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Research Article

General Science

Material Science

We investigate the profiles of off-diagonal components of static linear (*α _{xy}*,

Keywords quantum dot, impurity, Gaussian white noise, damped propagation, polarizability

Author info

1 Department of Chemistry, Brahmankhanda Basapara 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.

ReceivedSep 2 2014 AcceptedOct 18 2014 PublishedNov 6 2014

CitationGanguly J, Saha S, Ghosh M (2014) Modulation of off-diagonal
components of static linear and nonlinear polarizabilities of doped quantum
dots by coupled influence of noise and damped propagation of impurity. *Science
Postprint* **1**(1): e00036. doi:10.14340/spp.2014.11A0001.

Copyright©2014 The Authors. Science Postprint is 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.

FundingNo significant funding.

Competing interestNo relevant competing interests were disclosed.

Corresponding authorManas Ghosh

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

E-mailpcmg77@rediffmail.com

Quantum dots (QDs) are the final destination so far as miniaturization of semiconductor devices is concerned. They exhibit more prolific nonlinear optical effects than the bulk materials, and possess widespread applications in high-speed electro-optical modulators, far infrared photodetectors, semiconductor optical amplifiers, and so on. The study of optical properties of these devices endows us with a lot of important information about their energy spectrum, the Fermi surface of electrons, and the value of electronic effective mass. Undoubtedly, QDs have justified their worth as a high performance semiconductor optoelectronic device. However, QDs are often contaminated with dopants during their manufacture which immensely alter their properties. This kind of huge alteration takes place because of subtle interplay between the impurity potential and the dot confinement potential. A large number of investigations on doped QD 1–3 therefore conform with increasing need of deciphering their properties. Within the domain of optoelectronic applications, impurity driven modulation of linear and nonlinear optical properties has been found to be extremely important in photodetectors and in varieties of high-speed electro-optical devices 4. A plethora of many important works on both linear and nonlinear optical properties of these structures was therefore an obvious outcome 5–12.

External electric field has often been found to significantly highlight an important aspect related with confined impurities 13. The electric field changes the energy spectrum of the carrier and regulates the intensity output of the optoelectronic devices. Moreover, the electric field often diminishes the symmetry of the system and leads to generation of nonlinear optical properties. Among the nonlinear optical properties the second-order quantity produces the lowest-order nonlinear effect with magnitude usually larger than higher-order ones. Thus, the applied electric field turns out to be highly relevant so far as optical properties of doped QDs are concerned 4, 7, 14–25.

The performance of mesoscopic devices is largely affected by *Noise* specially when the size of the electronic system falls within nanometer scale 26. Quite often, noise perturbs the level of device performance 27 and hinders the manufacture of semiconductor heterostructure for numerous applications. Hastas et al. have demonstrated the relevance of noise in self-assembled *InAs* QDs embedded in *GaAs* 28. The noise can result *externally*, or it may be *intrinsic*. It is usually the change in the impurity configurations that gives rise to intrinsic noise 29. There are also some important works that deal with low-frequency noise and its applications to defect related optoelectronic properties of materials and structures 30–33. The phenomenon of *damping* in QD attracted a lot of attention owing to its importance in fundamental physics as well as nanoelectronic applications. The applications include manufacture of high quality single electron transistors 34, logic elements (quantum bits) 35, memory cells 36, and lasers based on QD heterostructures 37. These nanoelectronic devices contain doped semiconductor fragments and the functioning of these impurities come under direct influence of damping 38–41. This happens as a result of sensitive interaction between environmental excitation in the vicinity of dopants and the QD carriers 41. Consequently, we find a good number of studies on the QD electronic damping dynamics very much linked with environmental excitation 35, 42.

Later we have made some investigations on optical properties of QD insisted by damped propagation of dopant 43, 44 and also by *Gaussian white noise* 45, 46. In the present manuscript we have explored the *combined influence* of damped drift of dopant and noise on some of the *off-diagonal* components of *static linear (α _{xy}, α_{yx}), first nonlinear (second order) (β_{xyy}, β_{yxx}),* and

The model considers an electron subject to a harmonic confinement potential *V* (*x, y*) and a perpendicular magnetic field *B*. The confinement potential reads$V(x, y)
=\frac{1}{2} m^* \omega_0^2(x^2 + y^2)$, where *ω*_{0} is the harmonic confinement frequency. Under the effective mass approximation, the Hamiltonian of the system becomes

