Abstract

We make an extensive investigation of the profiles of a few diagonal and off-diagonal components of linear (αxx, αyy, αxy, and αyx), first nonlinear (βxxx, βyyy, βxyy, and βyxx), and second nonlinear (γxxxx, γyyyy, γxxyy, and γyyxx) polarizabilities of quantum dots under the influence of external pulsed field. The quantum dot is doped with repulsive Gaussian impurity. The number of pulse and the dopant location have been found to fabricate the said profiles in combination. The β components display greater delicacy in their profiles in comparison with the α and γ counterparts. The interplay between number of pulse and dopant site offers an intricate framework to attain stable, enhanced, and often maximized output of linear and nonlinear polarizabilities.

Keywords:quantum dot, impurity, polarizability, pulsed field, dopant location

Author and Article Information

1 Department of Chemistry, Bishnupur Ramananda College, India.

2 Department of Chemistry, Physical Chemistry Section, Visva Bharati University, India.

Recieved: Feb 6 2015 Accepted: Mar 19 2015 Published: Apr 1 2015

Citation: Saha S, Ghosh M (2015) Polarizabilities of impurity doped quantum dots under pulsed field. Science Postprint 1(2): e00046. doi:10.14340/spp.2015.04A0001.

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 interest.

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

Peer reviewers: Sezai ELAGÖZ¹, Reviewer A and B
1 Nanophotonic Center, Cumhuriyet Üniversity, Sivas-Turkey

Introduction

Quantum dots (QDs) exhibit rich nonlinear optical effects which are much more subtle than the bulk materials. This justifies their widespread applications in a variety of optical devices. However, QDs are often contaminated with dopants during their manufacture which noticeably alter their properties. The contamination introduces additional potential to the QD system which interplays with inherent QD confinement potential. The interplay comes out to be responsible for the dramatic change in various properties of QD. This has insisted a large number of investigations on doped QD 1-9. In view of optoelectronic applications, impurity guided modulation of linear and nonlinear optical properties is highly important in photodetectors and in several high-speed electro-optical devices 10. Naturally we find a huge collection of important works on both linear and nonlinear optical properties of these structures 10-29 .

External electric field often elucidates important aspects related with confined impurities. The electric field affects the energy spectrum of the carrier and monitors the performance of the optoelectronic devices. Moreover, the electric field often disrupts the symmetry of the system and favors emergence of nonlinear optical properties. Thus, the applied electric field deserves special attention in view of studying the optical properties of doped QDs 30-41.

Recently we have investigated the role of impurity on linear and nonlinear responses of 2-d QDs 42. In the present work we have explored some of the diagonal and off-diagonal components of linear (αxx, αyy, αxy, and αyx), second order (βxxx, βyyy, βxyy, and βyxx), and third order (γxxxx, γyyyy, γxxyy, and γyyxx) polarizabilities of quantum dots under the influence of external pulsed field. The diagonal and off-diagonal components are expected to behave differently under the aegis of pulsed field because of their varied interactions with the field. Of late Şahin made some important contribution to the third order optical property of a spherical QD and analyzed the role of impurity 16. The notable works of Karabulut and Baskoutas 24 and Yilmaz and Şahin 27 also deserve mention in related contexts which include the effects of electric field and impurity. The electromagnetic pulsed field has been applied along both x and y directions to the doped quantum dot (i.e. it is polarized in x and y directions). We have found that the number of pulses fed into the system from the external field (np) and the dopant location (r₀) play significant role in shaping the polarizability components 43. The role of dopant site has been critically explored because of its well-known influence in modulating the optical properties of doped heterostructures. In their notable works Karabulut and Baskoutas 24, and Baskoutas et al. 30 analyzed the importance of off-center impurities exploiting an accurate numerical method (PMM, potential morphing method). Very recently Khordad and Bahramiyan have made important work on how dopant position affects the optical properties of various QDs 28. The study reveals the outcome of combined influence of np and r₀ on the various polarizability components.

