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Research Article
General Science
Applied Physics

Influence of position-dependent effective mass on optical refractive index change of impurity doped quantum dots in presence of Gaussian white noise

Suvajit Pal1, Jayanta Ganguly2, Surajit Saha3, Manas Ghosh4

Abstract

We inspect the influence of position-dependent effective mass (PDEM) on the total optical refractive index change (RI) of impurity doped quantum dots (QDs) in presence and absence of noise. Noise has been introduced to the system additively and multiplicatively. The impurity potential is modeled by a Gaussian function and the noise applied being Gaussian white noise. A perpendicular magnetic field serves as a confinement and a static external electric field has been applied. The total RI change profiles have been followed as a function of incident photon energy for different values of PDEM. Always a comparison has been made with fixed effective mass (FEM) to understand the role of PDEM on total RI change profile. Using PDEM the said profile considerably deviates from that of FEM both in presence and absence of noise. The deviations also nicely reflect the outcome of interplay between noise and PDEM in designing the total RI change profiles. However, a switch from one mode of application of noise to another does not alter the total RI change profiles to any noticeable extent. The observations indicate the possibility of harnessing the total RI change profiles of doped QD systems exploiting noise and PDEM.

Keywords: quantum dot, impurity, refractive index change, Gaussian white noise, position-dependent effective mass

Author and Article Information

Affiliation 1 Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum 731124, West Bengal, India

2 Department of Chemistry, Brahmankhanda Basapara High School, Basapara, Birbhum 731215, West Bengal, India

3 Department of Chemistry, Bishnupur Ramananda College, Bishnupur, Bankura 722122, West Bengal, India

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

RecievedOct 13 2015 Accepted: Nov 30 2015 Published: Dec 17 2015

CitationPal S, Ganguly J, Saha S, Ghosh M (2015) Influence of position-dependent effective mass on optical refractive index change of impurity doped quantum dots in presence of Gaussian white noise. Science Postprint 1(2): e00055. doi:10.14340/spp.2015.12A0002.

Copyright©2015 The Authors. Science Postprint published by GH 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 support

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

Introduction

Low-dimensional semiconductor systems (LDSS) such as quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) display immensely large nonlinear optical (NLO) properties. Such amplified demonstration of nonlinear effects originates from the existence of quantum-confinement effect that characterizes LDSS and which happens to be much more intense in comparison with the bulk materials. As a result of such stringent confinement, LDSS experience small energy separation between the subband levels and large value of electric dipole matrix elements. These two factors favor achievement of resonance conditions. Landmark work on NLO properties of QWLs was first carried out by Ahn and Chuang [1]. Later on, the enhanced NLO properties of LDSS have initiated a cornucopia of investigations which deal with e.g. probing the electronic structure of mesoscopic media, application of electronic and optoelectronic devices in the infra-red region of the electromagnetic spectrum [2–6], exploring the area of integrated optics and optical communications [7, 8], and most importantly, understanding the fundamental physics.

As soon as an impurity (dopant) is introduced into LDSS, the dopant potential begins to interact with the confinement potential of LDSS. The said interaction almost invariably affects the energy level distribution which ultimately results into dramatic change in their electronic and optical properties. Thus, a controlled incorporation of dopant could be expedient in achieving desirable optical transitions. Such desirable optical transition assumes immense importance in manufacturing optoelectronic devices with tunable emission or transmission properties and ultranarrow spectral linewidths. This has largely augmented the scope of technological applications of LDSS. Furthermore, the nearness of the optical transition energy and the confinement strength (or the quantum size) provides a means of fine-tuning the resonance frequency. In consequence, optical properties of doped LDSS have observed exhaustive research activities [9–37].

Recently, we come across a good number of investigations which involve position-dependent effective mass (PDEM) of LDSS. PDEM leads to appreciable change in the binding energy of the doped system and thus affects the optical properties. Such change in the optical properties has instigated intense research activities on LDSS with spatially varying effective mass in recent years. In this respect the works of Rajashabala and Navaneethakrishnan [38–40], Peter and Navaneethakrishnan [41], Khordad [42], Qi et. al. [43], Peter [44], Li et. al. [45], and Naimi et. al. [46] deserve mention.

