Electron concentration

The free-electron concentration n can be approximated by

$${\displaystyle n=N_{c}\exp \left[-{\frac {E_{c}-E_{F}}{k_{\text{B}}T}}\right],}$$

where

• E_c is the energy of the conduction band,
• E_F is the energy of the Fermi level,
• k_B is the Boltzmann constant,
• T is the absolute temperature in kelvins,
• N_c is the effective density of states at the conduction band edge given by ${\textstyle N_{c}=2\left({\frac {2\pi m_{e}^{*}k_{\text{B}}T}{h^{2}}}\right)^{3/2}}$, with m*_e being the electron effective mass and h being Planck's constant.

Hole concentration

The free-hole concentration p is given by a similar formula

$${\displaystyle p=N_{v}\exp \left[-{\frac {E_{F}-E_{v}}{k_{\text{B}}T}}\right],}$$

where

• E_F is the energy of the Fermi level,
• E_v is the energy of the valence band,
• k_B is the Boltzmann constant,
• T is the absolute temperature in kelvins,
• N_v is the effective density of states at the valence band edge given by ${\textstyle N_{v}=2\left({\frac {2\pi m_{h}^{*}k_{\text{B}}T}{h^{2}}}\right)^{3/2}}$, with m*_h being the hole effective mass and h Planck's constant.

Mass action law

Using the carrier concentration equations given above, the mass action law can be stated as

$${\displaystyle np=N_{c}N_{v}\exp \left(-{\frac {E_{g}}{k_{\text{B}}T}}\right)=n_{i}^{2},}$$

where E_g is the band gap energy given by E_g = E_c − E_v. The above equation holds true even for lightly doped extrinsic semiconductors as the product ${\displaystyle np}$ is independent of doping concentration.