In electronics and

semiconductor A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...

physics, the law of mass action is a relation about the concentrations of free electrons and

electron hole In physics, chemistry, and electronic engineering, an electron hole (often simply called a hole) is a quasiparticle which is the lack of an electron at a position where one could exist in an atom or atomic lattice. Since in a normal atom or ...

s under

thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...

. It states that, under

, the product of the free electron concentration

n

and the free hole concentration

p

is equal to a constant square of intrinsic carrier concentration

n_

. The intrinsic carrier concentration is a function of temperature. The equation for the mass action law for

s is:

np = n_^

Carrier concentrations

In semiconductors, free electrons and

holes A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...

are the carriers that provide

conduction Conductor or conduction may refer to: Music * Conductor (music), a person who leads a musical ensemble, such as an orchestra. * Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas * Conduction, a type of structured f ...

. 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 Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermod ...

, giving the results below.

Electron concentration

The free-electron concentration ''n'' can be approximated by

n = N_c \exp\left \frac\right

where * ''E''_c is the energy of the

conduction band In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...

, * ''E''_F is the energy of the

Fermi level The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include the work required to remove ...

, * ''k''_B is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...

, * ''T'' is the absolute temperature in

kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

s, * ''N''_c is the effective density of states at the conduction band edge given by

N_c = 2\left(\frac\right)^

, 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

p = N_v \exp\left \frac\right

where * ''E''_F is the energy of the

, * ''E''_v is the energy of the

valence band In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...

, * ''k''_B is the

, * ''T'' is the absolute temperature in

s, * ''N''_v is the effective density of states at the valence band edge given by

N_v = 2\left(\frac\right)^

, 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

np = N_c N_v \exp\left(-\frac\right) = n_i^2,

where ''E''_g is the

band gap energy In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in ...

given by ''E''_g = ''E''_c − ''E''_v. The above equation holds true even for lightly doped extrinsic semiconductors as the product

np

is independent of doping concentration.

References

External links

Doping, Carrier Concentration, Mobility, and Conductivity

Semi-conductor tutorial
Electronic engineering Empirical laws {{Electronics-stub