# Impurity Modulated Static Linear and First  Nonlinear  Polarizabilities of Doped Quantum Dots

Authors: Nirmal Kr Datta; Manas Ghosh
Journal: ISRN Optics
 (2012)

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## Abstract

We explore the pattern of linear and first nonlinear optical (NLO) response of repulsive
impurity doped quantum dots harmonically confined in two dimensions. The dopant impurity
potential chosen assumes Gaussian form. The quantum dot is subject to a static electric field. For
some fixed values of transverse magnetic field strength (ωc), and harmonic confinement potential
(ω0), the influence of impurity strength (V0), impurity stretch (ξ), and impurity location (r0)
on the diagonal components of static linear (αxx and αyy), and the first NLO (βxxx and βyyy) responses of the dot are computed through linear variational route. The investigation reveals the
crucial roles played by the various impurity parameters in modulating these optical responses.
Interestingly, maximization in the first NLO responses has been observed for some particular
dopant location and at some particular value of spatial stretch of the dopant.


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## Body

## 1. Introduction

The study of impurity states in low-dimensional heterostructures has emerged as an important aspect to which many theoretical and experimental works have been dedicated. As a result, nowadays the researches on doped semiconductor devices come out to be ubiquitous [1]. The doped system has quantized properties making them ideal objects for scientific study and robust applications. The dopants modify the chemical potential of a material. The control of optoelectronic properties of a wide range of semiconductor devices [2–4] now turns out to be the most fascinating aspect of impurity doping in such materials.Miniaturization of semiconductor devices reaches the bottom of the avenue with the advent of so-called low-dimensional structures such as quantum dots (QDs), with QD’s new perspectives and delicacies in the field of impurity doping emerge, due to the mingling of new confinement sources with impurity related potentials [5]. Such confinement, coupled with the dopant location, can dramatically alter the electronic and optical properties of the system [6, 7].For quite sometimes, search for molecules and materials with high linear and nonlinear susceptibilities and ultrafast response time has been pursued all over the world [8–14]. The targeted molecules are important in communication technology, data storage, optical switching, and so forth [9, 11, 15]. As a result of numerous studies, certain broad designing clues have emerged. Based on these guidelines, most often one tries to design donor-acceptor molecules with large charge transfer from the donor to the acceptor moiety. NLO response is maximized for certain optimal combinations of charge transfer and hopping interactions and the length over which the charge is transferred [16–19]. Modulation in the molecular electronic parameters brought through subtle structural changes or change in substitution changes the electronic wave function and therefore the electron density distribution which ultimately shapes all the properties of atoms and molecules. It would be crucial to have at one’s disposal, systems, the electronic structures of which could be continuously tuned by adjusting suitable control parameters. In molecules, these variations are only discrete, having caused by either a change of substituent or by structural alteration or by both. Quantum dots are representatives of the systems where the electronic structural dispositions can be practically continuously varied as functions of a few of the system parameters. From the above discussion, it appears inevitable that variation in the electronic structure of the dot induced by the impurity could subtly shape all its properties. It is thus crucial to study the role of dopant impurity on the optical properties of the quantum dots. A thorough expedition through literature reveals some notable works on the optical properties of impurity doped quantum dots [20, 21]. Doped quantum dots, thus, beyond any doubt, possess the potential to exhibit high linear and nonlinear optical response properties and would thus be principally applied in the area of all-optical signal processing. A detailed investigation of various dopant parameters relevant to optical signal processing would continue to be a topic of active research. In what follows, of late, we have also studied the frequency dependent linear and nonlinear polarizabilities of doped quantum dots as functions of various dopant parameters [22, 23].In the present manuscript, we inspect the diagonal components of linear (αxx and αyy) and first nonlinear polarizabilities (βxxx and βyyy) of a repulsive impurity doped quantum dot subject to a static electric field. Following earlier works on the effects of a repulsive scatterer in multicarrier dots in the presence of magnetic field [24, 25], here we have considered that the QD is doped with a repulsive Gaussian impurity. When the impurity is doped in an on-center location, it does not destroy the inversion symmetry of the dot and consequently the emergence of β value is ruled out. However, nonzero α values are envisaged. At off-center dopant locations, however, emergence of both nonzero α and β values is observed owing to the destruction of inversion symmetry. We have found that, in conjunction with the dopant location, the strength and the spatial stretch of the dopant also affect the polarizability values quite prominently.


