Gate Voltage Tunable Second Harmonic Generation in Mono- and Bi-layer Black Phosphene

Black phosphorene (BP) has emerged as a promising platform for tunable nonlinear photonics due to its layer-dependent bandgap, high carrier mobility, and remarkable in-plane anisotropy. This study investigates the second-harmonic generation (SHG) of monolayer and bilayer BP under an external static electric field, with describing the electronic states by a tight-binding model and the dynamics by semiconductor Bloch equations. Our results reveal that BP exhibits large second-order nonlinear optical response along the armchair direction, with significant resonant enhancement when the incident photon energy approaches half of its bandgap. Under an applied electric field of $10^7$ V/m, the effective second-order nonlinear susceptibility of BP can be as large as $10^3$ pm/V, surpassing that of the conventional nonlinear crystal AgGaSe$_2$ by more than an order of magnitude. With respect to the static electric field induced by gate voltage, we discuss the relation between the electric-field-induced second harmonic (EFISH) generation and conventional SHG -- under lower gate voltage, the EFISH approach agrees well with the SHG solutions, whereas the former is no longer applicable under higher gate voltage. Specifically, as the increasing gate voltage, monolayer BP exhibits the bandgap expansion and the corresponding blue-shift in the SHG resonant peak. In contrast, bilayer BP undergoes a semiconductor-to-semimetal transition, forming Dirac cone and generating divergent SHG spectra as photon energy goes to zero. Additionally, the chemical potential allows for precise control over interband and intraband nonlinear responses. This work provides important theoretical foundations for the development of BP-based tunable nonlinear photonic devices and expands the application potential of anisotropic two-dimensional materials in nonlinear optics.