The effect of Woods–Saxon potential on envelope function,intersubband dispersion curves and group velocity of InAs/GaAsquantum dots with wetting layer

Ali Khaledi-Nasab a,b,c,n, M. Sabaeian b, M. Sahrai c, V. Fallahi a, M. Mohammad-Rezaee b

a Department of Laser and Optical Engineering, University of Bonab, 55517611678 Bonab, Iranb Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iranc Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran

H I G H L I G H T S

� The Schrödinger equation for InAs/GaAs quantum dots/wetting layersystem is solved numerically usinga finite element method (FEM).

� Woods–Saxon potential is applied toquantum dots as well as a constantfinite barrier between InAs quantumdot/wetting layer and GaAs matrix isconsidered for comparison.

� The envelop wavefunction for bothconstant and WS potentials is inves-tigated.

� The effect of Woods–Saxon potentialon linear susceptibility and group velo-city is explored for different radii ofdome-shaped InAs/GaAs quantum dot.

G R A P H I C A L A B S T R A C T

a r t i c l e i n f o

Article history:Received 16 August 2013Received in revised form27 January 2014Accepted 28 January 2014Available online 4 February 2014

Keywords:InAs/GaAs quantum dotWoods–Saxon potentialWetting layer

a b s t r a c t

In this study, one band Schrödinger equation for InAs/GaAs quantum dots coupled to their wetting layer wassolved numerically by using a finite element method (FEM). We have carried out the conduction of theWoods–Saxon (WS) potential in the quantum dots (QDs) as well as a constant finite barrier between InAsquantum dot/wetting layer and GaAs matrix is considered for comparison. It is found by WS potential that theenvelope functions never become completely localized inside the dot; therefore this delocalization leads tostrong alternations in absorption and dispersion profiles which is not negligible. Also, it is found that groupvelocity is affected by the WS potential and reveals remarkable blue-shift in comparison to the constant finitepotential.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Over the last decade, progress in nanofabrication has been thecenter of attention due to its contribution in practical and conceptual

areas such as physical, chemical and biological researches. Recentscientific reports have emphasized on the tremendous potentialsof semiconductor nanoparticles, the so called quantum dots (QDs).In QDs, the dimensions of the confinement are less than de Brogliewavelength of carriers, or equivalently, the nanocrystal diameter has tobe less than twice the Bohr radius excitons in the bulk materials [1].Owning to marvelous characteristics, QDs have found numerousapplications in a vast variety of fields such as cell biology (i.e. abilityto be conjugated to proteins) [2–5], nanoscale emitters [6,7], QD solar

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.01.0341386-9477 & 2014 Elsevier B.V. All rights reserved.

n Corresponding author at: University of Tabriz, Research Institute for AppliedPhysics and Astronomy, East Azerbaijan, Tabriz 51665163, Iran. Mobile: +98 916 9926029.

E-mail address: Ali.khaledi1989@gmail.com (A. Khaledi-Nasab).

Physica E 60 (2014) 42–49

cells [8,9], and quantum information processing [10,11]. Moreover, it isrevealed that QDs are proper labels for single-molecule experiments inlive cells [12]. Mainly, QD probes were promoted for their signatureoptical spectrum, the so called wide absorbance, narrow, and sym-metric emissionwith size-dependent peak positions [3]; moreover thesize tunable spectra are other interesting features of QDs [9,10].

With respect to new advantages on nanotechnology, the self-assembled QDs can be formed during the epitaxial growth byusing a vapor phase like molecular beam epitaxy (MBE) [13], metalorganic chemical vapor deposition (MOVPD) [14], Stranski–Kar-stanov (SK) method [15], and chemical synthesis like colloidalchemistry or electrochemistry [16]. QDs can be synthesized atinterface steps of thin quantumwells [17,18] or by self-assembly inthe SK growth mode during MBE [19,20]. Afterward the process isbeing driven by the strain resulting from the smaller latticeparameter of the matrix (barrier) compared to that of the dots;for instance in our case it is 7% for InAs dots in GaAs. The InAssamples used for optical spectroscopy are then covered by thebarrier material which is GaAs. In experiment the InAs dotscontain a remarkable amount of fraction x of Ga, leading to theformation of In1� xGaxAs dots. The SK is applicable to a largevariety of III–V and II–VI compounds [21]. A fascinating alternativefor fabricating GaAs or InAs QDs is provided by a technique whichis not strain driven, called molecular droplet epitaxy (MPE) [22].The high optical quality of GaAs droplet dots owing to newachievements has allowed investigations of carrier and nuclearspin dynamics [23,24]. Because of the carrier confinement poten-tials which are between tens and hundreds of meV, the sampleselaborated with the above techniques are suitable for variousoptical applications like optical spin manipulation [21].

