Ground State Confinement Energy of Quantum Dots and the Brus Equation: A Mathematical Approach

Abstract

The present study review the ground state confinement energy term in the Brus equation for the bandgap energy of a spherically shaped semiconductor quantum dot within the framework of effective mass approximation. Bandgap variation in a nanometer sized semiconductor is due to Confinement energy. A good estimate of the confinement energy is important for optoelectronic based applications of quantum dot.  The Schrodinger wave equation for a spherical nanoparticle in an infinite spherical potential well was solved in spherical polar coordinate system. Physical reasons in contrast to mathematical expediency were considered and solution obtained. The result reveals that the shift in the confinement energy is less than that predicted by the Brus equation as was adopted in most literatures. A “bird eye" view of the brus equation reveals that it is nothing but a Schrodinger equation modified to account for the effect of an electron-hole pair (exciton) confined to a nanometric spherical shaped semiconductor referred to as quantum dot. It is blind to the varied crystal structures that exist for semiconductors.