Incomplete ionization

The densities of ionized impurities are calculated in the context of Thomas-Fermi approximation with these formulas:

(16)\[N_\text{D}^+(\mathbf{x})=\sum_{i\in \text{Donors}}\frac{N_{\text{D},i}(\mathbf{x})}{1+g_{\text{D},i}\exp((E_{\text{F},n}(\mathbf{x})-E_{\text{D},i}(\mathbf{x}))/k_\text{B}T)}\]

(17)\[N_\text{A}^-(\mathbf{x})=\sum_{i\in \text{Acceptors}}\frac{N_{\text{A},i}(\mathbf{x})}{1+g_{\text{A},i}\exp((E_{\text{A},i}(\mathbf{x})-E_{\text{F},p}(\mathbf{x}))/k_\text{B}T)}\]

where the summation is over all different donor or acceptors, \(N_\text{D},N_\text{A}\) are the doping concentrations, \(g_\text{D},g_\text{A}\) are the degeneracy factors (\(g_\text{D}=2\) and \(g_\text{A}=4\) for shallow impurities), and \(E_{D},E_{A}\) are the energies of the neutral donor and acceptor impurities, respectively.

These energies of neutral impurities \(E_{\text{D},i},E_{\text{A},i}\) are determined by the ionization energies \(E_{\text{D},i}^\text{ion},E_{\text{A},i}^\text{ion}\) , the bulk conduction and valence band edges (including shifts due to strain) and the electrostatic potential.

(18)\[E_{\text{D},i}(\mathbf{x})=E_\text{c}(\mathbf{x})-e\phi(\mathbf{x})-E_{\text{D},i}^\text{ion}(\mathbf{x})\]

(19)\[E_{\text{A},i}(\mathbf{x})=E_\text{v}(\mathbf{x})-e\phi(\mathbf{x})+E_{\text{A}}^\text{ion}(\mathbf{x})\]

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Last update: 04/12/2024