Methods

The Hamiltonian representing a 2-d quantum dot with single carrier electron laterally confined (parabolic) in the x-y plane and containing a Gaussian impurity is given by

\begin{eqnarray} H_0 = H_0^\prime + V_{imp}, \end{eqnarray} (1)

where H₀^’ is the Hamiltonian in absence of impurity. Under the effective mass approximation it reads

\begin{eqnarray} H_0^\prime = \frac{1}{2m^*}\left[-i\hbar\nabla +\frac{e}{c}A \right]^2 + \frac{1}{2}m^* \omega_0^2(x^2+y^2), \end{eqnarray} (2)

where the confinement potential reads V(x, y) = 1/2 (m^* ω₀²(x² + y²)) with harmonic confinement frequency ω₀ and the effective mass m^*. The value of m^* has been chosen to be 0.067m₀ resembling GaAs quantum dots. We have set ħ = e = m₀ = a₀ = 1 and perform our calculations in atomic unit. The parabolic confinement potential has been utilized in the study of optical properties of doped QDs by Çakir et al. 17, 18. Recently Khordad and his coworkers introduced a new type of confinement potential for spherical QD's called Modified Gaussian Potential, MGP 47, 48. As an additional confinement a perpendicular magnetic field (B ~ mT) is also applied. In Landau gauge [A = (By, 0, 0)] (A being the vector potential), the Hamiltonian transforms to

\begin{eqnarray} H_0^\prime = -\frac{\hbar^2}{2m^*}(\frac{\partial ^2} {\partial x^2}+ \frac{\partial^2}{\partial y^2})+ \frac{1}{2}m^* \omega_0^2x^2+\frac{1}{2}m^*(\omega_0^2+ \omega_c^2)y^2 - i\hbar\omega_cy\frac{\partial}{\partial x}, \end{eqnarray} (3)

ωc = eB/m*c being the cyclotron frequency and Ω² = ω₀² +ωc² can be treated as the effective frequency in the y-direction. Vimp being the impurity (dopant) potential (Gaussian) 44-46 and is given by

\begin{eqnarray} V_{imp} = V_0 \; e^{-\xi \left[(x-x_0)^2+(y-y_0)^2 \right]}. \end{eqnarray} (4)

Positive values of ξ and V₀ indicate a repulsive impurity. V₀, (x₀, y₀), and ξ^-1 represent the impurity potential, the dopant coordinate, and the spatial stretch of impurity, respectively.

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 φn(px) and φm(qy) respectively, as

\begin{eqnarray} \psi(x, y)=\sum_{n, m}C_{n, m}{\phi_n(p x) \phi_m(q y)}, \end{eqnarray} (5)

where C n, m are the variational parameters and p =√m^*ω₀/ħ and q =√m^*Ω₀/ħ. 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 external pulsed field can be represented by

\begin{eqnarray} \varepsilon(t) = \varepsilon(0) S(t)\sin(\nu t). \end{eqnarray} (6)

ε(t) is the time-dependent field intensity modulated by a pulse-shape function S(t) where the pulse has a peak field strength ε(0), and a fixed frequency ν. The pulsed field is applied along both x and y directions. In the present work we have invoked a sinusoidal pulse given by

\begin{eqnarray} S(t) = \sin^2\left(\frac{\pi \; t}{T_p}\right), \end{eqnarray} (7)

where Tp stands for pulse duration time. Thus Tp, or equivalently np (the number of pulses), appears to be a key control parameter. The advantage of using the pulsed field is that the shape of the pulse i.e. the function S(t) can possess several analytical forms (experimentally obtainable to some extent). Thus, we can input energy to the system in a particular pattern. Moreover, by controlling the number of pulse externally it becomes possible to regulate the amount of energy input to the system. Since, depending upon form of S(t), energy input occurs in different patterns to the system the polarizabilities are expected to be affected. The shape function S(t) can assume sinusoidal, Gaussian, triangular, and saw-tooth appearances represented by following analytical expressions 43 :

