Phys. Status Solidi B 249, No. 3, 516–520 (2012) / DOI 10.1002/pssb.201100417 p s sb

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basic solid state physics

Ground state switching in InGaN/GaNquantum dot molecules

Stefan Schulz*,1 and Eoin P. O’Reilly1,2

1Photonics Theory Group, Tyndall National Institute, Lee Maltings, Cork, Ireland2Department of Physics, University College Cork, Ireland

Received 8 July 2011, revised 18 August 2011, accepted 14 November 2011

Published online 29 December 2011

Keywords built-in fields, nitrides, quantum dots, single-particle states, tight-binding

*Corresponding author: e-mail stefan.schulz@tyndall.ie, Phone: þ353 21 490 4175, Fax: þ353 21 490 4402

We present a detailed analysis of the electronic properties of

In0.25Ga0.75N/GaN quantum dot (QD) molecules using an sp3

tight-binding model, including strain and built-in fields. The

influence of the interdot separation is studied in detail. Our

analysis reveals that, even if we assume two identical QDs,

the molecular description of bonding and anti-bonding states

breaks down for the single-particle states. This behavior

originates from two effects, the strain field and the built-in

potential. The strain field interactions between the two QDs

suppress the ability to form bonding and anti-bonding hole

states. Additionally, the built-in potential along the c-axis leads

with increasing separation to a ground state switching between

the two QDs. This is related to the behavior of the built-in

potential in an isolated QD, where the potential drops back

quickly toward zero, and may even change sign outside a

strained c-plane QD.

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1 Introduction Stacks of semiconductor quantumdots (QDs) are widely used, for example, as the activeregion of QD laser structures [1] and have been proposed asentangled photon sources [2]. If the spacer layer thickness(slt) between the different QD layers is large, the QDs in thestack behave like isolated dots. However, if the distance issufficiently small, the dots can become coupled: the wavefunctions of individual dots in a stack can overlap due toquantum-mechanical tunneling, so that the dots are coupledelectronically, a situation useful for laser applications [3].

Nitride semiconductors have emerged over the last fewyears as important materials for blue and ultraviolet light-emitting devices [4]. For quantum information processingapplications, nitride-based QDs open new spectral regionsfor single-photon sources [5].Moreover, due to the reductionof the built-in field in a nitride-based QD compared to anitride-based quantum well (QW) of the same height andcomposition [6, 7], QD-based light-emitting diodes [8, 9]and laser structures [10], operating in the green spectralregion have been demonstrated recently and have shownsuperior performance than their QW based counterparts.

Therefore, understanding the mechanisms of interdotcoupling is important from the point of view of fundamentalproperties of stacks of nitride-basedQDs, and is crucial whencoupling is employed in device design. In this paper we

present a detailed analysis of the electronic structure of anIn0.25Ga0.75N/GaN QD molecule (QDM) based on a tight-binding (TB)model, and taking strain and electrostatic built-in fields into account. The influence of the barrier thicknessbetween the two QDs is studied in detail.

2 Theory To study the electronic properties of QDMsmade of In0.25Ga0.75N we apply an sp3 TB model [11, 12].The strain dependence of the TB-matrix elements is includedvia the Pikus-Bir Hamiltonian [13] as a site-diagonalcorrection. In doing so, the hydrostatic and uniaxialdeformation potentials are included directly without anyfitting procedure. This approach is similar to the straindependence of an eight-band k � p model [13]. The electro-static built-in potentials arising from spontaneous andpiezoelectric polarization are also included as site-diagonalcontributions in the TB-Hamiltonian. Strain and polarizationfields are calculated in the framework of a surface integralmethod [14].

3 Results In the following we discuss the electronicproperties of In0.25Ga0.75N QDMs. For the QD geometry weassume a lens-shaped structure, as suggested by experimen-tal data in Ref. [15]. To keep the analysis simple andtransparent, we consider here molecules made of two

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vertically stacked QDs of identical shape, size, andcomposition with varying interdot distance slt. We assumehere values of slt� 1, 2, 4.1, 6.2, and 8.3 nm, in accordancewith recent experimental findings on stacked InGaNQDs [16].

The average diameter of InGaNQDs scatters around 15–25 nm while the average height is approximately 2–6 nm[15–17]. Therefore, we assume a diameter of d� 19.2 nmand a height of h� 3.1 nm for both QDs. Since QD structureswith 15–25% In have been reported in the literature, weassume here an In content of 25%. The QDM is embedded ina large box of the GaN barrier material, which we refer to asthe surrounding matrix.

To analyze the electronic structure of In0.25Ga0.75N/GaNQDMs in detail we proceed in the following way. To studythe influence of strain and built-in fields separately, weswitch these contributions off in a first step. Subsequently weinvestigate the influence of the strain field on the electronicstructure on its own, neglecting the electrostatic built-inpotential. In a third step we include this contribution in thedescription, and study its impact on the electronic structure.

