Nonlinear Hall effect in monolayer phosphorene with broken inversion symmetry

Nonlinear Hall effect (NLHE), a new member of the family of Hall effects, in monolayer phosphorene is investigated. We find that phosphorene exhibits pronounced NLHE, arising from the dipole moment of the Berry curvature induced by the proximity effect that breaks the inversion symmetry of the system. Remarkably, the nonlinear Hall response exhibits central minimum with a width on the order of the band gap, followed by two resonance-like peaks. Interestingly, each resonance peak of the Hall response shifts in the negative region of the chemical potential which is consistent with the shift of valence and conduction bands in the energy spectrum of monolayer phosphorene. It is observed that the two peaks are asymmetric, originated from anisotropy in the band structure of phosphorene. It is shown that the NLHE is very sensitive to the band gap and temperature of the system. Moreover, we find that a phase transition occurs in the nonlinear Hall response and nonlinear spin Hall conductivity of the system under the influence of spin-orbit interaction, tuned by the strength of interaction and band gap induced in the energy spectrum of monolayer phosphorene with broken inversion symmetry.