Pisarenko's Formula for the Thermopower

Abstract

The thermopower $α$ (also known as the Seebeck coefficient) is one of the most fundamental material characteristics for understanding charge carrier transport in thermoelectric materials. Here, we revisit the Pisarenko formula for the thermopower, which was traditionally considered valid only for non-degenerate semiconductors. We demonstrate that regardless of the dominating scattering mechanism, the Pisarenko formula describes accurately enough the relationship between thermopower $α$ and charge carrier concentration $n$ beyond the non-degenerate limit. Moreover, the Pisarenko formula provides a simple thermopower-conductivity relation, $α= \pm \frac{k_{\mathrm{B}}}{e} (b - \ln σ)$, valid for materials with $α> 90$ $μ$V K$^{-1}$ when acoustic phonon scattering is predominant. This offers an alternative way to analyze electron transport when Hall measurements are difficult or inaccessible. Additionally, we show how the Pisarenko formula can be used to estimate the maximum power factor of a thermoelectric material from the weighted mobility of a single, not necessarily optimized, sample at any given temperature.

Figures (5)

• Figure 1: Effective mass $m^{\ast}_d$ as a function of the thermopower $\alpha$, calculated from the experimental data of Yb_xCo4Sb12 (Ref. tang2015) using the numerical solution (Eq. \ref{['Eq:conc_SPB']}, black symbols), and two analytical formulas, namely, for degenerate semiconductors (Eq. \ref{['Eq:alpha_deg']}, red symbols) and Pisarenko formula (Eq. \ref{['Eq:Pisarenko']}, blue symbols). Acoustic phonon scattering was assumed ($r = -1/2$). The vertical gray dotted line indicates the threshold $\alpha$ value ($\approx 155µV\per K$), below which Eq. \ref{['Eq:alpha_deg']} provides more accurate $m^{\ast}_d$ estimates. The black arrows indicate underestimation of $m^{\ast}_d$ when Eq. \ref{['Eq:alpha_deg']} is used for $\alpha > 155µV\per K$. The gray dashed curve is a guide to the eye to visualize the general trend.
• Figure 2: Thermopower $\alpha$ as a function of the chemical potential $\eta$, calculated using the numerical solution (Eq. \ref{['Eq:alpha_SPB']}, black solid curve), the degenerate limit (Eq. \ref{['Eq:alpha_SPBdeg']}, red dotted curve), the non-degenerate limit (Eq. \ref{['Eq:alpha_SPBnondeg']}, blue dotted curve), and two analytical formulas for degenerate (Eq. \ref{['Eq:alpha_deg']}, red dashed curve) and non-degenerate (Pisarenko formula, Eq. \ref{['Eq:Pisarenko']}, blue dashed curve) semiconductors within the acoustic phonon scattering approximation ($r = -1/2$). The upper panel shows $\delta$, the relative difference in $\alpha$ calculated using the corresponding formulas. The thin gray dotted lines indicate the threshold where $\delta$ between $\alpha$ values calculated from Eq.\ref{['Eq:alpha_deg']} and Eq.\ref{['Eq:Pisarenko']} is equal and reaches $\approx13%$, representing the limits of their applicability.
• Figure 3: A Jonker plot ($\alpha$ vs. log-scale $\sigma$), schematically displaying the thermopower-conductivity relation as predicted by the Pisarenko formula (Eq. \ref{['Eq:Pisarenko_sigma']}, white dashed line) and the full numerical solution (Eq. \ref{['Eq:alpha_SPB']}, light gray dotted curve). The slope of the Pisarenko line is $\pm k_{\mathrm{B}}/e$$\approx$$\pm 86.291µV\per K$ (positive slope for $n$-type semiconductors and a negative slope for $p$-type semiconductors), while its horizontal position is determined by the weighted mobility $\mu_w$ value.
• Figure 4: Experimental (a) absolute thermopower $|\alpha|$ and (b) power factor $\alpha^{2}\sigma$ as functions of electrical conductivity $\sigma$ for selected thermoelectric material systems at various temperatures, including La2CuO4 (1123K),su1990BiCuSeO (800K),zhao2010liu2015feng2020aCuGaTe2 (475K),shen2016ahmed2017Ba8Ga16Sn30 (500K),saiga2012PbTe (700K),pei2014Mg2(Si,Sn) (300K),liu2014Co4Sb12 (800K),shi2011tang2014 and FeNbSb (800K).fu2015 Dashed lines represent Pisarenko formula based predictions (Eqs. \ref{['Eq:Pisarenko_sigma']} and \ref{['Eq:PF_simple']}, respectively), calculated using different values of weighted mobility $\mu_w$ providing the best agreement with each experimental dataset. A portion of the literature data was retrieved from the StarryData2 database.katsura2019datakatsura2025
• Figure 5: Normalized power factor $(\alpha^{2}\sigma)/(\alpha^{2}\sigma)_{\mathrm{max}}$ as a function of electrical conductivity $\sigma$ calculated using the full numerical solution (black solid curve), the non-degenerate limit (blue dotted curve), the degenerate limit (red dashed curve), and the Pisarenko formula (Eq. \ref{['Eq:PF_simple']}, blue dashed curve) within the acoustic phonon scattering approximation ($r = -1/2$). The upper panel shows $\delta$, the relative difference in $(\alpha^{2}\sigma)/(\alpha^{2}\sigma)_{\mathrm{max}}$ calculated using the corresponding formulas.