Of late, we have made detailed discussions on the importance of noise in governing the performances of QD devices [47, 48]. In the present study we explore the influence of PDEM on the total change of optical refractive index (RI) (which is a combination of linear and the third-order nonlinear changes of RI) of doped QD relevant to transition between |ψ0〉 and |ψ1〉 states in presence of Gaussian white noise. The system under investigation is a 2-d QD (GaAs) consisting of single carrier electron under parabolic confinement in the x–y plane. The QD is doped with an impurity represented by a Gaussian potential in the presence of a perpendicular magnetic field which acts as an additional confinement. An external static electric field has been applied to the system. Gaussian white noise has been administered to the doped QD via two different pathways i.e. additive and multiplicative [47, 48]. The profiles of total RI change are followed as a function of frequency of incident radiation, simultaneously with fixed effective mass (FEM) and dopant position-dependent effective mass (PDEM). And, in the same context the role of mode of application of noise (additive/multiplicative) has also been addressed which reveals some interesting results.

Methods

The impurity doped QD Hamiltonian, subject to external static electric field (F) applied along x and y-directions and spatially δ-correlated Gaussian white noise (additive/multiplicative) can be written as

\begin{eqnarray} H_0 = H_0^\prime + V_{imp} + \vert e \vert F (x+y) + V_{noise}. \end{eqnarray} (1)

Under effective mass approximation, H0´represents the impurity-free 2-d quantum dot containing single carrier electron under lateral parabolic confinement in the x-y plane and in presence of a perpendicular magnetic field. V(x, y) = 1/2 m*ω02 (x2 + y2) is the confinement potential with ω0 as the harmonic confinement frequency. H0´ is therefore given by

\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)

m* represents the effective mass of the electron inside the QD material. Using Landau gauge [A = (By, 0, 0), where A is the vector potential and B is the magnetic field strength], H0´ reads

\begin{eqnarray} H_0^\prime = -\frac{\hbar^2}{2m^*}\left(\frac{\partial ^2} {\partial x^2}+ \frac{\partial^2}{\partial y^2}\right)+ \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* being the cyclotron frequency. Ω2 = ω02 +ωc2 can be viewed as the effective confinement frequency in the y-direction.

Vimp is the impurity (dopant) potential represented by a Gaussian function [47, 48] viz. Vimp = V0 e−γ [(x − x_0)^2+( y – y_0)^2]. Positive values for γ and V0 indicate repulsive impurity. (x0, y0) is the site of dopant incorporation, V0 is the strength of the dopant potential, and γ−1 represents the spatial spread of impurity potential. γ here behaves equivalently to that of static dielectric constant (ε) of the medium and can be written as γ = kε, where k is a constant. It is quite pertinent to mention here that Khordad and his co-workers used a new type of confinement potential for spherical QD's called Modified Gaussian Potential, MGP [49, 50]. The dopant location-dependent effective mass m*(r0) where r0 = √(x02+y02) is given by [38, 41]

\begin{eqnarray} \frac{1}{m^*(r_0)} = \frac{1}{m^*} + \left(1- \frac{1}{m^*} \right)\exp(-\beta r_0), \end{eqnarray} (4)

where β is a constant chosen to be 0.01 a.u. The choice of above form of PDEM indicates that the dopant is strongly bound to the dot confinement center as r0 → 0 i.e. for on-center dopants whereas m*(r0) becomes highly significant as r0 → ∞ i.e. for far off-center dopants. It needs to be mentioned that the effective mass depends on r0 in an exponential manner given by eqn. (4). However, such a variation of effective mass does not distort the Gaussian nature of impurity potential. What we have actually done is using different discrete values of r0 in eqn. (4) and calculating m* accordingly. Thus, the Gaussian nature of impurity potential is not hampered as at a particular situation only a single r0 value is introduced into eqn. (4) and also in the impurity potential.

