Binding Energy And Absorption of Donor Impurity In Spherical GaAs/AlxGa1-x As Quantum Dots With Konwent Potential


 In this study, the electronic and optical properties of single or core/shell quantum dots, which are formed depending on the parameters in the selected Konwent potential, are investigated. Namely, the effects of the size and geometric shapes of quantum dots on the binding energy of the on-center donor impurity, the total absorption coefficient and refractive index which are including transitions between the some confined states, and the electromagnetically induced transparency between the lowest six confined states related to the donor impurity are investigated. We have used the diagonalization method by choosing a wave function based on the Bessel and Spherical Harmonics orthonormal function to find the eigenvalues and eigenfunctions of the electron confined within the quantum dots which have different types mentioned above. To calculate the optical absorption coefficients and electromagnetically induced transparency related to shallow-donor impurity, a two- and three-level approach in the density matrix expansion is used, respectively.


I. INTRODUCTION
Quantum dots (QDs) are one of the most important structures of low dimensional quantum systems which are used to manufacturing high-performance microelectronic and optoelectronic devices [1][2][3]. Due to advances in material growth technologies such as molecular beam epitaxy and metal-organic chemical vapor deposition, the production of low-dimensional semiconductor systems which have varied shapes and sizes is became possible. The dependence of the electronic energy spectrum on the geometric shape of the confinement potential and size of QD is very strong. This dependency allows us to manipulate the electronic and hence, optical characteristics of QD. For this reason, QDs are very important topics for researchers both theoretically and experimentally. As well as the geometric form and size of the confinement potential of QDs [4][5][6][7][8][9], the externally applied electric, magnetic and intense laser fields [10][11][12][13][14][15][16][17][18][19] affect the electronic properties, which influences the optical properties of QD. In devices based on QDs, the effects caused by impurities are also quite important and thus there are a lot of researches involving impurity-related studies with a special focus on QDs [20][21][22][23][24][25]. In addition to the shape, size, and externally applied fields, the presence of impurities causes radical changes in the energy spectrum of the quantum dots and provides the desired optical transitions to be obtained with respect to the purpose. As known, controlled optical transitions are very important in the design of optoelectronic devices with tunable emission or transmission characteristics and ultra-narrow spectral line width. The confinement of charge carriers in QDs causes to increase in the energy states, dipole matrix elements and the oscillator strength, which are related to a red or blue shift at absorption peak positions and an increase in peak amplitudes. Therefore, QDs are materials with great optical response [26]. This case prompted researchers to investigate optical phenomena such as optical absorption [27,28], optical rectification [29][30][31], second and thirdorder harmonic generations [32][33][34] and refractive index changes [35][36][37]. Optical absorption coefficients (OACs) of QDs which including impurity have different confinement potentials have been previously studied by many authors [38][39][40][41][42][43][44][45][46][47][48][49][50]. Optical properties of QDs are investigated for: a cylindrical core-shell QD by Kria et al [38], the spherical sector-shaped QD by Mora-Ramos et al [39], a laser-driven two-dimensional disc-shaped QD by Kilic et al [40], a spherical QD by Stevanović et al [41], a typical ring-shaped elliptical QD by Ghajarpour and Karimi [42], core-shell ellipsoidal QDs by Shi and Yan [43], a mul-tilayer QD by Rahul et al [44], a lens-shaped QD with a finite confinement potential by Khordad et al [45], a parabolic QD by Hosseinpour et al [46], two QDs coupled laterally with a circular cross-sectional shape by Duque et al [47], a pyramidal QD by Niculescu and Bejan [48], a triangular two-dimensional QD by Kasapoglu et al [49], a QD with Woods-Saxon potential by Lu et al [50].
