Page 207 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
Since optical phonons have energies of about 50 meV and the conduction
band has a width of the order of 1 eV and more there, will be almost no
electrons able to overcome this barrier and turn back. Therefore, in a bulk
semiconductor crystal Bloch oscillation are not observed.
On the other hand, the above example tells us that all material properties
of electrons needed to describe fluxes of moments like the energy current
density or the particle current density are in this special case due to
phonon interaction, i.e., momentum transfer between the phonon system
and the electronic system. Since there is a dissipation channel opened for
the electronic system, these fluxes are termed irreversible. The term in
the BTE responsible for this process is the collision term, which we have
approximated in this section by a relaxation time. Therefore, all constitu-
tive laws for the current densities will be connected to this relaxation
time.

6.2 Local Equilibrium Description

As already pointed out in the previous section, we want to analyze what
kind of constitutive equations follow from the deviation from thermal
equilibrium if we apply a relaxation time approximation. Therefore, we
introduce the picture of local equilibrium. This means that at least locally
in position space a thermodynamic equilibrium distribution f ( kx t)
,,
0
exists to which f kx t,,( ) relaxes. Let us assume that the equilibrium dis-
tribution is given by a Fermi distribution

1
,,
f ( kx t) = ----------------------------------------------------------- (6.24)
0
,
(
–
exp  E k() µ x t)  1
---------------------------------- +
 kT x t) 
(
,
µ
We see that in (6.24) the chemical potential and the temperature T
appear time and space dependent.
204 Semiconductors for Micro and Nanosystem Technology