Effect of magnetic field, size and donor position on the absorption coefficients related a donor within the core/shell/shell quantum dot

In this study, linear, nonlinear and total optical absorption coefficients related a single shallow donor atom confined in semiconductor core/shell/shell quantum dot heterostructure are researched in detail within the compact density matrix formalism approximation. For this purpose, firstly, the energies and the wavefunctions are computed by the diagonalization method in the effective mass approach. Moreover, the effects of size modulation, donor position and magnetic field are analyzed. The numerical results indicate that the linear and nonlinear parts of the absorption coefficients related with intersubband 1s→1p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1s\rightarrow 1p$$\end{document} and 1p→1d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1p\rightarrow 1d$$\end{document} donor transitions undergo significant changes.


Introduction
It is generally accepted that the progress of electronic and opto-electronic devices depends on understanding the basic chemical and physical properties of low-dimensional structures (LDSs). In these LDSs, the geometric confinement limits the movement of charge carriers in space and it displays large changes in electrical and optical properties due to the occurrence of discrete energy distribution. Therefore, in recent years, intensive research activities have been conducted around the world on the behavior of matter at nanoscale. Although various devices related nanoscale particles have been developed, the properties of controlled nanoscale materials such as light emitting diodes, photo detectors and quantum dot (QD) single photon source are still the biggest problem of scientists.
Due to the last developments of semiconductor nanoelectronics, it has become possible to reduce dimensionality from bulk semiconductors to zero-dimensional semiconductor nanostructures (QDs). These nanostructures are very significant because their charge carrier motion is confined in all three directions, and therefore efficient control of the physical properties of these structures becomes possible. Proper adjustment of these physical properties makes QDs advantageous due to their potential application in the development of semiconductor optoelectronic devices. Therefore, some optical properties of semiconductor QDs such as dipole transition (Hasanirokh et al. 2019), oscillator strength (Naimi 2021), photoionization cross-section (Holovatsky et al. 2020), optical absorption coefficients (OACs) ) and refractive index changes (RICs) (Sedehi et al. 2021) have attracted great interest in theoretical studies in last years.
The present information on the optical properties of nanomaterials is the product of previous studies carried out by many scientists, and the optical properties of QDs have attracted much interest theoretically in last years. Conducted studies include researches on the effects of external perturbations such as electric field, magnetic field, intense laser field, pressure and temperature (Sargsian et al. 2021;Kasapoglu et al. 2015;Bouzaiene et al. 2016). The results indicate that these perturbations allow the possibility of adjusting the value of oscillator strength and the peak resonance frequencies, thus purports an chance to change the optical response.
The electronic and optical properties in a QD are affected by the asset of impurities (Sedehi et al. 2021). Some studies (Mocatta et al. 2011;Koenraad and Flatté 2011;Moraru et al. 2011) have showed that the realization of single dopant scheme in QDs is possible. This enables a broad range of potential optoelectronic devices. It is notable that due to the presence of impurity and other external factors, the computation of energies and wavefunctions is a work that takes remarkable theoretical working. Because of the complicated practical geometries, boundary conditions and effects of the surrounding environment, it is generally not feasible to find analytical solutions using familiar standard methods. To accomplish this obstacle, researchers have used different methods such as variational approximation (Shi et al. 2021), tight-binding model (Pérez-Conde and Bhattacharjee 2005), perturbation theory (Yakar et al. 2013) and diagonalization method .
In theoretical studies on QDs, it is usual to get a spherical shape and many works have focused on the optical properties of this type of structure Naimi 2021;Máthé et al. 2021;Naifar et al. 2020;Al et al. 2020). Thanks to the rapid development of crystal magnification procedures such as molecular beam epitaxy, metal organic chemical vapor deposition and electron lithography, and the advancement of chemical production processes, various dimensional new generation quantum nanostructures have been produced and one of which is core/shell/shell QD (CSSQD) (Dorfs et al. 2010;Maximov et al. 2001). These structures consist of semiconductor materials with different band alignment. They are spatially alligned in such a way that a larger bandwidth core, acting as a substrate, is covered by smaller bandwidth spherical shells (Fig. 1). The originality of these structures is that their physical properties can be regulated in a controlled manner, resulting in changing energy levels. Therefore, it is significant to examine the electronic and optical properties of CSSQDs.
