Ultrathin films of black phosphorus as suitable platforms for unambiguous observation of the orbital Hall effect

Phosphorene, a monolayer of black phosphorus, is a two-dimensional material that lacks a multivalley structure in the Brillouin zone and has negligible spin-orbit coupling. This makes it a promising candidate for investigating the orbital Hall effect independently of the valley or spin Hall effects. To model phosphorene, we utilized a density-functional-theory-derived tight-binding Hamiltonian, which is constructed with the pseudoatomic orbital projection method. For that purpose, we use the PAOFLOW code with a newly implemented internal basis that provides a fairly good description of the phosphorene conduction bands. By employing linear response theory, we show that phosphorene exhibits a sizable orbital Hall effect with strong anisotropy in the orbital Hall conductivity for the out-of-plane orbital angular momentum component. The magnitude and sign of the conductivity depend on the in-plane direction of the applied electric field. These distinctive features enable the observation of the orbital Hall effect in this material unambiguously. The effects of strain and of a perpendicularly applied electric field on the phosphorene orbital Hall response are also explored. We show that a supplementary electric field applied perpendicular to the phosphorene layer in its conductive regime gives rise to an induced in-plane orbital magnetization.