We study a quantum dot coupled to two semiconducting reservoirs, when the dot level and the electrochemical potential are both close to a band edge in the reservoirs. This is modelled with an exactly solvable Hamiltonian without interactions (the Fano-Anderson model). The model is known to show an abrupt transition as the dot-reservoir coupling is increased into the strong-coupling regime for a broad class of band structures. This transition involves an infinite-lifetime bound state appearing in the band gap. We find a signature of this transition in the continuum states of the model, visible as a discontinuous behaviour of the dots transmission function. This can result in the steady-state DC electric and thermoelectric responses having a very strong dependence on coupling close to critical coupling. We give examples where the conductances and the thermoelectric power factor exhibit huge peaks at critical coupling, while the thermoelectric figure of merit ZT grows as the coupling approaches critical coupling, with a small dip at critical coupling. The critical coupling is thus a sweet spot for such thermoelectric devices, as the power output is maximal at this point without a significant change of efficiency.
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