Polarizability and binding energy of a shallow donor in spherical quantum dot-quantum well (QD-QW)

The polarizability and the binding energy is estimated for a shallow donor confined to move in inhomogeneous quantum dots (CdS/HgS/CdS). In this work, the Hass variational method within the effective mass approximation in used in the case of an infinitely deep well. The polarizability and the binding energy depend on the inner and the outer radius of the QDQW, also it depends strongly on the donor position. It’s found that the stark effect is more important when the impurity is located at the center of the (QDQW) and becomes less important when the donor moves toward the extremities of the spherical layer. When the electric field increases, the binding energy and the polarizability decreases. Its effects is more pronounced when the impurity is placed on the center of the spherical layer and decrease when the donor move toward extremities of this spherical layer. We have demonstrated the existence of a critical value ( R 1 R 2 ) c r i which can be used to distinguish the tree dimension confinement from the spherical surface confinement and it’s may be important for the nanofabrication techniques.


Introduction
In this last decade the advances in nanofabrication technology have made it possible to manufacture low-dimensional semiconductor materials such as quantum well (QW), quantum well wire (QWW) and finally quantum dots (QDs) structure where the electron is confined in three directions. With their reduced dimensionality, new effects can be observed and studied. In particular, the binding energies of shallow donors are expected to become important while they are known to be low in bulk materials [1][2]. The semiconductor quantum dots (QDs) structure, often called artificial atoms, have great potential for device applications in the nanometer scale [3]. Therefore, in recent years, both experimental and theoretical studies of the electronic and optical properties of homogenous quantum dots (QDs) have attracted the considerable attention of many scientists in condensed matter physics and applied sciences [4][5][6]. In this structure, a new type of quantum dot (QD) structure, called a quantum dot-quantum well (QDQW) or inhomogeneous quantum dots (IQD), has been fabricated and studied. Theses structures experimentally are composed of two semiconductor materials one of which, with the smaller band gap, is embedded between a core and outer shell of the material with the larger band gap. Nowadays, a synthesized inhomogeneous spherical quantum dot with a central core and one or several layers of shells (such as CdSe/ZnS, InAs/ZnSe, CdS/PbS) called core-shell structure quantum dot or quantum dot-quantum well (QDQW) attracts many scientists' interests [7,8]. Xi Zhang and al [9] studied the third-order nonlinear susceptibilities associated with intersubband transition are theoretically calculated for ZnS/CdSe cylindrical quantum dot quantum well. X.N.Li u and D.Z.Yao [10] interested on the study for nonlinear optical properties of the CdSe/ZnS quantum dot quantum well (QDQW). Xing and al. [11] focus problem of bound polarons in quantum dot quantum well (QDQW) structures is studied theoretically. For more details on the chemical fabrication of this new artificial structure, we refer to references: CdSe/ZnS/CdSe/ZnS [12], In(Ga)As/GaAs [13] and InAs/ZnSe [14].
Understanding the Impurity states in these structures is an important problem in semiconductors technology. Many works have been devoted to the studies of impurity states in the quantum well, quantum wire and quantum dots. Recently, J.Silva and al [15] have calculated the donor binding energy as a function of the donor position in QD with an infinite spherical well and for different radius of the structure. They found that the donor binding energy decreases when the donor position increases reaching a minimum, as the donor position is equal to the radius of the QD. Recently the dipole polarizability of the harmonic oscillator of nanodot has been calculated analytically [16][17]. Many theoretical attempts have been made to understand the polarizability [18][19][20].
For QDQW, to the best of our knowledge no work has been done to treat the polarisability and evaluate the effect of the donors position. The quantum confined stark effect has been the subject of numerous experimental and theoretical investigations [21] which demonstrated that the binding energy are shifted to low energies in the presence of an uniform electric field. The effect of the latter is to change the energy levels in the QD and giving rise to a polarization of the distribution of charge carriers and has other energy transition between quantum states [22]. D Schoss et all are presented a theoretical and experimental study on the inhomogeneous quantum dots (CdS/HgS/CdS) [23].
In this work, we use a variational method to calculate the polarisability and the binding energy of an electron bound to a donor impurity confined to move in inhomogeneous quantum dots (CdS/HgS/CdS) in the strong and the moderate regime of confinement. We will examine the electric field and the position of a donor impurity effects on the binding energy and the polarizability. We will examine the effect of the size of the dots.

