Scaling of energy gaps in phosphorene nanoflakes

Electronic structure of phosphorene nanoflakes which consist of hundreds of phosphorus atoms are studied in the framework of unrestricted Hartree-Fock approach. On the base of Pariser-Parr-Pople model for electron-electron interactions, a simplified Bethe-Salpeter formalism is established for the calculation of excitation states of the system. Taking into account the electron-hole interaction in various dielectric environments, the optical gap of a triangular phosphorene nanoflake is shown to increase as the screening effect becomes stronger while its graphene counterpart exhibits just the opposite dependence. After confirming an exponential dependence of the optical gap on the effective dielectric constant, the quasiparticle and optical gaps are also found to obey an exponential scaling rule against the total number of atoms in the nanoflakes, respectively. By extrapolating the dependence on the size of the system, one is able to estimate the exciton binding energy of a monolayer phosphorene sheet on a SiO₂substrate to be 0.894 eV. The result is found to agree well with the previous experimental result of 0.9 eV.