\begin{eqnarray*} S(t) = \sin^2\left(\frac{\pi \; t}{T_p}\right),\;\;\; \text{Sinusoidal} \;\; \text{pulse} \\ S(t) = \exp{\left[-k(t-T_p)^2 \right]}, \;\;\; \text{Gaussian} \;\; \text{pulse} \\ S(t) = 1-\mid 1- \frac{2t}{T_p} \mid, \;\;\; \text{Triangular} \;\; \text{pulse}, \;\; \text{and} \\ S(t) = a \left(1-\frac{t}{T_p} \right), \;\;\; \text{Saw-tooth} \;\;\text{pulse}. \end{eqnarray*}

In the present work we have employed sinusoidal pulse. Figure 1 depicts the profiles of five consecutive sinusoidal pulses as a function of time.

Figure 1 The sinusoidal pulse profile

With the application of pulsed field the time-dependent Hamiltonian becomes

\begin{eqnarray} H(t)=H_0+V_1(t), \end{eqnarray} (8)

where

\begin{eqnarray} V_1(t) = -|e|\left[\varepsilon_x(t).x + \varepsilon_y(t).y \right] \end{eqnarray} (9)

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

\begin{eqnarray} \psi(x, y, t)=\sum_q a_q(t)\psi_q. \end{eqnarray} (10)

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

\begin{eqnarray} \alpha_{xx} \epsilon_x^2 = \frac{5}{2} \langle E(0) \rangle -\frac{4}{3} \left[\langle E(\epsilon_x) \rangle + \langle E(-\epsilon_x) \rangle \right]+\frac{1}{12} \left[\langle E(2\epsilon_x) \rangle + \langle E(-2\epsilon_x) \rangle \right], \end{eqnarray} (11)

and a similar expression for αyyε y².

\begin{eqnarray} \alpha_{xy}\epsilon_x\epsilon_y & = & \frac{1}{48} \left[E(2\epsilon_x, 2\epsilon_y)-E(2\epsilon_x, -2\epsilon_y)- E(-2\epsilon_x, 2\epsilon_y)+E(-2\epsilon_x, -2\epsilon_y)\right] \nonumber \\ && - \frac{1}{3}\left[E(\epsilon_x, \epsilon_y)- E(\epsilon_x, -\epsilon_y)-E(-\epsilon_x, \epsilon_y)+ E(-\epsilon_x, -\epsilon_y)\right] \end{eqnarray} (12)

and a similar expression for computing αyx component.

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

\begin{eqnarray} \beta_{xxx} \epsilon_x^3 = \left[E(\epsilon_x,0) -E(-\epsilon_x,0)\right]-\frac{1}{2} \left[E(2\epsilon_x,0)-E(-2\epsilon_x,0)\right]. \end{eqnarray} (13)

and a similar expression is used for computing βyyy component.

\begin{eqnarray} \beta_{xyy}\; \varepsilon_x \varepsilon_y^2 & = & \frac{1}{2}\left[E(-\varepsilon_x, -\varepsilon_y)- E(\varepsilon_x,\varepsilon_y)+ E(-\varepsilon_x, \varepsilon_y)-E(\varepsilon_x, -\varepsilon_y)\right] \nonumber \\ && + \left[E(\varepsilon_x, 0)-E(-\varepsilon_x, 0) \right] \end{eqnarray} (14)

and a similar expression for computing βyxx component.

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

\begin{eqnarray} \gamma_{xxxx}\; \epsilon_x^4 = 4\left[E(\epsilon_x) +E(-\epsilon_x)\right]- \left[E(2\epsilon_x) + E(-2\epsilon_x)\right]-6 E(0), \end{eqnarray} (15)

and a similar expression for computing γyyyy component.

\begin{eqnarray} \gamma_{xxyy}\; \epsilon_x^2\epsilon_y^2 &=& 2 \left[E(\epsilon_x) +E(-\epsilon_x)\right]+2 \left[E(\epsilon_y)+ E(-\epsilon_y)\right] \nonumber \\ && -\left[E(\epsilon_x, \epsilon_y)+E(-\epsilon_x, -\epsilon_y)+E(\epsilon_x, -\epsilon_y)+E(-\epsilon_x, \epsilon_y)\right]-4E(0) \end{eqnarray} (16)

and a similar expression for computing γyyxx component.

