Incomplete ionization

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

(2.2.16)$$N_{\text{D}}^{+}(\mathbf{x})=\sum _{i\in \text{Donors}}\frac{N_{\text{D},i}(\mathbf{x})}{1+g_{\text{D},i}\exp ((E_{\text{F},n}(\mathbf{x})-E_{\text{D},i}(\mathbf{x}))/k_{\text{B}}T)}$$

(2.2.17)$$N_{\text{A}}^{-}(\mathbf{x})=\sum _{i\in \text{Acceptors}}\frac{N_{\text{A},i}(\mathbf{x})}{1+g_{\text{A},i}\exp ((E_{\text{A},i}(\mathbf{x})-E_{\text{F},p}(\mathbf{x}))/k_{\text{B}}T)}$$

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.

(2.2.18)$$E_{\text{D},i}(\mathbf{x})=E_{\text{c}}(\mathbf{x})-e\varphi (\mathbf{x})-E_{\text{D},i}^{\text{ion}}(\mathbf{x})$$

(2.2.19)$$E_{\text{A},i}(\mathbf{x})=E_{\text{v}}(\mathbf{x})-e\varphi (\mathbf{x})+E_{\text{A}}^{\text{ion}}(\mathbf{x})$$

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Last update: 04/12/2024