Self-assembled strained pyramid-shaped InAs/GaAs quantum dots:The effects of wetting layer thickness on discrete andquasi-continuum levels

Mohammad Sabaeian n, Mohammadreza ShahzadehPhysics Department, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran

H I G H L I G H T S

� Three-dimensional pyramid-shapedInAs/GaAs quantum dots coupled totheir wetting layer were investigated.

� Schrodinger equation was solved insingle-band effective mass approxi-mation.

� Isosurfaces of probability density ofS-state, P-state, and quasi-continuumstate were simulated.

� Effects of quantum dot size andwetting layer thickness on energyeigenvalues of bound states andquasi-continuum states and theirtransition energies were studied.

� Polarization of the transitions wasdetermined.

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 6 February 2014Received in revised form11 March 2014Accepted 17 March 2014Available online 25 March 2014

Keywords:Quantum dotEnvelop functionPyramid-shapedWetting layer

a b s t r a c t

The effects of wetting layer thickness and quantum dot (QD) shape on S-state, P-state, and quasi-continuum energy levels of three dimensional strained pyramid-shaped QDs were investigated in theframework of single-band and effective mass approximation. The energy eigenvalues as well astransition energy of the non-zero transitions were calculated as a function of the pyramid height andWL thickness. The polarization of transitions was also studied. An overall red shift was revealed withincreasing the WL thickness which differs in value for different transitions. The results were shown to beapproved by experimental observations.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Quantum dots (QDs) have attracted tremendous attention dueto their potential applications in solar cells [1], biomedicine [2],photo-detectors [3], quantum cascade lasers [4], single-photon

sources [5,6], and quantum computing [7]. It has been found outthat the electronic and optical properties of QDs are extremelysensitive to size, stoichiometry, and shape which leads to a tunableabsorption-, emission-, and polarization-sensitive- spectra [8–10].This dependence comes from relatively small number of atomsforming such structures.

Self-assembled QDs are formed during epitaxial growth ofhetero-structures [11]. Three-dimensional islands start growingonce the active layer thickness reaches to a critical value. Strain-

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.03.0151386-9477/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: sabaeian@scu.ac.ir (M. Sabaeian).

Physica E 61 (2014) 62–68

driven growing layer can thereafter form QDs in a variety of shapes,depending on techniques and conditions used. The thickness of socalled wetting layer (WL) attached to QDs depends on the materialsand growth conditions [12,13]. Islanding transition during epitaxialgrowth of InGaAs QDs was discussed in Refs. [14,15].

Although the role of WL is significant in electronic and opticalproperties of QDs [16–25], there has been noticeably fewer researchon the effects of WL on QD properties [16]. In spite of discardingWLin theoretical works [26–34], self-assembled QDs cannot be grownwithout WL. This ignorance in theoretical works may arise frommathematical complexities in making analytical models.

The inevitable impact of wetting layer on QD properties such asoscillator strength and transition dipole moments has beenrevealed in experimental works [35–37]. It is known nowadaysthat the WL acts as a reservoir for charge carriers to escape intothe dots [17,38] as well as jumping between coupled QDs [7]. TheQD and WL/barrier states form a coupled system [39]. Wettinglayer has shown significant effects in QD lasers; it restricts themodulation response of QD based lasers [23] and affects the gain-current characteristic [40]. It also influences dark current in QDphoto-detectors [23].

Single-band Schrodinger equation in the framework of effectivemass approximation is widely used to calculate discrete as well asquasi-continuum states (known as WL states) of QDs at the Γ point[35,41–49]. The transitions between electronic states, however,depend strongly on the polarization of incident light. Narvaez et al.calculated the intersubband absorption spectrum of InGaAs QDs[50]. They concluded that for an InAs/GaAs quantum dot, theintersubband transition between ground state (S-state) and thefirst excited state (P-state) is purely in-plane polarized. Zhang et al.calculated the intersubband absorption for pyramid-shaped InAs/GaAs QDs [51]. They concluded that in-plane polarized transitionsare negligible from ground state to WL [51]. Sun et al. investigatedthe effects of WL thickness on the conduction- and valance-bandedges; However, a detailed study of energy eigenvalues as afunction of WL thickness and dot size was not carried out [52].Chen et al. measured two transitions related to heavy- and light-hole in the WL by adopting the reflectance difference spectroscopyto study the effects of the WL in the InAs/GaAs QDs [17,53]. Acomparison between GaAs/AlGaAs QDs grown by modified dropletepitaxy method without WL, InAs/GaAs QDs and GaAs/AlGaAs QDsgrown by the same method with tunable WL size was made bySanguinetti et al. [22,54]. They found strong influences of theexistence of WL on the optical properties of QDs [54]. Also,inspecting the effects of the WL in InGaAs QDs without WL grownby heterogeneous droplet epitaxy, revealed a crucial role for WL in anensemble of coupled QDs [18]. Sun et al. investigated the carriercapture into the WL in InAs/GaAs QDs using up-conversion spectro-scopy [55]. Brehm et al. investigated the actual onset of island-formation and critical WL thickness in the Ge/Si system and presenteda thermodynamic analysis to interpret their data to point out thesignificance of theWL [56]. Studies by Lee et al. [36] and Sun et al. [52]showed that the change in WL thickness widens the confiningpotential width without any significant change in potential depth.

