1. Introduction

Recent advances in nanofabrication technology have made it possible to fabricate nanostructures of different sizes and geometries [1–3]. Nanostructures have a wide range of applications including in nanomedicine [4, 5], optoelectronics [6, 7], energy physics [8–12], and gas sensing [13]. Now, even with utmost care and employing the most advanced techniques, it is not possible to fabricate nanostructures which are free of impurities. It may be advantageous, however, to introduce impurities into a nanostructure at the fabrication stage. The presence of such deliberately introduced impurities can lead to improved performance of nanodevices, for example, enhancement of electrical conductivity of semiconducting materials [14]. The impurity may actually be positively charged, in which case an electron may become

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

bound to it, thus forming an electron-hole pair. Photoionization is one of the useful probes for the particular nature of electron-impurity interactions in low-dimensional systems. In the process of photoionization, upon absorbing sufficiently enough energy from the irradiating electromagnetic field, the electron can break free from the impurity. In a sense, photoionization is the classical analog of the binding energy problem. Certainly, the subtlety in photoionization effects is in the variety of conditions in low-dimensional systems. These conditions include quantization of the electron's energy levels as well as the optical properties of the specimen.

binding energies in classical mechanics. The energies of the corresponding initial and final states are Ei and Ef , respectively. The system investigated here is a spherical quantum dot (SQD) of refractive index n and relative dielectric constant ε, which may be a GaAs material embedded in a Ga1-xAlxAs matrix, with a donor impurity embedded at its center. Now, one of the physical quantities that are useful in the description of this binding energy-like problem is called photoionization cross section. This quantity may be regarded as the probability that a bound electron can be liberated by some appropriate radiation per unit time per unit area,

σlm ¼ σoħω

in<sup>=</sup> 3E<sup>2</sup>

replaced by its Lorentzian equivalent given by

where this is the so-called Lorentzian linewidth.

following linear second-order differential equation

<sup>2</sup> dχð Þr dr � � <sup>þ</sup>

1 r2 d dr <sup>r</sup>

2.1. The electron's wave functions

X f

f j r ! <sup>j</sup><sup>i</sup> �D E � �

average electric field inside the quantum dot. Evaluation of the matrix elements for an SQD leads to the selection rules Δl ¼ �1 [21], that is, the allowed transitions are only those for which the l values of the final and initial states will be unity. In the investigations carried out here, the evaluations of the PCS are for transitions only between two electron's energy subbands. For purposes of computation, therefore, the Dirac delta function in Eq. (1) is

Now, in view of spherical symmetry, the solutions of the Schrödinger wave equation are sought in the general form Ψlmð Þ¼ r; θ;φ ClmYlmð Þ θ;φ χð Þr , where Clm the normalization constant, Ylmð Þ θ;φ the spherical harmonics of orbital momentum and magnetic quantum numbers l and m, respectively. The radial part of the total wave function, χð Þ r , is found to be the

kee<sup>2</sup>

The specific forms of the solutions of the differential equation described above depend on the particular electric confining potential considered. Here, the different radially dependent forms of the so-called intrinsic electric confinement potential of the spherical quantum dot, in turn, taken into account in solving Eq. (3) are (shown in Figure 1) (1) simple parabolic, (2) shifted

where μ is the effective mass of electron (of charge -e) and ke is the Coulomb constant.

<sup>ε</sup><sup>r</sup> � V rð Þ � � � l lð Þ <sup>þ</sup> <sup>1</sup>

r2 � �χð Þ¼ <sup>r</sup> <sup>0</sup> (3)

<sup>δ</sup> Ef � Ei � <sup>ħ</sup><sup>ω</sup> � � <sup>¼</sup> <sup>ħ</sup><sup>Γ</sup>

2μ <sup>ħ</sup><sup>2</sup> Elm <sup>þ</sup> � � � 2

� is the interaction integral coupling initial states to final states, <sup>α</sup>FS is the fine

! is the electron position vector. Finally, the amplitude of the PCS is

av<sup>ε</sup> � � in which Ein is the effective incident electric field and Eav the

<sup>δ</sup> Ef � Ei � <sup>ħ</sup><sup>ω</sup> � � (1)

Photoionization Cross Section in Low-Dimensional Systems

http://dx.doi.org/10.5772/intechopen.75736

107

Ef � Ei � <sup>ħ</sup><sup>ω</sup> � �<sup>2</sup> <sup>þ</sup> ð Þ <sup>ħ</sup><sup>Γ</sup> <sup>2</sup> , (2)

given by [15–20]

where f j r

! <sup>j</sup><sup>i</sup> �D E � �

structure constant and r

given by <sup>σ</sup><sup>o</sup> <sup>¼</sup> <sup>4</sup>π<sup>2</sup>αFSnE2

� �

In this regard, photoionization studies on nanostructures could offer insight into the electronimpurity interaction in a wide variety of conditions. These photoionization effects have fueled significant interest in the processes of photoionization in low-dimensional systems. The effects of geometry and hydrostatic pressure on photoionization cross section (PCS) have been reported in concentric double quantum rings [15]. The effect of applied electric field on photoionization cross section has also been probed in cone-like quantum dots [16]. The role that impurity position plays in modifying the PCS in a core/shell/shell quantum nanolayer [17] and a purely spherical quantum has been investigated [18]. Overall, it has been found that photoionization transitions are independent of the photon polarization for a centered impurity, while the transitions are dependent on the photon polarization when the impurity is offcentered. Influences of intense laser field and hydrostatic on PCS in pyramid-shaped quantum dots have also been reported [19]. There also have been studies of PCS in spherical core/shell zinc blende quantum structures under hydrostatic pressure and electric field [20].

In this chapter, the effect of geometry of confining electric potential on centered donor-related PCS in spherical quantum dots is investigated. The electric potentials considered are the parabolic, shifted parabolic, cup-like, and the hill-like potentials, all of which have a parabolic dependence on the radial distance of the spherical quantum dot. To start with, the Schrödinger equation is solved for the electron's eigenfunctions and energy eigenvalues within the effective mass approximation. It is emphasized that the treatment of photoionization process given here is limited only to isotropic media.