## 2. Method

We consider the energy eigenstates of an electron subject to a harmonic confinement potentialV(x,y) and a perpendicular magnetic field B, where V(x,y)=(1/2)m*ω02(x2+y2), ωc=eB/m*c, and Landau gauge [A=(By,0,0)] have been used. ω0, ωc, and A stand for harmonic confinement potential, cyclotron frequency (a measure of magnetic confinement offered by B), and vector potential, respectively. The Hamiltonian in our problem reads(1)H0′=-ℏ22m*(∂2∂x2+∂2∂y2)+12m*ω02x2,+12m*(ω02+ωc2)y2-iℏωcy∂∂x.
Define Ω2=ω02+ωc2 as the effective frequency in the y-direction. In real QDs, the electrons are confined in 3 dimensions that is, 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≪R1≃R2≃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)∈R2. A parabolic potential, V=(1/2)ω|x|2, is often used as a realistic and at the same time computationally convenient approximation. Assuming that the z-extension could be effectively considered zero, the electronic properties in these nanostructures have been successfully described within the model of the 1-electron motion in 2-d harmonic oscillator potential in the presence of a magnetic field [26–28]. Now, intrusion of impurity perturbation transforms the Hamiltonian to(2)H0(x,y,ωc,ω0)=H0′(x,y,ωc,ω0)+Vimp(x0,y0),
where Vimp(x0,y0)=Vimp(0)=V0e-ξ[(x-x0)2+(y-y0)2] with ξ>0 and V0>0 for repulsive impurity, and (x0,y0) denotes the position of the impurity center. In QD’s, the electrons visit an almost equipotential surface wherein they are set free. These electrons are carrier electrons. In reality, regular dot occurs rarely as the deformation of the boundary of QD’s modifies the effective potential of the carrier electrons to bunch of uneven pockets. Sometimes, these electrons play the role of defects themselves (repulsive impurity) [29]. V0 is a measure of the strength of impurity potential whereas ξ determines the extent of influence of the impurity potential. A large value of ξ indicates that the spatial extension of impurity potential is highly restricted whereas a small ξ accounts for spatially diffused one. Thus, a change in ξ in turn causes a change in the extent of dot-impurity overlap that affects the excitation pattern noticeably [22, 23]. The presence of repulsive scatterer simulates dopant with excess electrons. The choice of Gaussian impurity potential is not arbitrary as it has been exploited by several investigators [30–35].We write the trial wave functionψ(x,y) as a superposition of the product of harmonic oscillator eigenfunctions ϕn(αx) and ϕm(βy), respectively, as follows:(3)ψ(x,y)=∑n,mCn,mϕn(αx)ϕm(βy),
where Cn,m are the variational parameters and α=m*ω0/ℏ and β=m*Ω/ℏ. The matrix elements of Vimp are given by(4)(Vimp)n,m;n′,m′=V0〈ϕn(αx)ϕm(βy)|e-ξ[(x-x0)2+(y-y0)2]|ϕn′(αx)ϕm′(βy)〉=V0I1I2,
where(5)I1=〈ϕn(αx)|e-ξ(x-x0)2|ϕn′(αx)〉=A∫-∞+∞Hn(αx)Hn′(αx)e-α2x2e-ξ(x-x0)2dx,I2=〈ϕm(βy)|e-ξ(y-y0)2|ϕm′(βy)〉=B∫-∞+∞Hm(βy)Hm′(βy)e-β2y2e-ξ(y-y0)2dy,
with A=α/(2n+n′n!n′!π)1/2 and B=β/(2m+m′m!m′!π)1/2. With the transformations δ12=α2+ξ, δ22=β2+ξ, λ1=exp[-(ξx02(δ12-ξ))/δ12], and λ2=exp[-(ξy02(δ22-ξ))/δ22], one can write(6)I1=Aλ1∫-∞+∞Hn(α*u)Hn′(α*u)e-(u-ρ1)2du,I2=Bλ2∫-∞+∞Hm(β*v)Hm′(β*v)e-(v-ρ2)2dv,
where, ρ1=ξx0/δ1, ρ2=ξy0/δ2, α*=α/δ1, β*=β/δ2, u=δ1x, and v=δ2y. With the help of the standard integral [36], it is now easy to write(7)I1=D1∑k=0min⁡(n,n′)f(k,n,n′),I2=D2∑l=0min⁡(m,m′)g(l,m,m′),
where(8)f(k,n,n′)=2k⋅k!⋅Cnk⋅Cn′k⋅(1-α*2)(n+n′)/2-k⋅Hn+n′-2k(α1ρ1),g(l,m,m′)=2l⋅l!⋅Cml⋅Cm′l⋅(1-β*2)(m+m′)/2-l⋅Hm+m′-2l(β1ρ2),
with D1=Aλ1π1/2/δ1 and D2=Bλ2π1/2/δ2. Thus, finally, we obtain(9)(Vimp)n,m;n′,m′=V0⋅D1⋅D2⋅∑k=0min⁡(n,n′)∑l=0min⁡(m,m′)f(k,n,n′)⋅g(l,m,m′).Hn(x) stands for the Hermite polynomials of nth order. The pth eigenstate of the system in this representation can be written as(10)ψp(x,y)=∑ijCij,p{ϕi(αx)ϕj(βy)},
where i, j are the appropriate quantum numbers, respectively, and (ij) are composite indices specifying the direct product basis.In presence of a static electric field of strengthɛ, the effective Hamiltonian of the system (Heff) becomes(11)Heff=H0-{ɛx|e|x+ɛy|e|y}.
We have determined the values of 〈E(±ɛx,ɛy=0)〉, 〈E(±2ɛx,ɛy=0)〉, 〈E(ɛx=0,±ɛy)〉, and 〈E(ɛx=0,±2ɛy)〉 and we used the data to compute the direct components of polarizabilities of the dot by the following relations obtained by numerical differentiation [37–41]:(12)αxxɛx2=52〈E(0)〉-43[〈E(ɛx)〉+〈E(-ɛx)〉]+112[〈E(2ɛx)〉+〈E(-2ɛx)〉],
and a similar expression for αyyɛy2:(13)βxxxɛx3=[〈E(ɛx,0)〉-〈E(-ɛx,0)〉]-12[〈E(2ɛx,0)〉-〈E(-2ɛx,0)〉],
and a similar expression is used for computing βyyy component. In these expressions, 〈E(±ϵx)〉 represents average energy of the system when ɛy=0 and ɛx≠0. Analogously, we have also written 〈E(±ϵy)〉. The above relations indicate different combinations of electric field intensities and orientations.


