Semiconductor physics is of obvious importance, as it

We put the energy to zero at the top of the valence band and Eg at the bottom of the
conduction band. First, the density of states in the conduction band is of the free electron
form, and by integration of it (multiplied with the Fermi-Dirac function) we obtain the
electron concentration, n (H p. 116-118).
n = n
0
e
( μ−
Eg B
) / k T
,
where μ is the chemical potential. The prefactor n0 is weakly varying with temperature
(~T 3/2 ). The same argument is applied to hole states in the valence band, and analogously we
obtain the density of hole states and the hole concentration, p.
p =
p
0
e
−μ
/
k B
T
.
It is seen that the product np is exponentially dependent on the band gap divided by kBT.
B
np = n
0
p
0
e
−
Eg B
/ k T
It is a constant at a given temperature (H p. 119). The relations above hold always,
irrespective of whether the semiconductor is intrinsic or extrinsic (doped). In the case of
doped semiconductors it is just the chemical potential that is different and gives rise to
different values of n and p.
In the intrinsic case n=p (we denote this by ni) and an expression for the Fermi level, μ, is
easily obtained (H p. 119). At zero temperature it is in the middle of the band gap.
Rule of thumb: At T=300 K and if the effective masses of electrons and holes are equal to
the free electron mass, then n0=p0= 2.5 10 25 m -3 .
The conductivity is obtained as the sum of two free-electron like contributions from electrons
and holes, respectively. In semiconductors one defines electron and hole mobilities instead of
working with relaxation times. The mobility is defined as the magnitude of the drift velocity
divided by the electric field.
The electrical conductivity of an intrinsic semiconductor is obtained from the product of the
charge carrier concentration, the mobility, μe,h, and e.
σi = e (μe+μh) (n0p0) 1/2 exp (-Eg/2kBT)
Here n0 and p0 have very weak temperature dependences, proportional to T 3/2 . The
conductivity is thermally activated with activation energy equal to half the band gap.
3. Doping and extrinsic conductivity (H p. 121-125).
The number of conduction electrons in an intrinsic semiconductor decreases very quickly as
the band gap increases. For an insulator with band gap > 4eV, n will be less than 1 m -3 at room
temperature. This situation will never be reached in practice, however, since the presence of
impurities also affects the electron (and hole) concentration. It is very difficult to prepare
materials with impurity content less than 10 17 m -3 .