7.6 Quasi-fermi Levels
205
n(r) =N C F 1/2
F n (r) − E C
kT
(7.55a)
p(r) =N V F 1/2
−
F p (r) − E V
kT
.
(7.55b)
A quasi-Fermi level is sometimes called imref
5 and can also be denoted as E F n or E F p . We emphasize
that the quasi-Fermi levels are only a means to describe the local carrier density in a logarithmical way.
The quasi-Fermi levels can be obtained from the density via
F n =E C + kT ln
n
N C
(7.56a)
F p =E V − kT ln
p
N V
.
(7.56b)
The quasi-Fermi levels do not imply that the carrier distribution is actually a Fermi distribution. This
is generally no longer the case in thermodynamical nonequilibrium. However, in ‘well-behaved’ cases
the carrier distribution in nonequilibrium can be approximated locally as a Fermi distribution using a
local quasi-Fermi level and a local temperature, i.e.
f e (r, E) ∼ =
1
exp
E−F n (r)
kT (r)
+ 1
.
(7.57)
Using the quasi-Fermi levels, np is given by
n(r) p(r) = n
2
i exp
F n (r) − F p (r)
kT
.
(7.58)
We note that for an inhomogeneous semiconductor or a heterostructure (cf. Chap. 12), n i may also
depend on the spatial position. In the case of thermodynamic equilibrium the difference of the quasiFermi levels is zero, i.e. F n − F p = 0 and F n = F p = E F .
7.7 Deep Levels
For deep levels the short-range part of the potential determines the energy level. The long-range
Coulomb part will only lead to a correction. The term ‘deep level’ implies that the level is within the
band gap and far from the band edges. However, some deep levels (in the sense of the potential being
determined by the ion core) have energy levels close to the band edges or even within a band. Details
can be found in [267, 639–642].
The wavefunction is strongly localized. Thus, it cannot be composed of Bloch functions, as has
been done for the shallow levels for the effective-mass impurity. The localization in r space leads to a
delocalization in k space. Examples are Si:S, Si:Cu or InP:Fe, GaP:N, ZnTe:O. Deep levels can also
be due to intrinsic defects such as vacancies or antisite defects.
Due to the larger distance to the band edges, deep levels are not efficient at providing free electrons
or holes. Quite the opposite, they rather capture free carriers and thus lead to a reduction of conductivity.
5 W. Shockley had asked E. Fermi for permission to use his name reversed. Fermi was not too enthusiastic but granted
permission.
205
n(r) =N C F 1/2
F n (r) − E C
kT
(7.55a)
p(r) =N V F 1/2
−
F p (r) − E V
kT
.
(7.55b)
A quasi-Fermi level is sometimes called imref
5 and can also be denoted as E F n or E F p . We emphasize
that the quasi-Fermi levels are only a means to describe the local carrier density in a logarithmical way.
The quasi-Fermi levels can be obtained from the density via
F n =E C + kT ln
n
N C
(7.56a)
F p =E V − kT ln
p
N V
.
(7.56b)
The quasi-Fermi levels do not imply that the carrier distribution is actually a Fermi distribution. This
is generally no longer the case in thermodynamical nonequilibrium. However, in ‘well-behaved’ cases
the carrier distribution in nonequilibrium can be approximated locally as a Fermi distribution using a
local quasi-Fermi level and a local temperature, i.e.
f e (r, E) ∼ =
1
exp
E−F n (r)
kT (r)
+ 1
.
(7.57)
Using the quasi-Fermi levels, np is given by
n(r) p(r) = n
2
i exp
F n (r) − F p (r)
kT
.
(7.58)
We note that for an inhomogeneous semiconductor or a heterostructure (cf. Chap. 12), n i may also
depend on the spatial position. In the case of thermodynamic equilibrium the difference of the quasiFermi levels is zero, i.e. F n − F p = 0 and F n = F p = E F .
7.7 Deep Levels
For deep levels the short-range part of the potential determines the energy level. The long-range
Coulomb part will only lead to a correction. The term ‘deep level’ implies that the level is within the
band gap and far from the band edges. However, some deep levels (in the sense of the potential being
determined by the ion core) have energy levels close to the band edges or even within a band. Details
can be found in [267, 639–642].
The wavefunction is strongly localized. Thus, it cannot be composed of Bloch functions, as has
been done for the shallow levels for the effective-mass impurity. The localization in r space leads to a
delocalization in k space. Examples are Si:S, Si:Cu or InP:Fe, GaP:N, ZnTe:O. Deep levels can also
be due to intrinsic defects such as vacancies or antisite defects.
Due to the larger distance to the band edges, deep levels are not efficient at providing free electrons
or holes. Quite the opposite, they rather capture free carriers and thus lead to a reduction of conductivity.
5 W. Shockley had asked E. Fermi for permission to use his name reversed. Fermi was not too enthusiastic but granted
permission.