## 3. Results and Discussion

### 3.1. System Parameters

The model Hamiltonian (cf. (1)) can be made to represent a 2-d quantum dot with a single carrier electron [28, 42]. The form of the confinement potential indicates lateral electrostatic confinement of the electrons in the x-y plane. Thus, it is disklike which is normally fabricated by molecular beam epitaxy (MBE) technique. Because of unequal confinements in x and y-directions, the symmetry is nonhomogeneous. m* is the effective electronic mass appropriate for describing the motion of the electrons within the lattice of the material to be used. We have used m*=0.5m0 and  set ℏ=e=m0=a0=1. The radial position of dopant (r0) has been varied from 0.0 a.u. (on-center) to 70.71 a.u. (off-center) positions. In the linear variational calculation, we have used basis functions (cf. (3)) with n,m=0–20 for each of the directions (x,y). The direct product basis spans a space of (21×21) dimension. We have checked that the basis functions span the 2-d space effectively completely, at least with respect to representing the observables under investigation. We have made the convergence test with still greater number of basis functions.We have made some attempt to reasonably connect our theoretical parameters to the real life-doped QD. The parameterξ in the impurity potential can be correlated to 1/d2, where d is proportional to the width of the impurity potential [24, 25]. Table 1 gives the d values in nm corresponding to different ξ values to have a feelings of the actual extension of the impurity domain. The m* value that we have used (m*=0.5 a.u.) closely resembles Ge quantum dots (m*=0.55 a.u.). The maximum value of the dopant strength (V0) was limited to ~10-3 a.u. or 27.2meV and the applied magnetic field is of the order of milliTesla (mT). Here, we want to mention that our method is not strictly a perturbation theory in that sense where some perturbation is added to the original Hamiltonian through the agency of a perturbation parameter. In our work, we have exploited linear variation principle to determine the approximate eigenstates of the system as it was not possible to solve the time-independent Schrodinger equation containing impurity in the Hamiltonian as well as in the presence of static electric field. Thus, firstly, we have evaluated the approximate eigenstates of the system taking harmonic oscillator eigenfunctions as the basis function. This involves modulation of the linear combination coefficients. Obviously, the modulation was not arbitrary but was governed by the diagonalization of the Hamiltonian matrix containing the contributions from impurity potential and static electric field. Thus, the method is actually a basis function expansion technique with properly adjusted linear combination coefficients. The validity of the method lies in the fact that how far it can endure the increase in V0 values so that normalization is not hampered. We did not make any forceful normalization of the wave function and checked that inherent normalization is well maintained even with a V0 value as large as 1.0×10-3 a.u. for a very large basis size (21×21) indicating that our method is quite robust so far as stability of the method is concerned. The static electric field has an intensity of 2.57×104 Vm-1. We believe that these values might give some realization of real systems containing actual impurity.Table 1
The width of the impurity domain in nm for differentξ values.
ξ (a.u.)d (nm)0.0000114.070.00014.450.0011.410.010.440.10.14


### 3.2.  Aspects of Polarizability Components

#### 3.2.1.  Influence of Impurity Location

Figure1 depicts the variation of components of linear polarizability (αxx and αyy) in x and y directions as functions of radial position (r0) of the dopant. Both the components are found to decrease monotonically with increase in r0. An increase in r0 takes the dopant away from the dot confinement and consequently there occurs a reduction in dot-impurity overlap. As a result, the extent of repulsive interaction also diminishes which is reflected in the observed behaviors of linear polarizability components. From the plot, it has also been observed that the αyy value is lower than αxx component. This could be due to varied extent of confinement in x and y directions. A close look at (1) reveals that the effective harmonic frequency along y-axis is Ω=ω02+ωc2 while that along x-axis is ω0. Thus, the y-direction is under stronger confinement in comparison to the x-direction and naturally enjoys less dispersive character. Consequently, the linear polarizability components which are intimately connected with the dispersive nature of the system assume different values in x and y directions. It needs to be mentioned that macroscopic polarizability should be isotropic for a symmetric QD in a perpendicular magnetic field. In our investigation, we have found anisotropy in the values of αxx and αyy on a microscopic scale because of a nonzero ωc. We feel that on a macroscopic scale, the anisotropies arising out of various microscopic components may balance each other leading to no net anisotropy of the system. However, a more extensive and exhaustive computation is required for an in depth investigation.Figure 1
Plots ofαxx and αyy versus r0 with V0=1.0×10-6 a.u. and ξ=0.001 a.u. (i) for αxx and (ii) for αyy.We now investigate the diagonal components of first hyperpolarizability (βxxx and βyyy) of the doped dot. For the emergence of nonzero (β), doping of impurity at off-center position is essential. An impurity doping at on-center position does not destroy the inversion symmetry of the Hamiltonian and is therefore unable to generate non-zero β value. Both the β components exhibit maximization at a particular dopant location as the impurity is shifted away from the dot center (Figure 2). A shift of the dopant away from the dot confinement center is endowed with opposite consequences. On one hand, the intensity of dot confinement certainly decreases thereby favoring the high optical response. On the other hand, the said shift also decreases the extent of dot-impurity repulsive interaction and discourages emergence of high optical response. So far as generation of β values are concerned; it appears that the interplay of these two opposing factors are responsible for the aforesaid maximization. For high NLO response, therefore, the designer quantum dots appear to be not hard to find. The requirement being the controlled incorporation of the dopant at definite site to break the symmetry of the confinement potential. In this regard, we should mention that of late there are some excellent experiments which show the mechanism of dopant incorporation [43] and how such incorporations can be controlled [44].Figure 2
Plots of|βxxx| and |βyyy| versus r0 with V0=1.0×10-6 a.u. and ξ=0.001 a.u. (i) for |βxxx| and (ii) for |βyyy|.


