Size-dependent intersubband optical properties of dome-shaped InAs/GaAs quantum dots with wetting...

Size-dependent intersubband optical propertiesof dome-shaped InAs/GaAs quantum dots

with wetting layer

Mohammad Sabaeian* and Ali Khaledi-NasabPhysics Department, Faculty of Science, Shahid Chamran University, Ahvaz, Iran

*Corresponding author: [email protected]

Received 16 November 2011; revised 20 February 2012; accepted 13 April 2012;posted 26 April 2012 (Doc. ID 158136); published 18 June 2012

In this work, the effects of size and wetting layer (WL) on subband electronic envelop functions, eigen-energies, linear and nonlinear absorption coefficients, and refractive indices of a dome-shaped InAs/GaAsquantum dot (QD) were investigated. In our model, a dome of InAs QD with its WL embedded in a GaAsmatrix was considered. A finite height barrier potential at the InAs/GaAs interface was assumed.To calculate envelope functions and eigenenergies, the effective one-electronic-band Hamiltonian andelectron effective mass approximation were used. The linear and nonlinear optical properties were cal-culated by the density matrix formalism. © 2012 Optical Society of AmericaOCIS codes: 230.0230, 230.5590, 250.0250.

1. Introduction

Semiconductor quantum dot (QD) structures haveattracted tremendous attention due to their uniquephysical properties and their potential applicationsin microelectronic and optoelectronic devices [1–5]as well as life sciences and biotechnology [5–8].Due to their relatively higher efficiency comparedwith bulk, they have also found applications as solarcells [5–12]. Performance improvements such as athreshold current independent of temperature, zerolinewidth enhanced factor, and extremely high differ-ential gain have been achieved in QD-based lasers[13]. They have also provided the possibility of gen-eration of femtosecond pulses over a wide range ofwavelengths [14].

In zero-dimension structures, the free carriers areconfined to a small region by a so-called confinementpotential providing the quantization of electronicenergy states based on the size of the dots. Atom-like discrete energy levels occurred when confiningthe carriers in a nanoregion [15,16]. Photons with

appropriate energy can cause the intersubbandtransitions involving large electric dipole moments[17,18]. The optical properties such as refractiveindex, absorption coefficient, and absorption crosssection can be easily calculated once the linear andnonlinear susceptibilities of the QD are known.Large electric dipole matrix elements, along withsmall energy differences between subbands, enhancethe nonlinear contribution of dielectric constant, soone expects the light intensity to play a crucial rolein the optical properties of the dots. In this regard,the dot sizes can alter the values of electronic eigen-energies and their corresponding envelope functions.

Various shapes of QDs can be grown by theStranski–Krastanov (S–K) method [15,19–21]. Thesize and shape of such QDs depend on the growthconditions and the used techniques [22–24]. The op-toelectronic properties of an ensemble of QDs areaffected by the size distribution and the geometry ofthe dots [25–29].

The S–K method is essentially a self-organizedheteroepitaxial growth during molecular beam epi-taxy (MBE) [10,15]. In this technique, after a numberof lattice-mismatched atomic layers deposited on abarrier layer (substrate), accumulated strain energy

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4176 APPLIED OPTICS / Vol. 51, No. 18 / 20 June 2012

forces transition from layer to island growth. Thishappens when a so-called “wetting layer” (WL)reaches a critical thickness of 3 to 4 nm [30–32]. TheS–K approach, however, leads to a broad size distri-bution with inherent [15].

The residual strain energy in dot structure can af-fect the energy eigenvalues, envelope functions, andso the optical properties of QDs [22]. For example,Kim et al. observed a 90 nm blueshift in lowest-energy transition photoluminescence measurementsfor InGaS QDs grown on tensile-strained GaAsPcompared with similar structures utilizing GaAs bar-riers [22]. Tansu et al. used GaAsP barriers sur-rounding the highly strained InGaAsN quantumwell and realized high-performance lasers from 1170to 1400 nm wavelength regions [33]. We note that In-GaAsN material quantum wells have been realizedto be high-performance lasers by metalo-organicchemical vapor (MOCVD) and MBE methods [34].However, for somematerial systems such as the well-known InAs/GaAs system, the strain doesn’t play adominate role [35]. Furthermore, the strain effectscan be compensated using strain compensationlayers as demonstrated by Nuntawong et al. [23,36].The strain effect has also been compensated by Zhaoet al. [37] and Park et al. [38] in InGaN/AlGaN andInGaN/InGaN quantum wells, respectively.