The electronic and optical properties of QDs were illustrated bymany authors in the literature both theoretically and experimen-tally. The effect of size and wetting layer on intersubband electro-nic and linear and nonlinear optical properties of InAs/GaAs QDswere investigated by numerous works both theoretically andexperimentally [25–27]. Sandall et al. [28] took advantage ofIn(Ga)As QDs grown on a silicon substrate as a photo-detector.They measured dark currents in In(Ga)As on Si photodetectorsthree orders of magnitude lower than in Ge on Si. Terahertzgeneration from InAs/GaAs QDs was reported by Daghestaniet al. [29] as photoconductive antennae.

Recently Michael et al. [30] investigated the slowdown ofoptical pulses due to quantum coherence effects theoretically foran active material consisting of double InGaAs-based QD mole-cules; moreover the dependences of slowdown and shape of thepropagating probe pulses on lattice temperature and drive inten-sities were explored as well. In an interesting letter, E. Stinaff et al.[31] reported that asymmetric pair of InAs QDs tuned intoresonance by applying an electrical field so that a hole forms acoherent molecular function, also they have found that Coulombinteractions shift the molecular resonance of the charged excitonwith respect to single charge, enabling light-induced coupling ofthe QDs. S. Ramanathan et al. [32] have reported an experimentalresearch on the effect of tunnel coupling on the Quantum-Confined Stark Effect for excitons in InAs/GaAs coupled QDs,precisely they demonstrated the crucial effect separation barrierwhich leads to a remarkable escalation on zero-field dipolemoments and polarizability.

The InAs/GaAs QDs have great application in solar cell becausethe InAs/GaAs QDs solar cells in comparison to similar bulkstructures like GaAs p-i-n structures show an increase in photo-current [33] as well as significantly improved efficiency [34]. Aten-layer InAs/GaAs QD solar cells exhibit enhanced short circuitcurrent ðJscÞ and open circuit voltage ðVocÞ comparable to a controlGaAs p-i-n solar cell [35]. Fabrication of InAs/GaAs QD solar cells(QDSCs) with enhanced photocurrent and no degradation in open

Voc compared to a solar cell grown without QDs and composedsolely of wetting layers is being investigated by Guimard et al. [36].Moreover the InAs/GaAs QDs are very interesting materials in laserapplications; the different and efficient characteristics of InAs/GaAs QDs are being investigated in lasers monolithically grown onSi [37] and Ge [38] substrates, and high performance lasers [39].InAs/GaAs QDs are being used in variety of research and techno-logical areas including light emitting diode (LED) [40], intersub-band detectors [41], and photodetectors [42,43]. In the appli-cations of InAs/GaAs QDs the wavelength is specified for theapplications; in some application like LEDs, lasers and commu-nications it is between 1 μm and 1.3 μm [44–48].

In nuclear physics motion of the free electrons is a basicproblem which has a conclusive influence on the abundance ofmetallic clusters. The movement of electrons occurs in well-defined orbitals around the central nucleus and its mean fieldpotential (MFP). The MFP is being produced by positively chargedions and the rest of electrons, and in MFP the potential is describedby some free parameters such as depth, width, and slope ofpotential barrier, which must be fitted to experiments. The MFPis empirical and its example could be given by Woods–Saxonpotential [49]. Actually in order to study nucleon–nucleon scatter-ing, Woods and Saxon proposed a potential, today known asWoods–Saxon (WS) potential [50]. The WS potential is a reason-able potential for nuclear shell models and hence attracts lots ofattention and it is one of the important short-range potentials inphysics [51]. This potential is used to represent the distribution ofnuclear densities [52] as well as it is applied to numerous problemsin condensed matter physics, atomic physics, and chemical physics[51]; moreover adjusting the WS potential parameters has significantimpact on the optical properties [52]. The WS potential also is beingapplied to solve the Duffin–Kemmer–Petiau (DKP) equation which isthe first order relativistic wave equation and represents spin-0(scalar) and spin-1 (vectorial) particles [53].

