Importance of quantum correction for the quantitative simulation of photoexcited scanning tunneling spectra of semiconductor surfaces

Photoexcited scanning tunneling spectroscopy is a promising technique for the determination of carrier concentrations, surface photovoltages, and potentials of semiconductors with atomic spatial resolution. However, extraction of the desired quantities requires computation of the electrostatic potential induced by the proximity of the tip and the tunnel current. This calculation is based on an accurate solution of the Poisson as well as the continuity equations for the tip-vacuum-semiconductor system. For this purpose, the carrier current densities are modeled by classical drift and diffusion equations. However, for small tip radii and highly doped materials, the drift and diffusion transport model significantly overestimates a semiconductor's carrier concentration near the surface, making the quantification of physical properties impossible. In this paper, we apply quantum correction to the drift and diffusion model, in order to account for the so-called quantum compressibility, i.e., reduced compressibility of the carrier gas due to the Pauli principle, in the region of the tip-induced band bending. We compare carrier concentrations, potentials, and tunnel currents derived with and without quantum correction for $\mathrm{GaN}(10\overline{1}0)$ and GaAs(110) surfaces to demonstrate its necessity.