Point defects calculation

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In the context of periodic first-principles calculation of charged point defects in semiconductor, the defect formation energy, \(E^{q}_{f}\) of a point defect \(X\) with charge state \(q\) is defined as a function of the Fermi level \(E_F\) as:

\[ E_{f}^{q}(X, E_F) = E_{\mathrm{tot}}^{q}(X) - E_{\mathrm{tot}}(\mathrm{bulk}) + E_{\mathrm{corr}}^{q} \\ - \Sigma n_i \mu_i + q(E_F + E_{VBM}^{\mathrm{bulk}} + \Delta V_{q/\mathrm{bulk}}), \]

where, \(E_{\mathrm{tot}}^{q}(X)\) is the total energy of the defective supercell, \(E_{\mathrm{tot}}(\mathrm{bulk})\) is the total energy of the pristine supercell, \(n_i\) is the number of the \(i\)-th atoms being added to the pristine system to create the defective supercell (so \(n=-1\) for a vacancy), \(\mu_i\) is the chemical potential of the \(i\)-th atoms calculated using the same pseudopotential, \(E_{VBM}^{\mathrm{bulk}}\) is the valence band maximum of the pristine supercell, and \(\Delta V_{q/\mathrm{bulk}}\) is the potential alignment term between pristine bulk supercell and defective charged supercell.

Transition Levels

From now on, let us consider the example of Zn-substituted Cu impurity with nearest neighbor S vacancy defect comples in zinc blende ZnS, i.e., \((\mathrm{Cu}_{\mathrm{Zn}}\)-\(V_{\mathrm{S}})\) in ZnS.

Chemical potentials

It is customery to reference the chemical potentials to their bulk phases. In case of \(\mathrm{ZnS}\), there are two extreme growth conditions: the \(\mathrm{Zn}\)-rich and the \(\mathrm{Zn}\)-poor conditions. According to the thermodynamic stability constraint, we have:

\[ \mu_{\mathrm{ZnS}} = \mu_{\mathrm{Zn}}^{\mathrm{Zn}\text{-rich}} + \mu_{\mathrm{S}}^{\mathrm{Zn}\text{-rich}} , \]

or alternatively,

\[ \mu_{\mathrm{ZnS}} = \mu_{\mathrm{Zn}}^{\mathrm{Zn\text{-poor}}} + \mu_{\mathrm{S}}^{\mathrm{Zn\text{-poor}}}. \]

The \(\mathrm{Zn}\)-rich or in other words \(\mathrm{S}\)-poor limit of chemical potentials

\[ \mu_{\mathrm{Zn}} \;=\; \mu_{\mathrm{Zn}} - \mu_{\mathrm{Zn}}^0 \]

Heat of formation

For a material to be thermodynamically stable, the heat of formation \(\Delta H_f < 0\). In terms of the chemical potential, it is given as:

\[ \Delta H_f = \mu_{\mathrm{ZnS}} - \mu_{\mathrm{Zn}}^{0} - \mu_{\mathrm{S}}^{0}. \]

In practice, \(\mu_{\mathrm{ZnS}}\) is obtained as the total energy per formula unit, and the elemental (emphasized with a \(0\) superscript) chemical potentials \(\mu_{\mathrm{Zn}}^{0}\) and \(\mu_{\mathrm{S}}^{0}\) are obtained as the total energy per atom of bulk \(\mathrm{Zn}\) in the HCP structure and \(\mathrm{S_2}\) molecule in gas phase:

\[ \mu_{\mathrm{Zn}}^0 \;=\; E_{\mathrm{tot}}(\mathrm{Zn}), \]

\[ \mu_{\mathrm{S}}^0 \;=\; \frac{1}{2} E_{\mathrm{tot}}(\mathrm{S_2}). \]

\[ \mu_{\mathrm{S}} \;=\; \mu_{\mathrm{S}} - \mu_{\mathrm{Zn}}^0 \]

\[ \mu_{\mathrm{Zn}} \;\le\; \mu_{\mathrm{Zn}}^0 \]

\[ \mu_{\mathrm{S}} \;\le\; \mu_{\mathrm{S}}^0 \]