In addition to the electronic structure in semiconductor nanostructures, optical absorption at inter-subband transitions, especially nonlinear absorption, is a very attractive subject. Electromagnetically induced transparency (EIT) has potential applications in many fields from nonlinear optics to quantum information science, namely nonlinear optics, optical memories, quantum optics, quantum computation, quantum information sciences and optical switches by controlling the light speed in low-dimensional semiconductors and researches about EIT continue intensively because of the potential applications mentioned above [51][52][53][54][55][56][57]. EIT which provides the coherent control of the optical properties of materials is one of the most interesting effects of quantum optics and it is a coherent optical nonlinearity that makes a medium transparent in a narrow spectral range around an absorption line. This transparency window leads to slow down and/or stop light that is one of the most striking EIT effects [58]. As known quantum confinement is mainly dealt with the energy of confined charge carriers (electrons and/or holes) in low dimensional systems. The energy levels of electrons in these systems are discrete, not continuum as in the bulk materials. QDs have perfect linear and nonlinear optical properties due to discrete energy levels which result from strong confinement that making them excellent for optoelectronic applications [59][60][61][62], and thus QDs are excellent structures for investigating the EIT [63,64].
This study that we thought had not been studied before aims to examine the electronic and optical properties of GaAs/Al x Ga 1−x As QDs with Konwent potential, which transform into single or core/shell quantum dots depending on the changes in the structure parameters. The organization of the paper is the following: Section II contains the presentation of the theoretical framework, in Section III, we discuss the obtained results, and in Section IV the conclusions are given.

A. Energies and wavefunctions
In this study we are interested in investigating the states of an electron-impurity system confined in a spherical GaAs/Al x Ga 1−x As QD with confining potential described by the Konwent-like model. It is considered a shallow donor impurity located on the center of the structure and the effects of spatial variation of the effective mass are neglected. Once the wavefunctions and energies for the system have been obtained, without and with im-purity, we proceed to calculate, among others, the binding energy, the total absorption coefficient between the lowest three confined states, and the EIT between the lowest six confined states.
In the effective mass approach, the Hamiltonian for confined electron in KQD without impurity can be written as follows where p and m * are the momentum and the electron effective mass. V KP (r) is the confinement potential proposed by Konwent [65] and defined as where k is the parameter related to the dot-size in the radial direction, A is the structure parameter used to get with different QD-shapes, and V 0 = Q c ∆E g is the potential depth. Q c = 0.6 is the offset parameter of conduction band and ∆E g = 1.247 x eV [66] is the difference between the band gaps of Al x Ga 1−x As and GaAs materials, where the aluminum concentration is taken as x = 0.3. (color online) The r-dependence of the 3Dconfinement potential in a GaAs-Al0.3Ga0.7As Konwent-like spherical quantum dot. Results are for three values of the A-parameter with k = 2.5 nm and V0 = 0.225 eV (see Eq. (2)).
The r-dependence of the 3D-confinement potential in a GaAs-Al 0.3 Ga 0.7 As KQD for three values of the Aparameter is depicted in Fig. 1 (see Eq. (2)). As seen, the value of the A-parameter determines the shape of the potential. For A < 1, the system behaves as a core/shell QD where the potential height in the shell is regulated by the value of the A-parameter, reaching its maximum value when A → 0. In the limit A = 0, V KP (r) = V 0 . As the A-parameter presents a finite value, but close to zero, the core/shell-like behavior appears with a core where the potential is constant and a shell whose distance from the QD-center systematically tends to infinity. As the A-parameter increases, approaching A = 1, a combination of two factors occurs: i ) a systematic reduction of the potential barrier in the core region and ii ) a systematic reduction of the core radius. The central barrier in the core disappears when A = 1 and its null value remains nearly constant over space up to a certain extent of the radial coordinate that depends on the value of the k-parameter, which controls the effective size of the QD. For A > 1 and k-decreasing, the system turns into a QD with parabolic potential in which it is possible to observe a shift of the potential bottom towards positive energies. It is important to highlight that the height of the central barrier in the core is definitely controlled by the A-parameter.