Few studies [54–57] have been carried out on the effects of WSpotential in solving 1D position-dependent mass Schrödinger equa-tion, which calculated the function, transmission and reflectioncoefficients. Costa et al. [58] simulated the spatial confinement; theirmethodology enables the study on a wide range of systems andconfinement regimes by using variational methods to investigate WSpotential in a QD/atom system by varying two parameters in themodel potential. Investigation of two electrons confined in a QD withthe WS potential by using a numerical diagonalization method of theHamiltonian matrix within the effective mass frame was per-formed by Xie [54]. The scattering solutions of the 1D Schrö-dinger equation for WS potential within the position-dependentmass formalism were carried out by Arda et al. [57], and theyobtained the function, transmission and reflection coefficientsin terms of Heun's function. Lu et al. [55] studied the effects ofintense laser fields on intersubband nonlinear properties ofdonor impurities in a GaAs/AlGaAs QD with WS potential witheffective mass approximation. They have showed the depen-dence of linear and third order susceptibility on laser intensityand a few values for dot radius, as well as barrier height andconfinement barrier slope. Recently, Aytekin et al. [60] studiedthe effects of an applied electric field on the refractive indexchanges, optical rectification, and second and third harmonicgeneration in two dimensional QDs with WS potential in theframework of density matrix. It is demonstrated that opticalrectification susceptibility is strongly dependent on the appliedelectric field. The applied electric filed value also causes the secondharmonic generation susceptibility to shift the peak position. More-over, their calculations showed an increase in the third harmonicgeneration susceptibility by increasing the applied electric field.

WS potential has been used to explain the effect of interdiffusion on electronic states and light absorption [21–23],

A. Khaledi-Nasab et al. / Physica E 60 (2014) 42–49 43

excitonic states [24], and impurity states [25] in QDs, and electro-nic states of semiconductor super-lattices [26].

In this communication, the Schrödinger equation for dome-shaped InAs/GaAs QDs coupled to their wetting layer is solvednumerically. Both constant and WS potentials have been investi-gated by performing a comparison. Using envelope function anddensity matrix approach, the linear susceptibility and groupvelocity of InAs/GaAs QDs will be explored.

2. Theory and Model

InAs QDs and wetting layer grown on GaAs matrixes are theworkhorse system for optical simulations. InAs grows on GaAsinitially layer-by-layer, but after just 1.5 monolayers InAs QDs formand the QDs are subsequently capped by GaAs [59]. The envelopefunction in single QDs was calculated by solving Schrödingerequation within the effective mass approximation using a finiteelement method (FEM). Throughout this work, the InAs QDs areembedded on InAs wetting layer in a large GaAs matrix, as well asthe GaAs cap covering the QDs according to the SEM images of realgrown QDs [59]. In the following, a model procedure will bepresented that allows us to determine energy eigenvalue andassociated envelope function of any cylindrically symmetric QDsstructure. The starting point is the 1-band Schrödinger equation inthe effective mass approximation

�ℏ2

2∇

1m*

∇ψð x!Þ� �

þVeð x!Þψð x!Þ¼ Eψ ð x!Þ; ð1Þ

where ℏ is the well-known Planck's constant, mn is the electroneffective, ψ ð x!Þ is the electronic envelope function, E is theeigenenergy, and Veð x!Þ is the potential energy. Using cylindricalsymmetry for dome-shaped QD, the envelope function can beseparated. Thus the envelope function will consist of a part withcoordinate dependence (r and z) in addition to another angular-dependent ðφÞ unit. In cylindrical coordinates x!¼ ðz; r;ϕÞ,Veð x!Þ¼ Veðz; rÞ where z and r are the axial and radial coordinates,and ϕ is the azimuthal angle ranging from 0 to 2π. Cylindricalsymmetry allows us to use the separation of variables technique asfollows for the envelope function:

ψð x!Þ¼ χðz; rÞΦðϕÞ: ð2ÞAfter some mathematical treatment Eq. (1) will be reformu-

lated as

�ℏ2

2∂∂z

1mn

∂χðz; rÞ∂z

� �ΦðϕÞ�ℏ2

2r∂∂r

rmn

∂χðz; rÞ∂r

� �ΦðϕÞ

�ℏ2χðz; rÞ2mnr2

∂2Φ∂ϕ2 þVeðz; rÞχðz; rÞΦðϕÞ ¼ Eχðz; rÞΦðϕÞ ð3Þ

Here we divide Eq. (3) by ΦðϕÞχðz; rÞ=mnr2; therefore Eq. (3) isrearranged as

mnr2

χðz; rÞ�ℏ2

2∂∂z

1mn

∂χðz; rÞ∂z

� ��ℏ2

2r∂∂r

rmn

∂χðz; rÞ∂r

� � !

