A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites

TLDR

Large semiconductor crystals exhibit electronic properties governed by bulk band structure, whereas reducing the crystallite size introduces quantum confinement that strongly alters excited states, ionization potential, and electron affinity. The study investigates whether the photochemical redox potential of a carrier shifts with crystallite size. The authors model the charged crystallite using an effective‑mass approximation, dielectric‑polarization electrostatics, and carrier penetration outside the particle, predicting a size‑dependent shift on the order of \(h^{2}/(8\,m\,R^{2})\). The model predicts that 50‑Å crystallites can exhibit redox potential shifts of several tenths of an electron‑volt, with the carrier charge density concentrating near the surface when the effective mass is very small.

Abstract

Large semiconductor crystals have intrinsic electronic properties dependent upon the bulk band structure. As the crystal becomes small, a new regime is entered in which the electronic properties (excited states, ionization potential, electron affinity) should be strongly dependent upon the electron and hole in a confined space. We address the possibility of a shift in the photochemical redox potential of one carrier, as a function of crystallite size. As a semiquantitative guide, one might expect a shift on the order of h2/8em*R2 due to the kinetic energy of localization in the small crystallite. We model the elementary quantum mechanics of a charged crystallite using (a) the effective mass approximation, (b) an electrostatic potential for dielectric polarization, and (c) penetration of the carrier outside the crystallite in a cases of small effective mass. Shifts of several tenths of an eV appear possible in crystallites of diameter 50 Å. The carrier charge density reside near the crystallite surface if the effective mass is very small.