The term Vnoise represents the noise contribution to the Hamiltonian H0. It comprises of a spatially δ-correlated Gaussian white noise [f(x, y)] which assumes a Gaussian distribution (generated by Box-Muller algorithm) having strength ζ and is described by the set of conditions [47, 48]:

\begin{eqnarray} \langle f(x,y) \rangle = 0, \end{eqnarray} (5)

the zero average condition, and

\begin{eqnarray} \langle f(x,y) f(x^\prime,y^\prime) \rangle = 2 \zeta \delta \left((x,y)-(x^\prime,y^\prime)\right), \end{eqnarray} (6)

the spatial δ-correlation condition. In reality, there exist a variety of physical situations in which noise can be realized and bears interest. Noise can generate externally, or it may be intrinsic. It is usually the rearrangement of impurity configurations that gives rise to intrinsic noise [51]. Experimentally, noise can be generated by using a function generator (Hewlett-Packard 33120A) and its characteristics i.e. Gaussian distribution and zero mean can be maintained [52]. The Gaussian white noise can be applied to the system via two different modes (pathways), i.e., additive and multiplicative [47, 48]. These two different modes actually control the extent of system-noise interaction. Additive noise is a random term that does not undergo any kind of coupling with system coordinates. A multiplicative noise term is a random term that gets coupled with the system coordinates. The multiplicative character indicates that it depends on the instantaneous value of the variables of the system. It does not scale with system size and is not necessarily small [53, 54]. In case of additive white noise Vnoise becomes

\begin{eqnarray} V_{noise} = \lambda_1 f(x,y). \end{eqnarray} (7)

And with multiplicative noise we can write

\begin{eqnarray} V_{noise} = \lambda_2 f(x,y) (x+y). \end{eqnarray} (8)

The parameters λ1 and λ2 absorb in them all the neighboring influences in case of additive and multiplicative noise, respectively.

In order to solve the time-independent Schrödinger equation we have generated the sparse Hamiltonian matrix (H0) where the matrix elements involve the function ψ (x, y), constructed as a superposition of the products of harmonic oscillator eigenfunctions. In this context requisite number of basis functions have been used after performing the convergence test. And H0 is diagonalized in the direct product basis of harmonic oscillator eigenfunctions to obtain the energy levels and wave functions. The matrix element for the noise term has been determined numerically using the relation

\begin{eqnarray} \delta (\sqrt{x^2+y^2}-\sqrt{x'^2+y'^2}) &=& \frac{\exp\left[-(x^2+y^2 + x'^2 + y'^2)/2 \right]} {\sqrt{\pi}} \nonumber \\ & \times & \sum_{k=0}^\infty \frac{H_k(\sqrt{x^2+y^2}) H_k(\sqrt{x'^2+y'^2})}{2^k k!}, \end{eqnarray} (9)

where Hk(x) stands for the Hermite polynomial of kth order.

We now consider interaction between a polarized monochromatic electromagnetic field of angular frequency ν with an ensemble of QDs. If the wavelength of progressive electromagnetic wave is greater than the QD dimension, the amplitude of the wave may be regarded constant throughout QD and the aforesaid interaction can be realized under electric dipole approximation. Now, the electric field of incident optical wave can be expressed as [21]

\begin{eqnarray} E(t)= E(t)\hat{k} = \left[2\tilde{E}\cos{(\nu t)}\right] \hat{k}= \left(\tilde{E}e^{i\nu t} + \tilde{E}^*e^{-i\nu t}\right)\hat{k}. \end{eqnarray} (10)

The electronic polarization P(t) induced by the incident field E(t) is given by.

\begin{eqnarray} P(t) &=& \varepsilon_0 \chi^{(1)}_{\nu}\tilde{E} e^{i \nu t} + \varepsilon_0 \chi_0^{(2)} \tilde{E}^2 + \varepsilon_0 \chi^{(2)}_{2\nu}\tilde{E}^2 e^{2i \nu t} + \varepsilon_0 \chi^{(3)}_{\nu}\tilde{E}^2 \tilde{E}e^{i \nu t} \nonumber \\ &+& \varepsilon_0 \chi^{(3)}_{3\nu}\tilde{E}^3 e^{3i \nu t} +........ \end{eqnarray} (11)

