Abstract

We investigate the profiles of off-diagonal components of frequency-dependent linear (αxy and αyx), first nonlinear (βxyy and βyxx), and second nonlinear (γxxyy and γyyxx) polarizabilities of repulsive impurity doped quantum dots. The dopant impurity potential is modeled by a Gaussian function. The study puts emphasis on investigating the role of damping on the said polarizability components. From this perspective the dopant is considered to be migrating under damped condition which is intrinsically linear. The frequency-dependent polarizability components are then computed by exposing the doped system to an oscillating external electric field. The damping strength and the oscillation frequency design the polarizability components delicately. The delicacy becomes more pronounced for the off-diagonal β components in comparison with α and γ analogs. A variation of damping strength, however, reduces the additional delicacy of off-diagonal β components to a noticeable extent. The present study also highlights occasional deviations of profiles of the off-diagonal components from their diagonal analogs.

Keywords:quantum dot, impurity, frequency-dependent polarizability, damping, dopant propagation, off-diagonal components

Author and Article Information

1 Department of Chemistry, Hetampur Raj High School, West Bengal, India.

2 Department of Chemistry, Bishnupur Ramananda College, West Bengal, India.

3 Department of Chemistry, Physical Chemistry Section, Visva Bharati University, West Bengal, India.

Recieved: Nov 28 2014 Accepted: Dec 13 2014 Published: Jan 29 2015

Citation: Pal S, Saha S, Ghosh M (2015) Exploring off-diagonal frequency-dependent linear and nonlinear polarizabilities of quantum dot induced by damped drift of impurity. Science Postprint 1(2): e00043. doi:10.14340/spp.2015.01A0004.

Copyright: ©2014 The Authors. Science Postprint published by General Healthcare Inc. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs 2.1 Japan (CC BY-NC-ND 2.1 JP) License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Funding: No significant financial disclosure.

Competing interest: No competing interests.

Corresponding author: Manas Ghosh

Address: Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal, India.

E-mail: pcmg77@rediffmail.com

Introduction

The low-dimensional quantum systems, beyond any doubt, display more promising nonlinear optical effects than the bulk materials. Naturally, they have found widespread applications in high-speed electro-optical modulators, far infrared photodetectors, semiconductor optical amplifiers and so on. The optical effects also highlight the physics underlying energy spectrum, the Fermi surface of electrons, and the value of electronic effective mass. Among the low-dimensional quantum systems, quantum dots (QDs) are now identified as prolific semiconductor optoelectronic devices. However, QDs are often contaminated with dopants during their manufacture which abruptly alters their properties. A delicate interplay between the impurity potential and the QD confinement potential happens to be the main reason behind such dramatic change. A large number of investigations on doped QD 1-5 thus indicate the increasing need of deciphering their various properties. Within the purview of optoelectronic applications, impurity driven tuning of linear and nonlinear optical properties has been found to be immensely important in photodetectors and in several high-speed electro-optical devices 6. A visibly large number of important works on both linear and nonlinear optical properties of these structures was therefore an obvious outcome 6-25.

External electric field has often been found to reveal important aspects related with confined impurities. The electric field changes the energy spectrum of the carrier and controls the performance of the optoelectronic devices. Moreover, the electric field often annihilates the symmetry of the system and facilitates emergence of nonlinear optical properties. Thus, the applied electric field deserves special attention in view of understanding the optical properties of doped QDs 26-40.