þmnr2ðVeðz; rÞ�EÞ ¼ �ℏ2

2∂2Φ∂ϕ2

1ΦðϕÞ � �ℏ2

2l2 ð4Þ

It is straightforward to say that the solution of Eq. (4) could beproposed as

ΦðϕÞ ¼ eilϕ; ð5Þwhere l is a separation constant. The envelope function must be

single-valued under 2π rotation. In our modeling l can takes

values of l¼ 0; 71; 72;⋯:

�ℏ2

2∂∂z

1mn

∂χl∂z

� ��ℏ2

2r∂∂r

rmn

∂χl∂r

� �þ ℏ2n2

2mnr2þVeðz; rÞ

!χl ¼ Eχl ð6Þ

Due to existance of two different materials in the Schrödingerequation (in simplified or orginal form), there are discontiniouscoefficients under spatial derivative operators; hence physically, inthe modeling procedure the following boundary condition mustbe satisfied:

ψACðQ Þ; and �∇ψð x!Þmn

: n! !

QD=wetting layer

¼ �∇ψð x!Þmn

: n! !

matrixes

;

ð7Þ

where C is the class of continuous function, Q is the spatial ðz; rÞdomain of interest, and n is the outward normal vector. Theenvelope function solution for a stepwise constant effective masspotential is given by

ψnkðz; r;ϕÞ ¼ AnkZnkðzÞJnðkrÞeilϕ ð8Þ

Znk is acquired from the solution of

∂∂z

1meðzÞ

∂Znk

∂z

� �þ 2ℏ2 E�veðzÞ�

ℏ2k2

2meðzÞ

!Znk ¼ 0 ð9Þ

ψnk must satisfy the following condition:

∂ψnk∂r �KðJn� 1ðkrÞ� Jnþ 1 ðkrÞÞ

2JnðkrÞ ψnk ¼ 0; if JnðkrÞa0

ψnk ¼ 0; if Jn ¼ 0

8<: ð10Þ

Because

Jn�1 ðyÞ� Jnþ1ðyÞ ¼2∂JnðyÞ

∂yð11Þ

As it is obvious, the 3D Schrödinger equation was simplified to2D equation. In this regard, we considered an electron confinedin a two-dimensional QD and replaced VðrÞ by VWSðrÞ as

3nm

60nm

100n

m

GaAs Matrix

GaAs Matrix

InAs wetting layer

1

2

3

4

2’

5

6

7

8

InAs QD

= = 0

Fig. 1. The geometry and numbered boundaries of computational area ofInAs/GaAs QDs.

A. Khaledi-Nasab et al. / Physica E 60 (2014) 42–4944

WS-potential and given by

VWSðrÞ ¼V0

1þexpðR0�r=γÞ ð12Þ

where V0 is the potential depth with specific value for eachmaterial, R0 is the radius of QD, and γ denotes the confinementbarrier slope [60].

2.1. Boundary condition

In order to present more realistic solution, appropriate bound-ary conditions must be taken under consideration. Fig. 1 shows thegeometry, numbered boundaries as well as the dimensions ofdome-shaped InAs/GaAs QDs.

The confined electronic states are the main interest; hence,the Dirichlet boundary condition ðψ ¼ 0Þ is imposed for boundaries1 and 3. In order to satisfy the continuity condition ð�1=me∇ψ

ð x!Þ: n!ÞInAs ¼ ð�1=me∇ψð x!Þ: n!ÞGaAs (Eq. (8)) is applied for bound-aries 5, 6, 7, and 8.

