Band gap estimation from temperature dependent Seebeck measurement - Deviations from the 2e|S|_maxT_max relation

In characterizing thermoelectric materials, electrical and thermal transport measurements are often used to estimate electronic band structure properties such as the effective mass and band gap. The Goldsmid-Sharp band gap, E_g = 2e|S|_maxT_max, is a tool widely employed to estimate the band gap from temperature dependent Seebeck coefficient measurements. However, significant deviations of more than a factor of two are now known to occur. We find that this is when either the majority-to-minority weighted mobility ratio (A) becomes very different from 1.0 or as the band gap (E_g) becomes significantly smaller than 10 k_BT. For narrow gaps (E_g ≲ 6 k_BT), the Maxwell-Boltzmann statistics applied by Goldsmid-Sharp break down and Fermi-Dirac statistics are required. We generate a chart that can be used to quickly estimate the expected correction to the Goldsmid-Sharp band gap depending on A and S_max; however, additional errors can occur for S < 150 μV/K due to degenerate behavior.