Nonlinear corrections to the thermoelectric efficiency of a nanoscale device

We investigate the nonlinear thermoelectric transport in a generic nanoscale device connected to two side reservoirs at different temperatures ($T_L$ and $T_R$) and chemical potentials (${\mu }_L$ and ${\mu }_R$). We derive equations for the charge (electric) and heat (thermal) currents. These equations allow for the estimation of the second order contributions to the system’s thermoelectric response and the analytical derivation of the first nonlinear contributions to the system’s electric conductance ${\sigma }^{\left(2\right)}$, Seebeck coefficient $S^{\left(2\right)}$, and electronic thermal conductance ${\kappa }_{el}^{\left(2\right)}$. In the generation mode, when the system’s output power is positive ($P>0$), we estimate the maximum output power and efficiency of the system. The results are general and rely on generic dimensionless kinetic transport coefficients $K_n^p(\mu ,T)$ that depends on the system’s characteristic electronic transmission function $\tau \left(E\right)$. To outline the differences between the linear and nonlinear approximations we consider the particular case of a generalized Fano line-shape electronic transmission function and exactly calculate the dimensionless kinetic transport coefficients in terms of Hurwitz zeta functions and Bernoulli numbers. The output power efficiency of the system is estimated as function of the energy $\varepsilon =(E_d-\mu )/k_{B}T$ and broadening $\gamma ={\Gamma }_d/k_{B}T$ parameters. These results support the need for higher order terms in the theoretical analysis of the thermoelectric transport in nanoscale devices and allow for the optimization of the system’s properties for an efficient thermoelectric response.