In the presence of central donor impurity, the Hamiltonian is written as where Z = 0 (Z = 1) denotes the case without (with) the impurity, ε is the GaAs dielectric constant and r is the electron position. If the lengths are normalized to the effective Bohr radius [a B = 2 ε/(m * e 2 )] and the energies to the effective Rydberg [R * = 2 /(2m * a 2 B )], the dimensionless Hamiltonian is found as follows where ∇ 2 is the 3D-Laplace operator.
In order to find the energies and their corresponding wavefunctions, we have to solve the Schrödinger equation where, n, l, and m are the principal, angular momentum, and magnetic momentum quantum numbers. To solve this equation, the Hamiltonian matrix is diagonalized by a series expansion using the electronic wavefunctions of the infinite spherical QD as the basis. This series expansion is given in the form [67][68][69] ψ nlm (r, θ, φ) = j c nj lj ψ where c nl are the expansion coefficients and ψ (0) nlm (r, θ, φ) is the total wavefunction describing the motion of the electron without impurity and it is expressed as follows in the form of product of the radial and angular functions by virtue of the QD symmetry where ϕ (0) nl (r) and Y lm (θ, φ) are the radial and spherical harmonics wavefunctions, respectively. ϕ nl (r) is defined in terms of the a-radius of the infinite spherical QD by the following equation Here N is a normalization constant and j l are the spherical Bessel functions. Additionally, k nl is the n-th zero of the j l function. The binding energy (E b nlm ) is obtained from the difference between the energies in any nlm-states of the electron in the absence and presence of donor impurity and it is as defined by where, E nlm (Z = 0) and E nlm (Z = 1) are the energies without and with impurity in the KQD, respectively. We want to emphasize that in this study, we will focus our attention on states with n = 1 and m = 0. For the study of the OAC, we will use the first three states with l = 0, 1, 2. In the case of EIT, and due to the polarization of the two incident radiations, it will be necessary to extend the set to l = 0, 1, 2, 3, 4, 5. In Figs. 2 and 3 we present the wavefunction projections in the y = 0 plane for a confined electron in a GaAs/Al 0.3 Ga 0.7 As KQD. In particular, Fig. 2 compares the ground state wavefunctions in the absence and presence of the on-center impurity for three values of the A-parameter and two values of the k-parameter. In Fig.  3 the ground state and the first two excited states wavefunctions in the presence of impurity are shown for the same combinations of the A-and k-parameters. Note that for k = 2.5 nm, we have used a = 15 nm while for k = 5.0 nm the corresponding value is a = 25 nm. This ensures that the effects of the infinite barrier in the outer region of the shell on the energies are minimal. In both cases, with and without impurity, the upper limit of the sum in Eq. (6) has been chosen to guarantee a convergence of 0.01 meV in the first ten energy levels. In Fig. 2, the spherical character imposed by the potential can be observed in all the presented wavefunctions. For k = 5.0 nm and A = 0.2, it is clearly appreciated that the maximum of the wavefunction is located in the shell region and that the presence of the impurity generates an increase in the density of probability in the central region of the structure; this despite the presence of the potential barrier present in the core region. In the case of k = 2.5 nm and A = 0.2 the core/shell character is not very clear because a large part of the electronic wavefunction is located in the core region; this gives the little spatial extension of the structure. Consistent with the attractive potential associated with the impurity, in the all cases of Fig. 2, a reduction in the spatial extension of the wavefunctions is observed, compared to the case without impurity.
