3.5.5 Stratton's Approach

Stratton proposed to formulate the collision term using one single microscopic relaxation time. Usually, this relaxation time, $\ensuremath{{\tau}}$ , is modeled to be energy dependent. In order to obtain the fluxes on the right side of the governed flux equations directly, the weights for the derivation are chosen as

$$\ensuremath{X}= \left(1, \ensuremath{{\tau}}\ensuremath{\ensurema... ...\tau}}\ensuremath{\ensuremath{\mathcal{E}}\ensuremath{\mathitbf{p}}}\right) \,.$$ (3.36)

During the following derivation, several assumptions and simplifications are sequentially introduced. The assumptions made throughout the derivation of this transport model are as follows:

• Microscopic relaxation time approximation
• Time derivatives are neglected in the flux equations
• Product ansatz for the kinetic energy
• The tensor valued transport parameters are estimated by their traces
• Power-law approximation $\ensuremath{{\tau}}= \ensuremath{\tau_0} \left(\frac{\ensuremath{\mathcal{E}}}{\ensuremath{\ensuremath{\mathcal{E}}_0}}\right)^{{\ensuremath{r_\nu}}}$ for the microscopic relaxation time
• Parabolic band structure
• Linearized heated displaced Maxwellian distribution function for the closure relation
• $\ensuremath{\ensuremath{\mathitbf{k}}}$ -independent ${\ensuremath{r_\nu}}$

Subsections

• 3.5.5.1 Carrier Balance Equation
• 3.5.5.2 Particle Flux Equation
• 3.5.5.3 Energy Balance Equation
• 3.5.5.4 Energy Flux Equation

M. Wagner: Simulation of Thermoelectric Devices