Numerical investigation of the postgrowth intermixing effectson the optical properties of InAs/GaAs quantum dots

Manel Souaf a, Mourad Baira a, Bouraoui Ilahi a,b,n, Larbi Saxi a, Hassen Maaref aQ1

a UniversitéQ2 de Monastir, Laboratoire de Micro-optoélectronique et Nanostructures, Faculté des Sciences, 5019 Monastir, Tunisiab Department of Physics & Astronomy, College of Sciences, King Saud University, 11451 Riyadh, Saudi Arabia

a r t i c l e i n f o

Article history:Received 27 January 2014Received in revised form19 March 2014Accepted 14 April 2014

Keywords:Quantum dotsModelingIntermixingPhotoluminescence linewidth

a b s t r a c t

We report on a simple theoretical model allowing to investigate the rapid thermal annealing inducedquantum dots intermixing and consequent inhomogeneous broadening. In this model, where the 3DSchrod̈inger equation has been solved, by the orthonormal wave function expansion method, forstrained InAs QD, we assume a lens-shaped QD with a uniform indium composition and a constantaspect ratio during the intermixing process. The size and aspect ratio for as-grown InAs QD, have beenestimated by matching the calculated interband optical transition energies to the experimental photo-luminescence emission peaks from ground and excited states. The simulated results were correlatedwith photoluminescence data at various annealing temperatures. Keeping constant the QD aspect ratio,a good agreement has been found between experimental and calculated emission energies for differentindium atomic diffusion lengths. Small QDs are found to be more sensitive to the intermixing than largerQDs. This study allows also to calculate the full width at half maximum (FWHM) and compare it with theexperimental value. The theoretical calculations suggest that the origin of the inhomogeneous broad-ening is mainly related to the variation of the QDs size.

& 2014 Published by Elsevier B.V.

1. Introduction

Self-assembled InAs quantum dots (QDs) have been a subject ofextensive research due to their interesting systems for fundamen-tal physics and for the development of electronic and optoelec-tronic devices [1,2]. Post-growth compositional intermixingfurther offers new fields of applications including photonic inte-grated circuits and broadband light emitters and detectors [3,4].The intermixing of self-assembled InAs/GaAs QD formed by theStranski–Krastanov (S–K) growth method has been widely inves-tigated by several methods [1,5,7,8]. While the emission energyfrom intermixed QD has been successively tuned over a widerange [7,8], an inhomogeneous intermixing has been reported tooccur at a given intermixing degree [10–12]. Several theoreticaland experimental works have been carried out to investigate theintermixing effects on the QDs optical properties [6–21]. The lackof information concerning the real shape and indium compositionof the QDs presents a crucial part of the calculation [13].

Many numerical approaches dealing with the effect of inter-diffusion on the optical properties of QDs have been reported [14].

Gunawan et al. [21] and Djie et al. [15] have used different shapesof QDs including pyramidal, cubical and spherical QDs; they usedFick's law and momentum space methods to calculate the electro-nic structure. Petrov et al. [16] rather presented a theoretical andexperimental study using Fick's law. Maia et al. [17] and Osmanet al. [18] proposed a model for lens-shape, suggesting that theindium concentration varies linearly from the bottom (100%) tothe top (0%) of the intermixed QD.

In this paper we investigate theoretically the In–Ga inter-diffusion effect on the InAs QDs inhomogeneous broadening.We propose a simple model for a lens-shaped QDs [6,12,16]allowing to reproduce and explain the observed impact of thepost-growth intermixing on the PL properties of InAs/GaAs QDs.The proposed model is however general and can be used toinvestigate other QDs systems.

2. Theoretical approach

The calculation was carried out for a lens shaped InAs QDembedded in a large cylinder of a GaAs barrier material.In accordance with many previous theoretical studies, this geometryis the most realistic model to describe the three-dimensionalconfinement [1,5,21]. By modeling the intermixing effects, it ispossible to evaluate its influence on the QDs parameters such as

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Physica B

http://dx.doi.org/10.1016/j.physb.2014.04.0510921-4526/& 2014 Published by Elsevier B.V.

n Corresponding author at: Université de Monastir, Laboratoire deMicro-optoélectronique et Nanostructures, Faculté des Sciences, 5019 Monastir,Tunisia.