Some theoretical and experimental studies have been done on spherical core/shell/shell QD (CSSQD) structures before (Eren et al. 2021;Zeiri et al. 2019;Hayrapetyan et al. 2018;Ahmed and Christopher 2018;Hayrapetyan et al. 2015;Huang and Huang 2015;Jasmine et al. 2015;Pahomi and Cheche 2014;Poghosyan 2014;Hemdana et al. 2013;Wei et al. 2013;Kazaryan et al. 2012;Guo et al. 2011). Some of these studies include OACs: Hayrapetyan et al (Hayrapetyan et al. 2018) investigated the interband optical transitions caused by incident light polarized in the z-direction for the exciton in the CSSQD with threedimensional Winternitz-Smorodinsky confinement potential. In the absence and presence of a hydrogenic donor impurity in a CdTe∕Zn 0.2 Cd 0.8 Te∕ZnTe CSSQD, intersubband optical absorption, oscillator strength, and radiative lifetime between ground and excited states, as well as electronic properties, were investigated by Jasmine et al (Jasmine et al. 2015). Pahomi and Cheche (Pahomi and Cheche 2014) analyzed the effect of strain on the energy structure and linear optical absorption in CSSQDs.
However, there is no study in the literature on OACs caused by 1s → 1p and 1p → 1d single dopant transitions in GaAs∕Al x 1 Ga 1−x 1 As∕Al x 2 Ga 1−x 2 As CSSQD under the effect of a magnetic field. Therefore, in available theoretical study, linear, nonlinear and total OACs are investigated using density matrix formalism. The computation of the electronic state in the presence of both the donor impurity and the magnetic field is made by the diagonalization method. As a result of this research, it has been shown that the OACs of this structure are strictly dependent on external magnetic field, donor position and its geometric parameters such as core and shell thickness. It is hoped that this study will allow a lot elaborate specification of the donor states in the nanostructure researched.

Electron-impurity Hamiltonian and wave function
Since the optical behavior of nanoscale semiconductors is related to their electronic properties, the effects of varying size and external fields on the energy behavior of confined donor must first be studied.
Consider a hydrogen-like shallow donor located anywhere in the core region of a CSSQD nanocrystal with a spherically symmetrical design (Fig. 1). GaAs core material with wide band gap and radius a 1 is covered with Al x 1 Ga 1−x 1 As inner shell with a narrower band gap and shell thickness T s = a 2 − a 1 . The outer shell consists of Al x 2 Ga 1−x 2 As , which has a much narrower band gap and a thickness so large that quantum effects cannot be observed. The system is exposed to a magnetic field applied externally. In the effective mass approach, the Hamiltonian that describes the energy behavior of a single electron bound to the donor confined in such a system can be stated as where, the first term is the kinetic energy operator in the presence of the magnetic field, m * is the conduction band effective mass of the electron in the GaAs-core region (since the electron is mostly localized in the GaAs-core region, the effective mass of this region is used), e is the charge of electron and c is the light speed in free space. Also, � ⃗ A is the magnetic vector potential defined as � ⃗ The second term in the Eq. 1 represents the confinement potential and it is writen in the form where V 1,2 = 0.6(1247x 1,2 ) meV (Adachi 1985) is the conduction band offset between AlGaAs and GaAs and x 1 ( x 2 ) is the aluminum concentration in the inner (outer) shell.
The third term in the Eq. 1 shows the Coulomb potential energy between the donor impurity and the electron and it has the form where Z = 0 and Z = 1 correspond to the case without and with an impurity in the structure, respectively, is the dielectric constant, |⃗ r − ⃗ r d | is the electron-impurity distance, ⃗ r and ⃗ r d are the radial locations of the electron and the ionized impurity with respect to the center of the QD, respectively. It is assumed that the impurity is placed along the z-axis. In terms of spherical harmonics 1 | ⃗ r− ⃗ r d | term is expressed as (Yakar et al. 2013) where, and ( d and d ) are the polar angles of the electron (donor), and f (r) is At this point, to reduce numerical calculations, reduced units are used, describing as energy unit and a B = ℏ 2 m * e 2 as length unit and the Hamiltonian becomes where the dimensionless = eBa 2 B ℏc parameter defines the magnetic field strength.