In Fig. 3, where the ground state and the first two excited states are presented, in the presence of impurity, the following characteristics are observed: i ) as mentioned above, the ground state has spherical symmetry, that is, it corresponds to s-like states, where the wavefunctions are more spread out in space as k-increases and A-decreases, ii ) in all considered cases of A-and k-parameters, the first excited state has p z -like symmetry, corresponding to two antinodes located symmetrically on the z-axis and having the opposite sign; in this case the wave function is null at r = 0 (or z = 0), and iii ) the second excited state has d-like symmetry with two antinodes of the same sign located symmetrically on the z-axis and a thorus region of maximum amplitude, but opposite sign, which has axial symmetry with respect to the z-axis. Compared to the ground state, the first and second excited states have a greater spatial extension, and this extension increases with the increase (decrease) of the k-parameter (A-parameter). Considering z-polarized resonant incident radiation, and taking into account the symmetries of the wave functions, clearly the ψ 100 → ψ 110 and ψ 110 → ψ 120 transitions are allowed while the ψ 100 → ψ 120 transition is forbidden.

B. Optical absorption coefficient
Photoabsorption is the phenomenon of transition from low energy level to high energy level by absorption of a photon in any quantum system. When a photon with energy ω ≥ E j −E i is absorbed, a transition takes place between levels with energy E i and E j . The linear, thirdorder nonlinear, and total optical absorption coefficient (OAC) corresponding to these transitions are described as follows, respectively: and where ω is the angular frequency of the incident photon, µ is the vacuum magnetic permeability, ε r is the real part of the GaAs susceptibility, σ s is the electron density, is the non-diagonal matrix element known as the relaxation rate of the final and initial states, E ij = E j − E i is the transition energy between the initial and final states, I = 2 ε 0 n r c |Ẽ| 2 (where |Ẽ| is amplitude) is the intensity of the incident resonant field, n r is the GaAs refraction index, c is the vacuum light speed, and ε 0 = 8.85 × 10 −12 F/m is the vacuum permittivity. M ij is the dipole transition matrix element, and for incident resonant radiation polarized in the z-direction it is given as where ψ i and ψ j are the wavefunctions representing the initial and final states, respectively.
In accordance with the energy conservation law, every probable transition in the absorption spectrum can not be observed. In addition to the energy conservation, the angular momentum conservation and several symmetry rules have to also be provided. All permit-ted transitions are added in the dipole transition elements, and the cases where the dipole transition element is nonzero are sought. So, for a linearly polarized light interacting with the electric field polarized in the z-direction, only the z-component of the dipole transition element is nonzero, while the other components are zero. According to the property of spherical harmonics, only dipole transitions are allowed between cases that supply the rule ∆l = ±1. Also M jj − M ii is zero for the donor impurity at the center of QD. The resonance conditions are ω 1 = E 2 ij + ( Γ ij ) 2 and ω 3 = 1 3 E ij + 4 E 2 ij + 3 ( Γ ij ) 2 for linear and third-order nonlinear OACs, respectively. We define the following two-functions which are associated to the resonant conditions and C. Electromagnetically induced transparency In Fig. 4 we present the three energy level system in the Λ-configuration to study the impurity related EIT in KQD. The interlevel transition between the |a and |c (|b and |c ) states is dipolarly allowed through the probe (control) field while the transition between the |a and |b states is dipolarly prohibited (M ac = 0, M bc = 0, and M ab = 0). The parameters ω p and E p (ω c and E c ) are the frequency and strength of the probe (control) field. The probe and control lasers are detuned from the resonance frequencies by ∆ p = ω ca − ω p and ∆ c = ω cb − ω c . By using the rotating-wave approximation, the OAC (α(ω)) and refraction index (n T , RI) for a three-level system in the Λ-configuration interacting with two applied laser pulses (probe and control) are, respectively and where and Here Ω c = M ca · E p /(2 ) is the half of the Rabi frequency and γ ij is the decay rate between the i and j states composed from the radiative decay rate due to the spontaneous emission and the dephasing rate. For details, see Ref. [70].