Recently, the MOCVD method has been usedto grow nitride-based and arsenide-based QDs[12,39–42]. Kim et al. implemented this techniqueto growth InGaAsQDs onGaAsPmatrix [22]. Ee et al.usedMOCVD to grow InGaNQDs onGaNemitting atwavelength of 520 nm [12]. Depending on In concen-tration, the bandgap of InGaN QDs can cover thewhole range of the visible spectrum [35].

Although self-assembled QDs are grown experi-mentally on WLs [43,44], most simulations excludedthe WL for simplicity in their calculations. In fact,the coupling between electronic states in the QD andWL is essential to achieve better results [45–47].Both localized QD andWL states must be consideredas a coupled system. Matthews et al. experimentallyinvestigated the effect of WL on gain-current charac-teristics of InGaAs QD laser [13]. They showed thatthe population of WL states leads to a saturation ofpopulation inversion in QD states resulting gain sa-turation in QD laser. To confine the carriers in dotregion, some authors used higher bandgap matrixembedding QD [48–50]. The result was a higher pickgain.

As we mentioned earlier, in the MBE and MOCVDmethods, the existence of WL is essential to thegrowth of individual islands. In order to reduce theWL effects and fully control the QDs’ formation,the selective area epitaxy has been employed[51–55]. In particular, utilizing the diblock copolymerlithography has gained highly uniform InGaN-basedQDs with ultrahigh density on nanopatterned GaNtemplate [56].

Stier et al. proposed a model for the elastic,electronic, and linear optical properties of capped

pyramid-shaped InAs/GaAs QDs in the frame ofeight-band k · p theory [25]. Li et al. computed the en-ergy levels of an electron confined by disk-, ellipsoid-and conical-shaped InAs QDs embedded in GaAs[29]. Zhang et al. studied the wavelength shifts ofCdSe QDs dispersed in a polymer host causing thequantum size effect and electro-optic Stark effect ex-perimentally [57]. Seddik and Zorkani calculated theoptical absorption spectra of a hydrogenic donor im-purity in a spherical CdSe QD with infinite potentialconfinement in the presence of magnetic field usingvariation and perturbation methods within the effec-tive mass approximation [58]. Winkelnkemper et al.presented an eight-band k · p model to calculate theelectronic properties of wurtzite InxGa1−xN∕GaNQDs [35]. They considered the strain effects, piezo-electricity, pyroelectricity, spin-orbit and crystal-fieldsplitting in their model without considering the WL.Rostami et al. investigated the effect of centered de-fect on electrical and optical properties of sphericaland cubic QDs, the former analytically and the latternumerically [59]. Liu and Xu obtained analytical ex-pressions for envelope functions of an electron in acylindrical QD [60]. They then calculated the opticalabsorption coefficient and the variations in refractiveindex associated with intrasubband relaxations as afunction of incident photon energy, dot size, and Almole fraction β in AlβGa1−βAs material in the frameof density matrix formalism. Vahdani and Rezaei in-vestigated the effects of dot size and light intensityon the optical absorption coefficient and refractive in-dex changes of a parabolic QD [61]. Xie studied theeffect of a laser on the hydrogenic impurity in linearand third-order nonlinear absorption coefficient andrefractive index change of a disk-like QD [62]. Rezaeiet al. first solved the Schrödinger equation analyti-cally to calculate envelope function and eigenener-gies of a two-dimensional elliptic-shaped QD [63].They then used compact-density matrix formalismand the iterative method to investigate the effectof size and optical light intensity on linear and non-linear optical properties. Lu and Xie reported the im-purity and exciton effects on linear and nonlinearoptical properties of a disk-like QD under a magneticfield [64]. They used one-band effective mass theoryto solve the Schrödinger equation with the appropri-ate Hamiltonian. Xie then studied the nonlinear op-tical properties of a negative donor disk-like QD withGaussian confining potential [65]. Liang and Xie re-cently investigated the combined effects of hydro-static pressure and temperature on the opticalproperties of a hydrogenic impurity in a disk-shapedQD in the presence of an external electric field [66].

