Theoretical modeling of phosphorene, a two-dimensional material derived from black phosphorus, has been crucial in understanding its unique electronic, optical, and mechanical properties. Computational approaches, particularly density functional theory (DFT) and tight-binding (TB) methods, have provided predictive insights into its anisotropic behavior, strain response, and defect dynamics. These models have been validated against experimental data, offering a robust framework for exploring phosphorene’s potential in nanotechnology.

**Electronic Properties**
Phosphorene exhibits a direct bandgap that varies with layer thickness, ranging from approximately 0.3 eV (bulk) to 2.0 eV (monolayer). DFT calculations with hybrid functionals (e.g., HSE06) accurately predict this thickness-dependent transition, aligning with experimental optical absorption spectra. The band structure is highly anisotropic due to the puckered honeycomb lattice, leading to different effective masses along the armchair (lighter) and zigzag (heavier) directions. Tight-binding models parameterized from DFT results capture this anisotropy, reproducing the asymmetric carrier mobility observed in transport measurements. For instance, DFT-predicted hole mobilities exceed 10,000 cm²/Vs along the armchair direction, while experimental values range between 200–1,000 cm²/Vs due to substrate and defect effects.

Defects such as vacancies and adatoms significantly alter phosphorene’s electronic properties. DFT simulations reveal that single phosphorus vacancies introduce mid-gap states, reducing carrier lifetimes. Oxygen adsorption, a common environmental effect, dopes phosphorene p-type, consistent with experimental observations of increased hole concentration. Tight-binding models extended to include defect potentials qualitatively agree with these findings but require empirical adjustments for quantitative accuracy.

**Optical Properties**
Phosphorene’s optical absorption is polarization-dependent, with stronger light-matter interaction along the armchair direction. DFT-based GW-BSE (Bethe-Salpeter equation) calculations predict exciton binding energies of ~0.3–0.9 eV, explaining the large Stokes shift observed in photoluminescence (PL) spectra. The anisotropic optical conductivity, computed via DFT, matches differential reflectance measurements, showing higher absorption for in-plane polarized light along the armchair axis.

Strain engineering further modulates these properties. Uniaxial strain along the zigzag direction reduces the bandgap linearly (~100 meV per 1% strain), while biaxial strain induces a direct-to-indirect transition at ~5% compression. These predictions are corroborated by PL peak shifts under mechanical deformation. Tight-binding models, though less accurate for excitonic effects, efficiently map strain-dependent bandgap trends and are widely used for large-scale strain-optic simulations.

**Mechanical Properties**
Phosphorene’s puckered structure leads to exceptional mechanical anisotropy. DFT-calculated elastic constants predict a Young’s modulus of ~100 GPa (armchair) and ~30 GPa (zigzag), matching nanoindentation experiments. The Poisson’s ratio is negative (-0.03) along the out-of-plane direction under uniaxial zigzag strain, a unique feature confirmed by atomic force microscopy (AFM) measurements.

Fracture behavior is also directionally dependent. DFT simulations show that cracks propagate more easily along the zigzag direction due to lower bond-breaking energies, consistent with TEM observations of anisotropic tearing. Tight-binding models, while less precise for fracture dynamics, correctly rank the relative stability of different edge terminations (armchair edges are more stable than zigzag).

**Comparison with Experiments**
DFT and TB predictions generally align with experimental data but highlight areas where approximations limit accuracy. For example, DFT underestimates the bandgap without hybrid functionals, while TB models often oversimplify defect-induced disorder. Experimental carrier mobilities are lower than theoretical ideals due to extrinsic factors like phonon scattering and impurity screening, which are challenging to fully capture in simulations.

Strain responses predicted by DFT closely match Raman peak shifts under tension, but deviations occur at high strains (>7%) where anharmonic effects dominate. Optical anisotropy ratios (armchair/zigzag absorption) from GW-BSE agree with spectroscopy but require exact substrate corrections for quantitative fits.

**Conclusion**
Theoretical modeling of phosphorene has elucidated its anisotropic electronic, optical, and mechanical behavior, guiding experimental exploration. DFT provides high accuracy for property prediction but at high computational cost, while TB methods offer scalable insights for device-scale modeling. Both approaches underscore phosphorene’s versatility and inform strategies for strain engineering, defect mitigation, and optoelectronic integration. Future work may focus on improving defect dynamics models and incorporating environmental interactions for real-world applications.