Revisiting elementary quantum mechanics with the BenDaniel-Duke boundary condition

Abstract

The past decade has seen an upsurge of interest in quantum dots and nanostructures. These are essentially structures of size 1to10nm. We discuss a simplified model of a quantum dot in which we employ the effective mass approximation and the BenDaniel-Duke boundary condition. The latter accounts for the fact that the effective carrier mass inside the well, mi, is different from the mass outside, m0. The energies and the charge distributions in the nanostructure are determined by the mass discontinuity factor β=mi∕m0. The charge density at the interface becomes larger as β→0. We propose a novel quantum scale σ∼β2V0L2, where V0 is the barrier height and L is the well size. The scale σ represents a mass modified potential strength. An asymptotic analysis shows that the charge densities and energies of the carrier depend simply on σ. We also propose a definition of the penetration depth analogous to the one proposed by Garrett. Our study of tunneling reveals that the one-dimensional analog of the Ramsauer-Townsend resonances becomes increasingly sharp and well defined as β→0.

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