With attention to the symmetry of the problem the computa-tional area could be divided into two equal parts by line 20;therefore, boundary 2 will be replaced by 20 (dashed-dottedvertical line in Fig. 1). It is obvious for lZ1 the Dirichlet boundarycondition must be considered, and the reason is to guarantee thatthe expression ℏ2=2mel

2=r2χ l will not diverge at r¼ 0. On the otherhand, in special case, when l¼ 0, to ensure the presence of ∇

!ψ at

r¼ 0, the Neumann boundary condition must be imposed. In ourcomputational approach we will consider l¼ 1; hence we willapply the Dirichlet boundary condition. For relatively large valuesof r, the envelope function for InAs QDs/wetting layer and GaAsmatrixes, asymptotically approaches, either, the envelope func-tion for the pure wetting layer problem or, zero if the state istrue QD state; consequently Eq. (10) could be considered forboundary number 4. Finally based on mentioned symmetry weonly take half of Fig. 1 under consideration; in the other wordsFig. 1 is divided by 20 and the simulation carried out for thisgeometry.

Using a density matrix approach and perturbation theory, thelinear susceptibility for two-level system will be derived as [25,61]

χð1ÞðωÞ ¼ Nε0ℏ

M221

1ω21�ω� iγ21

þ 1ω21þωþ iγ21

� �� �ð13Þ

With respect to the probe frequency, the derivative of thedispersion has a major role in calculation of the group velocity.For sufficiently short propagation distances, pulse distortion is nottoo severe, and the motion of the pulse can be described by vggiven by vg ¼ c=ng , where ng ¼ nþω dn=dω . Therefore, the group

Table 1Energy eigenvalues (for ground and first excited states) and the transition dipolemoments for Woods–Saxon potential (WSP) and constant potential (CP).

R (nm) E0 (eV) E1 (eV) M12 (10�27)

WSP CP WSP CP WS CP

8 0.2664 0.276108 0.2906 0.292663 �0.0687 �0.07660013 0.1186 0.156374 0.2291 0.28252 �0.5476 �0.3346016 0.0878 0.115865 0.1705 0.222567 �0.6308 �0.5505020 0.0623 0.082379 0.1197 0.161054 �0.7202 �0.64270

Fig. 2. The ground state envelope function with constant potential (a) R¼8 nm and (b) R¼13 nm and Woods–Saxon potential (c) R¼8 nm and (d) R¼13 nm for a InAs/GaAsQD/wetting layer system. Woods–Saxon potential causes delocalization on the envelop function, where it shifts through wetting layer to the GaAs substrate.

A. Khaledi-Nasab et al. / Physica E 60 (2014) 42–49 45

velocity vg is expressed as [61–63]

Vg ¼c

1þχð1ÞðωÞþω dχð1Þ=dωð14Þ

As Eq. (6) implies, for a negligible real part of susceptibility, thegroup velocity can be significantly reduced via a steep positivedispersion [62,64,65]. Strong negative dispersion, on the otherhand, can lead to an increase in the group velocity and even anegative group velocity [66].

In real experiments, it is easier to measure the delay timewhich is given by

ΔT ¼ Lvg

� LC¼ ðng�1ÞL=C ð15Þ

where L is the length of optical medium and C is speed of light. Wewillwork with ng�1 which has direct correspondence to delay time.

3. Result and discussion

In order to carry out the numerical solution, V0 was set to 0 and0:697 eV for InAs and GaAs, respectively, and the electron effectivemass was set to 0:023 me for InAs and 0:069 me for GaAs [67].In WS potential the confinement barrier slope γ is set to 30 Å [55].In this study, the values of R0 ¼ 8; 13; 16 and 20 nm on WSpotential are used. To obtain the numerical results for χð1Þ andgroup index, the following parameters have been used: carrierdensity N¼ 3� 1022 atoms=cm2 [25], vacuum permittivityε0 ¼ 8:85� 10�12, relaxation rate γ21 ¼ 5� 1012 s�1 [25], speed

of light in vacuum c¼ 3� 108, transition frequencies ℏω21 ¼E2�E1 (as output of numerical simulation taken from Table 1),and transition dipole moments M21 ¼/ψ2j�erjψ1S which iscalculated by using numerical envelope function and homemadecodes of the Simpson integration method written in MATLAB.

Table 1 presents the result of energy eigenvalues, dipolemoments for both constant potential (CP) and WS potential(WSP). These data obtained directly from the numerical treatmentfor InAs/GaAs QDs by FEM solution for simplified form of Schrö-dinger equation (Eq. (6)). According to Table 1 the energy eigen-values for ground and first excited states take relatively smallervalues for WS potential in comparison to constant potential, whilethe transition dipole moment gets greater absolute values.

