PACS: 78.67Hc; 73.21La; 75.30
1. Introduction
There has been a tremendous improvement in research activities on the low dimensional semiconductor quantum dots due to the advanced fabrication techniques invented for the past few decades. The study of semiconductor quantum dots and nanocrystals has been of a great interest from the experimental and theoretical point of view in recent years 1,2,3,4,5,6,7,8,9,10,11,12,13. The origin of the interest lies in the size of quantization. The electron energy spectrum of an ideal quantum dot comprises a set of discrete levels. This makes the semiconductor quantum dot very important in the applications of optical and transport properties of semiconductors. The physical properties of the quantum dot are attractive not only from the fundamental scientific point of view, but also because of its potential application in the development of semiconductor optoelectronic devices 13.
Impurities in semiconductors can affect the electrical, optical, and transport properties. Understanding the nature of impurity states in semiconductor structures is a crucial problem. Usually impurities are classified as deep or shallow according to their ionization energy. Shallow impurities are defined as those impurities whose ionization energy is comparable or smaller than the thermal energy at room temperature. Shallow impurities are usually known as hydrogenic impurities since they are well described by the hydrogen atom model. The underlying assumptions behind this model are that the binding energy is small compared with the energy gap and the spatial extent of the wave function is larger than the lattice period. As a consequence, carriers have an energy close to the band edge, move with an energy-independent effective mass
The effects of hydrogenic impurity, hydrostatic pressure, temperature and geometrical parameters on optical absorption coefficients and refractive index of spherical quantum dots and raman scattering cross-sections have been reported by Karimi et al.24,25. Safarpour investigated the binding energy, and optical properties of an off center donor impurity in a quantum dot embedded in a nanowire and emphasized on how orientation and distance of impurity from center can serve as good factors in fabricating desired structures with specific electronic and optical properties 26. The effect of dot radius and parabolic potential on binding energy 1s-, 2p-, 3d- and 4f-states of a spherical quantum dot (QD) with parabolic potential has been recently reported 27,28 and the effect of electric field and magnetic field on the low lying states and optical properties of hydrogenic impurity has also been studied 29.
The presence of impurities in QD’s can significantly change the localization states. For simplicity and to protect the symmetric situation the impurity can be located at the center of the dot. An electron bounded to an impurity located at the center of quantum dot behaves like a bounded three-dimensional electron when the radius of the dot is very large. However, as the dot radius is reduced, spatial confinement becomes very important. Thus, spectroscopy tools provide information about the confining properties of electrons and holes bound to hydrogenic impurities in zero dimensional nanostructures.
The purpose of this work is to investigate hydrogenic impurities in spherical quantum dots characterized by the parabolic confining potentials, which have broader applications to realistic problems. The first part of the study contains the evaluation of the spectrum of hydrogenic impurity states and the second part involves study of optical properties. The emphasis is placed on the level energies, wave functions, binding energies, radial transition elements, and optical properties of hydrogenic impurities, varying with the confining potentials and the range of quantum dot. The Schrödinger equation is solved by finite difference method. We have calculated energy eigenvalues, eigenfunctions (
2. Theory
Within the framework of effective-mass approximation, the Hamiltonian of a center hydrogenic donor confined by a spherical QD with a parabolic potential can be written by
where the hydrogenic impurity is located at the center of the QD, e is the charge of the electron, r is the position vector of the electron originating from the center of the dot,
where
where
and
Using
The Schrödinger equation for the system
can be solved using method of separation of variables based on spherical symmetry, where
where N is the normalization coefficient,
Substituting Eq. (9) into the Schrödinger equation (7) and using separation of variables and simple mathematical steps the radial equation is written as:
The energy eigenvalue E nl for a particular energy state is solved numerically using the finite difference method as explained in the next section.
2.1 Energy spectra and radial matrix elements
The eigenenergies and the wavefunctions are obtained by solving the time independent Schrödinger equation for the system, using finite difference method. This is a numerical method for solving the partial differential equation (PDE) based on discretization of the Hamiltonian on a spatial grid. We have taken the grid points N = 1020 and the tolerance of 1.0e-6 is considered. Finite difference method is a simple method that gives useful and accurate results for a wide range of problems dealing with differential equations. In this method, continuous wave function of r is represented by a set of N discrete quantities and effective potential energy
is given by (N - 2) × (N - 2) diagonal matrix. Boundary condition used is u0 = uN = 0. For second order centered finite difference approximation
u(rj) is the eigenvector of the Schrödinger equation and
It is a (N - 2) × (N - 2) matrix. Hamiltonian matrix is [H] = [K] + [U eff], Schrödinger equation in matrix form becomes
where H is the Hamiltonian of the hydrogenic impurity in the semiconductor quantum dot(SQD) and E is the eigenvalue. The Hamiltonian H is reduced to tridiagonal matrix and is diagonalized using standard matlab subroutines to obtain the eigenvalues and the wavefunctions of a hydrogenic impurity in a SQD. This method has been implemented in various semiconductor heterostructures to obtain the unperturbed eigenvalues and wavefunctions 36,37.