In this work, the single-band Schrodinger equation will besolved for pyramid-shaped InAs/GaAs QD coupled to its WL usingfinite element method. A height-to-base ratio ðR¼ h=bÞ para-meter of 0.07 to 1 will be considered for pyramid-shaped QDs[48]. In order to shed light on the effects of WL on electronicproperties of QDs, four WL thicknesses of 0.5, 1, 1.5, and 2 nmwere considered along with a comparison with a model withoutWL [21,42,57,58]. The ground state (S-state), degenerate firstexcited states (P-state), and quasi-continuum states (wettinglayer states) will be considered and their corresponding energyvalues along with transition energies will be fully investigated asa function of the QD size.

2. Theory

The single-band Schrodinger equation in the framework ofeffective mass approximation is used to calculate electron energyeigenvalues.

The strain-included Hamiltonian of the QDs can be described as[59]:

H¼ �ℏ2

2∇

1mnðx; y; zÞ∇þV ðx; y; zÞ ð1Þ

where

mnðx; y; zÞ ¼mn

InAs; in QDmn

GaAs; elsewhere;

(ð2Þ

is effective mass with mn

InAs and mn

GaAs as the electron effectivemass in InAs and GaAs, respectively, and

Vðx; y; zÞ ¼0; in QD

ΔEC ; elsewhere;

(ð3Þ

is the strained potential barrier. ΔEC is the conduction band offsetbetween the InAs and GaAs in which when strain is taken intoaccount, the values of V ¼ ΔEC ¼ 500 meV [60,61] along witheffective masses of mn

InAs ¼ 0:04 me [61,62] and mn

GaAs ¼ 0:067me

[58,63] are used.A cross section of simulated system is illustrated in Fig. 1.

According to this figure, for the top and bottom boundaries (1 and 5),the Dirichlet boundary condition, ψ ¼ 0, and for other outer bound-aries (2, 3, 4, 6, 7, and 8) the Neumann boundary condition,n̂:ð∇ψÞ ¼ 0, is used. Also, the boundary condition of n̂� ð∇ψ=mnÞInAs ¼ n̂� ð∇ψ=mnÞGaAs is employed for interface boundaries,due to the potential finiteness [59].

In order to investigate the effects of QD size (or shape),different height-to-base ratio ðR¼ h=bÞ is considered in calcula-tions [48]. QDs considered in this work have a constant baselength of 14 nm with a varying height from 1 to 14 nm. Thecalculations will be carried out for WL thicknesses of 0.5, 1.0, 1.5,and 2 nm and the results will be compared to a systemwithout WL.

When the envelop functions were calculated by solving Eq. (1),the electron transition dipole moments are calculated using therelation Mif ¼ ⟨ψ f jð�e r!Þjψ i⟩, with i and j being the initial andthe final states. Transition dipole moments determine the allow-able transitions. Intersubband transitions in InAs/GaAs QDs can becategorized into three categories: in-plane polarized transitions,z-polarized transitions, and forbidden transitions. For all six

Fig. 1. Cross section of simulated QD.

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energy states, there are fifteen possible transitions. As it will beshown later in this work, some of these transitions vanish.

