  We construct a scattering theory of weakly nonlinear thermoelectric transport
through sub-micron scale conductors. The theory incorporates the leading
nonlinear contributions in temperature and voltage biases to the charge and
heat currents. Because of the finite capacitances of sub-micron scale
conducting circuits, fundamental conservation laws such as gauge invariance and
current conservation require special care to be preserved. We do this by
extending the approach of Christen and B\"uttiker [Europhys. Lett. 35, 523
(1996)] to coupled charge and heat transport. In this way we write relations
connecting nonlinear transport coefficients in a manner similar to Mott's
relation between the linear thermopower and the linear conductance. We derive
sum rules that nonlinear transport coefficients must satisfy to preserve gauge
invariance and current conservation. We illustrate our theory by calculating
the efficiency of heat engines and the coefficient of performance of
thermoelectric refrigerators based on quantum point contacts and resonant
tunneling barriers. We identify in particular rectification effects that
increase device performance.
