Computational Modeling of the Dielectric Function of Silicon Slabs with Varying Thickness

The dynamical dielectric function of a silicon : slab, in the region from near IR to UV light frequencies, is expected to vary with its thickness, and it is important to know whether its optical properties are similar to those of bulk silicon. Slabs of varying thickness are modeled starting from their atomic structure. Modeled Si(111) surfaces are terminated by hydrogen atoms to compensate the dangling bonds, and optical properties have been obtained for Si slabs with 4, 8, and 12 layers. Real and imaginary parts of the dielectric function are obtained from the polarization of the slab, expressed in terms of a delayed response function constructed from a reduced density matrix (RDM) which includes electronic dissipative effects due to coupling of photoexcited electrons to the substrate lattice and to electronic excitations. The related index of refraction and absorption coefficient have also been calculated from the above treatment. These optical properties are obtained using density functional theory (DFT) and plane wave basis sets to construct the equation of motion of a RDM, solved for steady light absorption. Both GGA (PBE) and hybrid (HSE) DFT exchange-correlation density functionals are employed to calculate the optical properties from the RDM. The imaginary part of the dielectric function is related to the light absorbance, and has been compared to measurements showing that better agreement is obtained with the HSE hybrid functional containing part of the exact short-range electronic exchange energy. We present a procedure by which one can reproduce the HSE results for the dielectric function from the computationally less expensive GGA PBE functional calculations, using a single photon energy shift parameter and results from PBE calculations. Our treatment shows that the onset of light absorption and strong diffraction are similar for thin slabs and bulk silicon, and that they have similar peak structure as functions of photon energy. Both properties increase with slab thickness at most photon energies. This makes silicon slabs reliable structures for photovoltaic applications.