Fig. 2 illustrates the ground state envelope function. Fig. 2(a)and (b) shows the envelope function with constant potentialðV ðrÞ ¼ V0Þ and Fig. 2(c) and (d) shows the envelope function withWS potential ðV ðrÞ ¼ VWSðrÞÞ for R¼8 and 13 nm, respectively. Theenvelope function for ground state moves slightly toward the GaAsmatrix for WS potential, while in the upper radii, the amount ofleakage envelope function stays relatively the same as a constantpotential. In the better words, in the ground state for both WSpotential and constant potential at R¼8 nm, we have shared quantumwell (wetting layer) and QD states; nevertheless for R¼13 nm the onlypure QD state remains on the envelope function profile. The effect ofWSP potential is small, leading the envelope function to take slightlybigger peak than constant potential.

Fig. 3 shows the distribution of first excited state envelopefunction for the constant potential (Fig. 3(a)–(d)) and WS potential(Fig. 3(e)–(h)). Fig. 3(a) shows that for R¼8 nm the envelope

Fig. 3. The first excited state envelop function with constant potential (a) R¼8 nm, (b) R¼13 nm, (c) R¼16 nm, and (d) R¼20 nm and Woods–Saxon potential (e) R¼8 nm,(f) R¼13 nm, (g) R¼16 nm, and (h) R¼20 nm for InAs/GaAs QD/wetting layer system. In constant potential the couple lobes are completely inside the InAs QD and WL whilefor Woods–Saxon potential the distribution of envelope function on GaAs matrixes could be observed. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

A. Khaledi-Nasab et al. / Physica E 60 (2014) 42–4946

function is distributed in wetting layer and QD. In the other wordsthe envelope function consists of a quantum well state, which is avalley, located wetting layer, and a confined QD state which is apeak on QD region. According to Fig. 3(b) at R¼13 nm the QD statechanges its phase from peak to valley, also the quantum well statemoves toward QD region and one can say that we have sharedstate between QD and wetting layer. As Fig. 3(c) and (d) shows theenvelope function leaves the wetting layer and it is localized insidethe dot for R¼16 and 20 nm. The phase of both lobes reverses tobecome peak for the top lobe and valley for bottom one. Accordingto Fig. 3(e) with WS potential, the distribution of envelopefunction is almost the same as constant potential; however, thefirst lobe shows a leakage to GaAs matrix albeit it is very slight andnegligible. According to Fig. 3(f), at R¼13 nm both lobes areinside the dot with considerable tail-shaped leakage toward theGaAs matrix. Moreover in comparison to the same dimensionwith constant potential, first of all there is no quantum well statebut only couple of QD states, secondly, the top and bottom lobes ofenvelope function are peak and valley respectively while theywere vice versa for constant potential, and finally the envelopefunction is not entirely confined inside the QD region owing toleakage to matrixes. Fig. 3(g) and (h) explains that for R¼16 and20 nm by increasing the radius of QD, the leakage of envelopefunction leakage becomes stronger and namely the delocalizationincreases dramatically. While in constant potential upper radiicorrespond to more localization of envelope function insidethe QD. Fig. 3(g) indicates that the overall profile of envelopewave function is similar to its mutual dimension with constantpotential but with significant amount of leakage. Furthermoreas Fig. 3(h) shows, the envelope function again experiences suddenchanges in the location of peaks (red area) and valley (blue area) incomparison to its same radius (Fig. 3(d)) constant potential. In otherwords in constant potential the red peak and blue valley werelocalized at the top and bottom of QD respectively; nevertheless itwould be vice versa for the same QD with WS potential.

Fig. 4 shows both real part and imaginary part of linearsusceptibility versus the probe field frequency for (a) R¼8 nmand (b) R¼13 nm, with constant and WS potential. According toFig. 4(a), the absorption and dispersion for constant potentialreveal the bigger peaks in comparison to WS potential. ForR¼8 nm around 10 � 1012 Hz movement on the position of thepeaks occurs. The amplitude of the peaks is relatively 0:005smaller in WS potential. Fig. 4(b) shows a big shift in amplitudeof the peaks between constant and WS potentials. As it has beenclearly observed in Fig. 3(a) and (f), at R¼13 nm for constantpotential, the envelope function has shared states in QD andwetting layer, while it has couple if QD states with leakage inWS potential. At R¼13 nm the peaks take relatively remarkablereduction when overall profile of linear susceptibility appears. Theamplitude of the peaks becomes around 1=4 of its value onconstant potential, and also around 20 � 1012 Hz movement inthe position of the peaks shows up.

