2-d Finite barrier rectangular quantum dots I: Schrodinger description

Engin Ata n, Doan Demirhan, Fevzi BykklPhysics Department, Faculty of Science, Ege University, Turkey

H I G H L I G H T S

Inseparable piecewise potential is usedas the connement potential of the2-d rectangular shape quantum dot.

Theoretical approach is based on thetransfer matrices along axes, and theparity and the rotation symmetries ofthe dot.

The transcendental energy relations ofthe dot are obtained as a 2-fold degen-erate and quantized by half of therotation angles.

G R A P H I C A L A B S T R A C T

Quantized transcendental energy relations of inseparable nite barrier rectangular quantum dot being2-d counterpart of ubiquitous 1-d potential-well are obtained.

a r t i c l e i n f o

Article history:Received 12 November 2013Received in revised form27 February 2014Accepted 3 March 2014Available online 2 May 2014

Keywords:Rectangular quantum dotInseparable piecewise constant potentialExact solutionTransfer matrix methodParity and rotation symmetries ofrectangular QD

a b s t r a c t

The bound state energy levels of a nite barrier rectangular shaped quantum dot are obtained usinga transfer matrix method and imposing parity and rotation symmetries of the dot on the wavefunction.Energy relations containing potential depth, dot size and effective masses are found in the 2-folddegenerate quantized transcendental equations.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In the range of length scale between bulk and atomic size thelaws of quantum mechanics rule; the nanostructures in whicha controllable number of carriers are conned in all spatialdimensions are called quantum dots [14,20,15]. As a quantumconnement structure, quantum dots have promising abilities to

test new physical effects and to develop new nano-electronicdevices [19], especially as articial atoms [11,33,13].

While the properties of quantum dots are determined by thegrowth methods as well as the shape, size and composition of thematerials, the electronic and optical properties depend on theconnement potential. Thus when making a theoretical descrip-tion of the quantum dot system it is essential to know theconnement potential. An exact, realistic one can be obtainedwhen a solution of Poisson equation is found with relativedielectric constant and charge density [12,17,4] which are func-tions of position and temperature. Since making some assump-tions gives the potential almost bowl like the harmonic oscillator

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.03.0021386-9477/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail addresses: enginata@hotmail.co.uk (E. Ata),

dogan.demirhan@ege.edu.tr (D. Demirhan),fevzi.buyukkilic@ege.edu.tr (F. Bykkl).

Physica E 62 (2014) 7175

is usually used for the connement [16,24]. It is a good approx-imation for self-assembled quantum dots and gives exact solutionsfor many-body calculations [11,18].

A simple model akin to a harmonic oscillator is piecewiseconstant potential with innite high barrier because a harmonicoscillator and innite high barrier potentials may be regardedas box problems with innite connement. Since the innite highbarrier potential considers the dot size and shape it differs fromthe parabolic one. Since it has exact solutions for almost all thegeometries in which the Schrodinger equation is separable it is awell-studied potential for a 2-d rectangular dot with an n-carrier[23], and in the presence of high magnetic elds [22], elliptic andelliptic cylinder [2], parabolic [28], lens [27], parabolic cylinder[30], triaxial ellipsoidal [29], spherical [7] shaped quantum dots.A comparison of energy levels of rectangular quantum experimentwith innite barrier calculation is given by Ref. [5].

Another simple intermediate model between the harmonicoscillator and the innite barrier is piecewise constant potentialwith a nite barrier height which leads to nite connement andtranscendental energy relation. Although the nite barrier givesmore realistic results than the innite barrier, it has been onlysolved for elliptic [25], disk [19,31], and lens [19] shaped quantumdots. The calculations for a rectangular shape have been made bythe nite element method [21], factorized series approximation[8], variational calculus of binding energies [32], and assuming theseparability [10].

While a 1-d counterpart is the most-known example, 2-drectangular shaped quantum dot energy levels do not have anexact treatment. We intend to give the energy relations of a single-particle conned in rectangular dot with inseparable piecewiseconstant potential

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