Quantum size effects in spherical semiconductor microcrystals

Abstract

The size dependence of the lowest electron-hole state in semiconductor microcrystals is calculated using the variational principle with a three-parameter Hylleraas-type wave function. For very small particles the Coulomb interaction may be treated as a perturbation.For larger particles the size dependence of the energy is much sharper than that expected in previous work.Wannier excitons in semiconductor rnicrocrystals provide an interesting spectroscopic system for the study of size quantization effects. Recently, several groups' have studied this phenomenon in colloidal spherical particles of CdS and spherical microcrystals of CuC1, CdS, CdSe, and CdS Se& ", etc. , in a glass matrix. Brus' has given a variational calculation for the size dependence of the electron-hole pair state while Efros and Efros' have calculated the spectra in some limiting cases. For a particle radius smaller than the Bohr radius ao of the exciton, the size quantization of the electron and hole band states dominates and the effect of the Coulomb attraction between the electron and hole can be treated as a perturbation. For very large particles, Efros and Efros conclude that the lowest energy of the electron-hole pair in the microcrystal is larger than that in the bulk due to size quantization of the center-of-mass motion. The size dependence of the measured energy levels can thus yield a good estimate of the total mass I &+m2 of the electron-hole pair. From a semiconductor-physics point of view this would be especially interesting because the total mass is sensitive to the heavier-particle mass, usually the hole, while the reduced mass involved in the bulk exciton binding energy is sensitive to the lighter-particle mass, usually the electron. Experimental results for CuCl particles give a total mass much smaller than that generally accepted.In this Brief Report we report an accurate variational calculation for the lowest electron-hole pair state. The dependence of the energy levels as well as the wave functions on the particle size are critically evaluated. Our calculations clearly indicate that the determination of the to-tal mass by this method is not satisfactory.In the effective-mass approximation the essential problem is to solve the Schrodinger equation for the envelope function f: AV' AV 2m ) 2m2 2 47TEOE7 )2 + &0 P(ri, r~) = EP(r&, r2), (1) where m~, mz and r&, r2 are the electron and hole masses and position vectors, respectively, and r~2 --~r , -rT he confining potential Vo is 0 inside the spherical crystallite of radius R and infinite outside this sphere, with the corresponding boundary condition Q(rl, r2)=0 for r, or r2)R .(2) For small particles the energy levels were calculated by treating the Coulomb interaction between the electron and hole as a perturbation.From single-particle wave functions linear combinations corresponding to zero angular momentum were formed. Using a finite set of these as the basis set, the matrix representation for the Hamiltonian in Eq. (1) was obtained and diagonalized. While the unperturbed level...

Binding energy of heavy excitons in spherical quantum dots under hydrostatic pressure

“…Brus [9] has given a variational calculation for the size dependence of the electron-hole pair state. Nair et al [10] calculated the lowest electronhole state in semiconductor microcrystals as a function of size, using the variational principle with a three-parameter Hylleraas wavefunction; for very small particles, they treated the Coulomb interaction as a perturbation and it was based on an infinite confinement potential. Kayanuma [11] made variational calculations and determined the ground-state energy for an exciton confined in a microcrystal with finite potential barriers.…”

Comparison of the Magnetic and Optical Properties of Wide‐Gap (II,Mn)VI Nanostructures Confined in Mesoporous Silica

“…[2,10,31] The occurring quantum confinement, which is mostly reflected by a blue-shift of the band-gap energy, has been discussed extensively. [30,[48][49][50][51][52][53][54] Other nanoscale semiconductor compounds, such as III-V semiconductors (e.g. GaAs or InP) or other II-VI semiconductors (e.g.…”

Evolution of the electronic structure with size in II-VI semiconductor nanocrystals

“…In most EMA calculations, the confining potentials for the electron and the hole have been assumed infinite. [14,16,17,18,19,20] Therefore, the electron and the hole wavefunctions vanish at and beyond the surface of the nanocrystal, without the possibility of any tunnelling. In the strong confinement regime, where R, the nanocrystal radius, is much smaller than a B , the Bohr exciton radius, Brus proposed [17] the following expression for the bandgap of the finite sized system…”