The impacts of electronic state hybridization on the binding energy of single phosphorus donor electrons in extremely downscaled silicon nanostructures

We present the density functional theory calculations of the binding energy of the Phosphorus (P) donor electrons in extremely downscaled single P-doped Silicon (Si) nanorods. In past studies, the binding energy of donor electrons was evaluated for the Si nanostructures as the difference between the ionization energy for the single P-doped Si nanostructures and the electron affinity for the un-doped Si nanostructures. This definition does not take into account the strong interaction of donor electron states and Si electron states explicitly at the conductive states and results in a monotonous increase in the binding energy by reducing the nanostructure's dimensions. In this paper, we introduce a new approach to evaluate the binding energy of donor electrons by combining the projected density of states (PDOS) analysis and three-dimensional analysis of associated electron wavefunctions. This enables us to clarify a gradual change of the spatial distribution of the 3D electron wavefunctions (3DWFs) from the donor electron ground state, which is fully localized around the P donor site to the first conductive state, which spreads over the outer Si nanorods contributing to current conduction. We found that the energy of the first conductive state is capped near the top of the atomistic effective potential at the donor site with respect to the surrounding Si atoms in nanorods smaller than about 27 a₀. This results in the binding energy of approximately 1.5 eV, which is virtually independent on the nanorod's dimensions. This fact signifies a good tolerance of the binding energy, which governs the operating temperature of the single dopant-based transistors in practice. We also conducted the computationally heavy transmission calculations of the single P-doped Si nanorods connected to the source and drain electrodes. The calculated transmission spectra are discussed in comparison with the atomistic effective potential distributions and the PDOS-3DWFs method.