Recently we have amply discussed the importance of damping in influencing the performances of QD devices 41-43. In these investigations we have explored the exclusive role played by damped propagation of dopant on diagonal components of static 41, frequency-dependent 41, 42, and off-diagonal components of static 43 linear and nonlinear polarizabilities of doped QD. The frequency-dependent optical response properties of these systems assume importance in view of designing devices with promising technological applications. In the present study we explore how drift of dopant, exclusively under damped condition affects the off-diagonal components of frequency-dependent linear (αxy, αyx), first nonlinear (second order) (βxyy, βyxx), and the second nonlinear (third order) (γxxyy, γyyxx) polarizabilities of doped QD. The present work is entirely different from our previous work 43. In the previous work we have computed static polarizabilities in presence of a static electric field. However, the present investigation deals with frequency-dependent polarizabilities where a time-dependent external electric field is employed whose frequency controls the optical properties. Thus, the present inspection is completely different from previous one in view of the nature of electric field and also in view of observed outcomes. As before , this time we also consider inherently linear propagation of dopant from a fixed dot confinement center getting continually damped and corresponding time- dependent modulation of spatial stretch of impurity (ξ⁻¹). An external sinusoidal electric field has been applied to QD with frequency varying over a range. The damped propagation of dopant reduces the inversion symmetry of the dot leading to the emergence of non-zero β value. The present enquiry reveals delicate interplay between damping strength, the effective confinement potential, and the frequency of the external field which ultimately designs the polarizability profiles.

Methods

Our model Hamiltonian represents a 2-d quantum dot with single carrier electron laterally confined (parabolic) in the x − y plane. The confinement potential reads $V(x, y) =\frac{1}{2} m^* \omega_0^2(x^2 + y^2)$, where ω0 is the harmonic confinement frequency. The parabolic confinement potential has found extensive usage in various studies on QDs 1, 3, 5, 9, 19, 28, particularly in the study of optical properties of doped QDs by Çakir et al. 12. A perpendicular magnetic field (B ~ mT in the present work) is also present as an additional confinement. Using the effective mass approximation we can write the Hamiltonian of the system as

In the above equation m^* stands for the effective electronic mass within the lattice of the material. The value of m^* has been chosen to be 0.067m0 representing GaAs quantum dots. We have set ħ = e = m0 = a0 = 1 and perform our calculations in atomic unit. In Landau gauge [A = (By, 0, 0)] (A being the vector potential), the Hamiltonian transforms to

$\omega_c=\frac{eB}{m^*c}$ being the cyclotron frequency. Ω² = ω0² + ωc² can be viewed as the effective frequency in the y-direction.

We now introduce impurity (dopant) to QD and the dopant is represented by a Gaussian potential 44-46. To be specific, in the present case we write the impurity potential as $V_{imp} = V_0 \; e^{-\xi \left[(x-x_0)^2+(y-y_0)^2 \right]}$. Choice of positive values for ξ and V0 gives rise to repulsive impurity. Among various parameters of impurity potential (x0, y0) denotes the dopant coordinate, V0 is a measure of strength of impurity potential, and ξ^-1 determines the spatial stretch of impurity potential. This needs to be mentioned that here the Gaussian potential is used to model the existence of impurity in 2-d QD. However, in the above references 44-46 the QDs are 3-d materials. In real QDs the electrons are confined in 3-dimensions i.e. the carriers are dynamically confined to zero dimensions. The confinement length scales R¹, R², and R³ can be different in three spatial directions, but typically R³ $\ll$ R¹ $\simeq$ R² $\simeq$ 100 nm. In models of such dots R³ is often taken to be strictly zero and the confinement in the other two directions is described by a potential V with V (x) → ∞ for |x| → ∞, x = (x¹, x²) $\in$R². In agreement with above discussion, Xi has beautifully mentioned that QDs are created mainly through producing a lateral confinement restricting the motion of the electrons, which are initially confined in a very narrow quantum well (QW), and they usually have the shape of a flat disk, with transverse dimensions considerably exceeding their thickness. The energy of single-electron excitations across the disk surpasses other characteristic energies of the system and the confined electrons can be regarded as 2-d 16. From this perspective the present impurity potential can be used as a realistic and at the same time computationally convenient approximation, assuming that the z-extension could be effectively considered zero. Recently Khordad and his coworkers introduced a new type of confinement potential for spherical QD’s called Modified Gaussian Potential, MGP 47, 48. The Hamiltonian of the doped system reads

We have employed a variational recipe to solve the time-independent Schrödinger equation and the trial function ψ(x, y) has been constructed as a superposition of the product of harmonic oscillator eigenfunctions 41-43 φn(px) and φm(qy) respectively, as

where Cn, m are the variational parameters and $p = \sqrt{\frac{m^* \omega_0}{\hbar}}$ and $q = \sqrt{\frac{m^* \Omega}{\hbar}}$. The general expressions for the matrix elements of H0’ and Vimp in the chosen basis have been derived 41-43. In the linear variational calculation, requisite number of basis functions have been exploited after performing the convergence test. And H0 is diagonalized in the direct product basis of harmonic oscillator eigenfunctions.

