Modeling the surface photovoltage of silicon slabs with varying thickness

The variation with thickness of the energy band gap and photovoltage at the surface of a thin semiconductor film are of great interest in connection with their surface electronic structure and optical properties. In this work, the change of a surface photovoltage (SPV) with the number of layers of a crystalline silicon slab is extracted from models based on their atomic structure. Electronic properties of photoexcited slabs are investigated using generalized gradient and hybrid density functionals, and plane wave basis sets. Si(1 1 1) surfaces have been terminated by hydrogen atoms to compensate for dangling bonds and have been described by large supercells with periodic boundary conditions. Calculations of the SPV of the Si slabs have been done in terms of the reduced density matrix of the photoactive electrons including dissipative effects due to their interaction with medium phonons and excitons. Surface photovoltages have been calculated for model Si slabs with 4-12, and 16 layers, to determine convergence trends versus slab thickness. Band gaps and the inverse of the SPVs have been found to scale nearly linearly with the inverse thickness of the slab, while the electronic density of states increases quadratically with thickness. Our calculations show the same trends as experimental values indicating band gap reduction and absorption enhancement for Si films of increasing thickness. Simple arguments on confined electronic structures have been used to explain the main effects of changes with slab thickness. A procedure involving shifted electron excitation energies is described to improve results from generalized gradient functionals so they can be in better agreement with the more accurate but also more computer intensive values from screened exchange hybrid functionals.