Multiple-probe electronic open boundaries with bad contacts

We revisit the weak-coupling limit of the hairy probes method for electronic open boundaries. In this limit, the electronic density matrix is approximately stationary. We exploit this fact to combine hairy probes with electron-phonon scattering at the level of low-order transitions between eigenstates of the electronic Hamiltonian. This provides a method that is computationally very efficient, and whose results can be interpreted quite straightforwardly. The resultant time-dependent hybrid method is illustrated through the numerical calibration of a thermoelectric nanoscale thermometer.