χν(1), χ0(2), χ2ν(2), χν(3), andχ3ν(3), are known as linear, nonlinear optical rectification (NOR), second harmonic generation (SHG), third-order, and third harmonic generation (THG) susceptibilities, respectively. In general, the electronic polarization of n-th order is given by

\begin{eqnarray} P^{(n)}(t) = \frac{1}{S}Tr\left(\rho^{(n)}qr\right), \end{eqnarray} (12)

where S is the area of interaction and Tr is the trace over the diagonal elements of the matrix (ρ(n)qr). ε0 being the vacuum permittivity. χ2ν(2) = 0 for spherically symmetric systems and thus these kind of systems fail to generate second-order optical interactions. It emerges only in non-centrosymmetric structures [4]. By means of density matrix approach and iterative procedure, considering optical transition between two states |ψ0〉 and |ψ1〉, the linear and the third-order nonlinear changes in RI are given by

\begin{eqnarray} \frac{\Delta n^{(1)}(\nu)}{n_r} = \frac{1}{2n_r^2 \varepsilon_0}. \frac{\sigma_s \vert M_{ij} \vert^2 \left(\Delta E_{ij} -\hbar \nu \right)} {\left(\Delta E_{ij} - \hbar \nu \right)^2+ \left(\hbar \Gamma_{ij}\right)^2}, \end{eqnarray} (13)

and

\begin{eqnarray} \frac{\Delta n^{(3)}(\nu)}{n_r} &=& - \frac{\mu c I} {4\varepsilon_0 n_r^3}. \frac{\sigma_s \vert M_{ij} \vert^2} {\left[\left(\Delta E_{ij} - \hbar \nu \right)^2 + \left(\hbar \Gamma_{ij}\right)^2\right]^2} \nonumber \\ & \times & \left[4 \left(\Delta E_{ij} - \hbar \nu \right) \vert M_{ij} \vert^2 -\frac{\left(M_{jj}-M_{ii}\right)^2} {\Delta E_{ij}^2 + \left(\hbar \Gamma_{ij}\right)^2}. \left(\Delta E_{ij} - \hbar \nu \right) \right. \nonumber \\ &\times& \left. \left\{\Delta E_{ij}\left(\Delta E_{ij} - \hbar \nu\right) - \left(\hbar \Gamma_{ij}\right)^2 \right\} - \left(\hbar \Gamma_{ij}\right)^2 \left(2 \Delta E_{ij} - \hbar \nu \right)\right]. \end{eqnarray} (14)

Finally, the total change in RI becomes

\begin{eqnarray} \frac{\Delta n(\nu)}{n_r} = \frac{\Delta n^{(1)}(\nu)}{n_r} + \frac{\Delta n^{(3)}(\nu)}{n_r}. \end{eqnarray} (15)

In the above equations Γij is the phenomenological relaxation rate, caused by the electron-phonon, electron-electron and other collision processes. The diagonal matrix element i.e. Γjj gives the relaxation rate of state |j〉 and Γjj = 1/τjj, whereτjj is the relaxation time of |j〉-th state. The off-diagonal matrix element i.e. Γij (=1/τij, i ≠ j) gives the relaxation rate of |i〉-th and |j〉-th states with relaxation time τij. ε0 is the vacuum permittivity. σs is the carrier density, Mij = |e|〈ψi | ^x + ^y |ψj〉, (i, j = 0, 1; ^x and ^y represent x-hat and y-hat, respectively) is the matrix elements of the dipole moment, ψi (ψj) are the eigenstates and ΔEij = Ei − Ej is the energy difference between these states. nr is the static component of refractive index (RI) in QD (= √εr, where εr is the static dielectric constant). c is the speed of light in free space, μ is the magnetic permeability of the system (= 1/ε0 c2), and I is the intensity of the electromagnetic field.

