  Valleytronic materials can provide new degrees of freedom to future
electronic devices. In this work, the concepts of the ferrovalley metal (FVM)
and valley gapless semiconductor (VGS) are proposed, which can be achieved in
valleytronic bilayer systems by electric-field tuning, where the interaction
between out-of-plane ferroelectricity and A-type antiferromagnetism can induce
layer-polarized anomalous valley Hall (LP-AVH) effect. The K and -K valleys of
FVM are both metallic, and electron and hole carriers simultaneously exist. In
the extreme case, the FVM can become VGS by analogizing spin gapless
semiconductor (SGS). Moreover, it is proposed that the valley splitting
enhancement and valley polarization reversal can be achieved by electric field
in valleytronic bilayer systems. Taking the bilayer $\mathrm{RuBr_2}$ as an
example, our proposal is confirmed by the first-principle calculations. The FVM
and VGS can be achieved in bilayer $\mathrm{RuBr_2}$ by applying electric
field. With appropriate electric field range, increasing electric field can
enhance valley splitting, and the valley polarization can be reversed by
flipping electric field direction. To effectively tune valley properties by
electric field in bilayer systems, the parent monolayer should possess
out-of-plane magnetization, and have large valley splitting. Our results shed
light on the possible role of electric field in tuning valleytronic bilayer
systems, and provide a way to design the ferrovalley-related material by
electric field.
