Advances in Condensed Matter Physics

Volume 2018, Article ID 5687272, 13 pages

https://doi.org/10.1155/2018/5687272

## Energy Spectrum of Shallow Donor in Elongated InAs/GaAs Volcano-Shaped Quantum Dot

^{1}Escuela de Física, Universidad Industrial de Santander, A. A. 678, Bucaramanga, Colombia^{2}Grupo de Investigación en Teoría de la Materia Condensada, Universidad de Magdalena, Santa Marta, Colombia

Correspondence should be addressed to L. F. García; moc.liamg@ragarfl

Received 24 January 2018; Accepted 26 February 2018; Published 4 April 2018

Academic Editor: Yuri Galperin

Copyright © 2018 L. F. García et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the effect of the nonparabolicity of the conduction band on the spectral properties of the shallow donor in elongated InAs/GaAs quantum dots with volcano-shaped profile in the framework of the Kane model by using a simple semiempiric relation between geometrical parameters of the quantum dot profile and the confinement potential, governing the in-plane electron’s movement. We represent the solution of the Schrödinger equation in the form of double Bessel-Fourier series expansion. We show that the nonparabolicity of dispersion of the conduction band, given by the Kane formula, conduces to significant lowering of the donor energies and to stronger confinement of the electron within the quantum dot. Calculated results for the energies as functions of the electric and magnetic fields for different quantum dot dimensions were compared with those obtained in the effective mass approximation. Our results exhibit a high sensibility of the probability density of electron distribution in volcano-shaped quantum dot to the variation of the external magnetic and electric fields.

#### 1. Introduction

Recent development of nanofabrication techniques such as chemical vapor deposition, liquid-phase, molecular beam, and advanced droplet epitaxy [1–6] offers an attractive route for designing of new semiconductor materials for possible applications in opto- and microelectronic devices. In addition, the electrical and optical properties of nanostructures may be modified by doping the shallow impurities, whose energy levels in these materials are very sensible to the variation of the impurity position, the nanostructure size, and dimension. A great deal of attention of investigators during previous two decades has been attracted to the study of the effect of the quantum confinement on the impurity states in various nanostructures, such as single, double, and multiple quantum wells [7–11], quantum-well wires [12, 13], and quantum dots (QDs) [14–18] by commonly using the effective mass approximation (EMA) and the variational method.

The morphology of nanostructures can be sufficiently complicated, and therefore the calculation by means of the variational method requires generally a lot of computational work related especially to the estimation of multidimensional integrals. In the case of QDs, which generally have a form of nonhomogeneous very thin layer, the numerical procedure can be simplified in the framework of the adiabatic approximation (AA), in which one can study separately the slow in-plane and fast transversal movements [19, 20]. Application of the AA allows us to reduce the actual 3D problem for impurity confined in a thin semiconductor layer to two-dimensional Schrödinger equation for the hydrogen atom placed in a field with an adiabatic potential. The adiabatic potential of QDs with different morphologies can be calculated numerically [21].

However, the applicability of two-dimensional models and the AA in many cases is doubtful for two reasons: possible nonparabolicity of the conduction band and strong tunneling of the electron outside the QD trough junctions. The first of them is essential when the gap between the valence and conduction bands is small and the mixing of these two bands is essential such as in InAs ( ~ 0.4 eV) or InSb ( ~ 0.5 eV), while the tunneling becomes essential when the barrier height in junctions and the layer thickness are relatively small. Calculations, realized previously in framework of the EMA [22] have demonstrated that in a thin InAs/GaAs quantum disk with the thickness < 4 nm, the electron can be located outside the disk with the probability being about 30%. Earlier, it has been proposed to replace the EMA by the Kane model [23] with a consistent energy-dependent effective mass in order to analyze these effects on the spectral properties of the electron and exciton confined in a thin InAs/GaAs QD by using finite element method [24, 25] and AA [26]. It has been shown that the nonparabolicity causes an essential lowering of the carriers’ energies reinforcing in this way their confinement inside SAQD and giving in this way an additional reason for applicability of 2D models for QDs with relatively small barrier heights and energy gaps.