Results and Discussion

Linear (α) and second nonlinear (γ) polarizability components:

Figure 2a depicts the profiles of αxx component as a function of np for on-center (r₀ = 0.0 a.u.), near off-center (r₀= 28.28 a.u.), and far off-center (r₀ = 70.71 a.u.) dopant locations, respectively. In these plots we have varied np from 1 to 20. This range of np is expected to reveal the influence of pulse number on various polarizability components. The plots exhibit nearly similar behavior as a function of np at all dopant locations. At all dopant locations αxx increases monotonically with np up to np ~ 8–9 after which the said component saturates with further increase in np. The behavior indicates steadily increasing input of energy to the system from the pulsed field up to np ~ 8–9. And the energy input stabilizes as soon as np exceeds the said threshold value. The dopant location, however, does not affect the general pattern of variation of linear diagonal polarizability components. The influence of dopant location is manifested only by an enhancement in the magnitude of αxx as we move from on-center to more and more off-center locations. The enhancement can be attributed to the increased dispersive character of the system which is in compliance with the gradually depleting confinement potential accompanying above movement. The αyy component displays nearly similar behavior.

Figure 2 Plots of α components vs np with (i) on-center, (ii) near off-center, and (iii) far off-center dopants. (a) for αxx and (b) for αxy

Figure 2b displays the similar plot for the off-diagonal αxy component. Firstly, we find a drastic reduction (by a factor of ~ 10³) in the value of the off-diagonal component in comparison with the diagonal counterpart. Moreover, the pattern of variation of the polarizability component shows significant departure from that of the diagonal one. Here the component falls persistently with increase in np up to np ~ 11 beyond which it saturates at all dopant locations. The behavior indicates that the energy input from the pulsed field is principally used up in enhancing the diagonal components, whereas the off-diagonal components receive only a meager amount of the input energy. This small energy input from the pulsed field even decreases with increase in np up to np ~ 11. Here also the dopant location does not affect the general pattern of variation of αxy component. However, the dependence of magnitude of αxy on dopant location is not as straightforward as that of diagonal components. In the present case αxy maximizes at near off-center location (r0 = 28.28 a.u.) unlike a steady enhancement as observed in the previous case. This indicates that the energy input from the pulsed field exhibits more sensitivity on dopant location for the off-diagonal linear polarizability components than the diagonal ones. The off-diagonal αyx component displays quite similar behavior.

Figure 3a depicts the similar profile for diagonal γxxxx component. γxxxx varies with np in a manner quite similar to that of αxx (Figure 2a). The other diagonal component γyyyy behaves similarly. Figure 3b delineates the off-diagonal γxxyy component and reveals noticeable dependence on dopant location. As before there occurs a ~ 10³ fold reduction in the magnitude of the off-diagonal component in comparison with the diagonal counterpart. The profile of γxxyy component resembles that of αxy (Figure 2b) at near (Figure 3b (ii)) and far off-center (Figure 3b(iii)) dopant locations. However, for an on-center dopant (Figure 3b(i)), γxxyy behaves similar to αxx component (Figure 2a). The observations thus again indicate greater sophistication in the pattern of variation of off-diagonal linear and second nonlinear polarizability components with np as the dopant location is changed. Whereas for an on-center dopant the pulsed field promotes emergence of both diagonal and off-diagonal components of second nonlinear polarizability side by side, they do so only for the former at the cost of the latter at other dopant sites. The other off-diagonal component γyyxx does not show any appreciable alteration in its behavior.