The dopant migration can be viewed in terms of dopant coordinates (x0(t), y0(t)). The correspondingtime-dependence of spatial stretch of dopant potential is given by $\xi(t) = \xi_0 \exp{\left \{- \eta \sqrt{x_0^2(t)+y_0^2(t)}\right\}}$, where η is a very small parameter and ξ0 is the initial value of ξ 41-43. In the present calculation we have used ξ0 = 0.001 a.u. and η = 3.1 × 10^-2 a.u., respectively. Now the time-dependent Hamiltonian reads

where

In the present work the inherent time-dependence of dopant propagation has been considered to be linear so that x0(t) = x0 + at, y0(t) = y0 + bt. The effect of damping has been modeled by introducing an exponentially decaying term phenomenologically through the parameter ξ (a measure of damping strength) giving rise to x0(t) = (x0 + at)e^-ζt and y0(t) = (y0 + bt) e^-ζt. This represents a faster fall of dopant propagation with increase in the damping strength incomparison with the undamped motion. The matrix element involving any two arbitrary eigenstates p and q of H0 due to V1(t) has been derived 41-43.

The external electric field V2(t) is now switched on with

where εx and εy are the field intensities along x and y directions and ν being the oscillation frequency. We have set ν to be equal to the separation between the ground and first excited states of the unperturbed Hamiltonian. Now the time-dependent Hamiltonian reads

The matrix elements due to V2(t) can be determined 41, 42.

The evolving wave function can now be described by a superposition of the eigenstates of H0, i.e.

and the time-dependent Schrödinger equation (TDSE) containing the evolving wave function has now been solved numerically by 6-th order Runge-Kutta-Fehlberg method with a time step size Δt = 0.01 a.u. and numerical stability of the integrator has been verified. The time-dependent superposition coefficients [aq(t)] has been used to calculate the time-average energy of the dot ‹E› 41, 42. We have determined the energy eigenvalues for various combinations of εx and εy and used them to compute some of the off-diagonal components of linear and nonlinear polarizabilities by the following relations obtained by numerical differentiation. For linear polarizability:

and a similar expression for computing αyx component.

The off-diagonal components of first nonlinear polarizability (second order/quadratic hyperpolarizability) are calculated from following expressions.

and a similar expression for computing βyxx component.

The off-diagonal components of second nonlinear polarizability (third order/cubic hyperpolarizability) are given by

and a similar expression for computing γyyxx component.

Results and Discussion

Role of oscillation frequency (ν):

We commence our discussion with the plot of αxy(ν) as a function of the frequency (ν) of oscillating field which was varied over a range on either sides of the ΔE01 energy gap of the unperturbed system (Figure 1) at ζ = 1.0 × 10^-5. αxy(ν) decreases monotonically with increase in ν. However, the diagonal components previously exhibited entirely reverse behavior 41. It thus appears that the energy input from the external field promotes the diagonal components of linear polarizability more and more with increase in ν at the cost of the off-diagonal ones. The αyx(ν) component exhibits nearly similar behavior.

Figure 1 Plot ofαxy component as a function of ν at ζ = 1.0 × 10^-5

Figures 2a and 2b reveal the pattern of variation of βxyy and βyxx as a function of ν at three different values of ζ viz. low (1.0 × 10^-7 a.u.), medium (1.0 × 10^-5 a.u.), and high (1.0 × 10^-3 a.u.), respectively. The βxyy component exhibits prominent maxima around ν ~ 4.0 × 10^-6 a.u. at all damping strengths which is very close to ΔE01 energy gap of the unperturbed system. The βyxx component, on the other hand, falls monotonically with increase in ν. The observation thus clearly reflects the inherent non-equivalence of two β components unlike α and γ components. While the near resonance condition causes maximum enhancement of dispersive and asymmetric nature of system toward βxyy component, the same condition persistently subsides the other β component.