3. Results

Six energy eigenvalues with corresponding envelop functionswere calculated by solving the Schrodinger equation. The stateswere labeled as jei⟩ where i¼ 1;2;…;6. The iso-surfaces of envelopfunctions are shown in Fig. 2 and the energy diagram is plotted inFig. 3. As Fig. 3 shows je2⟩ and je3⟩ are degenerate. Also, je4⟩, je5⟩and je6⟩ constitute a quasi-continuum set, i.e. states with closeenergies. Our calculations revealed that the transitions je2⟩-je1⟩,je3⟩-je1⟩, je4⟩-je2⟩, je5⟩-je2⟩, je6⟩-je2⟩, je4⟩-je3⟩, je5⟩-je3⟩, andje6⟩-je3⟩ are allowable and in-plane polarized. Only, a singletransition of je4⟩-je1⟩ is z-polarized. In this regard, two transitionsare of great importance. The first one is je2⟩-je1⟩ (or je3⟩-je1⟩)which is known as P-to-S transition [17,64]. The other one isje4⟩-je1⟩ which describes a WL-to-ground state transition (z-polarized polarization). The latter is of significance, since this isthe only direct transition from wetting layer to ground state andhas significant application in intersubband quantum dot devicessuch as quantum dot lasers and photo-detectors [38,65]. This z-polarized transition from WL to S-state is practically of greatimportance, because it can be easily interpreted as the transfer ofcharge carriers from the WL into the dot during the transition inz direction [66].

Allowable transitions obtained in this work, according to thecalculated transition dipole moments values, are in good agree-ment with – and strongly approved by – previous experimentaland theoretical works [43,51,66,67].

The envelop functions of first three states are mainly located inthe pyramid region and not in wetting layer. These are calledS-state (ground state) and Px- and Py-state (the first excited states)according to our notation of je1⟩, je2⟩, and je3⟩, respectively.

The ground state energies as a function of the height for QDswithout WL and with WL with thicknesses of 0.5, 1.0, 1.5, and2.0 nm are shown in Fig. 4. The first excited state energy asa function of height is represented in Fig. 5. As one can see fromFigs. 4 and 5, the energy eigenvalues decrease with increasing theQD height. Furthermore, as the WL thickness is increased, thevalues of energy decrease. The latter can be explained through theless confinement effect of the electron agreeing with Heisenberg'suncertainty principle.

Due to one dimensional confinement of electron in the (natural)quantum dot with WL, the WL-states form a quasi-continuumenergy band (Fig. 3 [57,65,68,69]. Three states expand mostly inthe WL. Variations of these WL-state energies with increasing theQD height reveal different trends. Size dependence of e4j i and e5j ienergy has been represented in Figs. 6 and 7, respectively. For thesestates, while the electron energy drops noticeably with increasingthe QD height for WL thickness of 0.5 nm, however, for 2 nm WL

Fig. 2. The isosurface of the envelop functions related to six energy eigen-values.

Fig. 3. The energy diagram of the energies of the system.

Fig. 4. Ground state energy as a function of QD height without WL and with WL ofthicknesses of 0.5, 1.0, 1.5, and 2.0 nm.

Fig. 5. First excited state energy as a function of QD height without WL and withWL of thicknesses of 0.5, 1.0, 1.5, and 2.0 nm.

M. Sabaeian, M. Shahzadeh / Physica E 61 (2014) 62–6864

thickness, the drop in energy with increasing the QD height isconsiderably reduced. The variations of e6j i state energy versus theQD height are illustrated in Fig. 8. As Fig. 8 shows, for this state the

variations of energy are moderate than two previous cases withincreasing the QD height. This is because of locating the envelopfunction mainly in wetting layer, such that with increasing the QDheight, no noticeable change in envelop function happens.

The reduction of energy values with increasing the WL thick-ness is attributed to the entrance of the envelop function into theWL as shown quantitatively in Fig. 9(a)–(d). In Fig. 9 for a pyramidheight of 3 nm, the envelop function of je1⟩ state is plotted for twoWL thicknesses of 0.5 nm, (Fig. 9a), and 2.0 nm, (Fig. 9b), and theenvelop function of je2⟩ state is shown for WL thicknesses of0.5 nm, (Fig. 9c), and 2.0 nm, (Fig. 9d). From Fig. 9 it can be seenthat when the WL thickness is increased, envelop function shiftsfrom WL towards the pyramid zone.

As the emitting wavelength of QDs is of great importance, inthis work we inspect the energy difference between the statesproportional to the emission frequency with the change of wettinglayer thickness. Fig. 10 shows the transition energy of je2⟩-je1⟩transition, which is an important transition, versus the height ofQD for several wetting layer thicknesses. According to this figure,first, the transition energy increases with QD height. Second, thetransition energy is decreased with increasing the wetting layerthickness. The thicker the wetting layer, the smoother the varia-tion of energy difference versus the QD height. In fact, when thewetting layer is increased, the effect of QD confinement is reduced.For je4⟩-je1⟩ transition energy, Fig. 11 represents a different trendshowing a critical height for QD which varies with changing thewetting layer thickness. For thin wetting layers, for example 0.5 or1 nm, the QD with small height to �7 nm, the je4⟩ state lies mainlyin wetting layer. When QD height is increased, the je4⟩ increases itsoverlapping with the pyramid region, leading to a decrease inenergy difference. Similar behavior is also seen for je4⟩-je2⟩transition representing in Fig. 12, except with a lower energydifference.

