182
7 Electronic Defect States
(a)
(b)
Fig. 7.2 a np for silicon at T = 300 K as a function of the position of the Fermi level. The valence-band edge E V is
chosen as E = 0. np is constant for the range of Fermi energies given by (7.13) (4kT ≈ 0.1 eV). b n, p and
√ np as a
function of the Fermi level
Within the Boltzmann approximation, the product of the electron and hole density is
n p = N V N C exp
−
E C − E V
kT
= N V N C exp
−
E g
kT
(7.12)
= 4
kT
2π 2
3
(m d,e m d,h )
3/2 exp
−
E g
kT
.
Thus, the product n p is independent of the position of the Fermi level, as long as the Boltzmann
approximation is fulfilled. This is the case when the Fermi level lies within the band gap and it is
sufficiently far away from the band edges, fulfilling about
E V + 4 kT < E F < E C − 4 kT .
(7.13)
The relation (7.12) is called the mass-action law.
In Fig. 7.2, the product np is shown for silicon over a wide range of Fermi energies. If E F is within
the band gap, np is essentially constant. If the Fermi level is in the valence or conduction band, np
decreases exponentially.
7.3 Intrinsic Conduction
First, we consider the conductivity of the intrinsic, i.e. an ideally pure, semiconductor. At T = 0 all
electrons are in the valence band, the conduction band is empty and thus the conductivity is zero (a
completely filled band cannot conduct current). Only at finite temperatures the electrons have a finite
probability to be in a conduction-band state and to contribute to the conductivity. Due to neutrality, the
electron and hole concentrations in the intrinsic semiconductors are the same, i.e. each electron in the
conduction band comes from the valence band,
− n + p = 0 ,
(7.14)
or n i = p i . Therefore