General formalism
We Consider a hydrogenic impurity located at the position r0 of a quantum dot-quantum wells (QDQW) made from [CdS (Core)/HgS(well)/CdS (Shell)] subjected to the action of an electric field outside F. The confining potential is assumed as a model of an infinite deep well. In the effective mass approximation, the Hamiltonian system is written: Where V(r) is the confining potential. It's written by: R1 et R2 denotes the inside and outside radius of the quantum dot-quantum wells, respectively. H0 is the Hamiltonian in the absence of the electric field given by: 0 is the dielectric constant and m* is the effective mass of the electron in the QDQW.
The Schrödinger equation expressed in spherical coordinates (r,,) is written as: We use a variational method approach to determine the ground state binding energy and the polarisability, We adopt the wave function given by : 0 is the wave function in the absence of the electric field (f = 0) given by: Where  and  are the variationals parameters (which takes into account the presence of the electric field). The energy is obtained by the minimization with respect to the variationals parameters: The binding energy Eb of the Impurity is given by: Where Et and ESub represents the ground-state energy of the structure QDQW with and without the impurity respectively. The polarizability can be calculated from the dipole moment and is defined as: We use the effective Bohr radius

Results and Discussion
In this section we will discuss the results obtained by applying the direct variational method for calculating the electric polarizability by the formula dipole moment P and the binding energy of a hydrogenic impurity in the absence and presence of an electric field in the quantum dot-quantum well (QDQW In our calculations, two extreme cases that present themselves: R1 = 0 (Homogeneous Quantum Dot ''HQD'' of radius R2) and R1→R2 for R2 fixed which corresponds to an infinitely thin spherical layer. We will present the results for the cases of a HQD and IQD.

Homogeneous Quantum Dot ''HQD''
In with decreasing the dot radii R2. It is due to the distribution of the electronic wave function. Therefore, the decrease of the geometric dimensions of the quantum dot creates a compression wave function and the electron is closest to the impurity. It is causes the increased binding energy. Our results are in good agreement with those found by Silva-Valencia and all [2,25].
In figure 3, we plotted the electric polarizability of an impurity associated with an electron according to the radius of the QD for different values of the electric field. In this figure, we note that the polarizability decreases with increasing the intensity of the electric field. Ours results show that the polarizability increases as the radius of the dot increases. We remark that the electric field effect is more pronounced for large dots. That we can explain that by for larges quantum dots, the electronic confinement is negligible and therefore the results tend to the case of semiconductor solid 3D. The intensity of the electric field increases the confinement of the electron and reduces the polarizability.

Inhomogeneous Quantum Dot ''IQD''
In figure 5, we present the binding energy Eb as a function the ratio (R1/R2) for two different regimes of confinement has R2 = 1a* and R2 = 2a* and for two values of electric field f = 0,4 and 0.8. We see that the binding energy depends strongly of the ratio We note that when the electric field increases, the binding energy decreases and we also note that the electric field effect becomes more pronounced when the ratio (R1/R2) tend to 1. We have reported in figure 6, the electric polarizability of the donor impurity according to the ratio R1/R2 of the quantum dot quantum well (QDQW) for four values of outer radii R2 and for an electric field strength (f = 0.2). This figure illustrates the competition between the geometric confinement (the size of QDQW) and containment of the electric field on the polarizability.
For larges values of the ratio R1/R2 (low spherical layer), the geometric confinement is important and the wave function of the electron is very localized. The electric polarizability of the impurity becomes less important and is relatively insensitive to the outer radii R2 of the QDQW. For low values of the ratio R1/R2 (large spherical layer), the effect of electric field predominates. The polarizability is strongly depend of the size of quantum dot-quantum well. We note that the polarizability decreases as the R1/R2 ratio increases.

Conclusion
In conclusion, we have studied the polarizability and the binding energy of an impurity located in Inhomogeneous Quantum Dots (IQD) in the presence of an electric field. It's clear that they strongly depend on the geometric confinement, the impurity position and the electric field. We have shown the existence of a critical value (R1/R2)crit which may be important for the nanofabrication techniques.