To get the energies and wavefunctions, the Schrödinger equation H nlm (r, , ) = E nlm nlm (r, , ) must be solved, where E nlm is the energy eigenvalue for certain quantum numbers (n: principal quantum number, l: angular momentum quantum number, m: magnetic momentum quantum number) and nlm (r, , ) is the wavefunction corresponding to this energy. The angular momentum quantum number l = 0, 1, 2, ... is marked by the familiar impression s, p, d, ... . The diagonalization method is used to resolve the Schrödinger equation and for this purpose, the electron wavefunctions of the infinite QD are taken as the basis function (Holovatsky et al. 2017(Holovatsky et al. , 2014(Holovatsky et al. , 2018 where c nlm are the expansion coefficients (the size of the expansion is adjusted by performing a convergence test for each energy state), (0) nlm (r, , ) is the total wavefunction describing the movement of the electron without the magnetic field and the donor impurity, and the exact solutions for an electron in an infinite radial potential are  where the radial wavefunction-(0) nl (r) is given as where N is the normalization constant, k nl is the nth root of the spherical Bessel function-j l and a 3 ( a 3 >> a 2 ) is the radius of the infinite spherical QD.

Optical absorption coefficients
The optical transition from a first state to a latest state by a photon is commonly known as photon absorption. For any transition in LDSs, these absorptions are thought to be important for investigating new optical properties. In a monochromatic electromagnetic field with a frequency , the possibility of transition between states i and j is known as the oscillator strength P ij expressed by the Fermi golden rule and it is expressed in the form (El Haouari et al. 2017) where, E ij = E j − E i is the energy difference between the first and latest states, while M ij is the dipole matrix element (DME) of the transition between the i and j states. The oscillator strength is highly dependent on the overlap of wavefunctions and the energy difference between states and thus it gives an idea of the dominant color of the emitted light.
OACs can be acquired by density matrix approximation and linear and nonlinear sensitivities for a two-level atomic system are given as (Ghanbari and Khordad 2021;Hashemi et al. 2021) and respectively, where, s is the electron density in the QD, 0 is the dielectric constant of the free space, ℏ is the incident photon energy, ℏ ij is Lorentzian line width, ij = 1∕T ij is the non-diagonal damping term known as the relaxation ratio between the first and latest states and it is described as the reverse of the relaxation time ( T ij ).
Sensitivity ( ) is associated with the OAC in the form where, indicates the magnetic susceptibility and r is the dielectric permeability of the material. In this way, the linear, nonlinear and total intersubband OACs are theoretically formulated as and respectively, where I = 2 0 n r c|Ẽ| 2 is the intensity of the linearly polarized electromagnetic field, Ẽ is the amplitude of this electromagnetic field and n r is the relative refractive index of the structure. Near the resonance, the second term of the parenthesis in Eq. 15 causes little effect in nonlinear OAC due to the small value of M jj − M ii and thus it can be neglacted. In this case, the resonance condition is ℏ = √ E 2 ij + (ℏ ij ) 2 for linear OAC and ℏ = 1 3 E ij + √ 4E 2 ij + 3(ℏ ij ) 2 for nonlinear OAC. Thus, the maximum values of linear and nonlinear OACs are given in the forms ( Assuming that electromagnetic radiation is linearly polarized throughout the z-axis, the DME is defined in a single electron system by where, i and j are wavefunctions of the first and latest states, respectively. In order for the DME to be different from zero, the selection rules Δl = ±1 and Δm = 0, ±1 must be provided. In this study, the intersubband transitions between only m = 0 states in the CSSQD will be discussed.

Numerical results and discussion
In this section, the effects of magnetic field, geometric confinement and donor position change on OACs will be discussed according to the spherical CSSQD nanostructure model outlined above. The values of the material input parameters taken into account in the computation are presented as follows: = 13.18 (since the electron is mostly localized in the GaAs-core region, the dielectric constant of this region is used), m * = 0.067m 0 (where m 0 = 9.10956 × 10 −31 kg is the free electron mass), I = 400 MW∕m 2 , s = 1 × 10 23 m −3 , T ij = 0.14 ps , n r = 3.2 , a B = 10.42 nm , Ryd = 5.23 meV ).
