Page 204 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Semi-Classical Boltzmann Transport Equation
The single terms are easily interpreted: first there is the change in energy
density in time, second we have an energy source term, which, with the
electric field as the force term F = – qE , is the Joule heating of elec-
trons, and third we have the divergence of the energy current density. The
term on the r.h.s. we interpret as an energy relaxation term.

From the balance equations we see that the momentum procedure as
applied to the BTE produces an infinite series of coupled equations: the
continuity equation (6.19) which contains the particle current density that
has its equation of motion given by the momentum balance (6.21) and
contains the momentum current density tensor as given in (6.15). One
part of the momentum current density tensor is the energy density, which
has it equation of motion given by the energy balance (6.22), which in
turn contains the energy current density, i.e., a moment of third order,
which has its own equation of motion coupled to higher moments, etc.

This is an infinite hierarchy of equations that overall correspond to the
BTE. Here the same arguments hold as given for the infinite series of
moments representing the distribution function. Let us assume that for
our purpose additional information about higher r-th moments does only
contribute to marginal changes in the description of our system. This
means that we may break the hierarchy at a certain point and discard or
approximate all higher order moments. Therefore, some knowledge
about approximating or modeling the current densities is needed.

6.1.5 Relaxation Time Approximation

Up to know we did not specify the scattering term of the BTE in more
detail than given by (6.5) or (6.6). The scattering or transition probabili-
ties W k →( k') must be calculated by taking into account the micro-
scopic interaction process between particles. This is where again
quantum mechanics enters. In this section we will do another approxima-
tion step which is in good agreement with the above–derived moment
equations. We assume the distribution function to be an equilibrium dis-

Semiconductors for Micro and Nanosystem Technology 201