Fig. 5 shows the absorption and dispersion profiles for(a) R¼16 nm and (b) R¼20 nm. It is obvious that the dissimilaritybetween the WS and constant potentials is relatively less thanR¼13 nm. Fig. 5 indicates that by rising the dimension of QD from16 nm (Fig. 5(a)) to 20 nm (Fig. 5(b)) the whole profile of linearsusceptibility moves toward lower frequencies. The resonancefrequency of the peaks for WS is 40� 1012 Hz bigger thanconstant potential. Indeed in both constant and WS potentials,the peak takes higher absolute values by increasing the radius.

One of the important parameter which has direct associationwith linear susceptibility is group velocity and its mutual para-meter group index. Fig. 6 depicts a comparison between groupindices for R¼8 nm. Both curves for WS and constant potentialsare subjected to negative delay time and consequently

superluminal light propagation. Obviously, the peak of constantpotential occurs in higher frequencies than that of WS potential.The movement is about 10� 1012 Hz toward higher frequencies.

Fig. 7 illustrates the group velocity for (a) constant and (b) WSpotentials for R¼13, 16, and 20 nm. According to Fig. 7(a) withincreasing the dot radius, blue-shift takes place. A blue-shift isaround 40� 1012 Hz between the resonance peaks. On the otherhand, the amplitude of superluminal peaks tends to have slightsmaller values by size escalation.

Fig. 7(b) shows that for R¼13 nm, the superluminal peak takesrelatively half of the amplitude value at 25� 1012 Hz higherfrequency of its corresponding peak on constant potential. Sur-prisingly at R¼16 nm and R¼20 nm the group index takes almostthe same values but in upper frequencies for WS potential. More-over in WS potential the amplitude of group index is the same forR¼16 nm and R¼20 nm, with red-shift for upper radii.

4. Conclusion

In this work, the effects of Wood–Saxson (WS) potential ondispersion curves and group velocity of a dome-shaped InAs/GaAsQD with its wetting layer were investigated. The single band

Fig. 4. Real and imaginary parts of linear susceptibility for (a) R¼8 nm and(b) R¼13 nm versus the probe filed frequency by constant and WS potentials.The transition frequency and transition dipole moments are chosen in accordanceto Table 1.

A. Khaledi-Nasab et al. / Physica E 60 (2014) 42–49 47

Schrodinger equation in effective mass approximation was used toexplore the envelope function and energy eigenvalues. A finiteelement method (FEM) was used to solve Schrodinger equation.It is found that WS potential causes a delocalization in the

envelope function for radius upper than R¼13 nm. In constantpotential, the envelope function is completely inside the wettinglayer and QD region; while by employing the WS potential, tail-shaped leakage of envelope function appears. This leakage sig-nificantly appears in the first excited state. It is also concluded thatthe localization of envelope function on constant potential leads tohigher absolute values of absorption/dispersion curves in allstudied models. For WS potential, the peak occurs at higherfrequencies than those of constant potential. Consequently, theWS potential causes blue-shift in the whole profile of linearsusceptibility and group index. In addition to blue-shift on groupvelocity, it is found that for R¼13 nm, the negative superluminalpeaks take relatively smaller values than those in WS potential.Having reduced absorption is beneficial in slow/fast light applica-tions. The overall results of this study show that the WS potentialcan be employed to reduce the absolute values of the peaks.Moreover engineering the optical properties of QDs is possible byemploying the WS potential.

Acknowledgment

The first author would like to appreciate the helpful discus-sion of Mr. Mehdi Kashisaz (Department of Chemical & Process

Fig. 6. The comparison of group velocity for R¼8 nm for constant and WSpotentials. The transition frequency and transition dipole moments are chosen inaccordance to Table 1.

Fig. 7. The group velocity for R¼13, 16 and 20 nm for (a) constant potential and(b) WS potential. The transition frequency and transition dipole moments arechosen in accordance to Table 1.

Fig. 5. Real and imaginary parts of linear susceptibility for (a) R¼16 nm and(b) R¼20 nm versus the probe filed frequency by constant and WS potentials. Thetransition frequency and transition dipole moments are chosen in accordance to Table 1.

A. Khaledi-Nasab et al. / Physica E 60 (2014) 42–4948

Engineering, Universiti Kebangssan Malaysia, 43600 UKM Bangi,Malaysia) in preparing the manuscript of this paper.

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