Confinement effects in supported vs. isolated quantum structures: A study of Si(001) films

Abstract

A conventional quantum well is a supported quantum structure whereas a free-standing film is an isolated quantum structure. The application of the effective mass particle-in-a-well approach (EMA) to quantum wells leads to the well-known quantization rule whereby the lowest, j = 0 quantum state is forbidden. The EMA identifies these forbidden states with the bottom of the potential wells at the band energy minima and maxima for each band n. Direct pseudopotential band structure calculations for free standing silicon films are compared here with such EMA solutions. While near the band energy minima the EMA wavefunctions agree with the results of the direct approach, they disagree completely near the band energy maxima. The standard EMA forbidden states Γ1v, Γ25′v, Γ25′v, I'25′v, and Δmin for n=1,2,3,4, and 5, respectively, are all potential well minimum states, whereas the forbidden states found in the direct calculation, Γ1v, X1v , X4v, X4v, Δmin and X1c for bands n=1,2,3,4,5 and 6, respectively, are all band minimum energy states. At the valence band maximum Γ25′v, direct calculations reveal a novel quantum state of constant envelope functions, whose energy does not vary with film's size. Such a "zero confinement state" is absent in the EMA quantum well problem.