# Derivation of electrical conductivity, Seebeck coefficient and thermal conductivity

1. ### Denver Dang

142
1. The problem statement, all variables and given/known data

Hi...

Don't know if it's actually homework, since it's not, but I hope it's okay to post in here.
I am looking for a paper/website/article of some sort, that might have the derivations of the above mentioned coefficients ?
It's for calculating the figure of merit:

$$Z=\frac{\sigma {{S}^{2}}}{\kappa }$$

2. Relevant equations

The equations if was hoping to maybe find some derivatins of is these three:

$$\sigma ={{e}^{2}}\int{d\varepsilon \left( -\frac{\partial {{f}_{0}}}{\partial \varepsilon } \right)\Xi \left( \varepsilon \right)}$$

$$S=\frac{e{{k}_{B}}}{\sigma }\int{d\varepsilon \left( -\frac{\partial {{f}_{0}}}{\partial \varepsilon } \right)\Xi \left( \varepsilon \right)}\frac{\varepsilon -\mu }{{{k}_{B}}T}$$

$${{\kappa }_{0}}={{k}_{B}}T\int{d\varepsilon \left( -\frac{\partial {{f}_{0}}}{\partial \varepsilon } \right)\Xi \left( \varepsilon \right)}{{\left[ \frac{\varepsilon -\mu }{{{k}_{B}}T} \right]}^{2}}$$

where:

$$$\Xi =\sum\limits_{\overrightarrow{k}}{{{\overrightarrow{v}}_{\overrightarrow{k}}}{{\overrightarrow{v}}_{\overrightarrow{k}}}{{\tau }_{\overrightarrow{k}}}}$$$

3. The attempt at a solution

Don't know if it is possible to find derivations of these, or somewhat similar, or I have to calculate it myself. But I just wanted to try.