3.5.7 Non-Diagonal Relaxation Time Ansatz

In the sequel, an alternative ansatz for the scattering operator is introduced. Instead of the commonly used relaxation time approximation, the stochastic part of the moments is modeled using an expansion of the scattering integrals into the odd moments of the distribution function [90,91]. Thus, the scattering integrals are represented as linear combinations of the fluxes derived. The weight set chosen for this ansatz is the same as in Bløtekjær's approach, so the left side of the Boltzmann transport equation can be expressed similarly. Using the deterministic parts of the particle and energy flux equations from Eqs. (3.86) and (3.96)

$$\ensuremath{\ensuremath{\mathitbf{F}}_0}= - \frac{1}{\mathrm{q}} ... ...{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\tilde{\varphi}}$$ (3.98)

$$\ensuremath{\ensuremath{\mathitbf{F}}_1}= -\frac{5}{2} \frac{1}{\... ...nsuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\tilde{\varphi}}\,,$$ (3.99)

the corresponding equations are formally expanded as

$$\ensuremath{\ensuremath{\mathitbf{F}}_0}= \ensuremath{Z_{00}}\ens... ..._\nu}+ \ensuremath{Z_{01}}\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}$$ (3.100)

$$\ensuremath{\ensuremath{\mathitbf{F}}_1}= \ensuremath{Z_{10}}\ens... ...u}+ \ensuremath{Z_{11}}\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}\,.$$ (3.101)

Since the actual quantities of interest are the particle current $\ensuremath{\ensuremath{\mathitbf{j}}_\nu}$ and the energy flux density $\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}$ , the coupled equations are formulated in order to explicitly express the particle and energy flux, respectively

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= \frac{\ensuremath{Z_{... ...\ensuremath{Z_{00}}\ensuremath{Z_{11}}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}}$$ (3.102)

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}= \frac{\ens... ...uremath{Z_{00}}\ensuremath{Z_{11}}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}} \,.$$ (3.103)

The fluxes $\ensuremath{\ensuremath{\mathitbf{F}}_0}$ and $\ensuremath{\ensuremath{\mathitbf{F}}_1}$ Eqs. (3.98) and (3.99) are inserted to (3.102) and thus the particle current equation reads

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= - \frac{\ensuremath{Z... ...math{\nu}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ln m^*$$ (3.104)

$$\quad - \frac{\ensuremath{Z_{11}}-\frac{5}{2}k_\ensuremath{\mathr... ...remath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T_\ensuremath{\nu}}\,.$$

Introducing the electrochemical potential (3.62), the current can be expressed as a linear combination of a $\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$ and a $\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T_\ensuremath{\nu}}$ expression

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= - \frac{\ensuremath{Z... ...ath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$$ (3.105)

$$\quad -\frac{ \left(\ensuremath{Z_{11}}-\frac{5}{2}k_\ensuremath{... ...remath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T_\ensuremath{\nu}}\,.$$

For the according energy flux equation, which is expressed analogously, the fluxes $\ensuremath{\ensuremath{\mathitbf{F}}_0}$ and $\ensuremath{\ensuremath{\mathitbf{F}}_1}$ Eqs. (3.98) and (3.99) are inserted to (3.103)

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}= - \frac{\f... ...math{\nu}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ln m^*$$ (3.106)

$$\quad - \frac{\frac{5}{2}k_\ensuremath{\mathrm{B}}\ensuremath{T_\... ...remath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T_\ensuremath{\nu}}\,.$$

With the electrochemical potential, it can be rewritten as

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}= - \frac{\f... ...ath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$$ (3.107)

$$\quad -\frac{\left(\frac{5}{2} k_\ensuremath{\mathrm{B}}\ensurema... ...remath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T_\ensuremath{\nu}}\,.$$

A coefficient comparison between the particle flux equation (3.105) and the according equation derived using Bløtekjær's concept of macroscopic relaxation times (3.88) enables the identification of several transport parameters

$$\ensuremath{\ensuremath{\mu}_\nu}= \frac{\ensuremath{Z_{11}}- \fr... ...uremath{Z_{00}}\ensuremath{Z_{11}}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}} \,,$$ (3.108)

$$\ensuremath{\ensuremath{\mu}_\nu^\mathrm{u}}= \frac{1}{\frac{5}{2... ...uremath{Z_{00}}\ensuremath{Z_{11}}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}} \,,$$ (3.109)

$$\ensuremath{\alpha}= -\ensuremath{\mathrm{s}_\nu}\frac{k_\ensurem... ..._\ensuremath{\mathrm{B}}\ensuremath{T_\ensuremath{\nu}}\ensuremath{Z_{01}}} \,.$$ (3.110)

Analogously to the extended ansatz for the stochastic part of Boltzmann's equation, these transport coefficients can be seen as an extension to the ones derived using Bløtekjær's ansatz. Assuming the cross coefficients $\ensuremath{Z_{01}}$ and $\ensuremath{Z_{10}}$ to be zero, the coefficients for Bløtekjær's approach carried out in Section 3.5.6 are obtained. Although the scattering parameters $\ensuremath{Z_{ij}}$ can be calculated using accurate physical models, this approach results in a very complicated description.

M. Wagner: Simulation of Thermoelectric Devices