Figure 2 Plot of off-diagonal β components as a function of ν at three different ζ values for (a) βxyy and (b) βyxx

Figure 3 depicts variation of γxxyy component with ν at three previous ζ values. At all ζ values the component exhibits saturation up to ν ~ 3.7 × 10^-6 a.u. The component declines sharply with further increase in ν till ν ~ 4.7 × 10^-6 a.u. beyond which the saturation reappears. The observation indicates not-so-straightforward role played by ν toward the off-diagonal component of third nonlinear polarizability. The low (ν ≤ 3.7 × 10^-6 a.u.) and high frequency (ν ≥ 4.7 × 10^-6 a.u.) regimes cause sort of balance between the oscillatory field and damping leading to saturation of the component. However, within a moderate frequency regime (3.7 × 10^-6 a.u. ≤ ν ≤ 4.7 × 10^-6 a.u.) damping completely offsets the influence of external field forcing sharp fall of γxxyy. Nearly similar behavior has also beenobserved for γyyxx component (figures not shown).

Figure 3 Plot of γxxyy component as a function of ν at three different ζ values

Role of damping strength (ζ):

We now focus on the response of off-diagonal linear and nonlinear polarizability components to a variation of damping strength. Figure 4 displays the variation of αxy with ζ as the oscillation frequency of external field has been kept at 5.0 × 10^-6 a.u. In the underdamped regime αxy falls severely with ζ until ζ ~ 4.7 × 10^-4. Such a drastic fall can be attributed to an enhanced confinement with increase in ζ which decreases the dispersive nature of the system thereby reducing αxy. Further increase in ζ over a wide range stabilizes αxy to a noticeable extent. Such increase in damping strength compels the dopant to nest very close to the dot confinement origin. The close residence invites strong dot-impurity repulsive interaction which may in turn promote the dispersive character of the system. The observed stability of αxy over a wide range of ζ indicates a balanced condition between these diverse influences. The observations being quite similar for αyx components we refrain from showing the figure.

Figure 4 Plot of αxy component as a function of ζ at ν = 5.0 × 10^-6 a.u.

Figures 5a and 5b represent the variation of βxyy and βyxx components as a function of ζ, respectively. A steady increase in the damping strength has been found to enhance both βxyy and βyxx components which culminate in saturation in the moderately damped and over damped domains. In both the cases the oscillation frequency has been kept fixed at ν = 5.0 × 10^-6 a.u. The observed increase in the magnitudes of both the components in the underdamped regime reflects dominance of dot-impurity repulsive interaction over damping. However, the observed saturation over a stretched variation of damping strength again indicates kind of balance between various factors that control the dispersive and asymmetric nature of the system as pointed earlier. It appears interesting to note that the basic non-equivalence in the two first nonlinear off-diagonal components is completely shrouded when damping strength is altered over a range. However, the said non-equivalence becomes quite perspicuous when oscillation frequency is varied (cf. figures 2a and 2b).

Figure 5 Plot of off-diagonal β components with variation of ζ at ν = 5.0 × 10^-6 a.u., for (a) βxyy and (b) βyxx

Figure 6 delineates the similar profile for γxxyy component at ν = 5.0 × 10^-6 a.u. The profile closely resembles the corresponding αxy plot (Figure 4) with greater smoothness. We can therefore infer that the damping strength behaves equivalently to αxy and γxxyy components, though to different extents. The γyyxx plot looks similar and hence not given for the brevity of the paper.

Figure 6 Plot of γxxyy component as a function of ζ at ν = 5.0 × 10^-6 a.u.

Conclusions

The frequency-dependent off-diagonal components of linear, first nonlinear, and second nonlinear polarizabilities of impurity doped quantum dots have been investigated with sincere emphasis on their dependence on damping strength. The off-diagonal components indeed have been found to be dependent on the oscillation frequency of external field and the damping strength. The off-diagonal β components, being mutually non-equivalent, generate more delicate behavior when oscillation frequency is varied in comparison to α and γ counterparts. However, a variation of damping strength completely screens the influence of above mentioned non-equivalence between the off-diagonal β components. The overall behavior of off-diagonal components exhibits noticeable departure from their diagonal analogs on several occasions 41. The results are thus quite interesting and could bear significance in the relevant field of research.

