On the Lorenz number of multiband materials

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 ${~}1.49×{}{10}^{{-}8}{{0.16em}{0ex}}W{{0.16em}{0ex}}{{Ω}}{{0.16em}{0ex}}{K}^{{-}2}$ for nondegenerate semiconductors and ${~}2.45×{}{10}^{{-}8}{{0.16em}{0ex}}W{{0.16em}{0ex}}{{Ω}}{{0.16em}{0ex}}{K}^{{-}2}$ 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 2 (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 multiband effects on the Lorenz number of real systems is demonstrated.