#### 3.2.2.  Influence of Impurity Strength

We have now variedV0 over a wide range to observe its influence on the polarizability components. Figure 3 exhibits the αxx and αyy profiles as function of dopant strength for on and two off-center (r0=42.43 a.u. and 70.71 a.u.) dopant locations. The plots reveal how a variation in the dopant strength can influence the polarizability components. We notice that at all dopant locations both the linear polarizability components exhibit a saturation at high dopant strength. Within extremely small dopant strength regime (V0→0), there occurs an initial uprise in the polarizability values with increase in V0. It seems that in this regime the dot-impurity interaction is very feeble. Small increase in dopant strength in this domain primarily causes some sort of development in the dot-impurity repulsive interaction and we observe an initial surge in the polarizability values. At large V0 the plots show some kind of saturation indicating a steady extent of dot-impurity interaction. As observed earlier now also we find that αyy values are somewhat lower in magnitude than αxx values. Moreover, from the plots, we find that while the values of a particular component for on and near off-center dopant locations (r0=42.43 a.u.) are in close proximity, the value drops substantially at far off-center dopant location (r0=70.71 a.u.) owing to a diminished dot-impurity interaction.Figure 3
Plots ofαxx and αyy versus V0 with ξ=0.001 a.u. at on and two off-center dopant locations: (i) for αxx at r0=0.0 a.u., (ii) for αxx at r0=42.43 a.u., (iii) for αxx at r0=70.71 a.u., (iv) for αyy at r0=0.0 a.u., (v) for αyy at r0=42.43 a.u., and (vi) for αyy at r0=70.71 a.u.A similar saturation behavior is also envisaged in|βxxx| and |βyyy| profiles as a function of V0 (Figure 4). Since the |β| components assume zero value at on-center dopant location the figure depicts their profiles at near off-center (r0=42.43 a.u.) and far off-center (r0=70.71 a.u.) dopant locations. Here, also we notice a lower magnitude of |βyyy| component in comparison to its counterpart along x-direction. Interestingly, each of the |β| component exhibits greater magnitude in the far off-center dopant location in comparison to that of near off-center position. The behavior happens to be quite obvious as already we have come across position dependent maximization in |β| components.Figure 4
Plots of|βxxx| and |βyyy| versus V0 with ξ=0.001 a.u. at two off-center dopant locations: (i) for |βxxx| at r0=42.43 a.u., (ii) for |βxxx| at r0=70.71 a.u., (iii) for |βyyy| at r0=42.43 a.u., and (iv) for |βyyy| at r0=70.71 a.u.


#### 3.2.3.  Influence of Impurity Spread

We now turn our attention towards inspecting the influence of spatial stretch of impurity (ξ) on the polarizability components for on and two off-center (r0=42.43 a.u. and 70.71 a.u.) dopant locations (Figure 5). The αxx and αyy components initially delineate sort of decrease with increase in ξ (in small ξ domain) but ultimately culminate in some steady value with further increase in ξ value. A small ξ value implies that the impurity potential is diffused over a long spatial region, so also its influence. In the very low ξ regime, this extreme diffusive nature of the impurity domain results in substantial dot-impurity interaction and consequently a large dispersive nature of the system. The polarizability components thus register somewhat large value. The scenario changes completely in the high ξ regime. Now, a spatially quenched impurity potential undergoes very weak overlap with the dot confinement center resulting in small dispersive nature of the system. This causes a fall in the linear polarizability values. At very high value of ξ, the spatial stretch of impurity becomes highly condensed and the decreasing dispersive nature of the system reaches its limit. The outcome being the steady values of the linear polarizability components. Also, as before, the αyy components are smaller in magnitude in comparison to their x-directional counterpart. Furthermore, for each components, the extent of fall in its magnitude with increase in ξ becomes progressively more pronounced with shift of the dopant from on- to off-center location. This is simply because of the fact that since a high ξ value makes dot-impurity overlap already weak, a shift of dopant to more off-center location further aggravates the situation and hastens the fall of polarizability. Peculiarly, for the intermediate off-center impurity positions, the curve is not monotonous. It seems difficult to understand why in this case we do not encounter an otherwise straightforward curve. It appears that at intermediate dopant locations, along with dot-impurity overlap, the extent of dot confinement also plays a role in shaping the polarizability components. At these locations, a reduced dot-impurity overlap decreases the polarizability values. On the other hand, because of considerable dot-impurity separation the dot confinement also becomes less stringent making the system more flexible which enhances the polarizability values. The competitive behavior between these two opposite influences could be the cause behind the departure of the aforesaid profile from an otherwise monotonous plot.Figure 5
Plots ofαxx and αyy versus ξ with V0=1.0×10-6 a.u. at on and two off-center dopant locations: (i) for αxx at r0=0.0 a.u., (ii) for αyy at r0=0.0 a.u., (iii) for αxx at r0=42.43 a.u., (iv) for αyy at r0=42.43 a.u., (v) for αxx at r0=70.71 a.u., and (vi) for αyy at r0=70.71 a.u.|βxxx| and |βyyy| components, on the other hand, exhibit prominent maximization as a function of ξ (Figure 6). Here, we present the corresponding profiles at near off-center (r0=42.43 a.u.) and far off-center (r0=70.71 a.u.) dopant locations and as usual the |β| components assume zero value at on-center dopant location. The continual decrease in dopant’s spatial stretch with increase in ξ value might have some contrasting consequences on |β| components. Although it reduces the confines of dot potential on the dopant but simultaneously the dot-impurity repulsive interaction is also diminished. We feel that a changeover in the mutual dominance of these two contrasting factors could give rise to maximization of |βyyy| components at some particular ξ value. As expected, here also we notice a lower magnitude of |βyyy| component in comparison to its counterpart along x-direction. Interestingly, each of the |β| component exhibits greater magnitude in the far off-center dopant location in comparison to that of near off-center position. The behavior happens to be quite obvious as already we have visualized position dependent maximization in |β| components.Figure 6
Plots of|βxxx| and |βyyy| versus ξ with V0=1.0×10-6 a.u. at two off-center dopant locations: (i) for |βxxx| at r0=42.43 a.u., (ii) for |βxxx| at r0=70.71 a.u., (iii) for |βyyy| at r0=42.43 a.u., and (iv) for |βyyy| at r0=70.71 a.u.