Author Contributions

All authors equally contributed the work.

References

1. Gülveren B, Atav Ü, Şahin M, Tomak M (2005) A parabolic quantum dot with N electrons and an impurity. Physica E 30: pp. 143–149. doi:10.1016/j.physe.2005.08.007.
2. Tąs H, Şahin M (2012) The electronic properties of core/shell/well/shell spherical quantum dot with and without a hydrogenic impurity. J. Appl. Phys. 111: 083702. doi:10.1063/1.3702874.
3. Sadeghi E, Avazpour A (2011) Binding energy of an off-center donor impurity in ellipsoidal quantum dot with parabolic confinement potential. Physica B 406(2): pp.241–244. doi:10.1016/j.physb.2010.10.051.
4. Barati M, Vahdani MRK, Rezaei G (2007) Lower-lying states of hydrogenic impurity in lens-shaped and semi-lens-shaped quantum dots. J. Phys.: Condens. Matter 19(13): 136208. doi:10.1088/0953-8984/19/13/136208.
5. Yakar Y, Çakír B, Özmen A (2013) Off-center hydrogenic impurity in spherical quantum dot with parabolic potential. Superlattice. Microst. 60: pp. 389–397. doi:10.1016/j.spmi.2013.05.015.
6. Vahdani MRK, Rezaei G (2009) Linear and nonlinear optical properties of a hydrogenic donor in lens-shaped quantum dots. Phys. Lett. A 373(34): pp.3079–3084. doi:10.1016/j.physleta.2009.06.042.
7. Kumar KM, Peter AJ, Lee CW (2012) Optical properties of a hydrogenic impurity in a confined Zn1-xCdxSe/ZnSe quantum dot. Superlattice.Microst. 51(1): pp. 184–193.doi:10.1016/j.spmi.2011.11.012.
8. Karabulut I, Baskoutas S (2008) Linear and nonlinear optical absorption coefficients and refractive index changes in spherical quantum dots: Effects of impurities, electric field, size, and optical intensity. J.Appl. Phys. 103: 073512. doi:10.1063/1.2904860.
9. Yakar Y, Çakir B, Özmen A (2010) Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential. Optics Commun. 283(9): pp. 1795–1800. doi:10.1016/j.optcom.2009.12.027.
10. Chen T, Xie W, Liang S (2013) Optical and electronic properties of a two-dimensional quantum dot with an impurity J. Lumin. 139: pp. 64–68. doi:10.1016/j.jlumin.2013.02.030.
11. Şahin M (2009) Third-order nonlinear optical properties of a one- and two-electron spherical quantum dot with and without a hydrogenic impurity. J. Appl. Phys.: p. 063710. doi:10.1063/1.3225100.
12. Çakir BY, Yakar YA, Özmen A, Özgür Sezer M, Şahin M (2010) Linear and nonlinear optical absorption coefficients and binding energy of aspherical quantum dot. Superlattice. Microst. 47(4): pp. 556–566. doi:10.1016/j.spmi.2009.12.002.
13. Xie W (2010) Impurity effects on optical property of a spherical quantum dot in the presence of an electric field. Physica B 405(16): pp. 3436–3440. doi:10.1016/j.physb.2010.05.019.
14. Xie W (2010) Optical properties an off-center hydrogenic impurity in a spherical quantum dot with Gaussian potential. Superlattice. Microst. 48(2): pp. 239–247. doi:10.1016/j.spmi.2010.04.015.
15. Xie W (2013) Third-order nonlinear optical susceptibility of a donor in elliptical quantum dots. Superlattice. Microst. 53: pp. 49–54. doi:10.1016/j.spmi.2012.09.009.
16. Xie W (2008) Nonlinear optical properties of a hydrogenic donor quantum dot. Phys. Lett. A 372(33): pp.5498–5500. doi:10.1016/j.physleta.2008.06.059.
17. Baskoutas S, Paspalakis E, Terzis AF (2006) Effects of excitons in nonlinear optical rectification in semi parabolic quantum dots. Phys. Rev. B 74: 153306. doi:10.1103/PhysRevB.74.153306.