Recently, advanced technics have enabled the fabrication quantum dot structures with a wide diversity of almost ideal morphologies but with inherent imperfections, reflected in experimentally found photoluminescence spectra [27–31]. The structures generally show a good circular symmetry, but with an inherent slight elongation along of one of the directions and a noticeable asymmetry in the thickness of the layer breaking the rotational symmetry, which complicate the theoretical analysis of the photoluminescence spectrum. Below we propose a simple numerical procedure, which allows us to analyze the energy spectrum of a shallow donor placed in an elongated InAs/GaAs volcano-shaped quantum dot, in the framework of the Kane model and by using a nonvariational method. Moreover, our procedure permits studying the effect of the electric and magnetic fields applied, respectively, perpendicularly and parallel to the crystal growth direction. In order to calculate the energies of lower levels as functions of the magnetic field, we use an interpolative relationship between the adiabatic potential of the InAs layer and the local thickness found in [26], considering a model with finite barrier height in the junctions of the InAs QD surrounded by the GaAs matrix and taking into account the nonparabolicity of the conduction bands in both materials. We obtain the energies and wave functions of reduced two-dimensional Schrödinger equation by using the double Fourier-Bessel series expansion method.

The organization of this paper is as follows: In Section 2, we present the theoretical model and explain our calculation method of the energies, the charge distribution, the density of energy states, and the electric and magnetic momenta for on- and off-axis donors confined in an elongated InAs/GaAs volcano-shaped quantum dot. Numerical results are presented in Section 3 and some conclusions in Section 4.

#### 2. Theory

In order to study the energy spectrum of shallow donor confined in volcano-shaped InAs/GaAs QR, taking into account strong nonparabolicity of the InAs conduction band, we use below a simple version of the Kane model, proposed in [23, 24]. In this version the electron in a semiconductor bulk is considered as a free particle with the energy-dependent effective mass placed in a band with the parabolic dispersion. In addition, it is assumed that the material parameters in QDs have mismatches in junctions and they are characterized by spatial-dependent band-gap energy and effective mass. The corresponding nonlinear Schrödinger equation for the electron released from the donor with the position vector in a homogeneous magnetic field applied along the -axis and a homogeneous electric field in the Kane model has the form [23, 24]Here is the dielectric constant of the InAs, is vector potential, is the confinement potential equal to zero inside the QD layer and to ; otherwise, defines the profile of QD, and is the energy- and position-dependent effective mass, defined aswhere is free electron mass, is Kane’s momentum matrix element, is the band gap, is the spin–orbit splitting of the valence band, and are eigenvalues of the nonlinear wave equation (1). We simulate the* InAs*/*GaAs* ring by using the material parameters for* InAs* inside the rings being = 0.42 eV, = 0.38 eV, and . The parameters for* GaAs* are = 1.52 eV, = 0.34 eV, and ; the mismatch of the band gap at the junction is equal to [27–31]. We use below, as dimensionless units, the effective Bohr radius nm, the effective Rydberg eV, and (kV/cm) as units of magnetic and electric fields and the dimensionless magnetic field strength, respectively, with being the dielectric constant of* InAs *material.

Taking into account the strong anisotropy of the nanostructure, one can use the adiabatic approximation, in which fast transverse and slow in-plane movements may be considered separately, representing the electron wave function as follows:The transverse wave function with fixed polar coordinates satisfies additional Ben Daniel Duke boundary condition at the bottom of the layer with the coordinate and at each point of the top of the layer . One can find the lowest energy for such movement that is used then as the adiabatic potential for the in-plane electron movement by solving a transcendental equation [26] and obtaining the in-plane adiabatic confinement potential related to the layer thickness as follows:

To simulate a uniform volcano-like QR we use below the parameterized function, which in polar coordinates defines the variation of the thickness of the layer: Here is the height of the rim and is the effective width of the volcano, related to the central hole as follows:We choose the dependence of the oval-shaped rim radius with the eccentricity on the polar coordinate asIn Figure 1 we display profiles of radial cross sections of the model for different central hole heights, and Figure 2 shows the 3D images corresponding to the ratio at the upper row, the ratio at the lower row, and the eccentricities = 0, 0.5, and 1.0.