III. RESULTS AND DISCUSSION
In this section, the effects of potential parameters on the binding energies, OACs, and EIT in the  The Fig. 5 shows the changes of energy without impurity (Z = 0) and with impurity (Z = 1) and binding energy as a function of the A-parameter for two different k-values in a KQD. Since the electron confinement increases due to the transformation of the structure from core/shell KQD towards a single KQD with increasing the A-parameter, the uncorrelated and correlated energies are increasing functions of the A-parameter. Furthermore, all energies mentioned above decreases with the k-parameter since the electron confinement decreases. Furthermore, with the addition of impurity to the structure, the negative Coulomb potential between the electron and the impurity causes a reduction in energy. Finally, it is seen that the impurity-free and impuritycontaining energies increase gradually towards the higher states (E 100 < E 110 < E 120 ). We can see that the effect of impurity is systematically reduced as one passes from the ground state to the first excited state and subsequently to the second excited state. It is clear that the excited states are more extended in space and thus the interaction with the impurity becomes weaker. This explains the reason why for the binding energy of the three states shown, E B 100 > E B 110 > E B 120 is satisfied. Note the trend exhibited by the binding energy for the three states shown. They are increasing functions of the Aparameter. However, it can be seen that for large values of A-parameter there is a tendency to reach constant values. This is explained by the fact that for a sufficiently large A value, the system has the tendency to reach the limit of a strongly confined zero-dimensional hydrogen system. The binding energy grows as A-increases due to the fact that the infinite potential barrier, located on the outside of the KQD, approaches towards the central region of the structure, pushing the electronic wavefunction towards the impurity position. With this, there is a reduction in the average electron-impurity distance, which leads to a gradual increase in the Coulomb interaction and consequently to an increase in the binding energy. The energy transitions (a1, a2), the reduced dipole matrix elements (b1, b2), the Ω-function (c1, c2, see Eq. (14)), and the Γ-function (d1, d2, see Eq. (15)) in the GaAs/Al 0.3 Ga 0.7 As Konwent-like spherical QD as a function of the A-parameter are depicted in Fig. 6. The results are for k = 2.5 nm (upper row) and k = 5.0 nm (lower row) with on-center shallow-donor. Solid lines are considering transitions from the ground state to the first excited state (ψ 100 → ψ 110 ) whereas dashed lines are considering transitions from the first excited state to the second excited state (ψ 110 → ψ 120 ). From Figs. 4(a1, a2), it is observed that all the transition energies are increasing functions of the A-parameter. This is in perfect agreement with the fact that as the confinement of the carriers increases due to the effective reduction in the KQD-size, the separation between the energy levels also increases, as can be seen in Fig. 6, and hence the consequent increase of transition energies. It is clear that ∆E 12 > ∆E 0 12 because the presence of the impurity affects mainly the ground state with a reduction in energy. Comparing the first and second excited states, the impurity produces a more noticeable effect on the first one, hence the reason why ∆E 23 > ∆E 0 23 is explained. Comparing Figs. 6(a1) and 6(a2), it can be seen in the first case that the calculated variation of the A-parameter generates a change in the transition energy of 0.5 eV while in the second one the variation is of the order of 0.2 eV. The larger the structure, the closer the energy states and the smaller the transition energy. Taking into account that the increase in the A-parameter translates into an effective decrease in the KQD-size, this explains the reason why in panels 6(b1, b2) the reduced dipole matrix element is always a decreasing function of the A-parameter. The larger the original structure (which corresponds to the one whose k-parameter is the largest), the more noticeable is the variation of M 0 ij and M ij . While in Fig. 6(b1) the variation exhibited by the matrix elements is of the order of 2.5 nm, in Fig. 6(b2) the changes exceed 5.0 nm. In the calculation of the Ω-and Γ-functions, according to Eqs. (14) and (15), it is evident that these are dominated by the behavior of the dipole