18. Zeng Z, Garoufalis CS, Terzis AF, Baskoutas S (2013) Linear and nonlinear optical properties of ZnS/ZnO core shell quantum dots: Effect of shell thickness, impurity, and dielectric environment. J. Appl.Phys. 114: 023510. doi.org/10.1063/1.4813094.
19. Peter AJ (2006) Polarizabilities of shallow donors in spherical quantum dots with parabolic confinement. Phys. Lett. A 355(1): pp. 59–62. doi:10.1016/j.physleta.2006.01.107.
20. Karabulut I, Baskoutas S (2009) Second and third harmonic generation susceptibilities of spherical quantum dots: Effects of impurities, electric field and size. J. Comput. Theor. Nanosci. 6(1): pp. 153–156. doi:10.1166/jctn.2009.1020.
21. Rezaei G, Vahdani MRK, Vaseghi B (2011) Nonlinear optical properties of a hydrogenic impurity in an ellipsoidal finite potential quantum dot. Current Appl.Phys. 11(2): pp. 176–181. doi:10.1016/j.cap.2010.07.002.
22. Yilmaz S, Şahin M (2010) Third-order nonlinear absorption spectra of an impurity in a sphericalquantum dot with different confining potential. Phys.Status Solidi. B 247(2): pp. 371–374. doi:10.1002/pssb.200945491.
23. Tiutiunnyk A, Tulupenko V, Mora-Ramos ME, Kasapoglu E, Ungan F, Sari H et al. (2014) Electron-related optical responses in triangular quantum dots. Physica E 60: pp. 127–132. doi:10.1016/j.physe.2014.02.017.
24. Chen T, Xie W, Liang S (2012) Nonlinear optical properties in a quantum well with the hyperbolic confinement potential. Physica B 407(2): pp. 263–267. doi:10.1016/j.physb.2011.10.044.
25. Xie W, Liang S (2011) Optical properties of a donor impurity in a two-dimensional quantum pseudo dot. Physica B 406(24): pp. 4657–4660. doi:10.1016/j.physb.2011.09.053.
26. Baskoutas S, Paspalakis E, Terzis AF (2007) Electronic structure and nonlinear optical rectification in a quantum dot: effects of impurities and external electric field. J. Phys: Cond. Mat. 19(39): 395024. doi:10.1088/0953-8984/19/39/395024.
27. Narayanan M, Peter AJ (2012) Electric field induced exciton binding energy and its non-linear optical properties in a narrow InSb/InGaxSb1-x quantum dot. Superlattice. Microst. 51(4): pp.486–496. doi:10.1016/j.spmi.2012.01.012.
28. Murillo G, Porras-Montenegro N (2000) Effects of an electric field on the binding energy of a donor impurity in a spherical GaAs(Ga,Al)As quantum dot with parabolic confinement. Phys. Status Solidi B 220(1): pp.187–190. doi:10.1002/1521-3951(200007)220:1<187::AID-PSSB187>3.0.CO;2-D.
29. Sadeghi E (2011) Electric field and impurity effects on optical property of a three-dimensional quantum dot: A combinational potential scheme. Superlattice.Microst. 50(4): pp. 331–339. doi:10.1016/j.spmi.2011.07.011.
30. Xie W, Xie Q (2009) Electric field effects of hydrogenic impurity states in a disc-like quantum dot.Physica B 404(12–13): pp. 1625–1628. doi:10.1016/j.physb.2009.01.037.
31. Duque CM, Barseghyan MG, Duque CA (2010) Donor impurity in vertically-coupled quantum dots under hydrostatic pressure and applied electric field. Eur. Phys.J. B 73(3): pp. 309–319. doi:10.1140/epjb/e2009-00433-7.
32. Duque CM, Barseghyan MG, Duque CA (2009) Hydrogenic impurity binding energy in vertically coupled GaAs/Ga1-xAlxAs quantum-dots under hydrostatic pressure and applied electric field. Physica B 404(23–24): pp.5177–5180. doi:10.1016/j.physb.2009.08.292.