E-mail address: bilahi@ksu.edu.sa (B. Ilahi).

Please cite this article as: M. Souaf, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.04.051i

Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

composition, size, and optical transition energies. Accordingly, wehave used a simple theoretical model allowing to calculate theoptical transition energies at different annealing temperatures andto estimate the In diffusion length (see Fig. 1).

To calculate the carrier's confinement energies and their wavefunctions in InAs QD, the three-dimensional Schrodinger equationhas been numerically solved using the orthonormal wave functionexpansion method in cylindrical coordinates [22]. The model canbe described as follow:

� ∇ℏ2

2mnðr;φ; zÞ∇ !

ψðr;φ; zÞþVðr;φ; zÞψðr;φ; zÞ ¼ Eψðr;φ; zÞ ð1Þ

where mn is the electron or hole effective mass. E and ψ(r,φ,z) arethe quantized energy levels and the corresponding wave function,respectively. We have assumed the wave function as a linearcombination of the expansion basis functions.

ψðr;ϕ; zÞ ¼∑n;m;lAnmlψnmlðr;ϕ; zÞ ð2Þ

With n, m, and l are the positive integers, the orthogonal periodicfunctions ψmnl and the coefficients Anml must be determined. Tosolve the exact single-particle states we calculate the matrixelement of the Hamiltonian operator between individual ortho-gonal periodic functions. The matrix elements are evaluated fromthe Hamiltonian equation H:

Hnmln'm'l' ¼∭Ω �ψn

n'm'l' ∇ℏ2

2mn∇

!ψnmlþψn

n'm'l'Vψnml

!d Ω ð3Þ

The impact of strain on the carrier confinement due to thelattice mismatch between InAs and GaAs materials is adopted forthe calculation. Hydrostatic and uniaxial strains are induced, it canbe written as following [17,23]:

εh ¼ ðεxxþεyyþεzzÞ ð4Þand

εb ¼ εzz�12

εyyþεxx� � ð5Þ

Where εb and εh are the hydrostatics and uniaxiale strainrespectively.

εxx ¼ εyy ¼aGaAs�aInAs

aInAsð6Þ

εzz ¼ �2C12

C11εxx ð7Þ

the lattice mismatch between InAs and GaAs. The strained effecton conduction band is δEc¼ acnεh and for valence band is

δEv¼ av εxxþεyyþεzz� ��1

2bðεbÞ ð8Þ

then, the strained band gap (Eg) can be written as following:

EgInAs_str ¼ EgInAs�unstrþδ Ec�δ Ev ð9ÞWhere EgInAs�unstr is the unstrained InAs band gap energy.

However, during the annealing process, indium atoms outdiffuse from the central core of InAs QD to the barrier due to theIn–Ga interdiffusion effect. This may change considerably thechemical composition of QDs from pure InAs QD to a randomInXGa(1�X)As alloy. To simplify the calculation, we hypothesize auniform In composition profile from the center to the border of theQD and we set a fixed aspect ratio during the intermixing process[17,18]. We define the lens shaped QD volume before annealing:

VQD ¼ hQDπ

244h2QDþ3b2QD

� �ð10Þ

The corresponding QD volume after intermixing is as follows:

V 0QD ¼ h0QDπ

244h'QD2þ3b0QD2� �

ð11Þ

The indium composition can be derived as follows:

XIn ¼VQD

V 0QD

ð12Þ

where hQD, h'QD, bQD, and b'QD are the height and the base of non-interdiffused and interdiffused QDs respectively. This change ofcomposition in the QD affects the lattice mismatch. As a result, thestrain between GaAs and InXGa(1�X)As QD is reduced witchinfluences the QD material's band gap, and carriers' effectivemasses. To take into account this change in our calculation, wehave used Vegard's law:

aInxGa1� xAs ¼ xmInAsþð1�xÞmGaAs ð13Þ

EgInxGa1� xAs ¼ xEgInAsþð1�xÞEgGaAs ð14Þ

mn

InxGa1� xAs ¼ x:mn

InAsþð1�xÞ:mn

GaAs ð15Þ

3. Experiments

To correlate the proposed model with experiments we usedsingle layer of self-organized InAs QDs sample grown by MolecularBeam Epitaxy (MBE) on semi-insulating (0 0 1) GaAs substrate attemperature of 500 1C. The QDs have been formed by depositing2.4 ml of InAs followed by 50 nm GaAs barrier. For the intermixingprocedure, the samples were protected by SiO2 layer prior to therapid thermal annealing for 30 s at the temperatures of 650 1C,750 1C and 800 1C. More details on the growth process andcharacterization of these samples can be found elsewhere [7,28].

