  Here, we investigate the maximum power and corresponding efficiency of
thermoelectric generators through devising a set of protocols for the
isothermal and adiabatic processes of thermoelectricity to build a Carnot-like
thermoelectric cycle, with the analysis based on fluctuation theorem (FT).
First of all, the Carnot efficiency can be readily obtained for the
quasi-static thermoelectric cycle, with the vanishing power. Moreover, the
maximum power-efficiency pair of the finite-time thermoelectric cycle is
derived, which is found to have the identical form to that of Brownian motors
characterized by the stochastic thermodynamics. However, it is of significant
discrepancy compared to the linear-irreversible and
endoreversible-thermodynamics-based formulations. The distinction compared to
the linear-irreversible-thermodynamics case could result from the difference in
the definitions of Peltier and Seebeck coefficients in the thermoelectric
cycle. As for the endoreversible thermodynamics, we argue the applicability of
endoreversibility could be questionable for analyzing the thermoelectric cycle
here, due to the incompatibility of the endoreversible hypothesis that
attributes the irreversibility to finite heat transfer with thermal reservoirs,
though the distinction of the mathematical expressions can vanish with the
assumption that the ratio of thermoelectric power factors at the high and low
temperatures is equal to the square root of the temperature ratio (this
condition could significantly deviate from the practical case).
