Mass action law (electronics)

For other uses, see Mass action.

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

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

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

n=N_{c}{\text{ 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 $\textstyle N_{c}=2\left({\frac {2\pi m_{e}^{*}kT}{h^{2}}}\right)^{{3/2}}$ , 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

p=N_{v}{\text{ 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 $\textstyle N_{v}=2\left({\frac {2\pi m_{h}^{*}kT}{h^{2}}}\right)^{{3/2}}$ , 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

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

where E_g is the bandgap energy given by E_g = E_c − E_v

References

↑ S, Salivahanan; N. Suresh Kumar (2011). Electronic Devices & Circuits. India: Tata McGraw Hill Education Pvt Ltd. p. 1.14. ISBN 0-07-070267-5.

External links

This article is issued from Wikipedia - version of the 7/13/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.