## 3.1. System Parameters

The model Hamiltonian (cf. (1)) can be made to represent a 2-d quantum dot with a single carrier electron [28, 42]. The form of the confinement potential indicates lateral electrostatic confinement of the electrons in the x-y plane. Thus, it is disklike which is normally fabricated by molecular beam epitaxy (MBE) technique. Because of unequal confinements in x and y-directions, the symmetry is nonhomogeneous. m* is the effective electronic mass appropriate for describing the motion of the electrons within the lattice of the material to be used. We have used m*=0.5m0 and  set ℏ=e=m0=a0=1. The radial position of dopant (r0) has been varied from 0.0 a.u. (on-center) to 70.71 a.u. (off-center) positions. In the linear variational calculation, we have used basis functions (cf. (3)) with n,m=0–20 for each of the directions (x,y). The direct product basis spans a space of (21×21) dimension. We have checked that the basis functions span the 2-d space effectively completely, at least with respect to representing the observables under investigation. We have made the convergence test with still greater number of basis functions.We have made some attempt to reasonably connect our theoretical parameters to the real life-doped QD. The parameterξ in the impurity potential can be correlated to 1/d2, where d is proportional to the width of the impurity potential [24, 25]. Table 1 gives the d values in nm corresponding to different ξ values to have a feelings of the actual extension of the impurity domain. The m* value that we have used (m*=0.5 a.u.) closely resembles Ge quantum dots (m*=0.55 a.u.). The maximum value of the dopant strength (V0) was limited to ~10-3 a.u. or 27.2meV and the applied magnetic field is of the order of milliTesla (mT). Here, we want to mention that our method is not strictly a perturbation theory in that sense where some perturbation is added to the original Hamiltonian through the agency of a perturbation parameter. In our work, we have exploited linear variation principle to determine the approximate eigenstates of the system as it was not possible to solve the time-independent Schrodinger equation containing impurity in the Hamiltonian as well as in the presence of static electric field. Thus, firstly, we have evaluated the approximate eigenstates of the system taking harmonic oscillator eigenfunctions as the basis function. This involves modulation of the linear combination coefficients. Obviously, the modulation was not arbitrary but was governed by the diagonalization of the Hamiltonian matrix containing the contributions from impurity potential and static electric field. Thus, the method is actually a basis function expansion technique with properly adjusted linear combination coefficients. The validity of the method lies in the fact that how far it can endure the increase in V0 values so that normalization is not hampered. We did not make any forceful normalization of the wave function and checked that inherent normalization is well maintained even with a V0 value as large as 1.0×10-3 a.u. for a very large basis size (21×21) indicating that our method is quite robust so far as stability of the method is concerned. The static electric field has an intensity of 2.57×104 Vm-1. We believe that these values might give some realization of real systems containing actual impurity.Table 1
The width of the impurity domain in nm for differentξ values.
ξ (a.u.)d (nm)0.0000114.070.00014.450.0011.410.010.440.10.14


## 3.2.  Aspects of Polarizability Components

### 3.2.1.  Influence of Impurity Location

Figure1 depicts the variation of components of linear polarizability (αxx and αyy) in x and y directions as functions of radial position (r0) of the dopant. Both the components are found to decrease monotonically with increase in r0. An increase in r0 takes the dopant away from the dot confinement and consequently there occurs a reduction in dot-impurity overlap. As a result, the extent of repulsive interaction also diminishes which is reflected in the observed behaviors of linear polarizability components. From the plot, it has also been observed that the αyy value is lower than αxx component. This could be due to varied extent of confinement in x and y directions. A close look at (1) reveals that the effective harmonic frequency along y-axis is Ω=ω02+ωc2 while that along x-axis is ω0. Thus, the y-direction is under stronger confinement in comparison to the x-direction and naturally enjoys less dispersive character. Consequently, the linear polarizability components which are intimately connected with the dispersive nature of the system assume different values in x and y directions. It needs to be mentioned that macroscopic polarizability should be isotropic for a symmetric QD in a perpendicular magnetic field. In our investigation, we have found anisotropy in the values of αxx and αyy on a microscopic scale because of a nonzero ωc. We feel that on a macroscopic scale, the anisotropies arising out of various microscopic components may balance each other leading to no net anisotropy of the system. However, a more extensive and exhaustive computation is required for an in depth investigation.Figure 1
Plots ofαxx and αyy versus r0 with V0=1.0×10-6 a.u. and ξ=0.001 a.u. (i) for αxx and (ii) for αyy.We now investigate the diagonal components of first hyperpolarizability (βxxx and βyyy) of the doped dot. For the emergence of nonzero (β), doping of impurity at off-center position is essential. An impurity doping at on-center position does not destroy the inversion symmetry of the Hamiltonian and is therefore unable to generate non-zero β value. Both the β components exhibit maximization at a particular dopant location as the impurity is shifted away from the dot center (Figure 2). A shift of the dopant away from the dot confinement center is endowed with opposite consequences. On one hand, the intensity of dot confinement certainly decreases thereby favoring the high optical response. On the other hand, the said shift also decreases the extent of dot-impurity repulsive interaction and discourages emergence of high optical response. So far as generation of β values are concerned; it appears that the interplay of these two opposing factors are responsible for the aforesaid maximization. For high NLO response, therefore, the designer quantum dots appear to be not hard to find. The requirement being the controlled incorporation of the dopant at definite site to break the symmetry of the confinement potential. In this regard, we should mention that of late there are some excellent experiments which show the mechanism of dopant incorporation [43] and how such incorporations can be controlled [44].Figure 2
Plots of|βxxx| and |βyyy| versus r0 with V0=1.0×10-6 a.u. and ξ=0.001 a.u. (i) for |βxxx| and (ii) for |βyyy|.