In Fig. 2, the change of linear, nonlinear and total OACs related to 1s → 1p and 1p → 1d transitions for the central donor atom in the absence of a magnetic field are plotted as function of the incident photon energy considering various core/shell sizes. As can be seen from the Eqs. 17 and 18, OACs reach the maximum when ℏ ≈ E ij , and the peak amplitudes of linear and nonlinear OACs are proportional to Ω 1 and Ω 2 , respectively. In Fig. 2a, the transition energy (TE) decreases as a result of closing the energy difference between the 1s and 1p states due to the increase in the size of the core material, while the related wave functions more overlap. The competitive influence between these two factors induces the peak amplitude of linear OAC to be senseless to change in core size. However, the fourth power of the DME in the nonlinear OAC is more dominant and thus the peak magnitude of the nonlinear OAC is increased. In addition, it can be noticed that the nonlinear OAC has a large effect on the total OAC, thus the peak magnitude of the total OAC is decreased. It is quite clear that the increase in core size improves the peak magnitudes of both linear and nonlinear OACs associated with the 1p → 1d transition. Because the growing core size pushes down the tunneling effect and hence the electron in the 1d state is chiefly confined in the core region, the wave functions of the 1p and 1d states overlap, thus the peak magnitudes of the OACs increase. However, the peak amplitudes of OACs depend on the TE as (17) (1) well as the overlap of the wavefunctions. In contrast to the increasing overlap of wavefunctions, the TE incurs a decrease with growing core size. Here, it can be said that the DME is more dominant than the TE. Other than this, as the nonlinear OAC is affected more by the core size than the linear one, the peak magnitude of the total OAC is decreased. Also, the peak positions of the OACs show a redshift with the increase in core size. This is because the energy difference between the 1p and 1d states in the large core size is smaller than the energy difference in the small core size. All these changes correspond exactly to the values in Table 1.  In Fig. 2b, no change is observed in the peak magnitudes and positions of the OACs related to the 1s → 1p transition when the shell thickness increases. Because the change in shell thickness is not affect the probability distributions and energy levels of 1s and 1p states. As can be seen from Table 1, there is no significant change in the TE nor in the Ω 1 and Ω 2 factors. The peak amplitudes of the OACs related to the 1p → 1d transition decrease and the peak positions show a redshift with growing shell thickness. Because, although the probability density of the 1p state does not change with the growing in the shell thickness, the probability density of the 1d state is localized in a wider region within the structure, so the DME is decreased. As a combined result of both the decreasing of the DME and the redshift of the TE, the Ω 1 and Ω 2 multipliers and consequently the peak amplitudes of the OACs are decreased.
To examine the influence of the position of donor atom in the core region, OACs are presented in Fig. 3 against the incident photon energy for different positions of the impurity without a magnetic field. The peak magnitude of the linear OAC for the 1s → 1p transition is greatest when the donor atom is at the center of QD, but it remains constant outside the center. On the other hand, the peak magnitude of the nonlinear OAC is greatest when the impurity is at the core/shell boundary, but it remains constant at other positions. However, the peak locations of the OACs show a redshift as the impurity draws away from the dot center, due to the convergence of 1s and 1p energy levels. As can be understood from these changes, although the DME increases at the core/shell boundary, the peak magnitude of the linear OAC does not vary due to competition between the DME and the TE. However, the peak magnitude of the nonlinear OAC rises due to the fact that it is proportional to the fourth power of the DME. Since the peak amplitudes of the nonlinear OAC are smaller than the linear ones, the peak amplitudes of the total OAC show an equivalent variation with the linear ones except for a small decrease in amplitude. The same is true for the 1p → 1d transition. However, the peak locations of the OACs for the 1p → 1d transition is shifted first slightly blue and then red as the donor atom draws away from the QD center. This is because of the fact that the energy difference between the 1p and 1d levels before rises and after that decreases as the donor atom draws away from the QD center. While the peak magnitude of the linear OAC is not affected by the impurity location, the peak amplitude of the nonlinear OAC is increased at the core/shell boundary, so it can be said that the DME is larger at the core/shell boundary. All changes described are consistent with the values in Table 2.  Fig. 4 Linear, nonlinear and total OACs associated with 1s → 1p (red) and 1p → 1d (blue) on-center impurity transitions in the GaAs∕Al 0.1 Ga 0.9 As∕Al 0.4 Ga 0.6 As CSSQD as a function of incident photon energy in the absence (solid) and presence (dashed) of the magnetic field for several core/shell sizes To understand the influence of the magnetic field, linear, nonlinear and total contributions to the OACs as a function of the incident photon energy in the absence and presence of the magnetic field for different core/shell sizes are drawn in Fig. 4. It is supposed that the donor atom is placed at the QD center and the entire structure is under the effect of the magnetic field. Regardless of the core/shell sizes, it can be understood that the peak locations of the OACs undergo a blueshift under the influence of the magnetic field. This is because the magnetic field creates an additional spatial confinement. That is, with