In this paper, we report the dependency of the in-tersubband optical properties of dome-shaped InAsQDs embedded in GaAs matrix on dome radius andradiation intensity. A finite height potential barrierappropriate for a more realistic experimental situa-tion is employed. For QDs with a WL, the eigenener-gies, envelope functions, electric dipole momentmatrix components, relative linear and nonlinear

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refractive indices changes (RLRICs and RNRICs),and linear and nonlinear absorption coefficients(LACs and NACs) are all calculated versus the domeradius and light intensity. The WL zone, which isusually ignored in the previous articles [25,28,29,44–47,57–66] is fully adopted in this work, andits effects on optoelectronic quantities are investi-gated in detail. Our model does not take the strainfield effect into account. Although our approach isapplied for III-V semiconductor optoelectronics de-vices, it can be applied very well on other significantoptoelectronic materials, such as III-nitride semi-conductors, to generate short wavelengths. Thesematerials, which are usually used to generate 420to 500 nm wavelengths, have found applications forsolid-state lighting and diode lasers, medicine, opti-cal storage, terahertz photonics, power electronics,thermoelectricity, and solar cells [67–73]. Also, ourmodeling can also be used for other systems, usingdilute nitride-based materials, such as InGaAsN at1.3 to 1.4 μm [33,34,74] and GaInNAsSb at 1.5 μm[50,74–76] with high performance as diode lasers.

2. Envelope Functions and Eigenenergies ofDome-Shape Quantum Dot

In the k · p approximation, the envelope function thatmodulates the periodic part of the Bloch function atk � 0, can describe the electron wave function in QDs[77]. Consider a dome-shaped QD with cylindricalsymmetry whose Schrödinger wave equation in one-band envelope function formalism is given by

−h2

8π2m� ∇2Ψ�r� � V�r�Ψ�r� � EΨ�r�; (1)

where m� is the electron effective mass having twodifferent values in the QD region and in the sur-rounding medium, ψ�r� is the electronic envelopefunction, and E is the eigenenergy. Cylindrical sym-metry allows us to use the separation of variablestechnique as follows:

Ψ�r� � χ�r; z�Θ�φ�; (2)

where r, z, and φ are variables of cylindrical coordi-nates. Substituting Eq. (2) in Eq. (1) and dividingboth sides by χ�r; z�Θ�φ� gives

1Θ

dΘdφ2 � −l2; (3)

−mer2h

8π21χl

�∂

∂z

�1me

∂χl∂z

�� 1

r∂

∂r

�rme

∂χ∂r

��mer2�V −E�

� −h2

8π l2; (4)

where l is a separation constant. As the envelopefunction must be single-valued under 2π rotation,the φ-part can be expressed as Θ�φ� ∝ exp�ilφ�,

leading to integer values of l � 0;�1;�2;….Equation (4) can be rearranged as

−h2

8π2�∂

∂z

�1me

∂χl∂z

��1

r∂

∂r

�rme

∂χl∂r

��

h2

8π2me

1

r2�V

�χl

�Eχl; (5)

which has the general form of

∇:�−c∇χl� � aχl � β∇χl � Elχl; (6)

with β � −h2

8π2me

1r, a � h2

π2me

l2

r2� V ; and c � h2

8π2me.

A. Boundary Conditions

To solve Eq. (6) and obtain the envelope functionsand energy eigenvalues, certain boundary conditionsare imposed. The simulation region and its bound-aries are shown in Fig. 1. According to Fig. 1, forboundaries 1 (top) and 5 (bottom) the condition ofχl � 0 is considered, while for boundaries 2, 3, 4, 6,7, and 8 the condition of n:�∇χl� � 0 is used, in whichn is the outward unit vector. For interface boundariesof InAs and GaAs, the condition of n:�∇χl∕m��GaAs �n:�∇χl∕m��InAs is adopted because of the finiteness ofthe potential barrier.

