Essay About New Numerical Method And Exciton Energy Of Cds Quantum Dots

Exciton Energy of Cds Quantum Dots
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Studying the quantum size eects on the exciton energy of two-dimensional CdS quantum
dots in the single band eective mass approximation for both electron and hole, we
use a nite connement in the x{y plane and we assume also that the induced charge is
spread along a very thin interface in the z-direction. Solving the SchrЁodinger equation
with a new numerical method, which is called potential morphing method, we obtain
the corresponding energy within the self-consistent Hartree scheme. Excellent agreement
is obtained with the experimental values of exciton energies for various sizes of
CdS quantum dots in the strong, medium and weak connement limit.
Keywords: Quantum Dots; excitons and related phenomena; solutions of wave equations:
bound states.
PACS numbers: 73.63.Kv, 71.35.-y, 03.65.Ge.
1. Introduction
Semiconductors quantum dots (QDs) have been a subject of intense theoretical and
experimental interest in the last few years.1 Contrary to bulk semiconductors where
the energy is a function of the momentum, in (small) QDs all the bands disappear
and are replaced by discrete levels.2
The increasing experimental capability in the fabrication of QDs for a wide
spectra of sizes, drives the experimental and theoretical interest for the overall
understanding of these systems since, due to quantum connement, the electronic
and optical properties are strongly dot-size dependent.1{3
Many theoretical studies have been devoted to the study of the quantum size
eects on the exciton energy in QDs. Most of them are related to studies in
Corresponding author. E-mail: [email protected]
4093
4094 S. Baskoutas et al.
the eective mass approximation (EMA) with perturbation method,4 variational
method,5{11 matrix diagonalization method,12;13 as well as to studies in dierent
procedures such as the tight-binding method14;15 and the pseudopotential
method.16 It is also well known that in the determination of the observed con-
nement eects in very small QDs (when their radius is much smaller than the
corresponding exciton Bohr radius R aB, the so called strong connement approximation
SCA), single band EMA is not valida;b any more.15;17;18 However we
must emphasize at this point the fact that in most cases the EMA results were
obtained under the assumption of perfect connement (innite barriers, IEMA) for
both electron and hole. Several papers9;20 with variational and perturbation methods
pointed out that this extreme hard-wall constraint is the main reason for the
disagreement between theory and experiment and carried out calculations assuming
an incomplete connement (nite barriers, FEMA) for the electron and hole.
The output of such calculations is in a very good quantitative agreement with that
obtained for the excitonic properties calculated using other methods.20
Recently P. G. Bolcatto et al.21 assuming at rst that FEMA is a qualitative,
flexible and versalite theoretical tool for the study of the excitonic properties of QDs
and taking into account the dielectric mismatch between QD and the surrounding
matrix, proposed the existence of an interface with a width of 2, which surrounds
the dot (for innite connement ! 0). Actually, the induced charge is spread
along this interface and the divergence in the self-polarization energy dissapears.c