Journal of Nanomaterials

Volume 2015 (2015), Article ID 937310, 9 pages

http://dx.doi.org/10.1155/2015/937310

## Barrier Thickness and Hydrostatic Pressure Effects on Hydrogenic Impurity States in Wurtzite GaN/Al_{x}Ga_{1−x}N Strained Quantum Dots

^{1}College of Science, Hebei United University, Tangshan 063000, China^{2}College of Light Industry, Hebei United University, Tangshan 063000, China

Received 6 January 2015; Accepted 17 March 2015

Academic Editor: J. David Carey

Copyright © 2015 Guangxin Wang 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

Within the framework of the effective mass approximation, barrier thickness and hydrostatic pressure effects on the ground-state binding energy of hydrogenic impurity are investigated in wurtzite (WZ) GaN/Al_{x}Ga_{1−x}N strained quantum dots (QDs) by means of a variational approach. The hydrostatic pressure dependence of physical parameters such as electron effective mass, energy band gaps, lattice constants, and dielectric constants is considered in the calculations. Numerical results show that the donor binding energy for any impurity position increases when the hydrostatic pressure increases. The donor binding energy for the impurity located at the central of the QD increases firstly and then begins to drop quickly with the decrease of QD radius (height) in strong built-in electric fields. Moreover, the influence of barrier thickness along the QD growth direction and Al concentration on donor binding energy is also investigated. In addition, we also found that impurity positions have great influence on the donor binding energy.

#### 1. Introduction

Recently, much attention has been paid to wide-band gap wurtzite (WZ) GaN/AlGaN quantum heterostructures due to their promising applications in optoelectronic devices such as light-emitting devices (LEDs) and laser diodes (LDs) [1–3]. Doping impurities in GaN-based confined systems is an effective method for controlling the electronic and optical properties of optoelectronic devices. It is well known that the hydrostatic pressure applied on a GaN-based semiconductor material can not only modify the parameters, such as the band gaps, the potential barriers, the conduction effective masses, the static dielectric constants, and the lattice constants, but also change the dimension of the low-dimensional systems, which is associated with the fractional change in the volume. Moreover, the strong built-in electric field induced by the spontaneous and piezoelectric polarizations also affects obviously the distribution of the carrier wave function in WZ nitride-based quantum heterostructures. According to the above characteristics, various studies concerning impurity states are reported in GaN-based nanostructures such as quantum wells [4–8], quantum well wires [9, 10], and (double) quantum dots [11–17]. In these references, considering the strong built-in electric field and/or hydrostatic pressure effects, the hydrogenic donor impurity states are mainly discussed in a radial infinite confinement potential barrier. Their results demonstrate that quantum structure size, strong built-in electric field, and hydrostatic pressure have a significant influence on the donor impurity binding energies, but there are few reports involved in barrier thickness effects on the donor binding energy in WZ GaN/Al_{x}Ga_{1−x}N strained QD in finite potential barrier to date.

To further demonstrate the barrier thickness effect on impurity states in a WZ GaN/AlGaN strained QD, we calculate the donor binding energy of hydrogenic impurity in a WZ GaN/AlGaN QD under a strong built-in electric field by means of a variational approach in the finite confinement potential. In our calculation, effective mass of the electron, dielectric constants, phonon frequencies, energy gaps, sizes (radius and height) of QD, and piezoelectric polarizations are considered as a function of hydrostatic pressure. The paper is organized as follows. In Section 2, we describe the theoretical framework. Then, the pressure and the strain coefficients of GaN and Al_{x}Ga_{1−x}N are discussed in Section 3. In Section 4, the numerical results are discussed. Finally, conclusions are drawn from the present study in Section 5.

#### 2. Theoretical Framework

In Figure 1, the schematic view of a cylindrical WZ GaN/Al_{x}Ga_{1−x}N QD is depicted, with a detailed description of the different dimensions of the QD (dot radius , height , and barrier thickness ). Additionally, the GaN quantum dot is embedded in an Al_{x}Ga_{1−x}N host matrix material, and the -axis is defined to be the growth direction. Within the frame of the effective mass approximation, the Hamiltonian for a hydrogenic donor impurity in cylindrical WZ GaN/Al_{x}Ga_{1−x}N QD under the influence of hydrostatic pressure can be written [4] aswhere denotes the position vector of the electron (impurity ion), is the absolute value of the electron charge, is the permittivity of free space, and is the pressure-dependent effective mean relative dielectric constant of GaN and Al_{x}Ga_{1−x}N materials. The Hamiltonian is given by [11]where and are the pressure- and strain-dependent effective masses of electron along and perpendicular to the -direction. And is the electron confinement potential due to the band offset () and is given by [13]In (2),* F*(*p*) is the pressure-dependent built-in electric field (BEF) in the finitely thick barrier layer for WZ GaN/Al_{x}Ga_{1−x}N QD. The values of the BEF along the growth direction in the well () and barrier () that result from the difference in the total electric polarizations in each region are given by simple formulas [14]:where , and , are the thicknesses and electronic dielectric constants for GaN and Al_{x}Ga_{1−x}N materials, , and , are the spontaneous and piezoelectric polarizations for GaN and Al_{x}Ga_{1−x}N materials, and the bowing parameter is chosen as −0.019 C/m^{2} [14]:The wave function of the electron confined in the WZ GaN/Al_{x}Ga_{1−x}N QD can be written as where the constants ( and ) are determined by the continuity of the derivative of the radial wave function at the QD boundary and is the electron -component angular momentum quantum number. The radial wave function of the electron can be obtained using the Bessel function and the modified Bessel function . Wave function can be expressed by means of the Airy functions Ai and Bi [16]: Here . The coefficients and can also be obtained by the transfer matrix methods [16].