3.5.11 Phenomenological Approach
Besides the systematic approach carried out in the preceding sections, the
thermoelectric behavior of semiconductors can also be explained by an approach
based on phenomenological irreversible thermodynamics
[54,96].
For the non-isothermal case, besides the gradient of the electrochemical
potential a temperature gradient yields an additional driving force which is
known as the Seebeck effect in the particle current relations
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{n}}= \ensuremath{\ensurema...
...uremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\right)$](img517.png) |
|
|
(3.162) |
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{p}}= - \ensuremath{\ensure...
...ath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\right) \,,$](img518.png) |
|
|
(3.163) |
which are similar to the systematically derived equations. The corresponding
heat fluxes are identified as
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{n}^\mathrm{q}}= \mathrm{q}...
...}}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$](img519.png) |
|
|
(3.164) |
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{p}^\mathrm{q}}= \mathrm{q}...
...}}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$](img520.png) |
|
|
(3.165) |
by the application of Onsager's reciprocity theorem. The total energy flux
of all three subsystems is written as
![$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_\mathrm{tot}^\mathrm{u}}= -...
...suremath{\Phi}_{\ensuremath{\nu}}}\ensuremath{\ensuremath{\mathitbf{j}}_{p}}\,.$](img521.png) |
(3.166) |
The differential energy densities of electrons and holes are derived using
Maxwell's relations [9] as
![$\displaystyle \ensuremath{\mathrm{d}}\ensuremath{u_\ensuremath{n}}= \ensuremath...
...math{n},\ensuremath{p}}- \ensuremath{\ensuremath{\Phi}_{\ensuremath{n}}}\right)$](img522.png) |
|
|
(3.167) |
![$\displaystyle \ensuremath{\mathrm{d}}\ensuremath{u_\ensuremath{p}}= \ensuremath...
...{n},\ensuremath{p}}- \ensuremath{\ensuremath{\Phi}_{\ensuremath{p}}}\right) \,.$](img523.png) |
|
|
(3.168) |
Inserting the accumulated differential energy densities for the electron, hole,
and lattice subsystems into the energy balance equation
![$\displaystyle \ensuremath{\ensuremath{\partial_{t} \ensuremath{u_\mathrm{tot}}}...
...remath{\cdot}}\ensuremath{\ensuremath{\mathitbf{j}}_\mathrm{tot}^\mathrm{u}}= 0$](img524.png) |
(3.169) |
yields the heat conduction equation
![$\displaystyle \ensuremath{c_{\mathrm{tot}}}\ensuremath{\ensuremath{\partial_{t}...
...th{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\right) + \ensuremath{H}$](img505.png) |
(3.170) |
accounting for the conservation of total energy with the heat source term
With the electrochemical potential (3.88) and the Seebeck
coefficient (3.153) for Maxwell-Boltzmann statistics, the heat-source term can be
rewritten to
which can be finally simplified for the static case because of the vanishing
-term in (3.37) to
The final result is equal to the heat-source term of the systematically
derived heat-flow equation in Section 3.5.10 under several simplifying
assumptions, when the electrochemical potential as well as the Seebeck
coefficient are expressed by the corresponding relations for Maxwell-Boltzmann statistics.
M. Wagner: Simulation of Thermoelectric Devices