Slow and fast absorption saturation of black phosphorus: experiment and modelling

According to the excited-state decay time, a slow saturable absorber model was employed to obtain the ground-state absorption and excited-state absorption cross sections σ_gs and σ_es of black phosphorus (BP) nanosheets, which are (1.25 ± 0.07) × 10^-16 cm² and (2.97 ± 0.21) × 10^-17 cm² at 515 nm, and (5.95 ± 0.17) × 10^-18 cm² and (5.19 ± 0.71) × 10^-19 cm² at 1030 nm, respectively. In comparison, the σ_gs and σ_es of MoS₂ and graphene were also obtained with the same model. The ratio σ_es/σ_gs of BP is the smallest (only 0.24 ± 0.03 at 515 nm and 0.09 ± 0.01 at 1030 nm) among the three two-dimensional layered materials both at 515 nm and 1030 nm. Relatively large ground-state transition probability and weak loss of excited-state absorption result in good saturable absorption performance. In comparison with MoS₂ and graphene, the much lower saturation intensity of BP can be well explained from this perspective.