B. Envelope Functions and Energy Eigenvalues

To solve the differential Eq. (6) numerically, weadopted the finite element method. To carry out oursimulations, we divided the whole simulation areainto 7360 triangle elements. For one quarter of QDwith its WL 1040 elements and for upper region ofQD and lower region of QD (i.e., GaAs matrix), 3504and 2816 elements have been employed, respectively.The effective electron mass was set to 0.023me forInAs and 0.069me for GaAs [45], whereme is the freeelectron mass. Also, the height of potential barrierwas set as V � 0.697 eV. Throughout our calcula-tions, the thickness of WL corresponding to refer-ences [15,31] has been set as 3 nm. Figures 2(a) to2(f) show the ground and first excited state envelop

Fig. 1. Simulation area with numbered boundaries. The thick-ness of the WL has been set as 3 nm.


functions of dome-shaped QD for three values ofdome radii: r � 3 nm [Fig. 2(a) and 2(b)], r � 7 nm[Figs. 2(c) and 2(d)], and r � 15 nm [Figs. 2(e) and2(f)]. The results show localization of ground stateenvelope function in WL for r � 2 nm to r � 6 nm.With increasing the dome radius, it then exits fromthe WL and expands over dome region. Figure 2(c)shows the ground state envelope function emergingfrom the WL. For the excited state, however, thishappens in two steps. At r ≈ 5 nm, one lobe exits,and then at r ≈ 11 nm, the other lobe relaxes its over-lap to WL. This effect actually will impose its impacton electric dipole moments and relevant opticalproperties.

Figure 3 shows the ground state (square) and firstexcited state (circle) energy eigenvalues versus thedome radius. For ground state, the eigenenergiesare almost constant with dome radius increase upto r � 7 nm, where the envelope function startsemerging from WL. The eigenvalues curve thendrops exponentially as dome radius increase. The si-milar exponentially decaying behavior is also seenfor the first excited state eigenenergies showingtwo plateau segments, which can be attributed totwo steps emerging of envelope function from WL.

In Fig. 4, we plotted the eigenenergies difference,ΔE, between the ground and the first excited statesversus the dome radius, the quantity that is propor-tional to resonance frequency. TheWL role is obviousespecially for radii lower than r ≈ 11 nm.

3. Optical Properties of Dome-Shaped Quantum Dot

A. Definitions

Let a linear x-polarized monochromatic electric fieldpropagate along the z direction as follows:

~E�z; t� � E0 iei�kz−ωt� � C:C; (7)

where k � Nω∕c is the complex propagation constantand ω is the angular frequency [16]. N � n� ini isthe complex refractive index with its n and ni as realand imaginary parts, respectively. The refractive in-dex can be calculated through the effective suscept-ibility as follows:

N ��1� χeff �ω�

p≈ 1� 1

2χeff �ω�; (8)

where effective susceptibility is defined as χeff �ω� �χ�1� � χ�2��ω� ~E� χ�3��ω� ~E2. χ�1�, χ�2� and χ�3� are the

Fig. 2. (Color online) Normalized ground state (left) and first ex-cited state (right) envelop functions for (a) and (b) r � 3 nm, (c) and(d) r � 7 nm, and (e) and (f) r � 15 nm.

Fig. 3. (Color online) Ground state (cubic) and first excited state(circle) energy eigenvalues against the dome radius.

Fig. 4. Difference between ground and first excited state energiesagainst the dome radius.


linear, second order, and third order susceptibilities,respectively. The relative refractive index changesand absorption coefficient of the medium can becalculated as [61]

Δnn

� n − 1n

� 12Re

�χeff �ω�n

�; (9)

α � 2niωc

� ω��μϵR

rIm�ϵ0χeff �ω��; (10)

respectively, where μ is the vacuum permeabilityand ϵR is the real part of permittivity. From Eqs. (9)and (10), one can conclude that the calculation ofoptical properties implies the calculation of linearand nonlinear susceptibilities.

