Exact Thermoelectric Transport Coefficients and Figure of Merit for Graphene Photothermoelectric Devices from a Finite Zeta-Function Mott Series

The standard Mott formula is widely used to describe thermoelectric transport, but it becomes less accurate when the temperature is not much smaller than the Fermi energy. In this work, we develop an all-orders extension of the Mott approach using a series of Riemann zeta functions. We show that when the transport function is a polynomial, the series ends after a finite number of terms, giving exact results within the model. We apply this method to graphene photothermoelectric devices using a quadratic conductivity model. The results provide closed-form expressions for the Seebeck coefficient, Lorenz ratio, and electronic figure of merit. The analysis shows that the Seebeck coefficient reaches a maximum instead of increasing indefinitely, while the Wiedemann-Franz law can be significantly violated at higher temperatures. We also find that disorder reduces the thermoelectric performance and that the electronic figure of merit has an upper limit in the clean graphene model. Finally, we discuss the effect of radiative heat transport on the figure of merit. These results provide a simple analytical way to study graphene thermoelectric transport beyond the usual low-temperature Mott approximation.