Figure 3 Plots of γ components vs np with (i) on-center, (ii) near off-center, and (iii) far off-center dopants. (a) for γxxxx and (b) for γxxyy

First nonlinear (β) polarizability components:

The inversion symmetry of the Hamiltonian (cf. equation 3) is preserved in the presence of an on-center dopant which annihilates the emergence of all β components. The β components begin to emerge for off-center dopants. Figure 4a and 4b represent the profiles of diagonal βxxx and βyyy components as a function of np for near off-center (r₀ = 28.28 a.u.) dopant, respectively. βxxx exhibits a minimization at np ~ 6 (Figure 4a) while βyyy exhibits maximization nearly at same value of pulse number (Figure 4b). Figure 4c and 4d display the similar profiles for the off-diagonal βxyy and βyxx components at same dopant location, respectively. βxyy component decreases steadily with np up to np ~ 15 and saturates thereafter (Figure 4c). βyxx component, on the other hand, increases monotonically up to np ~ 8 after which we observe saturation (Figure 4d). Unlike α and γ components, here we envisage a ~ 10 times enhancement in the magnitudes of off-diagonal components in comparison with their diagonal analogs. Thus, we notice that the β components behave quite differently from α and γ components as np is varied and offer greater subtlety. The pulsed field thus modulates the second-order polarizability more delicately than the linear and third-order polarizabilities.

Figure 4 Plots of β components vs np for a near off-center dopant. a) βxxx, b) βyyy, c) βxyy, and d) βyxx

In view of a comprehensive discussion we now study the profiles of β components for a far off-center dopant. Figure 5a and 5b depict the profiles of diagonal βxxx and βyyy components with variation of np for a far off-center (r₀ = 70.71 a.u.) dopant, respectively. Both βxxx and βyyy increase steadily till np ~ 9 and then saturate permanently. Figure 5c and 5d reveal the plots of corresponding off-diagonal components βxyy and βyxx, respectively. βyxx varies with np in a manner resembling the diagonal components (Figure 5d); and the saturation occurs at np ~ 10. Only the βxyy component behaves differently from the other three and undergoes maximization at np ~ 6. We could therefore infer that the extent of diversity among the various β components drops for a far off-center dopant in comparison with a near one. The said drop occurs because of lesser extent of dot-impurity interaction for a far off-center dopant.

Figure 5 Plots of β components vs np for a far off-center dopant. a) βxxx, b) βyyy, c) βxyy, and d) βyxx

Thus, it comes out that both dopant location and the number of pulses affect the polarizability profiles with significant delicacy. Particularly, the importance of dopant site in the present work conforms to other notable works which highlight the contribution of dopant location in designing various properties of mesoscopic systems. In this context the works of Sadeghi and Avazpour 4, 5, Yakar et al. 7, Xie 9, Karabulut and Baskoutas 24, Khordad and Bahramiyan 28, and Baskoutas and his co-workers 30 deserve proper mention.

Conclusions

A few diagonal and off-diagonal components of linear, first nonlinear, and second nonlinear polarizabilities of impurity doped quantum dots have been investigated under the influence of a pulsed field. The number of pulses fed into the system plays an important role in modulating the polarizability profiles. Added to this, the dopant location also noticeably fabricates the said profiles. The first nonlinear polarizability components exhibit greater delicacy in their profiles under the combined influence of number of pulses and dopant location in comparison with the linear and second nonlinear counterparts. The said combined influence offers a roadmap to achieve enhanced, maximized and stable linear and nonlinear polarizability of doped QD systems which could bear significance so far as optical properties of these systems are concerned.

It needs to be mentioned that although the present work focuses on a particular optical property i.e. polarizability, absorption spectra is also another important optical property that indeed demands sincere exploration. We therefore cited a good number of references that include the study of absorption spectra in doped QD system 10–15, 17–19, 23–25, 27–29. We hope to explore the effect of pulsed field on absorption spectra in our future communications.

Author Contributions

All authors equally contributed the work.

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