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I would strongly recommend this resource: http://www.tf.uni-kiel.de/matwis/amat/def_en/ It is a good complete on line introduction to defects theory in crystals. Some key points listed below.

The concentration of point defects $c_{PD}$ at equilibrium can be expressed by an Arrhenius formula: $$c_{PD}=A \cdot e^{-\frac{G_F}{kT}}=A \cdot e^{-\frac{S_F}{kT}}\cdot e^{-\frac{H_F}{kT}}$$ where $G_F$ is the free energy of formation, $H_F$ the enthalpy of formation and $S_F$ the entropy of formation. In solids $H_F$ can be confused with the energy of formation, $S_F$ is associated the variation of the vibration entropy of the crystal due to the introduction of a point defect. Formation enthalpies, and thus formation energies can be obtained by: $$ H_F(D) = E_{tot}(D) - E_{tot}(X)+\sum_i n_i \mu_i +q (E_F -E_V) $$

  • $E_{tot}(D)$ and $E_{tot}(X)$: total energies of the system with and without the defect D.
  • $n_i$: number of atom removed from the system, a negative value denote the addition of one atom.
  • $\mu_i$: chemical potential of the atom $i$. This should consider the chemical environment at the equilibrium condition (which reservoir is to take into account).
  • $E_F -E_V$: Fermi energy relative to the valence band maximum.
  • $q (E_F -E_V)$: energy change due to the exchange of electrons and holes with the carrier reservoirs.

For the equilibrium concentration in the multiple defects case an additional entropy term comes from the standard configuration entropy which can be calculated from the Boltzmann formula $$S=k_B \cdot \ln P$$ where P is the number of configurations for the same macrostate.

I would strongly recommend this resource: http://www.tf.uni-kiel.de/matwis/amat/def_en/ It is a good complete on line introduction to defects theory in crystals.

I would strongly recommend this resource: http://www.tf.uni-kiel.de/matwis/amat/def_en/ It is a good complete on line introduction to defects theory in crystals. Some key points listed below.

The concentration of point defects $c_{PD}$ at equilibrium can be expressed by an Arrhenius formula: $$c_{PD}=A \cdot e^{-\frac{G_F}{kT}}=A \cdot e^{-\frac{S_F}{kT}}\cdot e^{-\frac{H_F}{kT}}$$ where $G_F$ is the free energy of formation, $H_F$ the enthalpy of formation and $S_F$ the entropy of formation. In solids $H_F$ can be confused with the energy of formation, $S_F$ is associated the variation of the vibration entropy of the crystal due to the introduction of a point defect. Formation enthalpies, and thus formation energies can be obtained by: $$ H_F(D) = E_{tot}(D) - E_{tot}(X)+\sum_i n_i \mu_i +q (E_F -E_V) $$

  • $E_{tot}(D)$ and $E_{tot}(X)$: total energies of the system with and without the defect D.
  • $n_i$: number of atom removed from the system, a negative value denote the addition of one atom.
  • $\mu_i$: chemical potential of the atom $i$. This should consider the chemical environment at the equilibrium condition (which reservoir is to take into account).
  • $E_F -E_V$: Fermi energy relative to the valence band maximum.
  • $q (E_F -E_V)$: energy change due to the exchange of electrons and holes with the carrier reservoirs.

For the equilibrium concentration in the multiple defects case an additional entropy term comes from the standard configuration entropy which can be calculated from the Boltzmann formula $$S=k_B \cdot \ln P$$ where P is the number of configurations for the same macrostate.

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I would strongly recommend this resource: http://www.tf.uni-kiel.de/matwis/amat/def_en/ It is a good complete on line introduction to defects theory in crystals.