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

$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{n}}= \ensuremath{\ensurema...
...uremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\right)$     (3.162)
$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{p}}= - \ensuremath{\ensure...
...ath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\right) \,,$     (3.163)

which are similar to the systematically derived equations. The corresponding heat fluxes are identified as
$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{n}^\mathrm{q}}= \mathrm{q}...
...}}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$     (3.164)
$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_{p}^\mathrm{q}}= \mathrm{q}...
...}}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$     (3.165)

by the application of Onsager's reciprocity theorem. The total energy flux of all three subsystems is written as

$\displaystyle \ensuremath{\ensuremath{\mathitbf{j}}_\mathrm{tot}^\mathrm{u}}= -...
...suremath{\Phi}_{\ensuremath{\nu}}}\ensuremath{\ensuremath{\mathitbf{j}}_{p}}\,.$ (3.166)

The differential energy densities of electrons and holes are derived using Maxwell's relations [9] as
$\displaystyle \ensuremath{\mathrm{d}}\ensuremath{u_\ensuremath{n}}= \ensuremath...
...math{n},\ensuremath{p}}- \ensuremath{\ensuremath{\Phi}_{\ensuremath{n}}}\right)$     (3.167)
$\displaystyle \ensuremath{\mathrm{d}}\ensuremath{u_\ensuremath{p}}= \ensuremath...
...{n},\ensuremath{p}}- \ensuremath{\ensuremath{\Phi}_{\ensuremath{p}}}\right) \,.$     (3.168)

Inserting the accumulated differential energy densities for the electron, hole, and lattice subsystems into the energy balance equation

$\displaystyle \ensuremath{\ensuremath{\partial_{t} \ensuremath{u_\mathrm{tot}}}...
...remath{\cdot}}\ensuremath{\ensuremath{\mathitbf{j}}_\mathrm{tot}^\mathrm{u}}= 0$ (3.169)

yields the heat conduction equation

$\displaystyle \ensuremath{c_{\mathrm{tot}}}\ensuremath{\ensuremath{\partial_{t}...
...th{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\right) + \ensuremath{H}$ (3.170)

accounting for the conservation of total energy with the heat source term
$\displaystyle \ensuremath{H}= \mathrm{q}\frac{\ensuremath{\left\vert \ensuremat...
...} \right\vert}^2}{\ensuremath{\ensuremath{\mu}_{\ensuremath{p}}}\ensuremath{p}}$     (3.171)
$\displaystyle \quad + \mathrm{q}\left( \ensuremath{T}\left( \ensuremath{\partia...
...ath{p}} + \ensuremath{\ensuremath{\Phi}_{\ensuremath{p}}}\right) \ensuremath{G}$      
$\displaystyle \quad + \mathrm{q}\ensuremath{T}\left( \left( \ensuremath{\partia...
...bf{\nabla_{\!r}}}}\ensuremath{\cdot}}\ensuremath{\ensuremath{\mathitbf{j}}_{p}}$      
$\displaystyle \quad + \mathrm{q}\ensuremath{T}\left( \ensuremath{\ensuremath{\m...
...f{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\alpha}_{\ensuremath{p}}}
\right) \,.$      

With the electrochemical potential (3.88) and the Seebeck coefficient (3.153) for Maxwell-Boltzmann statistics, the heat-source term can be rewritten to
$\displaystyle \ensuremath{H}= \mathrm{q}\frac{\ensuremath{\left\vert \ensuremat...
...} \right\vert}^2}{\ensuremath{\ensuremath{\mu}_{\ensuremath{p}}}\ensuremath{p}}$     (3.172)
$\displaystyle \quad + \left( \frac{5}{2} k_\ensuremath{\mathrm{B}}\ensuremath{T}\right) \ensuremath{R}$      
$\displaystyle \quad + \mathrm{q}\left( \ensuremath{\ensuremath{\alpha}_{\ensure...
...bf{\nabla_{\!r}}}}\ensuremath{\cdot}}\ensuremath{\ensuremath{\mathitbf{j}}_{p}}$      
$\displaystyle \quad + \mathrm{q}\ensuremath{T}\left( \ensuremath{\ensuremath{\m...
...tbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\alpha}_{\ensuremath{p}}}\right)\,,$      

which can be finally simplified for the static case because of the vanishing $ \ensuremath{\ensuremath{\partial_{t} \ensuremath{\nu}}} $ -term in (3.37) to
$\displaystyle \ensuremath{H}= \mathrm{q}\frac{\ensuremath{\left\vert \ensuremat...
...} \right\vert}^2}{\ensuremath{\ensuremath{\mu}_{\ensuremath{p}}}\ensuremath{p}}$     (3.173)
$\displaystyle \quad + \mathrm{q}\left( \ensuremath{T}\left( \ensuremath{\ensure...
...emath{\ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{g}}}}\right) \ensuremath{G}$      
$\displaystyle \quad + \mathrm{q}\ensuremath{T}\left( \ensuremath{\ensuremath{\m...
...f{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\alpha}_{\ensuremath{p}}}
\right) \,.$      

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