\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}
(1)

*m** is the effective electronic mass within the lattice of the material considered. We have taken *m** = 0.067*m*_{0} and set $\hbar$ = *e* = *m*_{0} = *a*_{0} = 1. This value of *m** closely resembles *GaAs* quantum dots. In Landau gauge [*A* = (*By*, 0, 0)] (*A* being the vector potential), the Hamiltonian reads

\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}
(2)

*ω*_{c} = *eB/m*c* being the cyclotron frequency (equivalent to magnetic confinement offered by *B*). The magnetic field in atomic unit corresponds to a field strength of miliTesla (mT) order. We may define Ω^{2} =*ω*_{0}^{2} + *ω*_{c}^{2} as the effective frequency in the *y*-direction. The model Hamiltonian [cf. eq. (2)] represents a 2-d quantum dot with a single carrier electron 47 with lateral confinement (parabolic) of the electrons in the *x-y* plane. The parabolic confinement potential often serves as an appropriate representative of the potential in semiconductor structures 1, 3, 8, 18, 20 and has been actually invoked in the study of optical properties of doped QDs 11.

In the present problem we have considered that the QD is doped with a repulsive Gaussian impurity 48-52. Incorporating the impurity potential to the Hamiltonian [cf. eq. (2)] we obtain

\begin{eqnarray}
H_0(x, y, \omega_c, \omega_0)= H_0^\prime(x,y,\omega_c,
\omega_0)+ V_{imp}(x_0, y_0),
\end{eqnarray}
(3)

where $V_{imp}(x_0, y_0) = V_{imp} (0) = V_0 \;
e^{-\xi_0 \left[(x-x_0)^2+(y-y_0)^2 \right]}$ with
*ξ*_{0} > 0 and *V*_{0} > 0 for repulsive impurity, and (*x*_{0}, *y*_{0}) denotes the coordinate of the impurity center. *V*_{0} is a measure of the strength of impurity potential whereas *ξ*_{0}^{-1} determines the spatial stretch of the impurity potential.

Thus, the problem boils down to modeling the energy eigenvalues and eigenvectors of the two dimensional Hamiltonian *H*_{0}:

\begin{eqnarray}
H_0\psi_n(x, y) = E_n\psi_n(x, y).
\end{eqnarray}
(4)

Equation (4) turns out to be the energy eigenvalue equation of a two dimensional harmonic oscillator as *ω*_{c} (i.e., *B*) → 0 and *V*_{0} → 0. Naturally, we seek diagonalization of *H*_{0} in the direct product basis of harmonic oscillator eigenfunctions. The time-independent Schrödinger equation has been solved using variational method expressing the trial wave function *ψ*(*x, y*) in the product basis of harmonic oscillator eigenfunctions 43, 44 *ϕ*_{n} (*αx*) and *ϕ*_{m} (*βy*) respectively, as

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

where *C _{n, 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

The dopant propagation can be realized in terms of time-dependent dopant coordinates

\begin{eqnarray}
H(t)= \left[H_0 - V_{imp}(0)\right] + V_1(t),
\end{eqnarray}
(6)

where

\begin{eqnarray}
V_1(t) = V_0 \; e^{-\gamma(t) \left[\left\{x-x_0(t)
\right\}^2+ \left\{y-y_0(t)\right\}^2 \right]}.
\end{eqnarray}
(7)

In the present work the intrinsic time-dependence of dopant propagation has been considered to be linear for convenient handling of the problem so that *x*_{0}(*t*) = *x*_{0} + *at*, *y*_{0}(*t*) = *y*_{0} + *bt* 43-44. The hindrance offered by damping to the dopant propagation has been modeled by introducing an exponentially decaying term phenomenologically through the parameter *ζ* (a measure of damping strength) giving rise to *x*_{0}(*t*) = (*x*_{0} + *at*) *e** ^{-ζt}* and