Results and Discussion

The calculations are performed using the following parameters: ε = 12.4, m* = 0.067 m0, where m0 is the free electron mass, ε0 = 8.8542 x 10−12 F m−1, τ = 0.14 ps, nr = 3.2, ħω0 = 2.72 meV, F = 100 KV/cm, B = 1.0 T, ζ= 1.0 x 10−8, V0 = 272 meV, I = 1.5 x 1010 W/m2, σs = 5.0 x 1024 m−3, and r0 = 0.0 nm. The parameters are suitable for GaAs QDs. Some of the above parameters are readily available in a number of articles. For example, we cite two of them [21, 27].

Figure 1 evinces the variation of Δn(ν)/nr with incident photon energy ħν for FEM (m* = 0.067 m0) and PDEM [m*(r0)] corresponding to three different values of dopant locations i.e. on-center (r0 = 0.0 nm), near off-center (r0 = 0.4 nm), and far off-center (r0 = 0.75 nm) in absence of noise. The plots corresponding to FEM and PDEM are represented by dashed and solid lines, respectively. For FEM, it becomes conspicuous from the figure that an increase in r0 (i.e. shift of dopant from on to more off-center locations) shifts the total RI change peak positions to smaller frequency (red-shift) because of decrease in the energy level separation. Moreover, the total RI change peak exhibits a position-dependent maximization at r0 = 0.4 nm [18] indicating maximum overlap between the eigenstates concerned.

Figure 1: Plots of Δn(ν)/nr vs ħν at three different r0 values for FEM: (i) r0 = 0.0 nm, (ii) r0 = 0.4 nm, (iii) r0 = 0.75 nm, and for PDEM: (iv) r0 = 0.0 nm, (v) r0 = 0.4 nm, (vi) r0 = 0.75 nm under noise-free condition

Similar profiles for PDEM reveal an overall shift of the total RI change peaks to a smaller frequency domain w.r.t. FEM case. In case of PDEM the total RI change peaks display red-shift as before with increase in r0. However, unlike previous case, here the peak heights decrease steadily with increase in r0 and do not display any maximization. The observations thus suggest that, by an large, PDEM causes a fall in the energy interval between the relevant eigenstates w.r.t. FEM case. Furthermore, a shift of dopant from on to more off-center locations diminishes the said energy interval in case of PDEM just like FEM. However, above shift also causes a steady decline in the extent of overlap between the concerned eigenstates in case of PDEM.

Figure 2 displays the similar profiles in presence of additive noise. Here also the RI profiles considering PDEM appear on the lower energy side w.r.t. RI profiles considering FEM. This again indicates that even in presence of additive noise PDEM reduces the energy level separation between the eigenstates in comparison with FEM case. We have found that both for FEM and PDEM the total RI change peaks exhibit small blue-shift as the dopant is shifted from on-center to near off-center location followed by a faint red-shift with further passage to far off-center locations. The observations thus suggest small enhancement in ΔE01 energy gap under the sway of additive noise accompanying a dopant shift from on-center to near off-center locations followed by a nominal reduction in the said energy gap for far off-center dopants. This general trend of variation of energy separation against dopant location is found to be valid both in case of FEM and PDEM. The difference in the total RI change profiles considering FEM and PDEM is actually reflected through the peak heights. Whereas, quite similar to noise-free situation, the peak height exhibits maximization for a near off-center dopant (r0 = 0.4 nm) in case of FEM, the said maximization occurs for a far off-center dopant (r0 = 0.75 nm) considering PDEM. It can therefore be concluded that, in presence of additive noise, maximum overlap between the eigenstates involved takes place for near and far off-center dopants considering FEM and PDEM, respectively.