matrix elements. Then, Fig.  6(d1) is in line with the decreasing behavior of the curves in Fig. 6(b1). The same can be seen in Figs. 6(c2) and 6(d2) with respect to that shown in Fig. 6(b2). However, the behavior of Fig. 6(c1) shows that the Ω-function comes from a competitive effect between the transition energy, which is always increasing with the A-parameter, and the dipole matrix elements, that are always a decreasing functions of the A-parameter. However, note that in Fig. 6(c1)  In Figs. 7 the total OAC is presented as a function of the incident photon energy in the presence of the impurity atom for k = 2.5 nm (a) and k = 5.0 nm (b) and several A-values. Here, since the third order nonlinear OAC is very small, the total OAC gives an equivalent result with the linear one, so we prefer to show only the total OAC. The peak positions of the total OACs for the transitions corresponding to the case with impurities shift to the blue with the increase of A-parameter. Since the energy difference between the levels of interest increases with the A-parameter, the peak positions of the total OAC shift to blue. We also see that while the peak amplitude of the total OAC is almost constant for single KQD (A > 1), it decreases for the core/shell KQD (A = 0.1). This is a result related to the Ω-factor given in Figs. 6(c1, c2), where the resonant peak of linear OAC is directly proportional to the Ω-factor (see Eqs. (10) and (14)). Although the wavefunctions overlap better in the core/shell structure (see Figs. 2 and 3), the Ω-factor decreases due to the decrease in the transition energy. According with the results in Fig. 6, the presence and absence of the impurity does not create a noticeable change in absorption amplitudes. However, the presence of impurity causes an increase in transition energies as seen in Figs. 6(a1, a2), and thus the peak positions of the total OAC shift to the blue. Finally, it should be noted that the peak positions of the total OAC associated with the ψ 110 → ψ 120 transition always occur at higher energies than those associated with the ψ 100 → ψ 110 transition. Because, as seen in Figs. 6(a1, a2), ∆E 23 transition energy is greater than ∆E 12 transition energy for all parameter values. At the same time, the peak amplitudes of the total OAC corresponding to the ψ 110 → ψ 120 transitions are always greater than those of the ψ 100 → ψ 110 transitions. As a result of the better overlap of the wave function of the ψ 110 state with the wave function of the ψ 120 state, the Ω-factor is larger for the ψ 110 → ψ 120 transition (see Figs. 6(c1, c2)).
Next, we turn our attention to the impurity-related EIT in a GaAs/Al 0.3 Ga 0.7 As KQD. We have chosen a QD with k = 5.0 nm and considering on-center impurity. Our results are presented as a function of the A-parameter and probe field energy. In Fig. 8(a), the energies of the six lowest states are presented as a function of the A-parameter. In the inset of Fig. 8(a), we report the transition energy between the ground state and the 5th excited state (E 61 ). In Fig.  8(b) our results for the reduced dipole matrix element are depicted for the 1 → 6, 3 → 6, and 1 → 6 transitions considering z-polarized probe and control incident fields. Consistent with what has already been explained above, the increasing character of the energies for the all states shown in Fig. 8(a) corresponds to an increase in the confinement of the charge carriers as the effective size of the heterostructure decreases due to the increase of the A-parameter. In what follows, for convenience, the wavefunctions and energies of the confined states will be identified simply as ψ 1 , ψ 2 , ψ 3 , ... and E 1 , E 2 , E 3 , .... The higher the energy of a state, the more extended in space is its wavefunction, which has the consequence that the growth rate with the A-parameter is higher for the ψ 6 state than for the ψ 1 state. This has the final effect that the transition energy E 61 grows linearly with the A-parameter, as shown by the inset in Fig. 8(a). The reduction of the spatial extent of the structure as the A-parameter increases explains why the reduced dipole matrix elements in Fig. 8(b) decrease as the A-parameter increases. Again, the more extended the wave functions are in space, the more sensitive is the variation of the reduced dipole matrix element to changes in the A-value, a situation that is evident when comparing M 36 and M 16 . It is clear that M 13 = 0 because the wavefunctions of the ψ 1 and ψ 3 states have the same symmetry.