The noise comprises of random term [

\begin{eqnarray}
\langle \sigma(t) \rangle = 0,
\end{eqnarray}
(8)

the zero average condition, and

\begin{eqnarray}
\langle \sigma(t) \sigma(t^\prime) \rangle = 2 \mu
\delta(t-t^\prime),
\end{eqnarray}
(9)

the two-time correlation condition with a negligible correlation time, *μ* being the noise strength. The highly uctuating term *σ*(*t*) is called white noise because of a monotonous spectrum in frequency space, like that of *white* light. The noise strength *μ* becomes simply a measure of fluctuation intensity 53. We have invoked Box-Muller algorithm to generate *σ*(*t*). Consideration of noise [*σ*(*t*)] which is generated *internally* allows us envisage a coupling between itself and damping through the coupling parameter *λ*. The coupling modifies the drift of the dopant as *x*_{0}(*t*) = (*x*_{0} + *at*) *e ^{-ζt}* ・ {1 +

A variation of

The matrix element involving any two arbitrary eigenstates

\begin{eqnarray}
V_{r, s}^{imp}(t) &=& \langle \psi_r(x, y) \vert V_1(t) \vert
\psi_s(x, y) \rangle
\nonumber \\
&=& \sum_{nm} \sum_{n^\prime m^\prime} C_{nm, r}^* C_{n^\prime
m^\prime, s} \langle \phi_n(p x) \phi_m(q y) \vert V_1(t)
\vert \phi_{n^\prime}(p x) \phi_{m^\prime}(q y) \rangle
\nonumber \\
&=& V_0(t) \sum_{j=1}^{16} V_{r, s}^j(t)
\end{eqnarray}
(10)

However, the expressions of various related terms 43, 44 are modified and given in Appendix A. Using above relations definite expressions of $V_{r, s}^{imp}(t)$ can be obtained [cf. eq. (10)]. The external static electric field *V*_{2} of strength ϵ is now switched on where

\begin{eqnarray}
V_2 = -\{\epsilon_x|e|x+\epsilon_y|e|y\},
\end{eqnarray}
(11)

*ϵ _{x}* and

\begin{eqnarray}
H(t)= \left[H_0 - V_{imp}(0)\right] + V_1(t)+ V_2.
\end{eqnarray}
(12)

The matrix elements due to *V*_{2} reads

\begin{eqnarray*}
\left(V_2\right)_{n, m; n^\prime, m^\prime} &=&
\langle \psi_p(x, y) \vert V_2 \vert \psi_q(x, y) \rangle
\nonumber \\
&=& \sum_{nm} \sum_{n^\prime m^\prime} C_{nm, p}^*
C_{n^\prime m^\prime, q} \langle \phi_n(\alpha x)
\phi_m(\beta y) \vert V_2\vert \phi_{n^\prime}(\alpha x)
\phi_{m^\prime}(\beta y) \rangle \nonumber \\
&=& \epsilon_x \sum_{nm} \sum_{n^\prime m^\prime}
C_{nm, p}^* C_{n^\prime m^\prime, q} \left[\frac{1}{\alpha}
\left\{\sqrt{\frac{n^\prime+1}{2}}\delta_{n^\prime+1,n}+
\sqrt{\frac{n^\prime}{2}}\delta_{n^\prime-1,n}\right\}
\delta_{m,m^\prime} \right] \nonumber \\
& + & \epsilon_y \sum_{nm} \sum_{n^\prime
m^\prime} C_{nm, p}^* C_{n^\prime m^\prime, q} \left[\frac{1}
{\beta}\left\{\sqrt{\frac{m^\prime+1}{2}}\delta_{m^\prime+1,
m}+ \sqrt{\frac{m^\prime}{2}}\delta_{m^\prime-1,m}\right\}
\delta_{n,n^\prime} \right]
\end{eqnarray*}

The time-dependent wave function can now be described by a superposition of the eigenstates of *H*_{0}, i.e.

\begin{eqnarray}
\psi(x, y, t)=\sum_q a_q(t)\psi_q,
\end{eqnarray}
(13)

and the time-dependent Schrödinger equation (TDSE) containing the evolving wave function [cf. eq. (13)] has now been solved numerically by 6-th order Runge-Kutta-Fehlberg method with a time step size *△t* = 0.01 a.u. and numerical stability of the integrator has been verified (see Appendix B). The numerical solution of TDSE gives the time-dependent superposition coefficients with the initial conditions *a _{p}*(0) = 1,

The energy of the system *E* displays typical dynamics when represented as

\begin{eqnarray}
E(t)=\sum_k E _k (0)\;P_k(t),
\end{eqnarray}
(14)

where *E _{k}*(0) is the energy of

\begin{eqnarray}
\langle E \rangle = \frac{1}{T} \int_0 ^T E(t)dt,
\end{eqnarray}
(15)

where *T* is the final time up to which the dynamic evolution of the system is monitored. We have determined the energy eigenvalues for various combinations of *ϵ _{x}* and

\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}
(16)

and a similar expression is used for computing *α _{yx}* component.