In addition we have calculated optical properties like polarizability, oscillator strengths and susceptibility for a hydrogenic impurity in SQD atom under the effect of parabolic potential.
2.2 Polarizability
When a confined system is exposed to static electric field, the electron cloud gets displaced from its equilibrium position. This induces polarization, which is an important microscopic quantity because of its established relationship with another experimentally significant macroscopic property called refractive index. The electric dipole polarizability materializes the second order response of the system in a weak electric field. From perturbation theory, using the second order energy correction, the polarizability is obtained from the sum over states formula. The static 2 l -pole polarizability is given by:
Where l = 1, 2, 3 for dipole, quadrupole and octupole cases, respectively.
2.3. Oscillator strength
Another physical quantity of practical importance in the study on the optical properties is the dimensionless oscillator strength P fi , which is defined by
where
for 1s-2p transition, l´ = 1, l = 0, and m´ = m = 0 the matrix element
and
2.4. Magnetic Susceptibility
It is fully known that the magnetic susceptibility has a key aspect on quantum mechanics, magnetism, and optics. The magnetic susceptibility indicates the degree of magnetization of a material in response to an external magnetic field. Substances with a negative magnetic susceptibility are called diamagnetic. The diamagnetic susceptibility is given by 38
In the next section we have presented the results of our calculations.
3. Results and Discussion
In this study we report a detailed theoretical investigation of the hydrogenic impurity in a spherical quantum dot under parabolic confinement. Effective atomic units are used throughout the paper. Length is expressed in terms of effective Bohr radius
In Fig. 1 the behaviour of radial wave function u
nl
(r) of ground and excited states has been shown for different values of harmonic potential frequency
In Fig. (2) we have shown how energies of ground state and other excited states vary as a function of the frequency
In Fig. (3) we explore the effect of increase in impurity strength (
In Fig. 4 we have shown how energies of ground state and other excited states vary as a function of radius of the dot (r
0) for different values of frequency of parabolic potential (
In Fig. (5) we have calculated the binding energies of different states as a function of radius of the dot (r0). The binding energy of a hydrogenic impurity is known as the difference between the energy states without impurity (
Next, we have calculated the radial matrix elements which are defined as
In Fig. 7 we have reported variation of radial matrix elements of ground state and other excited states vary as a function of dot radius at different frequencies and at different impurity strength. Figure 7(a) shows the radial matrix elements as function of dot radius in the absence of impurity
The next set of figures is for the study of optical properties.
We have calculated the 2l -pole static polarizability of the
system for l = 1 for various harmonic potential frequencies as a
function of r
0. The effect of parabolic potential on such polarizabilities has been
explored by considering four values of
The diamagnetic susceptibility of a hydrogenic impurity as a function of r 0 has been shown in Fig. 9. It is observed from the figure that the diamagnetic susceptibility decreases from a maximum value as the radius increases. For higher frequency of harmonic potential the diamagnetic susceptibility decreases more slowly. This indicates that there is a strong influence of the confining potential and dimensions of the dot on the diamagnetic susceptibility. Similar results using different shapes and potentials have been reported by 42,43.
In Fig. 10 we have investigated how the oscillator strengths (1s-2p) varies with the dot radius and study how these get modified with the presence or absence of impurity at different frequencies of harmonic potential. From figure one can easily see that at the given radius (about a = 2a *), P fi displays a maximum and obtains the major portion of 1 (larger than 0.965), so at this given radius the other transition probabilities have a very small portion of 1 and tend toward 0. In addition as the dot radius increases the oscillator strength increases and reaches a constant value at large r 0. While the energy difference decreases with increasing dot radius, the overlapping grows. As a result, the oscillator strength is to be fixed at a constant value in large QDs. In large QD radii, the wave functions of the states, especially that of the 1s state, are localized near the center of the dot because of the attractive Coulomb potential of the impurity. And hence there is some limitation on the overlapping and the dipole matrix element has a constant value. These results are in good agreement with different studies 44.
4. Summary and Conclusion
We investigated the energy spectra, wave functions, binding energies, radial matrix elements of the ground and excited states and optical properties of hydrogenic impurity of the spherical QD with parabolic potential. Energy eigenvalues strongly depend on dot radius and parabolic potential parameters. Removal of degeneracy is observed with the increase of dot radius. Spectra of the system change drastically with the parabolic confinement. In addition, calculated results have shown that the existence of an impurity has a great effect on the energy spectra. The radial matrix elements and optical properties of the system show dependence on dot radius and parabolic confinement. We believe that our study makes an important contribution to the literature. The theoretical investigation of the optical properties in a spherical QD will lead to a better understanding of the properties of QDs. Such theoretical studies may have profound consequences for practical application of the optoelectronic devices and in optical communication.