  Optical and electronic properties of black phosphorus strongly depend on the
number of layers and type of stacking. Using first-principles calculations
within the framework of density functional theory, we investigate the
electronic properties of bilayer black phosphorus with an interlayer twist
angle of 90$^\circ$. These calculations are complemented with a simple
$\vec{k}\cdot\vec{p}$ model which is able to capture most of the low energy
features and is valid for arbitrary twist angles. The electronic spectrum of
90$^\circ$ twisted bilayer black phosphorus is found to be x-y isotropic in
contrast to the monolayer. However x-y anisotropy, and a partial return to
monolayer-like behavior, particularly in the valence band, can be induced by an
external out-of-plane electric field. Moreover, the preferred hole effective
mass can be rotated by 90$^\circ$ simply by changing the direction of the
applied electric field. In particular, a +0.4 (-0.4) V/{\AA} out-of-plane
electric field results in a $\sim$60\% increase in the hole effective mass
along the y (x) axis and enhances the $m^*_{y}/m^*_{x}$ ($m^*_{x}/m^*_{y}$)
ratio as much as by a factor of 40. Our DFT and $\vec{k}\cdot\vec{p}$
simulations clearly indicate that the twist angle in combination with an
appropriate gate voltage is a novel way to tune the electronic and optical
properties of bilayer phosphorus and it gives us a new degree of freedom to
engineer the properties of black phosphorus based devices.
