Model For A Biexciton

Model for a Biexciton in a Lateral

Quantum Dot Based on Exact Solutions

for the Hookean H2 Molecule.

I. Theoretical Aspects

E. V. LUDEÑA,1 L. ECHEVARRÍA,2 J. M. UGALDE,3 X. LOPEZ,3

A. CORELLA-MADUEÑO4

1Centro de Química, Instituto Venezolano de Investigaciones Científicas, IVIC,

Apartado 21827 Caracas, Venezuela

2Departamento de Química, Universidad Simón Bolívar, Sartenejas, Caracas, Venezuela

3Kimika Fakultatea, Euskal Herriko Unibertsitatea and Donostia International Physics Center,

Posta Kutxa 1072, 20080 Donostia, Euskadi, Spain

4Departamento de Física, Universidad de Sonora, Apartado Postal 1626 Hermosillo, Sonora, México

Received 15 March 2010; accepted 20 April 2010

Published online 13 January 2011 in Wiley Online Library (wileyonlinelibrary.com).

DOI 10.1002/qua.22818

ABSTRACT: We present here a model for the non-Born-Oppenheimer description of

the biexcitonic complex X2(eehh) trapped in laterally-coupled quantum dot system. We

define a zeroth-order Hamiltonian which allows us, under certain conditions on the

masses and coupling constants, to decouple the problem.We show that the zeroth-order

wavefunctions and energies can be described using the exact solutions for the Hookean

model for the H2 molecule [Ludeña et al., J Chem Phys 2005, 123, 024102]. We also apply

this Hookean model to the description of the excitonic complexes X+

2 (ehh), X−

(eeh), and to

the single exciton X(eh) and analyze the dependence of the total non-Born-Oppenheimer

zeroth-order energies, and binding energies for these sytems with the mass ratio σ. The

general features of the results obtained using this Hookean model agree quite well with

those of more elaborate calculations. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem 111:

1808–1818, 2011

Key words: biexcitons; lateral quantum dots; Hookean model;

non-Born-Oppenheimer; H2 molecule

Correspondence to: E. V. Ludeña; e-mail: popluabe@yahoo.es

International Journal of Quantum Chemistry, Vol 111, 1808–1818 (2011)

© 2011 Wiley Periodicals, Inc.

MODEL FOR A BIEXCITON

1. Introduction

Coupled semiconductor quantum dots are

novel structures that have attracted much

attention because of their potential applications in

quantum cryptography and quantum computation

[1–5]. Much progress has been achieved in the fabrication

and control of properties of vertically stacked

quantum dots [6–9] as well as of laterally coupled

ones [10–15].Afirst-principle’s theoretical treatment

of these structures (comprising over 5,000 atoms) has

been carried by Jaskolski et at. [16]. For a review

of application of simpler models to lateral quantum

dots in a magnetic field, see Helle et al. [17, 18].

The effects of exciton splitting of two laterally

coupled quantum dots under the influence of an

electric field are of particular interest [19]. This is

due to the important role that an electrically tuned

lateral quantum dot could play in the development

of controllable quantum gates, which are the basic

elements for the realization of quantum computers

[15, 20–22]. On the other hand, it has been experimentally

demonstrated that the type of coupling of

lateral quantum dots has a definite expression in the

photoluminiscence spectrumand, hence, it would be

desirable to attain understanding of this phenomenon

for the purpose of having a voltage-controlled

emission of non-classical light [15, 23]. The latter is

an essential feature in many areas having to do with

the development of opto-electronic devices [24].

The importance of excitons for determining the

optical properties in semiconductor nanocrystals is

already well known [25, 26]. The presence of biexciton

states in semiconductor quantum dots was

experimentally demonstrated (and theoretically corroborated)

by Hu in 1990 [24]. In a single quantum

dot, different pathways leading to the formation of

excitons have been studied [27] and coherent optical

control of biexcitons has been experimentally

achieved [28]. Calculations have been performed to

determine the binding energies of excitonic complexes

as a function of the quantum dot height [29]

and length [30]. Also, the size of quantum dots in

which biexcitons are generated under a two-photon

resonant excitation, has been determined [31].

In a coupled lateral quantum dot, the important

interactions are of the exciton–exciton type [32]. Evidence

for the coupling of quantum dots has been

obtained through the study of the splitting of exciton

emission [33]. Excitons, which generally cannot

be made to attain long lifetimes in bulk semiconductors,

can be made to be stable precisely in such

structures. This effect is essentially the result of

the spatial separation of the coupled quantum

dots. Excitons have been treated as composite particles

obeying Bose statistics [34–37]. But due to

the fact that their building blocks (electrons and

holes) are Fermions, they have also been treated

as Fermions [38–45]. A crucial aspect in theoretical

approaches to exciton interaction is the determination

of an effective exciton–exciton potential

which take into account the Fermionic exchange and

Coulomb forces. An important improvement in this

respect, which goes beyond treatments based on the

Hartree-Fock approximation was the introduction

of a Heitler and London type treatment by Okumura

[46]. Biexcitons in coupled quantum dots have

been considered as possible sources of entangled

photons [47].

The description of a two-exciton system X2(eehh)

requires a four particle Hamiltonian involving two

electrons and two holes [see Eq. (1) below], where

the electrons and the holes are characterized by their

respective effective masses me and mh, and interact

through effective Coulomb potentials (i.e., modified

through a particular dielectric constant). In general,

a proper solution to this equation leads to a non-

Born-Oppenheimer description of this four particle

system. A simpler approach, which leads, nonetheless

to an improvement over the Heitler-London type

treatment of Okumura, was deviced by Schindler

[32]. In this approach, an exact numerical solution

...