The linear and nonlinear susceptibilities can becalculated through the quantum density matrixformalism [78]. According to this formalism, the lin-ear susceptibility is calculated through the followingformula:

χ�1� � σℏε0

jM21j2ω21 − ω − iγ21

; (11)

where σ is the carrier density, M21 � hψ2j − ezjψ1i isthe off-diagonal component of electric dipole momentmatrix, ω21 � �E2 − E1�∕ℏ is the transition angularfrequency, and γ21 is the damping rate for off-diagonalelements of density matrix. Also, the second-ordersusceptibility due to optical rectification and third-order susceptibility are given respectively by [78]

χ�2�0 � σjM21j2ε0ℏ2��ω21 − ω�2 � γ212�

�2γ21

�M22

γ22−M11

γ11

�

� 2�M22 −M11�ω221 � γ221

�ω21�ω21 − ω� − γ221��; (12)

χ�3��ω��−σjM21j2

ε0ℏ3�ω21−ω−iγ21�

×�

4jM12j2�ω21−ω�2�γ221

−�M22−M11�2

�ω21−iγ12��ω21−ω−iγ12�

�:

(13)

B. Results

In order to calculate the dipole moment matrices thatare the major part of susceptibility calculations, weadopted Simpson’s double numerical integrationmethod in a homemade code programmed in Maple.We set the carrier density and the damping rate toσ�3×1022 m−3 and γ21 � 5 ps−1, respectively [16,61].Also, the refractive index of GaAs was set to n � 3.2.According to Boyd’s notation [78], the relationbetween field intensity and electric field amplitudeis expressed as I � 2nε0cjEj2.

Figure 5 shows the absolute value of the diagonalelements of dipole moment matrix, i.e.,M11 � jhψ1j −ezjψ1ij and M22 � jhψ2j − ezjψ2ij, versus the domeradius. According to this figure, from r � 2 nm tor � 6 nm, with increasing the dome radius, M11 mo-ment does not show noticeable change and remainsapproximately constant. This is due totally to locali-zation of envelope function in the WL region [seeFig. 2(a)]. After r � 6 nm, the ground state envelopefunction comes out from the WL, so the dipole mo-ment increases as the dome region expands. The in-creasing behavior of ground state dipole momentcontinues to bigger values of dome radius. However,for M22, the increasing in its value takes place intwo steps: From r � 2 nm to r � 5 nm, where the ex-cited state envelope function is totally localized inWL, M22 is small. In the second step, according toFig. 2(d), in the interval of r � 6 nm to r � 10 nm,one lobe of two parts excited state envelope functioncomes out from the WL, leading to a small rise inM22. After r � 10 nm, the excited state envelopefunction totally comes out from the WL and so withincreasing the dome radius and expanding the envel-ope function in dome region, M22 increases. In thelarger dome radii, M22 is bigger than M11.

Figure 6 shows the off-diagonal element of dipolemoment matrix, M21, versus the dome radius reveal-ing smaller values relative toM11 andM22 by one or-der of magnitude. It is clear that the behavior of M12is somehow more complicated than M11 and M12.This can be explained as follows: From r � 2 nm toapproximately r � 5 nm, both the ground and excitedstates lie in WL. So increasing the dome radius hasno noticeable effect on envelope functions. Then afterr � 5 nm to r � 8 nm, one part of excited state envel-ope function comes out from WL, whereas groundstate does not. This decreases the ground-first ex-cited states overlapping and leads to a fall in M12.From r � 8 nm to r � 10 nm, where the ground state

Fig. 5. (Color online) Absolute value of diagonal elements ofdipole moment matrix, M11 (cube) and M22 (circle), against thedome radius.


envelope function starts emerging from the WL, M12increases. After r � 10 nm, which both ground andfirst excited states exit completely from the WL,the dipole moment grows with increasing the domeradius.

