2.6. Doping

2.6.1. Ionization

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

\[\begin{split}\begin{aligned} N_\text{D}^+(\mathbf{r})&=\sum_{i}\frac{N_{\text{D},i}(\mathbf{r})}{1+g_{\text{D},i}\exp((E_{\text{F},n}(\mathbf{r})-E_{\text{D},i}(\mathbf{r}))/k_\text{B}T)} \\ N_\text{A}^-(\mathbf{r})&=\sum_{i}\frac{N_{\text{A},i}(\mathbf{r})}{1+g_{\text{A},i}\exp((E_{\text{A},i}(\mathbf{r})-E_{\text{F},p}(\mathbf{r}))/k_\text{B}T)} \end{aligned}\end{split}\]

where the summations are over all donor or acceptors species, \(N_\text{D},\,N_\text{A}\) 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.

\[\begin{split}\begin{aligned} E_{\text{D},i}(\mathbf{r})&=E_\text{c,0}(\mathbf{r})-e\phi(\mathbf{r})-E_{\text{D},i}^\text{ion}(\mathbf{r}) \\ E_{\text{A},i}(\mathbf{r})&=E_\text{v,0}(\mathbf{r})-e\phi(\mathbf{r})+E_{\text{A},i}^\text{ion}(\mathbf{r}), \end{aligned}\end{split}\]

where \(E_\text{c,0}(\mathbf{r})\) and \(E_\text{v,0}(\mathbf{r})\) are conduction and valence band edges including strain and temperature effects but no electrostatic potential, respectively, and \(\phi(\mathbf{r})\) is the electrostatic potential.

2.6.2. Activation Energies

Table 2.6.2.1 Donor levels (n-type) in units of \(eV\) relative to conduction band edge

Donor Name

Energy

Source

n-As-in-Si

0.054

n-As-in-Si

0.049

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

n-P-in-Si

0.045

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

n-Sb-in-Si

0.039

n-N-in-Si

0.045

n-As-in-Ge

0.013

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

n-P-in-Ge

0.012

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

n-N-in-SiC

0.10

n-Si-in-GaAs

0.0058

n-Si-in-AlAs

0.007

300 K, Landolt-Boernstein

n-Si-in-Al0.27Ga0.73As

0.006

Landolt-Boernstein

More parameters can be found here

Table 2.6.2.2 Acceptor levels (p-type) in units of \(eV\) relative to valence band edge

Acceptor Name

Energy

Source

p-In-in-Si

0.16

p-B-in-Si

0.045

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

p-Al-in-Si

0.057

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

p-B-in-Ge

0.010

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

p-Al-in-Ge

0.010

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York (1972)

p-Al-in-SiC

0.20

p-C-in-GaAs

0.027

Landolt-Boernstein 1982

More parameters can be found here

Last update: 2025-09-18