### 3.2.2.  Influence of Impurity Strength

We have now variedV0 over a wide range to observe its influence on the polarizability components. Figure 3 exhibits the αxx and αyy profiles as function of dopant strength for on and two off-center (r0=42.43 a.u. and 70.71 a.u.) dopant locations. The plots reveal how a variation in the dopant strength can influence the polarizability components. We notice that at all dopant locations both the linear polarizability components exhibit a saturation at high dopant strength. Within extremely small dopant strength regime (V0→0), there occurs an initial uprise in the polarizability values with increase in V0. It seems that in this regime the dot-impurity interaction is very feeble. Small increase in dopant strength in this domain primarily causes some sort of development in the dot-impurity repulsive interaction and we observe an initial surge in the polarizability values. At large V0 the plots show some kind of saturation indicating a steady extent of dot-impurity interaction. As observed earlier now also we find that αyy values are somewhat lower in magnitude than αxx values. Moreover, from the plots, we find that while the values of a particular component for on and near off-center dopant locations (r0=42.43 a.u.) are in close proximity, the value drops substantially at far off-center dopant location (r0=70.71 a.u.) owing to a diminished dot-impurity interaction.Figure 3
Plots ofαxx and αyy versus V0 with ξ=0.001 a.u. at on and two off-center dopant locations: (i) for αxx at r0=0.0 a.u., (ii) for αxx at r0=42.43 a.u., (iii) for αxx at r0=70.71 a.u., (iv) for αyy at r0=0.0 a.u., (v) for αyy at r0=42.43 a.u., and (vi) for αyy at r0=70.71 a.u.A similar saturation behavior is also envisaged in|βxxx| and |βyyy| profiles as a function of V0 (Figure 4). Since the |β| components assume zero value at on-center dopant location the figure depicts their profiles at near off-center (r0=42.43 a.u.) and far off-center (r0=70.71 a.u.) dopant locations. Here, also we notice a lower magnitude of |βyyy| component in comparison to its counterpart along x-direction. Interestingly, each of the |β| component exhibits greater magnitude in the far off-center dopant location in comparison to that of near off-center position. The behavior happens to be quite obvious as already we have come across position dependent maximization in |β| components.Figure 4
Plots of|βxxx| and |βyyy| versus V0 with ξ=0.001 a.u. at two off-center dopant locations: (i) for |βxxx| at r0=42.43 a.u., (ii) for |βxxx| at r0=70.71 a.u., (iii) for |βyyy| at r0=42.43 a.u., and (iv) for |βyyy| at r0=70.71 a.u.


### 3.2.3.  Influence of Impurity Spread

We now turn our attention towards inspecting the influence of spatial stretch of impurity (ξ) on the polarizability components for on and two off-center (r0=42.43 a.u. and 70.71 a.u.) dopant locations (Figure 5). The αxx and αyy components initially delineate sort of decrease with increase in ξ (in small ξ domain) but ultimately culminate in some steady value with further increase in ξ value. A small ξ value implies that the impurity potential is diffused over a long spatial region, so also its influence. In the very low ξ regime, this extreme diffusive nature of the impurity domain results in substantial dot-impurity interaction and consequently a large dispersive nature of the system. The polarizability components thus register somewhat large value. The scenario changes completely in the high ξ regime. Now, a spatially quenched impurity potential undergoes very weak overlap with the dot confinement center resulting in small dispersive nature of the system. This causes a fall in the linear polarizability values. At very high value of ξ, the spatial stretch of impurity becomes highly condensed and the decreasing dispersive nature of the system reaches its limit. The outcome being the steady values of the linear polarizability components. Also, as before, the αyy components are smaller in magnitude in comparison to their x-directional counterpart. Furthermore, for each components, the extent of fall in its magnitude with increase in ξ becomes progressively more pronounced with shift of the dopant from on- to off-center location. This is simply because of the fact that since a high ξ value makes dot-impurity overlap already weak, a shift of dopant to more off-center location further aggravates the situation and hastens the fall of polarizability. Peculiarly, for the intermediate off-center impurity positions, the curve is not monotonous. It seems difficult to understand why in this case we do not encounter an otherwise straightforward curve. It appears that at intermediate dopant locations, along with dot-impurity overlap, the extent of dot confinement also plays a role in shaping the polarizability components. At these locations, a reduced dot-impurity overlap decreases the polarizability values. On the other hand, because of considerable dot-impurity separation the dot confinement also becomes less stringent making the system more flexible which enhances the polarizability values. The competitive behavior between these two opposite influences could be the cause behind the departure of the aforesaid profile from an otherwise monotonous plot.Figure 5
Plots ofαxx and αyy versus ξ with V0=1.0×10-6 a.u. at on and two off-center dopant locations: (i) for αxx at r0=0.0 a.u., (ii) for αyy at r0=0.0 a.u., (iii) for αxx at r0=42.43 a.u., (iv) for αyy at r0=42.43 a.u., (v) for αxx at r0=70.71 a.u., and (vi) for αyy at r0=70.71 a.u.|βxxx| and |βyyy| components, on the other hand, exhibit prominent maximization as a function of ξ (Figure 6). Here, we present the corresponding profiles at near off-center (r0=42.43 a.u.) and far off-center (r0=70.71 a.u.) dopant locations and as usual the |β| components assume zero value at on-center dopant location. The continual decrease in dopant’s spatial stretch with increase in ξ value might have some contrasting consequences on |β| components. Although it reduces the confines of dot potential on the dopant but simultaneously the dot-impurity repulsive interaction is also diminished. We feel that a changeover in the mutual dominance of these two contrasting factors could give rise to maximization of |βyyy| components at some particular ξ value. As expected, here also we notice a lower magnitude of |βyyy| component in comparison to its counterpart along x-direction. Interestingly, each of the |β| component exhibits greater magnitude in the far off-center dopant location in comparison to that of near off-center position. The behavior happens to be quite obvious as already we have visualized position dependent maximization in |β| components.Figure 6
Plots of|βxxx| and |βyyy| versus ξ with V0=1.0×10-6 a.u. at two off-center dopant locations: (i) for |βxxx| at r0=42.43 a.u., (ii) for |βxxx| at r0=70.71 a.u., (iii) for |βyyy| at r0=42.43 a.u., and (iv) for |βyyy| at r0=70.71 a.u.


