Effect of a superconducting lead on heat generation in a quantum dot

By using the nonequilibrium Keldysh Green's functions, we study the heat generation in a quantum dot coupled to a normal and a superconducting lead. It is found that the magnitude of the superconducting energy gap plays a crucial role in the characteristics of heat generation. For example, the local heating induced by phonon assisted Andreev reflection increases hundredfold as the gap grows from below half phonon energy 0.5ω0 to above 0.5ω0. Another example is the heat in the QD can be taken away efficiently only when the gap is larger than one phonon energy. It is also found that at low temperature (T≪ω0) there exists a threshold bias voltage of half one phonon energy.