{Zr}$ can be determined by a method similar to that demonstrated in section 2.2.2. The allowed range for Zr is bracketed by the formation of metallic Zr and the stoichiometric compound ZrSi$_{2}$. So $\mu_{Zr}$ is bracketed by $(\mu_{Zr\:(bulk)}+H_{f\:ZrSi_{2}}<\mu_{Zr}<\mu_{Zr\:(bulk)})$. In the case that the defect atom is sitting in a compound semiconductor, AB, the compounds A$_{x}$Zr$_{y}$, B$_{x}$Zr$_{y}$, and A$_{x}$B$_{y}$Zr$_{z}$ must be compared to determine which phase brackets the allowed value for $\mu_{Zr}$.

Charging of the defect is simulated by adding or subtracting electrons from the supercell. The addition or removal of electrons is accounted for by adding or subtracting the energy of electrons from a reservoir of free electron gas. The chemical potential of a free electron gas is the fermi energy of the gas. The value of E$_{fermi}$ is not an independent variable. The value is solved from the atomic system configuration. It is noteworthy that a higher dopant concentration, such as Zr donors, means a higher E$_{fermi}$ and thereby a higher energy of formation of dopant atoms, and therefore a lower dopant concentration. This negative feedback loop has the consequence that the concentration of dopant defects is less sensitive to uncertainties in the formation energy calculations. [11,9]

The addition of charged defects complicates not only the constitutive equation but also the ab initio calculations themselves. Most modern day ab initio type calculations are based on pseudopotential-density functional theory methods. [12,13] The construction is based on a momentum space formalism and the exchange-correlation energy is approximated by the local density approximation. [14] Within these methods it is known that there are two terms in the momentum space formalism that mandate charge neutrality. In addition the local density approximation also requires a charge neutral cell. Van de Walle explains in reference [15] the modifications to modern ab initio software necessary to accurately calculate energies of charged supercells. The details will not be presented here.