Mass action law (electronics)

The free-electron concentration n can be approximated by

${\displaystyle n=N_{c}\exp \left[-{\frac {(E_{c}-E_{F})}{kT}}\right],}$

where

E_c is the energy of the conduction band,

E_F is the energy of the Fermi level,

k is the Boltzmann constant,

T is the temperature in kelvins,

N_c is the effective density of states at the conduction band edge given by ${\displaystyle \textstyle N_{c}=2\left({\frac {2\pi m_{e}^{*}kT}{h^{2}}}\right)^{3/2}}$, with m*_e being the electron effective mass and h being Planck's constant.

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

${\displaystyle p=N_{v}\exp \left[-{\frac {(E_{F}-E_{v})}{kT}}\right],}$

where

E_F is the energy of the Fermi level,

E_v is the energy of the valence band,

k is the Boltzmann constant,

T is the temperature in kelvins,

N_v is the effective density of states at the valence band edge given by ${\displaystyle \textstyle N_{v}=2\left({\frac {2\pi m_{h}^{*}kT}{h^{2}}}\right)^{3/2}}$, with m*_h being the hole effective mass and h Planck's constant.

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

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

where E_g is the band gap energy given by E_g = E_c − E_v.