Deriving Coefficients for Calculating Figure of Merit

[Denver Dang]

Homework Statement

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:

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

Homework 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:

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

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.

Thanks in advance.


Regards.

 

[mark726]

Hello,
Thank you for posting your question in this forum. It is absolutely okay to post here and I will do my best to assist you.

The figure of merit you are referring to is commonly used in the field of thermoelectrics and is known as the thermoelectric figure of merit (ZT). The ZT value is a measure of the efficiency of a thermoelectric material in converting heat to electricity.

To answer your question, there are several papers and articles that discuss the derivation and calculation of the ZT value. One of the most commonly cited papers is "Thermoelectric figure of merit of a one-dimensional conductor" by M. S. Dresselhaus et al. (Phys. Rev. B 47, 12976 – Published 1 May 1993). This paper provides a detailed derivation of the ZT value for a one-dimensional conductor.

Additionally, there are several textbooks and online resources that discuss the derivation of the ZT value and its components. Some recommended resources include "Introduction to Thermoelectricity" by Julian Goldsmid and "Thermoelectrics Handbook: Macro to Nano" edited by D.M. Rowe. You can also find helpful materials on websites such as ResearchGate and arXiv.

I hope this helps in your search for the derivation of the ZT value. If you have any further questions, please do not hesitate to ask. Best of luck.

Regards,