# Mass action law (electronics)

Under thermal equilibrium the product of the free electron concentration ${\displaystyle n}$ and the free hole concentration ${\displaystyle p}$ is equal to a constant equal to the square of intrinsic carrier concentration ${\displaystyle n_{i}}$. The intrinsic carrier concentration is a function of temperature.

The equation for the mass action law for semiconductors is:[1]

${\displaystyle np=n_{i}^{2}}$

## Carrier Concentrations

In semiconductors, free electrons and holes are the carriers that provide conduction. For cases where the number of carriers are much less than the number of band states, the carrier concentrations can be approximated by using Boltzmann statistics, giving the results below.

### Electron Concentration

The free electron concentration n can be approximated by

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

where

• Ec is the energy of the conduction band
• EF is the energy of the Fermi level
• k is the Boltzmann constant
• T is the temperature in Kelvins
• Nc 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 the planck constant.

### Hole Concentration

The free hole concentration p is given by a similar formula

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

where

• EF is the energy of the Fermi level
• Ev is the energy of the valence band
• k is the Boltzmann constant
• T is the temperature in Kelvins
• Nv 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 being the planck constant.

### Mass Action Law

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

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

where Eg is the bandgap energy given by Eg = EcEv