Fig. 2 presents the normalized 12 K PL spectra taken from theas-grown sample and samples annealed at different temperatures.The PL spectra have been recorded at 12 K for two excitationpower densities showing that the observed multiple peaks arisefrom the ground and excited states emission energies. The PLpeaks emission energies appear to be insensitive to the annealingprocess below 700 1C. With increasing the annealing temperaturebeyond 700 1C, the atomic interdiffusion considerably affects theQD PL properties [7,24,25]. The PL peaks position exhibits a blueshift around 180 MeV at Ta¼800 1C. The energy spacing betweenthe ground state and the first excited state emissions drops from67 to 33 MeV [8]. The observed blueshift is accompanied by an

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Fig. 1. Schematic model of a lens-shaped interdiffused InAs QD for different indiumcompositions.

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increase of the PL linewidth broadening up to an annealingtemperature of 750 1C. For higher intermixing degree, the PLlinewidth tends to be recovered.

4. Results and discussion

Fig. 3a shows the experimental and the simulation result of theground state and the exited states emission energies as a functionof indium composition. The calculated result for non-intermixedQDs suggests an initial InAs QDs height around: hQDE4.7 nm andradius RQDE17 nm. The deduced aspect ratio of the as-grown lensshaped QD is α¼0.14. With preserving the same aspect ratioduring the interdiffusion process and supposing a uniform Indiumprofile, intermixing process increases the QD size and decreasesthe In concentration. We found an In composition around 80% forTa¼750 1C and about 60% for Ta¼800 1C which is in agreementwith literature reports [16,21,27]. The numerical calculation showsa large blue shift of the emission energies and a reduction of theintersublevels spacing (ΔE) as observed by PL measurements. Thelateral and the vertical diffusion lengths induced by intermixingare set as adjustable parameters in our calculation insuring aconstant aspect ratio independently of the intermixing degree.It can be deduced by calculating the difference between heightsand radius of an as-grown and intermixed InAs QD (ΔhQD andΔRQD). The calculated values for ΔhQD and ΔRQD are reported inFig. 3b as a function of the annealing temperature. The calculatedvalues for ΔhQD and ΔRQD are reported in Fig. 3b as a function ofthe annealing temperature. The lateral diffusion length(verticaldiffusion length) is estimated to be about 1.5 nm for Ta¼750 1C(0.42 nm) and 3.5 nm for 800 1C (0.98 nm). It is worth mentioningthat, in our model, the vertical and lateral diffusion lengths are nottreated separately since they are interdependent through the fixedQD aspect ratio.

Our results are comparable to those obtained by using Fick'slaw [13,16,21]. Indeed, we have compared our results with theexperimental energy levels in the model presented by Hsu et al. [20]

and with the theoretical result presented by Petrov et al. [16] andfound good agreement despite the approximation used in our Model.These results clearly indicate that our simplified model succeeds inaccurate reproducing of the experimental results.

A striking feature, widely reported for post-growth intermixedQDs is the occurrence of an inhomogeneous broadening of the PLlinewidth for a given intermixing degree [7,10,17,26]. Accordingly,despite the narrow as-grown QDs size distribution, the PL resultsshow that the FWHM increases with the annealing temperaturereaching a maximum around 750 1C, followed by a shrinkage forhigher intermixing degree. This behavior is likely to be induced byan inhomogeneous intermixing accentuated by the fluctuation ofQDs size and surrounding strain. In order to understand theimpact of QDs size variation during intermixing process [19], wehave calculated the differential energy shift (ETa�EAsgrown) of theground state transition energy for three different QDs heights:h¼3 nm, h¼4 nm and h¼6 nm as a function of annealing tem-perature (the same aspect ratio is preserved).