Figure 2: Plots of Δn(ν)/nr vs ħν at three different r0 values for FEM: (i) r0 = 0.0 nm, (ii) r0 = 0.4 nm, (iii) r0 = 0.75 nm, and for PDEM: (iv) r0 = 0.0 nm, (v) r0 = 0.4 nm, (vi) r0 = 0.75 nm in presence of additive noise

In presence of multiplicative noise, the total RI change profile qualitatively resembles its additive analogue with minor difference in their magnitudes both for FEM and PDEM (Figure 3). However, presence of noise (both additive and multiplicative) amplifies the magnitude of total RI change in comparison with noise-free situation both in case of FEM and PDEM.

Figure 3: Plots of Δn(ν)/nr vs ħν at three different r0 values for FEM: (i) r0 = 0.0 nm, (ii) r0 = 0.4 nm, (iii) r0 = 0.75 nm, and for PDEM: (iv) r0 = 0.0 nm, (v) r0 = 0.4 nm, (vi) r0 = 0.75 nm in presence of multiplicative noise

In order to get a more vivid picture of the nature of change of total RI change peaks we now plot the peak heights as a function of PDEM (Fiure 4a). The plots exhibit that the said peaks undergo maximization at m*(r0) ~ 0.7 m0 in absence of noise indicating maximum overlap between the pertinent eigenstates (Fiure 4a(i)). The plots in presence of noise (both additive and multiplicative) are nearly similar and depict minimization of total RI change peak almost at the same value of m*(r0) as in case of noise-free condition (Figure 4a(ii), (iii)). Presence of noise, therefore, just reverses the extent of overlap between the eigenstates at this typical value of m*(r0). Figure 4b, on the other hand, delineates the plot of ħνmax against PDEM and provides a good impression of how the total RI change peaks shift with change in PDEM. The plots reveal a steady blue-shift of total RI change peaks with increase in PDEM indicating parallel increase in energy interval between the relevant eigenstates in absence of noise (Figure 4b(i)). Presence of noise (both additive and multiplicative) causes a blue-shift of total RI change peaks up to m*(r0) ~ 0.5 m0 beyond which the said peaks undergo red-shift with further increase in m*(r0) (Figure 4b(ii), (iii)). The observations thus suggest an initial enhancement of aforesaid energy interval up to m*(r0) ~ 0.5 m0 beyond which the interval diminishes in presence of noise. The plots also clearly reveal amplification in the magnitude of total RI change peak intensity in presence of noise over that of noise-free situation.

Figure 4: Plots of (a) [Δn(ν)/nr]max vs m(r0) and (b) ħνmax vs m*(r0): (i) under noise-free condition, (ii) in presence of additive noise, and (iii) in presence of multiplicative noise

Conclusions

The total optical refractive index change of impurity doped QD has been investigated in presence and absence of noise with fixed and position-dependent effective mass. The total RI change profiles have been monitored as a function of incident photon energy with FEM and for different values of PDEM. On the whole, with PDEM the total RI change peaks appear at a low energy regime in comparison with FEM both in presence and absence of noise. Apart from this, exploiting PDEM invites noticeable differences in the features of total RI change profiles from that using FEM. Those features chiefly include the height and the nature of shift of total RI change peaks as the doped system is subjected to external field. Presence of noise also brings about enhancement of total RI changes peak intensity over that of noise-free situation. And the nature of difference (between the total RI change profiles corresponding to FEM and PDEM) is also modified in presence of noise indicating sort of interplay between noise and PDEM. However, a change in the mode of application of noise (additive/multiplicative) does not produce any perceptible signature on the total RI change profiles both for FEM and PDEM. The findings could have relevance in the related field of research.

Author contributions

Conceived and designed the work: Pal S, Ganguly J, Saha S, Ghosh M
Acquired the data: Pal S, Ganguly J, Saha S, Ghosh M
Analyzed and/or interpreted the data: Pal S, Ganguly J, Saha S, Ghosh M
Drafted the work: Pal S, Ganguly J, Saha S, Ghosh M
Revised and approved the work: Pal S, Ganguly J, Saha S, Ghosh M

Acknowledgements

The authors S. P, J. G., S. S. and M. G. thank D. S. T-F. I. S. T (Govt. of India) and U. G. C.-S. A. P (Govt. of India) for support.

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