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:
n
p
=
n
i
2
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.
The free electron concentration n can be approximated by
n
=
N
c
exp
[
−
(
E
c
−
E
F
)
k
T
]
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
N
c
=
2
(
2
π
m
e
∗
k
T
h
2
)
3
/
2
, with m*e being the electron effective mass and h being the planck constant.
The free hole concentration p is given by a similar formula
p
=
N
v
exp
[
−
(
E
F
−
E
v
)
k
T
]
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
N
v
=
2
(
2
π
m
h
∗
k
T
h
2
)
3
/
2
, with m*h being the hole effective mass and h being the planck constant.
Using the carrier concentration equations given above, the mass action law can then be stated as
n
p
=
N
c
N
v
exp
(
−
E
g
k
T
)
=
n
i
2
where Eg is the bandgap energy given by Eg = Ec − Ev
