  We characterize the current-carrying nonequilibrium steady-state in a
single-band Hubbard model confronted with a static electric field in the
presence of quenched disorder. As disorder is not expected to dissipate the
extra energy injected by the field, optical phonons assisted by a fermionic
heat bath serve as dissipation channels for the current-induced Joule heat
generated by the accelerated electrons. In a purely electronic setup, the
disorder-induced dephasing cannot contribute states within the gap but only
smear out the edges of the Hubbard bands. When phonons are taken into account,
the different nature of disorder-induced dephasing and phonon-related
dissipation becomes clear. We show that although both disorder and
electron-phonon interaction enhance the current at off-resonant fields,
disorder effects play a marginal role since they cannot provide in-gap states
which are instead brought about by phonons and represent the privileged
relaxation pathway for excited electrons.