## 3.2.1.  Influence of Impurity Location

Figure1 depicts the variation of components of linear polarizability (αxx and αyy) in x and y directions as functions of radial position (r0) of the dopant. Both the components are found to decrease monotonically with increase in r0. An increase in r0 takes the dopant away from the dot confinement and consequently there occurs a reduction in dot-impurity overlap. As a result, the extent of repulsive interaction also diminishes which is reflected in the observed behaviors of linear polarizability components. From the plot, it has also been observed that the αyy value is lower than αxx component. This could be due to varied extent of confinement in x and y directions. A close look at (1) reveals that the effective harmonic frequency along y-axis is Ω=ω02+ωc2 while that along x-axis is ω0. Thus, the y-direction is under stronger confinement in comparison to the x-direction and naturally enjoys less dispersive character. Consequently, the linear polarizability components which are intimately connected with the dispersive nature of the system assume different values in x and y directions. It needs to be mentioned that macroscopic polarizability should be isotropic for a symmetric QD in a perpendicular magnetic field. In our investigation, we have found anisotropy in the values of αxx and αyy on a microscopic scale because of a nonzero ωc. We feel that on a macroscopic scale, the anisotropies arising out of various microscopic components may balance each other leading to no net anisotropy of the system. However, a more extensive and exhaustive computation is required for an in depth investigation.Figure 1
Plots ofαxx and αyy versus r0 with V0=1.0×10-6 a.u. and ξ=0.001 a.u. (i) for αxx and (ii) for αyy.We now investigate the diagonal components of first hyperpolarizability (βxxx and βyyy) of the doped dot. For the emergence of nonzero (β), doping of impurity at off-center position is essential. An impurity doping at on-center position does not destroy the inversion symmetry of the Hamiltonian and is therefore unable to generate non-zero β value. Both the β components exhibit maximization at a particular dopant location as the impurity is shifted away from the dot center (Figure 2). A shift of the dopant away from the dot confinement center is endowed with opposite consequences. On one hand, the intensity of dot confinement certainly decreases thereby favoring the high optical response. On the other hand, the said shift also decreases the extent of dot-impurity repulsive interaction and discourages emergence of high optical response. So far as generation of β values are concerned; it appears that the interplay of these two opposing factors are responsible for the aforesaid maximization. For high NLO response, therefore, the designer quantum dots appear to be not hard to find. The requirement being the controlled incorporation of the dopant at definite site to break the symmetry of the confinement potential. In this regard, we should mention that of late there are some excellent experiments which show the mechanism of dopant incorporation [43] and how such incorporations can be controlled [44].Figure 2
Plots of|βxxx| and |βyyy| versus r0 with V0=1.0×10-6 a.u. and ξ=0.001 a.u. (i) for |βxxx| and (ii) for |βyyy|.


## 3.2.2.  Influence of Impurity Strength

We have now variedV0 over a wide range to observe its influence on the polarizability components. Figure 3 exhibits the αxx and αyy profiles as function of dopant strength for on and two off-center (r0=42.43 a.u. and 70.71 a.u.) dopant locations. The plots reveal how a variation in the dopant strength can influence the polarizability components. We notice that at all dopant locations both the linear polarizability components exhibit a saturation at high dopant strength. Within extremely small dopant strength regime (V0→0), there occurs an initial uprise in the polarizability values with increase in V0. It seems that in this regime the dot-impurity interaction is very feeble. Small increase in dopant strength in this domain primarily causes some sort of development in the dot-impurity repulsive interaction and we observe an initial surge in the polarizability values. At large V0 the plots show some kind of saturation indicating a steady extent of dot-impurity interaction. As observed earlier now also we find that αyy values are somewhat lower in magnitude than αxx values. Moreover, from the plots, we find that while the values of a particular component for on and near off-center dopant locations (r0=42.43 a.u.) are in close proximity, the value drops substantially at far off-center dopant location (r0=70.71 a.u.) owing to a diminished dot-impurity interaction.Figure 3
Plots ofαxx and αyy versus V0 with ξ=0.001 a.u. at on and two off-center dopant locations: (i) for αxx at r0=0.0 a.u., (ii) for αxx at r0=42.43 a.u., (iii) for αxx at r0=70.71 a.u., (iv) for αyy at r0=0.0 a.u., (v) for αyy at r0=42.43 a.u., and (vi) for αyy at r0=70.71 a.u.A similar saturation behavior is also envisaged in|βxxx| and |βyyy| profiles as a function of V0 (Figure 4). Since the |β| components assume zero value at on-center dopant location the figure depicts their profiles at near off-center (r0=42.43 a.u.) and far off-center (r0=70.71 a.u.) dopant locations. Here, also we notice a lower magnitude of |βyyy| component in comparison to its counterpart along x-direction. Interestingly, each of the |β| component exhibits greater magnitude in the far off-center dopant location in comparison to that of near off-center position. The behavior happens to be quite obvious as already we have come across position dependent maximization in |β| components.Figure 4
Plots of|βxxx| and |βyyy| versus V0 with ξ=0.001 a.u. at two off-center dopant locations: (i) for |βxxx| at r0=42.43 a.u., (ii) for |βxxx| at r0=70.71 a.u., (iii) for |βyyy| at r0=42.43 a.u., and (iv) for |βyyy| at r0=70.71 a.u.


