Quantum-size effects of interacting electrons and holes in semiconductor microcrystals with spherical shape

TLDR

The study theoretically investigates quantum‑size effects of electron–hole systems confined in semiconductor microcrystals using a spherical‑dielectric continuum model. The authors solve the eigenvalue problem numerically with Ritz’s variational technique, classifying the lowest‑energy state into exciton confinement, individual particle confinement, and an intermediate regime depending on the ratio of quantum‑well radius to exciton Bohr radius. They find that for R/a_B* ≥ 4 the lowest exciton energy shift follows the rigid‑sphere model, that the interband oscillator strength varies sharply between 2 and 4, and that the absorption spectrum evolves accordingly, matching experimental observations.

Abstract

Quantum-size effects of an electron-hole system confined in microcrystals of semiconductors are studied theoretically with the spherical-dielectric continuum model. An extensive numerical calculation for the eigenvalue problem is carried out by Ritz's variational technique. The motional state of the lowest level is classified into three regimes: the regime of exciton confinement for R/${a}_{B}^{\mathrm{*}}$\ensuremath{\gtrsim}4, the regime of individual particle confinement for R/${a}_{B}^{\mathrm{*}}$\ensuremath{\lesssim}2, and the intermediate regime for 2\ensuremath{\lesssim}R/${a}_{B}^{\mathrm{*}}$\ensuremath{\lesssim}4, where R is the radius of the quantum well and ${a}_{B}^{\mathrm{*}}$ is the exciton Bohr radius. In the region R/${a}_{B}^{\mathrm{*}}$\ensuremath{\gtrsim}4, the high-energy shift of the lowest exciton state is described by the rigid-sphere model of the exciton quite well, which takes into account the spatial extension of the relative motion of the electron and the hole. The oscillator strength of the interband optical transition changes dramatically across the region 2\ensuremath{\lesssim}R/${a}_{B}^{\mathrm{*}}$\ensuremath{\lesssim}4. The metamorphosis of the absorption spectrum is shown as a function of R/${a}_{B}^{\mathrm{*}}$ and compared with the experimental data.