142
G. Lutz and R. Klanner
Electrons bound in one of the localized donor states may be emitted into the
conduction band by thermal excitation with a probability ε n , thereby ionizing
donors. Ionized donors may also capture electrons out of the conduction band. This
process is described by a capture cross section σ n . In thermal equilibrium these
two processes have to balance each other. That condition allows to derive a relation
between emission probability ε n and capture cross section σ n :
ε n = σ n ν th n n i exp ((E d − E i ) /kT ) ,
(5.2)
with ν th n thermal velocity of electrons in the conduction band, n i intrinsic carrier
concentration, E d donor energy level, E i intrinsic energy (Fermi level for an intrinsic
semiconductor). This relation is valid more generally and can be applied to nonequilibrium conditions.
Electrons in the conduction band and holes in the valence band can move freely
within the crystal lattice, their movement being only retarded by scattering on
imperfections of the lattice. These imperfections may be due to lattice defects,
doping atoms replacing regular atoms of the crystal (substitutional dopands) and
distortions of the lattice due to thermal vibrations. The simplified way of describing
these effects uses the assumption that charge carriers are accelerated by the electric
field and lose all previous history at each scattering, starting with random thermal
velocity again.
The movement due to the electric field is described by the drift velocity that for
low fields can be assumed to be proportional to the electric field:
ν n =
−qτ c /m n
E = −μ n E,
ν p =
qτ c /m p
E = μ p E,
(5.3)
with ν n , ν p , μ n , μ p being the drift velocities and low-field mobilities of electrons
and holes, respectively, q elementary charge, τ c average time between collisions,
m n , m p effective masses of electrons and holes, and E electric field. For a high
electric fields τ c decreases and the drift velocity saturates.
At very high electric field electrons and holes may acquire sufficient energy in
between collisions to generate additional electron hole pairs. This avalanche process
can be the cause for an electrical breakdown of devices. It may also be used as an
intrinsic amplification process in order to get sufficiently high signals from very
small ionization.
For inhomogeneous carrier distributions charge carriers will preferably diffuse
from high concentrations to regions of lower concentrations. This diffusion mechanism is described by
F n = −D n ∇n, F p = −D p ∇p.
(5.4)
G. Lutz and R. Klanner
Electrons bound in one of the localized donor states may be emitted into the
conduction band by thermal excitation with a probability ε n , thereby ionizing
donors. Ionized donors may also capture electrons out of the conduction band. This
process is described by a capture cross section σ n . In thermal equilibrium these
two processes have to balance each other. That condition allows to derive a relation
between emission probability ε n and capture cross section σ n :
ε n = σ n ν th n n i exp ((E d − E i ) /kT ) ,
(5.2)
with ν th n thermal velocity of electrons in the conduction band, n i intrinsic carrier
concentration, E d donor energy level, E i intrinsic energy (Fermi level for an intrinsic
semiconductor). This relation is valid more generally and can be applied to nonequilibrium conditions.
Electrons in the conduction band and holes in the valence band can move freely
within the crystal lattice, their movement being only retarded by scattering on
imperfections of the lattice. These imperfections may be due to lattice defects,
doping atoms replacing regular atoms of the crystal (substitutional dopands) and
distortions of the lattice due to thermal vibrations. The simplified way of describing
these effects uses the assumption that charge carriers are accelerated by the electric
field and lose all previous history at each scattering, starting with random thermal
velocity again.
The movement due to the electric field is described by the drift velocity that for
low fields can be assumed to be proportional to the electric field:
ν n =
−qτ c /m n
E = −μ n E,
ν p =
qτ c /m p
E = μ p E,
(5.3)
with ν n , ν p , μ n , μ p being the drift velocities and low-field mobilities of electrons
and holes, respectively, q elementary charge, τ c average time between collisions,
m n , m p effective masses of electrons and holes, and E electric field. For a high
electric fields τ c decreases and the drift velocity saturates.
At very high electric field electrons and holes may acquire sufficient energy in
between collisions to generate additional electron hole pairs. This avalanche process
can be the cause for an electrical breakdown of devices. It may also be used as an
intrinsic amplification process in order to get sufficiently high signals from very
small ionization.
For inhomogeneous carrier distributions charge carriers will preferably diffuse
from high concentrations to regions of lower concentrations. This diffusion mechanism is described by
F n = −D n ∇n, F p = −D p ∇p.
(5.4)