33. Kasapoglu E, Ungan F, Sari H, Sökmen I, Mora-Ramos ME, Duque CA (2014) Donor impuritystates and related optical responses in triangular quantumdots under applied electric field. Superlattice. Microst. 73: pp. 171–184. doi:10.1016/j.spmi.2014.05.023.
34. Duque CM, Mora-Ramos MF, Duque CA (2011) Hydrostatic pressure and electric field effects andnonlinear optical rectification of confined excitons in spherical quantum dots. Superlattice. Microst. 49(3): pp. 264–268. doi:10.1016/j.spmi.2010.06.008.
35. López SY, Porras-Montenegro N, Duque CA (2009) Excitons in coupled quantum dots: hydrostatic pressure and electric field effects. Phys. Status Solidi B 246(3): pp. 630–634. doi:10.1002/pssb.200880536.
36. Zeng Z, Garoufalis CS, Baskoutas S (2012) Combination effects of tilted electric and magnetic fields on donor binding energy in a GaAs/AlGaAs cylindrical quantum dot. J. Phys. D: Appl. Phys. 45(23) pp.235102. doi:10.1088/0022-3727/45/23/235102.
37. Zeng Z, Garoufalis CS, Baskoutas S, Terzis AF (2012) Stark effect of donor binding energy in a self-assembled GaAs quantum dot subjected to a tilted electric field. Phys. Lett. A 376(42–43): pp. 2712–2716. doi:10.1016/j.physleta.2012.07.032.
38. Kirak M, Yilmaz S, Şahin M, Gençasian M (2011) The electric field effects on the binding energies and the nonlinear optical properties of a donor impurity in a spherical quantum dot J. Appl. Phys. 109(9): 094309. doi:10.1063/1.3582137.
39. Liang S, Xie W, Lu L (2011) Electron Raman scattering of a hydrogenic impurity in a disc-shaped quantum dot. Physica B 406(3): pp. 652–655. doi:10.1016/j.physb.2010.11.072.
40. Liang S, Xie W, Li X, Shen H (2011) Photoionization and binding energy of a donor impurity in a quantum dot under an electric field: Effects of the hydrostatic pressure and temperature. Superlattice.Microst. 49(6) pp. 623–631. doi:10.1016/j.spmi.2011.03.013.
41. Pal S, Ghosh M (2014) Influence of damped propagation of dopant on the static and frequency-dependent third nonlinear polarizability of quantum dot. Chem. Phys. 608: pp. 284–288. doi:10.1016/j.cplett.2014.06.024.
42. Pal S, Ghosh M (2014) Influence of damping on the frequency-dependent polarizabilities of doped quantum dot. Superlattice. Microst. 73: pp. 239–255. doi:10.1016/j.spmi.2014.05.033.
43. Pal S, Saha S, Ghosh M (2014) Influence ofdamped propagation of dopant on the off-diagonal componentsof static linear and nonlinear polarizabilities of quantumdots. J. Mod. Phys. Appl. 2014: 8.
44. Adamowski J, Kwaśniowski A, Szafran B (2005) LO-phonon-induced screening of electron-electroninteraction in D- centres and quantum dots. J. Phys: Cond.Mat. 17(28) pp. 4489–4500. doi:10.1088/0953-8984/17/28/008.
45. Bednarek S, Szafran B, Lis K, Adamowski J (2003) Modeling of electronic properties of electrostatic quantum dots Phys. Rev. B 68: 155333doi:10.1103/PhysRevB.68.155333.
46. Szafran B, Bednarek S, Adamowski J (2001) Parity symmetry and energy spectrum of excitons in coupled self-assembled quantum dots. Phys. Rev. B 64: 125301. doi:10.1103/PhysRevB.64.125301.
47. Gharaati A, Khordad R (2010) A new confinement potential in spherical quantum dots: ModifiedGaussian potential. Superlattice. Microst. 48(3) pp. 276–287. doi:10.1016/j.spmi.2010.06.014.
48. Khordad R (2013) Use of modified Gaussian potential to study an exciton in a spherical quantum dot. Superlattice. Microst. 54: pp. 7–15. doi:10.1016/j.spmi.2012.10.014.