The behavior of je5⟩-je2⟩ transition energy, however, is quitedifferent from those of je3⟩-je2⟩ and je4⟩-je2⟩ transitions, whichshows a monotonic increase in transition energy with increasingthe WL thickness (Fig. 13). For je6⟩-je2⟩ transition energy, similartrend but somehow moderate is seen in Fig. 14.

Figs. 10–14 clearly reveal a red shift in transition wavelength(i.e. a decrease in transition energy) when the WL thickness isincreased. Such effect has been reported from experimental mea-surements by Sanguinetti et al. [54], Choi et al. [70], Xu et al. [71], andKim et al. [72].

By calculating the relative change of ground state energybetween QDs with the height of 1 nm and that with 14 nm, i.e.E1 nm�E14 nm=E1 nm � 100, for QDs without WL, and QDs with WLthicknesses of 0.5, 1, 1.5, and 2 nm, it was revealed that increasingthe WL thickness makes this quantity smaller (Fig. 15).

At the end of this article, in order to show the variations ofquasi-continuum energy band for several WL thicknesses, Fig. 16 isplotted. The figure shows the trend of energy variations of thisband in which with decreasing the WL thickness, variationsbecomes steeper. je4⟩ state which has a noticeable portion ofenvelop function in pyramid zone, in particular, shows a moresensitive behavior to pyramid height than the other two.

The results obtained in this work indicate that the effects ofchanging the WL thickness are more dominant in the transitionsconcerned with WL states than that of P-to-S transition which ismainly concerned with states located mostly in dot region. Thiscan be attributed to the subtle difference between increasing dotsize and the WL thickness. When the dot height is increased, thevolume of the 3D confined structure increases leading to entranceof envelop function from the WL towards the dot [73]. Onthe contrary, when the WL thickness is increased, the size of the1D confined structure increases leading to different results dis-cussed in the text. Since both S- and P-state discussed in this work

Fig. 6. Variations of je4⟩’s energy with increasing QD height for four WL thicknessesof 0.5, 1.0, 1.5, and 2.0 nm.

Fig. 7. Variations of je5⟩’s energy with increasing QD height for four WL thicknessesof 0.5, 1.0, 1.5, and 2.0 nm.

Fig. 8. Variations of je6⟩’s energy with increasing QD height for four WL thicknessesof 0.5, 1.0, 1.5, and 2.0 nm.

M. Sabaeian, M. Shahzadeh / Physica E 61 (2014) 62–68 65

are bound states and mainly located in the dot region, increasingthe WL thickness does not severely change the properties of P-to-Stransition; but –as it can be seen in this work- is able to change theproperties of transitions from WL states.

4. Conclusion

Six energy eigenvalues categorized in S-state, P-state, andquasi-continuum states, along with their mutual transition energieswere calculated for a natural pyramid-shaped QD with wetting layer.

Fig. 9. The envelop function (a) and (b) ψ1 ¼ r!je1⟩D

, (c) and (d) ψ2 ¼ r!je2⟩D

for QD height of 3 nm with WL thickness of 0.5 nm and 2.0 nm.

Fig. 10. je2⟩-je1⟩ Transition energy as a function of QD height for four WLthicknesses of 0.5, 1.0, 1.5, and 2.0 nm. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. je4⟩-je1⟩ Transition energy as a function of QD height for four WLthicknesses of 0.5, 1.0, 1.5, and 2.0 nm. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

M. Sabaeian, M. Shahzadeh / Physica E 61 (2014) 62–6866

The height as well as WL thickness was allowed to vary to cover allpossible growth results. All non-zero in-plane- and z-polarizedtransitions were inspected. The impact of WL thickness on energy

and transition energy were investigated showing a red shift oftransition frequency with increasing this parameter. As InAs/GaAsQDs are affected from strain, calculations were carried out consider-ing strained-potential.

Acknowledgement

The Authors would like to thank Shahid Chamran University ofAhvaz, Iran for financial support.

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