  We study the spectral and transport properties of a Mott insulator driven by
a static electric field into a non-equilibrium steady state. For the
dissipation, we consider two mechanisms: Wide-band fermion reservoirs and
phonons included within the Migdal approximation. The electron correlations are
treated via non-equilibrium dynamical mean field theory with an impurity solver
suitable for strong correlations. We find that dissipation via phonons is
limited to restricted ranges of field values around Wannier-Stark resonances.
To cover the full range of field strengths, we allow for a small coupling with
fermionic baths, which stabilizes the solution. When considering both
dissipation mechanisms, we find that phonons enhance the current for field
strengths close to half of the gap while lowering it at the gap resonance as
compared to the purely electronic dissipation used by Murakami and Werner
[arXiv:1804.08257]. Once phonons are the only dissipation mechanism, the
current in the metallic phase is almost one order of magnitude smaller than the
typical values obtained by coupling to a fermionic bath. In this case, the
transport regime is characterized by an accumulation of charge in the upper
Hubbard band described by two effective chemical potentials.