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

\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}
(17)

and a similar expression is used for computing *β _{yxx}* component.

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

\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}

(18)

(18)

and a similar expression is used for computing *γ _{yyxx}* component.

In this section we would like to discuss the variations of off-diagonal components of static linear (*α _{xy}*,

Figure 1 displays the variation of

Figure 2 depicts the similar profile for the *β _{xyy}* and

Figure 3 displays the variation of *γ _{xxyy}* with

The static off-diagonal linear and nonlinear polarizability components of impurity doped quantum dots under the combined influence of damped propagation of dopant and noise reveal some noteworthy aspects. In practice, the components are found to be strongly dependent on the noise-damping coupling strength. The noise-damping coupling parameter (*λ*) emerges as the chief tool with which the dispersive nature of the system can be tuned. With static electric field, the overall influence of noise-damping interaction is manifested through the emergence of sharp rise, sharp fall, and kind of steady behavior of off-diagonal polarizability components. We can therefore infer that it is the extent of coupling between damping and noise that ultimately tailors the profiles of off-diagonal polarizability components. However, the tailoring manifestly depends on the order of polarizability and can design the polarizability profile in diverse manners. The results appear interesting and expected to be important in the related field of research. It needs to be mentioned that there are no experimental data available so far to verify our model considerations. It will be a challenge in future to await verification.

Ganguly S, Saha S and Ghosh M have equivalently contributed the work.

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$\xi(t) = \xi_0. \; \exp(-\eta \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}). \; \exp(-\zeta t). \left\{1+ \lambda \sigma(t)\right\}$, where $\Lambda_1 = x_0^2+y_0^2$, $\Lambda_2 = 2(a x_0+b y_0)$, and $\Lambda_3 = a^2+b^2$. The relevant $w(t)$ and $v(t)$ functions 43, 44 grossly look like $w_1(t)=\frac{\xi(t)} {p^2+\xi(t)}$, $w_2(t)= \frac{\xi(t)} {q^2+\xi(t)}$, $v_1(t) = p x_0(t) \sqrt{\frac{\xi(t)}{p^2+\xi(t)}}$, and $v_2(t) = q y_0(t) \sqrt{\frac{\xi(t)}{q^2+\xi(t)}}$.

The modified expressions of corresponding derivatives read as follows 43, 44:

\begin{eqnarray*} w_1^{\prime}(t) &=& \frac{\eta p^2 \xi(t) e^{-\zeta t}}{\left[p^2+ \xi(t)\right]^2}. \left[\zeta \left\{1+ \lambda \sigma(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*} \begin{eqnarray*} w_2^{\prime}(t) &=& \frac{\eta q^2 \xi(t) e^{-\zeta t}}{\left[q^2+ \xi(t)\right]^2}. \left[\zeta \left\{1+ \lambda \xi(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*} \begin{eqnarray*} v_1^{\prime}(t) &=& \frac{p \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\sqrt{p^2+ \xi(t)}} \left[(x_0 + at) \lambda \sigma^{\prime}(t) + a\left\{1+ \lambda \sigma(t) \right\} - \zeta(x_0+ at)\left\{1+ \lambda \sigma(t) \right\} \right] \nonumber \\ &-& \frac{1}{2}\frac{p^3 \eta x_0(t) \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\left\{p^2+ \xi(t)\right\}^{\frac{3}{2}}} \left[\zeta \left\{1+ \lambda \sigma(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*}and

\begin{eqnarray*} v_2^{\prime}(t) &=& \frac{q \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\sqrt{q^2+ \xi(t)}} \left[(y_0 + at) \lambda \sigma^{\prime}(t) + a\left\{1+ \lambda \sigma(t) \right\} - \zeta(y_0+ at)\left\{1+ \lambda \sigma(t) \right\} \right] \nonumber \\ &-& \frac{1}{2}\frac{q^3 \eta y_0(t) \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\left\{q^2+ \xi(t)\right\}^{\frac{3}{2}}} \left[\zeta \left\{1+ \lambda \sigma(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*}Evaluation

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