Figure 7 shows the relative linear refractive indexchanges (RLRIC),Δn�1�∕n, as a function of photon en-ergy for several dome radii. This quantity indeeddoes not depend on light intensity. From this figurewe can see by increasing the dome radius, the RLRICresonates at lower photon energies while its height isincreasing. Figure 8 shows the resonant peak heightsof RLRIC versus the dome radius. This figure revealsan approximately constant value for RLRIC from r �2 nm to r � 10 nm, then a rapid change nearly r �11 nm where both ground and first excited statesexit from WL. For bigger radii, the variations of re-sonance RLRIC versus the dome radius is approxi-mately linear.

Figure 9 shows the linear absorption coefficient(LAC), α�1�, as a function of photon energy for fivedome radii bigger than r � 10 nm. Similar to theRLRIC case, the LAC also does not depend on lightintensity. By increasing the dome radius fromr � 10 nm, a dramatic change occurs for the reso-nant peak heights in the radial interval of r �10 nm up to r � 12 nm effect can be seen well fromFig. 10, where the height of LAC resonant peakshave been plotted versus the dome radius. The var-iations of resonant LAC versus the radius, before r �10 nm and after r � 12 nm are moderate.

Figure 11 shows the variations of relative refrac-tive index change due to second-order optical rectifi-cation versus the photon energy. The solid curvesdenote Δn�2�∕n for various dome radii of r � 12 nm,r � 14 nm, r � 16 nm, r � 18 nm, and r � 20 nm.The light intensity is taken to be constant as I �0.1 MW∕cm2. The results corresponding to higher

Fig. 6. Off-diagonal dipole moment element, M21, against thedome radius.

Fig. 7. RLRIC versus the photon energy for various dome radii ofr � 12 nm, r � 14 nm, r � 16 nm, r � 18 nm, and r � 20 nm.

Fig. 8. Heights of resonant peaks of RLRIC, Δn�1��ω � ω21�∕n,versus the dome radius.

Fig. 9. LAC versus the photon energy for various dome radii ofr � 12 nm (solid), r � 14 nm (dashed), r � 16 nm (dotted), r �18 nm (dashed–dotted), and r � 20 nm (dashed–dotted–dotted).


light intensities have been plotted by differentstyled curves. The dashed, dashed–dotted, anddotted curves stand for I � 0.15 MW∕cm2, I �0.2 MW∕cm2 and I � 0.3 MW∕cm2, respectively.From these curves it can be seen that the resonancefrequency shifts toward the lower values when thedome radius increases. Moreover, the heights of reso-nance peaks are growing by light intensity increas-ing. Figure 12 shows the height of the resonancepeaks of Δn�2�∕n versus the dome radius for severallight intensities. According to these curves, for a ra-dius below r � 10 nm, the peaks are approximatelythe same even for different light intensities, butabove 10 nm the peak height variations behave lin-early with the dome radius variations. The curveslops become larger when the light intensityincreases.

The optical rectification did not show any portionin absorption coefficient, due to purely reality of itscorresponding susceptibility, χ�2�0 .

Figure 13 shows the variations of relative third-order refractive index change (RTRIC) versus thephoton energy. Similar to the optical rectificationcase, to clarify thedomesize and light intensity effectson RTRIC, different styled curves have been used.From the right to the left, solid curves show RTRICfor r� 12 nm, r� 14 nm, r� 16 nm, r� 18 nm, andr � 20 nm, respectively, while the light intensity istaken to be constant as I � 0.1 MW∕cm2. For otherintensities, other styled curves have been usedas follow: the dashed curves for I � 0.15 MW∕cm2,the dashed–dotted curves for I � 0.2 MW∕cm2, anddotted curves for I � 0.3 MW∕cm2. For each group,the resonance frequency shifts toward the lower

Fig. 10. Heights of resonant peaks of LAC versus the domeradius.

Fig. 11. Relative refractive index change due to optical rectifica-tion versus photon energy for several light intensities of I �0.1 MW∕cm2 (solid curves), I � 0.15 MW∕cm2 (dashed curves),I � 0.2 MW∕cm2 (dashed–dotted curves), and I � 0.3 MW∕cm2

(dotted curves).

