Simplified approach to estimate Lorenz number using experimental Seebeck coefficient for non parabolic band

Reduction of lattice thermal conductivity ($\kappa_L$) is one of the most effective ways of improving thermoelectric properties. However extraction of $\kappa_L$ from the total measured thermal conductivity can be misleading if Lorenz ($L$) number is not estimated correctly. The $\kappa_L$ is obtained using Wiedemann-Franz law which estimates electronic part of thermal conductivity $\kappa_e$ = $L$$\sigma$T where, $\sigma$ and T are electrical conductivity and temperature. The $\kappa_L$ is then estimated as $\kappa_L$ = $\kappa_T$ - $L$$\sigma$T. For the metallic system the Lorenz number has universal value of 2.44 $\times$ 10$^{-8}$ W$\Omega$K$^{-2}$ (degenerate limit), but for no-degenerate semiconductors, the value can deviate significantly for acoustic phonon scattering, the most common scattering mechanism for thermoelectric above room temperatures. Up till now, $L$ is estimated by solving a series of equation derived form Boltzmann transport equations. For the single parabolic band (SPB) an equation was proposed to estimate $L$ directly from the experimental Seebeck coefficient. However using SPB model will lead to overestimation of $L$ in case of low band gap semiconductors which result in underestimation of $\kappa_L$ sometimes even negative $\kappa_L$. In this letter we propose a simpler equation to estimate $L$ for a non parabolic band. Experimental Seebeck coefficient, band gap($E_g$), and Temperature ($T$) are the main inputs in the equation which nearly eliminates the need of solving multiple Fermi integrals besides giving accurate values of $L$.