# On the Lorenz number of multi-band materials

## TL;DR

This paper investigates how multi-band effects in semiconductors can cause significant deviations in the Lorenz number from classical values, impacting thermoelectric material analysis and thermal conductivity measurements.

## Contribution

It provides analytical expressions and case studies demonstrating large Lorenz number deviations in multi-band semiconductors like PbTe and SnSe.

## Key findings

- Deviations up to a factor of two in unipolar systems.
- Order of magnitude deviations in bipolar systems.
- Analytical tools for quantifying Lorenz number variations.

## Abstract

There are many exotic scenarios where the Lorenz number of the Wiedemann-Franz law is known to deviate from expected values. However, in conventional semiconductor systems, it is assumed to vary between the values of ~1.49x10^{-8} W {\Omega} K^{-2} for non-degenerate semiconductors and ~2.45x10^{-8} W {\Omega} K^{-2} for degenerate semiconductors or metals. Knowledge of the Lorenz number is important in many situations, such as in the design of thermoelectric materials and in the experimental determination of the lattice thermal conductivity. Here we show that, even in the simple case of two and three band semiconductors, it is possible to obtain substantial deviations of a factor of two (or in the case of a bipolar system with a Fermi level near the midgap, even orders of magnitude) from expectation. In addition to identifying the sources of deviation in unipolar and bipolar two-band systems, a number of analytical expressions useful for quantifying the size of the effect are derived. As representative case-studies, a three-band model of the materials of lead telluride (PbTe) and tin sellenide (SnSe), which are important thermoelectric materials, is also developed and the size of possible Lorenz number variations in these materials explored. Thus, the consequence of multi-band effects on the Lorenz number of real systems is demonstrated.