In Figs. 9 (a) and (b), our results for the on-center shallow-donor impurity related total OAC and RI in the KQD (with k = 5.0 nm) are presented as a function of the incident probe photon energy for several values of the A-parameter taking into account the ψ 1 , ψ 3 , and ψ 6 states in the three-levels Λ configuration. Dashed lines are for Ω c = 0, in which case the OAC presents only one resonant structure located at E p ∼ E 61 = E 6 − E 1 (where E p is the incident probe photon energy). The position of the resonant peak shows a blueshift as the A-strength increases in complete agreement with the dependence shown with said parameter on the transition energy reported in the inset of Fig. 8(a). The systematic fall of M 16 in Fig. 8(b), explains the reason why there is an attenuation of the magnitude of the resonant peaks in Fig. 9(a) as A-increases. Consistent with only a resonant structure for the OAC when Ω c = 0, the results presented in Fig. 9(b) for the RI show that it becomes null at E p = E 61 and that concerning this energy value, the RI shows two resonant structures that have inversion symmetry. For Ω c = 0 and taking a constant value of the A-parameter, the magnitudes of the resonant structures in the OAC and RI are controlled by the γ 61parameter. When Ω c = 0, the control field is activated, and at that moment, the EIT effect appears. Precisely at the same value where the OAC presented the maximum when Ω c = 0, in the case of a finite Ω c value, the OAC is now suppressed, and it is replaced by two resonant structures symmetrically located concerning zero absorption and whose separation is controlled by the Ω c magnitude. The magnitude of these structures is not the same, the resonant peak located at higher energy is higher. This is associated with the linear term with the probe photon energy that controls the OAC. When the EIT is activated, two resonant structures appear in the OAC, in the same position as these, the RI replicates the results presented with Ω c = 0. In each resonant peak of the OAC, the RI becomes null with two structures that satisfy the inversion symmetry. It is important to note that for the case Ω c = 0, the magnitude of the resonant structures in the OAC and RI is controlled simultaneously by the γ 61 and γ 31 parameters, with the dominant effect associated with γ 31 . For a high enough value of γ 31 , the RI can go from four to just two resonant structures.

IV. CONCLUSIONS
In this study, the electronic and optical properties of GaAs/Al 0.3 Ga 0.7 As QD with Konwent potential, which transform into single or core/shell QDs depending on the changes in the structure parameters are investigated. Our results include: i ) Uncorrelated (Z = 0) and correlated (Z = 1) energies of the electron as the functions of the shape (A-parameter) and size (k-parameter) of KQDs, ii ) the binding energies for shallow donor impurity located on the center of the QDs, iii ) the total OACs for the allowed transitions between the lowest three confined states, and iv ) the impurity-related EIT between the lowest six confined states, respectively. Due to the strong quantum confinement caused by the change in the structure parameter in the potential, that is, the changes in size and geometric shape, it is seen that the QDs under investigation have a significant effect on the energy and optical spectrum. Moreover, it is seen that with selected realistic confinement potential and a special selection of the structure parameters, the optical response of the system can be tailored in a controllable manner. In this context, the results obtained indicate that quantum dots are highly important structures for optoelectronic applications.

Author contributions statement
The contributions of the authors are as follows: E. B. Al: was responsible for numerical calculations, figures and writing of the manuscript. E. Kasapoglu: proposed the problem and responsible for writing and editing the comments. H. Sari: was responsible for the verification of the solution method. I. Sökmen: proposed the solution method. C. A. Duque: was responsible for writing and editing the comments.
Competing interests statement: The authors do not have any financial and non-financial competing interests statement.
Data availability statement: All the files with tables, figures, and codes are available. The corresponding author will provide all the files in case they are requested.
Consent to Participate (Ethics): All co-authors participated actively in this research and agree with the final version of the manuscript.
Consent to Publish (Ethics): All co-authors agree with the final version of the manuscript and give their consent for it to be published, accepting all the ethical standards of the journal. The co-authors certify that this manuscript has not been published elsewhere and that it is not under consideration for publication in another journal. All relevant information has been placed in the manuscript in such a way that we guarantee the reproducibility of our research by other researchers.