In the following, the derivation of the particle flux equation is performed
analogously to Stratton's approach. The starting point is again the Boltzmann transport equation
with a general weight
in the form of equation (3.30). Inserting
as
yields
![$\displaystyle \underbrace{\vphantom{\frac{1}{\hbar}}\ensuremath{\ensuremath{\en...
...le \! \rangle}}{\ensuremath{{\tau_\ensuremath{\ensuremath{\mathitbf{j}}}}}} \,.$](img392.png) |
(3.80) |
Application of the trace approximation of tensor valued expressions
(3.39) as well as a product ansatz for the energy as presented in
equation (3.40) on term (i) results in
![$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\en...
... \! \langle \ensuremath{\mathcal{E}}\ensuremath{\gamma} \rangle \! \rangle} \,.$](img393.png) |
(3.81) |
The Poisson bracket within the average in (ii) has to be expanded using
(A.6). Furthermore, the inverse product rule (B.3) is used
to transform the
- term in the first term. The product ansatz
for the energy as well as the trace approximation result in
The third term (iii) can be handled in a straight-forward manner. Assembly
of all three terms leads to
![$\displaystyle \frac{2}{3} \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_...
...le \! \rangle}}{\ensuremath{{\tau_\ensuremath{\ensuremath{\mathitbf{j}}}}}} \,.$](img399.png) |
(3.83) |
Assuming parabolic bands and a heated, displaced Maxwellian,
becomes
unity and the average reads
![$\displaystyle \ensuremath{\langle \! \langle \ensuremath{\mathcal{E}} \rangle \...
...{3}{2} \ensuremath{\nu}k_\ensuremath{\mathrm{B}}\ensuremath{T_\ensuremath{\nu}}$](img401.png) |
(3.84) |
normalized with the carrier concentration (3.54). Summation over all
parts as well as the mobility definition consistent with the homogeneous case
![$\displaystyle \ensuremath{\ensuremath{\mu}_\nu}= \frac{\mathrm{q}\ensuremath{{\tau_\ensuremath{\ensuremath{\mathitbf{j}}}}}}{m^*}$](img402.png) |
(3.85) |
yields the final form of the particle current equation
![$\displaystyle - \frac{\ensuremath{\langle \! \langle \ensuremath{\ensuremath{\m...
...{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\tilde{\varphi}}$](img403.png) |
|
|
(3.86) |
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_\nu}= -\frac{k_\ensuremath{...
...nsuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\tilde{\varphi}}\,.$](img351.png) |
|
|
(3.87) |
Rewriting the particle current equation with the electrochemical potential
defined in Eqs. (3.60) and (3.62) results in
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_\nu}= - \ensuremath{\mathrm...
...remath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T_\ensuremath{\nu}}\,.$](img404.png) |
(3.88) |
M. Wagner: Simulation of Thermoelectric Devices