Effects of Size and Shape on Electronic States of Quantum Dots

Abstract

A strained-modified, single-band, constant-potential three-dimensional model is formulated to study the dependence of electronic states of InAs/ GaAs quantum dots ͑QDs͒ on shape and size variation. The QD shapes considered are ͑i͒ cuboid, ͑ii͒ cylindrical, ͑iii͒ pyramidal, ͑iv͒ conical, and ͑v͒ lens shaped. Size variations include ͑i͒ QD volume ͑ii͒ QD base length, and ͑iii͒ QD height, taking into account aspect ratio variation. Isovolume QD shapes with narrow tips were found to have higher ground-state energies than those with broad tips, and this is attributed to the smaller effective volume. The volume, base length, and height dependencies were obtained and found to tally well with both experimental results and advanced calculations. Hence, upon growth parameter variation, this can provide an alternative to confirm whether the change to the size of the uncapped QDs implies a similar change to the capped ones. Ground-state energy as function of aspect ratio does not follow a monotonic trend. Owing to the competing effect of a decrease in base length and an increase in height, the energy trend exhibits a sharp decrease to an optimum aspect ratio, followed by gentle, almost linear increase. The optimum aspect ratio varies among shapes and is predicted to be smaller for shapes with broad tips. The effective volume ratio of both shapes ͑V eff,CUBOID / V eff,PYRAMID ͒ was determined, and found to vary with aspect ratio. Furthermore, a "cross-over" of lens-shaped QD from "lower energy" to "higher energy" group is predicted due to significant shape transition.

Impurity Binding Energyin Quantum Dots with Parabolic Confinement in the Presence of Electric Field

“…where cylindrical coordinates (ρ, θ) are the distance and angle describing the position of an electron in the plane perpendicular to the electric field; F is external electric field applied in the z-direction; m* is the effective mass of the electron, V(ρ,z)=1/2m*ω 2 Because of the axial symmetry of the system, angular momentum projection onto the z axis is conserved L z =ℏ m (m=0, 1, 2, …-magnetic quantum number), and their eigenfunctions exp(imz) determine the dependence of the unknown electron wave function versus an angle:…”

“…Since Bastard calculated [1] for the very first time the impurity binding energy in QW many works on this subject have been published. Over the last three decades, new calculation methods have been developed to determine the binding energy of IS in QWs and QDs, by assuming the form of confinement potential [2][3][4][5] and the features of the band structure [5][6][7][8][9]. The form of the confinement potential is a significant characteristic.…”

Computational modeling and analysis of quantum dots

“…We also employ the finite barrier model as the infinite barrier EMA model with 1/R 2 scaling of the kinetic term has been shown to be inaccurate by over-estimating the band gaps for sizes where the confinement are significant [Sapra et al, 2003, Kayanuma andMomiji, 1990]. The finite barrier has been shown to be accurate and widely adopted with the single-particle finite well effective mass approximation model [Pellegrini et al, 2005, Zhang and Sharma, 2005, Ngo et al, 2006. Thus the finite depth/width square well EMA (FWEMA) method [Pellegrini et al, 2005, Baskoutas and Terzis, 2008, Marin et al, 1998] and the numerical finite element approach are used here to solve the Schrödinger equation,…”

“…Using numerical simulations, Fonoberov and Pokatilov [2002] Li et al [2001a] fitted the transition energies with volume, ∆E g ≈ V −γ o , to obtain the value of γ of 0.38 for disk shape, 0.35 for ellipsoidal lens shape, 0.31 for cut spherical lens shape, and 0.27 for cone shape. Ngo et al [2006] introduced the effective volume by replacing…”

“…Pryor [1999] indicated that the island volume matters more than shape. Ngo et al [2006] reported that the effective volume of the pyramidal shape is smaller than that of the cuboid shape with the same physical volume, and those with narrow tips (e.g., pyramidal shape) have a higher energy state compared to those with broad tips. This was attributed to smaller effective volume in quantum dots with narrow tips.…”