Fig. 12. (Color online) Relative refractive index change due tooptical rectification at resonate frequency, Δn�2��ω � ω21�∕n,versus dome radius for various intensities.

Fig. 13. RTRIC against the versus photon energy for several lightintensities of I � 0.1 MW∕cm2 (solid curves), I � 0.15 MW∕cm2

(dashed curves), I � 0.2 MW∕cm2 (dashed–dotted curves), and I �0.3 MW∕cm2 (dashed curves). From the right to the left, everygroup of curves stands for dome radius of r � 12 nm, r � 14 nm,r � 16 nm, r � 18 nm, and r � 20 nm.


photon energieswith increasing the dome radius. Theheight of resonant peaks grows with increasing thelight intensity.

Figure 14 shows the peaks heights of Δn�3�∕nversus the dome radius for several light intensities.The curves’ behavior is similar to those of Δn�2�∕nexcept that the third-order case is smaller by twoorders of magnitude (see Fig. 12).

Finally, we investigated the dome size and lightintensity effects on the third-order absorption coeffi-cient (TAC). The results are presented in Figs. 15and 16. In Fig. 15, we have plotted the TAC versusthe photon energy for several intensities ofI�0.1MW∕cm2 (solid curves), I � 0.15 MW∕cm2

(dashed curves), I � 0.2 MW∕cm2 (dashed–dottedcurves), and I � 0.3 MW∕cm2 (dotted curves). Fromthe right to the left, every group stands for radiir � 12 nm, r � 14 nm, r � 16 nm, r � 18 nm, andr � 20 nm, respectively. From these curves we see

that the contribution of third order in absorption coef-ficient is negative. The peaks’ heights versus thedome radius have been plotted in Fig. 16. The differ-ent styled curves were used for various intensities.Approximately, above a radius of r � 12 nm, thevariations of peaks’ height versus the dome radiushave a linear behavior.

4. Conclusion

In this work, firstly the effect of the size of a dome-shaped QD on its energy eigenvalues, envelope func-tions, and elements of dipole moment matrices wereinvestigated. Secondly, the effects of the size and thelight intensity on linear and nonlinear relative re-fractive indices and absorption coefficients were stu-died in detail. The effect of the WL, which is usuallyignored, was pointed out for small-radius domes. Ajump in electronic and optical properties was seenonce the dome radius becomes larger. This behaviorwas attributed to envelope function exit from theWL region. For the ground state envelop function be-yond approximately r ≈ 5 nm and for the first excitedstate one, beyond approximately r ≈ 10 nm, the corre-sponding eigenenergies show exponential decayingbehavior with dome radius increasing. The elementsof dipole moment matrices, however, show a linearbehavior after r ≈ 10 nm.

For optical properties, the heights of the resonantpeaks of the relative linear and nonlinear refractiveindices changes, Δn�1;2;3�∕n, along with third-orderabsorption coefficient (with its negative sign), showlinear variations versus the dome radius in all lightintensities for radii above r ≈ 10 nm, whereas the lin-ear absorption coefficient remains almost constantfor all dome radii.

The authors would like to thank ProfessorHamid Nadgaran (University of Shiraz), Professor I.Kazeminezhad (Shahid Chamran University ofAhvaz), and Dr. M. R. K. Vahdai (Shiraz University)for their helpful discussions.

Fig. 14. (Color online) RTRIC at resonance frequency,Δn�3��ω � ω21�∕n, versus the dome radius for various intensities.

Fig. 15. TAC against the photon energy for several light intensi-ties of I � 0.1 MW∕cm2 (solid curves), I � 0.15 MW∕cm2 (dashedcurves), I � 0.2 MW∕cm2 (dashed–dotted curves), and I �0.3 MW∕cm2 (dotted curves) and several dome radii. From right toleft: r � 12 nm, r � 14 nm, r � 16 nm, r � 18 nm, and r � 20 nm.

Fig. 16. (Color online) TAC at resonant frequency α�3��ω � ω21�versus the dome radius for various intensities.


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