the effect of the magnetic field, the spatial distribution of the electron wavefunction becomes weaker and the cyclotron radius decreases. As a result, the TE rises as given in Table 3 and the OACs shift to blue. For the core/shell sizes in the left column given in Fig. 4, it should be noted that with the influence of the magnetic field, there is a significant increment in the peak amplitudes of the OACs associated with the 1p → 1d transition. This is because both the TE and the DME increase with the effect of the magnetic field. Also, since the peak magnitude of the nonlinear OAC is small compared to the linear one, the peak magnitude of the total OAC is nearly the same as the linear one. Other than this, the small increase in TE for the 1s → 1p transition is balanced by the small decrement in the DME and consequently the OAC peak amplitudes are not change. For the core/shell sizes in the right column given in Fig. 4, with the effect of the magnetic field, the peak magnitude of the linear OAC associated with the 1p → 1d transition increases, while the peak magnitude of the nonlinear OAC decreases. This case can be explained as follows: As emphasized before, linear and nonlinear OACs are related to the factors Ω 1 and Ω 2 in the Eqs. 17 and 18, respectively. These factors depend on the magnetic field as they are associated with the DME and the TE. Therefore, it can be predicted that there is a rivalry between the DME and the TE. As the TE rises with the effect of the magnetic field, the DME declines. While the TE is dominant in the amplitude change of the linear OAC, the DME is dominant in the amplitude change of the nonlinear OAC because of the fourth power of the DME. Also, numerical calculations show that the magnetic field causes a little change in the peak magnitude of nonlinear OAC compared to linear one. For this reason, the magnitude of the total OAC is dominated by the change in the linear OAC. On the other hand, for the 1s → 1p transition, the decrement in the DME is more dominant under the effect of the magnetic field and as a result, the peak magnitudes of the OACs decline. Since the decrease in the peak Table 3 In the absence and presence of the magnetic field, the TEs and the values of some factors ( Ω 1 and Ω 2 ) for different core/shell sizes. The values are for x 1 = 0.1 , x 2 = 0.4 and r d = 0 magnitude of the linear OAC is greater than the decrease in the peak magnitude of the nonlinear OAC, the peak magnitude of the total OAC is also decreased. All results are in numbness with the values in Table 3. Finally, OACs as a function of photon energy for several x 1 concentrations are introduced in Fig. 5 in the absence and presence of the magnetic field. The figure shows that increasing x 1 concentration shifts the peak locations of OACs to blue. This is because the TE increases with increasing x 1 concentration because of the quantum size effect. Besides, it has been found that the peak amplitudes of linear OACs are virtually unchanged with increasing x 1 concentration. This is because with increasing x 1 concentration, as the TE increases, the overlap of the wavefunctions of the first and latest states decreases. The peak amplitudes of the nonlinear OAC show a small decrease due to the increasing x 1 concentration. This is because the structure becomes more symmetrical as a deeper confinement potential is created with increasing x 1 concentration. On the other hand, the influence of the magnetic field is very small due to the small core/shell sizes. Note that the effect of the magnetic field is most evident for the 1p → 1d transition for concentration x 1 = 0.1 . Because the energy of the 1d state is greater than the depth of the confinement potential for the concentration x 1 = 0.1 , thus it is more influenced by the magnetic field as it is localized in a larger region. All these explanations can be understood by looking at the values in Table 4.

Conclusions
In this theoretical study, the linear, nonlinear and total OACs of a single impurity atom confined in a spherical type CSSQD are extensively investigated by considering the effects of core/shell sizes, donor position and magnetic field. In this context, energies and wavefunctions are acquired by diagonalization method. The results obtained show the following: The peak locations of OACs are shifted to red with increasing core/ shell sizes. Peak amplitudes of linear, nonlinear and total OACs related to 1s → 1p and 1p → 1d transitions give different responses to increased core/shell sizes due to the reasons explained above. The peak positions of the OACs related to the 1s → 1p transition shifts clearly red as the donor atom shifts from the center of QD to the core/shell boundary, while the peak magnitudes of the linear and nonlinear OACs give maximum values at different donor positions. The peak locations and amplitudes of the OACs related to the 1p → 1d transition give quite complex responses to the change in donor position. For the core/shell sizes examined, the magnetic field shifts the peak locations of the OACs to blue. In addition, the magnetic field mostly decreases the peak amplitudes of the OACs. In the absence and presence of the magnetic field, the increase of the x 1 concentration has no effect on the peak magnitudes of the OACs, while it shifts the peak positions clearly to blue. As a result, it is concluded that the absorption and resonance TEs of the CSSQD can be easily controlled by setting the core/shell sizes, donor position and magnetic field. Besides, it has been found that nonlinear OAC is significant and it should be considered in the study of optical properties of donor atom in the CSSQD. Furthermore, the theoretical results obtained here may contribute to experimental works and ensure a kind of approximate forming for realistic applications such as optoelectronic devices and optical communications.
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