Validity of Wiedemann-Franz law in thermoelectric half Heusler compounds

There is renewed interest in the field of thermoelectrics for power generation. Several promising thermoelectrics are half-Heusler narrow band gap semiconductors. The efficiency of thermoelectric energy conversion depends on the transport coefficients through the figure of merit $ZT=\sigma S^2T/\kappa $. For large \textit{ZT} , it is necessary to decrease the total thermal conductivity ($\kappa =\kappa _l +\kappa _{el} )$ as well as increase the Seebeck coefficient ($S)$ and the electrical conductivity ($\sigma )$To determine $\kappa _l $ experimentally, one usually subtracts the electronic thermal conductivity ($\kappa _{el} )$ from measured $\kappa $, using the Wiedemann-Franz law ($\kappa _{el} =L_0 \sigma T$, $L_0 =2.45\times 10^{-8}W\Omega /K^2)$. To examine the validity of this law in half-Heusler compounds, we have chosen HfCoS as an example. We have calculated the electronic transport coefficients by employing \textit{ab-initio} electronic structure method and the Boltzmann transport equation in HfCoSb. We calculate $\kappa _{el} $at constant current \textbf{J }($\kappa _{el,J} )$ and constant electric filed \textbf{E }($\kappa _{el,E} )$ where $\kappa _{el,J} =\kappa _{el,E} -T\sigma S^2$ which shows a significant deviation from values obtained with Wiedemann-Franz law. $\kappa _{el,J} $ is much smaller than $\kappa _{el,E} $at low carrier concentrations ($n)$ and/or at high temperatures ($T)$ and the ratio $\kappa _{el,J} /\kappa _{el,E} \to 1$ at high $n$ and/or low $T$.