Exciton Energy of Cds Quantum Dots

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 connement 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 connement 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 connement, 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: bask@des.upatras.gr

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 connement 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 connement (innite 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 connement (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 innite connement  ! 0). Actually, the induced charge is spread

along this interface and the divergence in the self-polarization energy dissapears.c

The induced charge may have an opposite sign in comparison with the source charge

(in our case electron or hole), if the source charge is placed in a zone with lower

dielectric constant.21

Meanwhile, as we have already discussed above, all these studies are based either

on perturbation theory or variational theory. At rst glance, it is well known that

perturbation theory uses SCA in order to be valid. Therefore, an obvious question

that arises is: What about the validity of perturbation theory in the medium (R 

aB) or weak connement limit (R  aB)? As far as the variational method is

concerned, it is well known that it does not solve SchrЁodinger equation but only

minimizes the energy.22

In the present paper in order to overcome these general problems of these two

methods, we will use a dierent procedure for the solution of the time independent

SchrЁodinger equation. We will use a recently developed technique23 which is called

aAlthough matrix diagonalization methods give good results in EMA, in general they suer from

the fact that the kinetic energy contributions dominate over the Coulomb contributions in such a

way that corrections to the strong connement may become inaccurate.12

bAmong the best eorts2;8 (dierent from EMA) that have devoted in order to overcome the

discrepancy between the theory