  The non-Hermitian skin effect (NHSE) is a significant phenomenon observed in
non-Hermitian systems under open boundary conditions, where the extensive bulk
eigenstates tend to accumulate at the lattice edges. In this article, we
investigate how an electric field affects the localization properties in a
non-Hermitian mosaic Stark lattice, exploring the interplay between the Stark
localization, mobility edge (ME), and the NHSE induced by nonreciprocity. We
analytically obtain the Lyapunov exponent and the phase transition points as
well as numerically calculate the density distributions and the spectral
winding number. We reveal that in the nonreciprocal Stark lattice with the
mosaic periodic parameter $\kappa=1$, there exists a critical electric field
strength that describes the transition of the existence-nonexistence of NHSE
and is inversely proportional to the lattice size. This transition is
consistent with the real-complex transition and topological transition
characterized by spectral winding number under periodic boundary conditions. In
the strong fields, the Wannier-Stark ladder is recovered, and the Stark
localization is sufficient to suppress the NHSE. When the mosaic period
$\kappa=2$, we show that the system manifests an exact non-Hermitian ME and the
skin states are still existing in the strong fields, in contrast to the
gigantic field can restrain the NHSE in the $\kappa=1$ case. Moreover, we
further study the expansion dynamics of an initially localized state and
dynamically probe the existence of the NHSE and the non-Hermitian ME. These
results could help us to control the NHSE and the non-Hermitian ME by using
electric fields in the disorder-free systems.