The calculated results depicted in Fig. 4 assert that smaller QDsundergo more differential blue shift at lower annealing tempera-tures than larger QDs. Therefore, small QDs are more sensitive tothe variation of the In atom diffusion length at low intermixingregime when compared the larger size QDs. Taking the example ofTa¼750 1C, the energy shift is about 85 MeV for h¼6 nm, and103 MeV for h¼4 nm, with a difference around 18 MeV. In themeanwhile for annealing temperature beyond 800 1C, the energyshift is gradually decreasing when increasing the annealingtemperature leading to a reduced linewidth broadening. The totalemission energy blueshift vary from 184 MeV to 200 MeV whenthe QDs height is reduced from 6 nm to 4 nm. The difference

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Fig. 2. Normalized 12 K PL spectra obtained for two excitation power density:4 mW/cm2 (dashed lines) and 400 mW/cm2 (solid lines) from the as-grown andintermixed InAs QDs at different annealing temperatures.

Fig. 3. Experimental and calculated electron–hole transition energies for inter-mixed lens-shaped InAs QD: (a) as a function of indium composition and (b) as afunction of the vertical and lateral diffusion length.

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between the intermixing induced blue shifts calculated for differ-ent QDs sizes has comparable variation as the PL FWHM. Conse-quently, the QDs size dispersion appears to be a cause for theobserved increase in the linewidth inhomogeneous broadening(see the inset of Fig.4). Furthermore, we have simulated the PLFWHM evolution as a function of the annealing temperature forthe ground state emission peak and compared with the evolutionof experimental one. The estimation of the as-grown InAs QDs sizecorresponding to the extreme values of the emission peak's FWHMhas been performed by adjusting the emission energies using theQDs size as a fitting parameter. As shown by Fig. 5, keepingconstant the aspect ratio (α¼0.14), the QDs height is estimated tobe 4.9 nm for the lower-limit (LL) of FWHM, and around 4.4 nm forthe upper limit (UL) of the FWHM.

Using the previously derived diffusion lengths, we have simu-lated the theoretical variation of the PL FWHM as a function of theannealing temperature. The results are reported in Fig. 6 wherethe experimental and simulated PL linewidth variations are givenas a function of the annealing temperatures. The results show acomparable behavior. The FWMHexp of as-grown QD sample isabout 28 MeV and it increased up to 44 MeV after an annealingtemperature of 750 1C. Numerically driven FWHM also increaseswith increasing the intermixing degree getting its maximum at thesame annealing temperature (around 33 MeV). For higher anneal-ing temperature both FWHMexp and FWHMcal decreases down to33 MeV and 31 MeV respectively.

The simulation results suggest that the dominant parametergoverning the inhomogeneous broadening is related to the variation

of the QDs size. In fact, small QDs are more sensitive to smalldiffusion length than larger size QDs giving rise to more energy shift.The difference in magnitude between the simulated and the experi-mental PL FWHM values is likely to be induced by the point defectsdistribution around a given QDs and its position with respect toneighboring QDs. These dependent parameters, typical of self-formed QDs are hard to be taken into account due to their hazardousnature. The variation of QD size during intermixing process is thekey parameter that controls the sharpness of the PL linewidth.

5. Conclusion

We reported on simple theoretical model for intermixedInAs/GaAs QDs allowing an accurate simulation of the effects ofintermixing on the photoluminescence properties. In this model,the aspect ratio is supposed to be independent of the intermixingdegree and the intermixed QD's composition is considered to beuniform. Our results allow us to estimate the directional depen-dent atomic diffusion length through correlation with experimen-tal PL data.

Additionally, the proposed model has been successively used tostudy the origin of the InAs QDs inhomogeneous broadeninginduced by intermixing. Theoretical results using our numericalapproach prove a good agreement with experiments and showthat the origin of the inhomogeneous broadening is related to thevariation of QDs size.

Acknowledgment

This project Q3was supported Q4by King Saud University, Deanshipof Scientific Research, College of Sciences Research center.

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Fig. 4. Variation of the intermixing induced energy shift (ETa�EAs-grown) as afunction of annealing temperature (1C). The inset show the evolution of thedifferential shift simulated for QDs with different heights h¼3 nm, h¼4 nm, andh¼6 nm.

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Fig. 6. Experimental (circles) and simulated (triangles) of the PL FWHM as afunction of the annealing temperature.

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