## 3.2.3.  Influence of Impurity Spread

We now turn our attention towards inspecting the influence of spatial stretch of impurity (ξ) on the polarizability components for on and two off-center (r0=42.43 a.u. and 70.71 a.u.) dopant locations (Figure 5). The αxx and αyy components initially delineate sort of decrease with increase in ξ (in small ξ domain) but ultimately culminate in some steady value with further increase in ξ value. A small ξ value implies that the impurity potential is diffused over a long spatial region, so also its influence. In the very low ξ regime, this extreme diffusive nature of the impurity domain results in substantial dot-impurity interaction and consequently a large dispersive nature of the system. The polarizability components thus register somewhat large value. The scenario changes completely in the high ξ regime. Now, a spatially quenched impurity potential undergoes very weak overlap with the dot confinement center resulting in small dispersive nature of the system. This causes a fall in the linear polarizability values. At very high value of ξ, the spatial stretch of impurity becomes highly condensed and the decreasing dispersive nature of the system reaches its limit. The outcome being the steady values of the linear polarizability components. Also, as before, the αyy components are smaller in magnitude in comparison to their x-directional counterpart. Furthermore, for each components, the extent of fall in its magnitude with increase in ξ becomes progressively more pronounced with shift of the dopant from on- to off-center location. This is simply because of the fact that since a high ξ value makes dot-impurity overlap already weak, a shift of dopant to more off-center location further aggravates the situation and hastens the fall of polarizability. Peculiarly, for the intermediate off-center impurity positions, the curve is not monotonous. It seems difficult to understand why in this case we do not encounter an otherwise straightforward curve. It appears that at intermediate dopant locations, along with dot-impurity overlap, the extent of dot confinement also plays a role in shaping the polarizability components. At these locations, a reduced dot-impurity overlap decreases the polarizability values. On the other hand, because of considerable dot-impurity separation the dot confinement also becomes less stringent making the system more flexible which enhances the polarizability values. The competitive behavior between these two opposite influences could be the cause behind the departure of the aforesaid profile from an otherwise monotonous plot.Figure 5
Plots ofαxx and αyy versus ξ with V0=1.0×10-6 a.u. at on and two off-center dopant locations: (i) for αxx at r0=0.0 a.u., (ii) for αyy at r0=0.0 a.u., (iii) for αxx at r0=42.43 a.u., (iv) for αyy at r0=42.43 a.u., (v) for αxx at r0=70.71 a.u., and (vi) for αyy at r0=70.71 a.u.|βxxx| and |βyyy| components, on the other hand, exhibit prominent maximization as a function of ξ (Figure 6). Here, we present the corresponding profiles at near off-center (r0=42.43 a.u.) and far off-center (r0=70.71 a.u.) dopant locations and as usual the |β| components assume zero value at on-center dopant location. The continual decrease in dopant’s spatial stretch with increase in ξ value might have some contrasting consequences on |β| components. Although it reduces the confines of dot potential on the dopant but simultaneously the dot-impurity repulsive interaction is also diminished. We feel that a changeover in the mutual dominance of these two contrasting factors could give rise to maximization of |βyyy| components at some particular ξ value. As expected, here also we notice a lower magnitude of |βyyy| component in comparison to its counterpart along x-direction. Interestingly, each of the |β| component exhibits greater magnitude in the far off-center dopant location in comparison to that of near off-center position. The behavior happens to be quite obvious as already we have visualized position dependent maximization in |β| components.Figure 6
Plots of|βxxx| and |βyyy| versus ξ with V0=1.0×10-6 a.u. at two off-center dopant locations: (i) for |βxxx| at r0=42.43 a.u., (ii) for |βxxx| at r0=70.71 a.u., (iii) for |βyyy| at r0=42.43 a.u., and (iv) for |βyyy| at r0=70.71 a.u.


## 4.  Conclusions

The diagonal components of linear and first nonlinear polarizabilities of repulsive impurity doped quantum dots subject to a static electric field reveal intriguing features. For an in-depth analysis, we examine the roles played by impurity spread, impurity strength and most importantly impurity location meticulously on these components. We have found that the linear polarizability components decrease with increase in separation of dopant location from that of dot confinement center and also with an increase in spatial shrinkage of dopant potential. On the other hand, we envisage an increase in the said components with impurity strength. However, in all cases finally some steady behavior has been observed andy direction has been found to be under more stringent confinement than the x direction. The first nonlinear polarizability components evince maximization at some particular dopant location and for some particular value of spatial spread of dopant in absolute sense. The maximization appears to be due to the conflict between two opposing factors that foment and hinder the effective confinement of the dot on the dopant. However, as a function of dopant strength, we observe a persistent increase in |β| values in absolute sense which culminates in a steady magnitude. We expect that the results obtained could have important implications in optical applications of quantum dot nanodevices.At the fag end of the discussion, it appears to be quite significant to highlight the new findings in the present investigation in the light of the results of [23]. First of all, we want to mention that in the present study we have explored the role of impurity location rigorously which was absent from our earlier study on frequency-dependent polarizability (FDP) [23]. Secondly, in [23], we observed that in the limitingly small dopant strength domain an increase in dopant strength causes an initial lull in the FDP values. However, as the dopant strength is increased as bit FDP values begin to increase smoothly with dopant strength culminating in a steady value. In the present study, although we have found some kind of similar trend at high V0 limit, but behavior is absolutely different in small V0 domain. The role of dopant’s spatial spread comes out to be quite distinct in the two investigations. In the earlier study, we found that squeezing the spatial expansion of the dopant domain results in persistent increase in FDP values whereas in the present enquiry a reverse behavior has been envisaged. We hope to explore a rigorous investigation on the frequency-dependent and static polarizability components on a comparative basis in near future.In the present investigation, we did not consider the influence of size on the optical properties. Although in principle the dot wave function can stretch up to infinity but in practice, it actually terminates at some finite values. Thus, the size effect would be important at length scales within the actual termination of wave function.It is quite expected that donor and acceptor impurity would exhibit distinct impacts on the NLO properties. Recently, Hazra et al. have investigated the role of donor and acceptor impurities in a slightly different context. It needs further study to precisely understand the distinct roles of acceptor and donor impurity pertinent to the present investigation.



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*Source: 847532-2012-05-22.xml*