3.1 Electronic structure without strain andbuilt-in field In this section we analyze the electronicstructure of the QDMs when we artificially switch off strainand built-in fields. Energies of the first two bound electron(e1, e2) and hole (h1, h2) single-particle states are shown inFig. 1(a) and (b), respectively. Here, the electron and holelevels form bonding and anti-bonding molecular orbitals.Therefore, the single-particle energies split symmetricallyaround the electron and hole ground state energy of theisolated In0.25Ga0.75N/GaNQD. This behavior is similar to areal diatomic molecule (e.g., H2). However, one can inferfrom Fig. 1 that electronic coupling between the dots is onlypossible for small slt values (slt< 4 nm). This is related totwo facts. Firstly, due to the high effective hole masses in thenitride system, h1 and h2 are strongly confined in the QD.

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Figure 1 (online color at: www.pss-b.com) TB single-particleenergies in the absence of strain and built-in fields as a functionof the slt. The reference energy for our results is set to the unstrainedvalence band maximum of GaN.

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Therefore small spacer layer thicknesses are required toachieve coupling between these states. Secondly, eventhough the electron effective mass is much lower, the largeconduction band offset prevents an effective interdotcoupling for the ground state wave functions.

3.2 Influence of strain on the electronicstructure Figure 2 shows the single-particle energies ofthe first two bound electron and the first three bound holestates in the In0.25Ga0.75N/GaNQDM as a function of the slt,when taking only the strain field into account.

Looking at the electron single-particle energies in thecase of large slt, we find that e1 and e2 are degenerate. Forclosely spaced QDs (slt� 1–2 nm), the states e1 and e2 formbonding and anti-bonding states, similar to the situation inthe absence of the strain field. The main difference to theresults in the previous section is that e1 and e2 are shifted tohigher energies due to hydrostatic strain in the system.

The situation is completely different for the hole states.Here, the first two bound hole states h1 and h2 do not formbonding and anti-bonding states. When looking at prob-ability densities of the hole wave functions (not shown) onefinds, that for closely spaced QDs h1, h2, and h3 are localizedon the lower QD. This behavior is therefore similar to thebehavior in a heteronuclear diatomic molecule (e.g., HF).This finding can be explained by the following two reasons.Firstly, the strain between theQDsmodifies the valence bandoffsets, leading to an increased effective barrier between thedots. Therefore, tunneling and the ability to form bondingand anti-bonding states is suppressed for the hole states. Inaddition to this effect, the QDM lacks a center of inversion.Therefore, the strain state of the upper dot is different to thestrain state of the lower QD. The localization of the holestates on the lower QD can be understood when looking atthe strain state of an isolated QD. In an isolated lens-shapedQD, the region at the base of the dot experiences a strongbiaxial compressive strain,with amuch smaller biaxial strain

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Figure 2 (online color at: www.pss-b.com) Same as in Fig. 1 butthis time in the presence of strain but in the absence of the built-infield.

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518 S. Schulz and E. P. O’Reilly: Ground state switching in InGaN/GaN quantum dot moleculesp

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Electron Ground State e1

slt ≈ 1 nm slt ≈ 2 nm slt ≈ 4.1 nm slt ≈ 6.2 nm slt ≈ 8.3 nm

Hole Ground State h1

slt ≈ 1 nm slt ≈ 2 nm slt ≈ 4.1 nm slt ≈ 6.2 nm slt ≈ 8.3 nm

Figure 4 (onlinecolorat:www.pss-b.com)Probabilitydensitiesofthe electron and hole single-particle wave functions e1 and h1,respectively, for different spacer layer thicknesses slt. The QDgeometry is shown in light gray and the blue and red probabilitydensity isosurfacescorrespond to10and50%of themaximumvalue.

at the top of the dot. The barrier material above the QDexperiences a biaxial strain of opposite sign to that at the dotbase. Therefore, for closely spaced QDs, the bottom of theupper dot experiences a reduced biaxial strain compared tothe bottom of the lower dot. Because these biaxialcompressively strained regions are more favorable for(heavy-) hole states, one could expect that the hole statesare localized at the bottom of the lower QD. This is exactlythe situation we find here for the first few bound hole states.

3.3 Influence of built-in potentials on theelectronic structure In this section we analyze theinfluence of the total (spontaneousþ piezoelectric) built-inpotential ftot on the electronic structure of the In0.25Ga0.75NQDMs. Here we now take both strain and built-in potentialinto account. As discussed in Ref. [18], when modeling theelectronic structure of coupled c-plane nitride-based QDs,the sign of the shear strain-related piezoelectric coefficiente15 becomes important, since it affects the behavior of ftotabove and below an isolated QD. In the literature positive aswell as negative values are reported [19]. Here, we havechosen e15< 0, in accordance with recent experimental [20]and theoretical [21, 22] findings.

Figure 3(a) shows, as a function of slt, the single-particleenergies for the first three bound electron states e1, e2, and e3while Fig. 3(b) depicts the first three bound hole states, h1, h2,and h3. As expected from the quantum confined Stark effect,the electron states are shifted to lower energieswhile the holestates are shifted to higher energies, compared to thesituation without the built-in field (cf. Fig. 2). Moreover,when looking at Fig. 3, we find that the molecular-likedescription of the single-particle states breaks down for bothelectrons and holes. Additionally, an interesting feature inthe energy spectrum for electrons and holes is, that aroundslt� 2 nm,we observe a kink in the spectrum. Evenwhenftotbreaks the symmetry between the twoQDs, onemight expectthat the ground state energies for electrons and holes

Figure 3 (online color at: www.pss-b.com) The same as in Fig. 1but this time taking strain and built-in field into account.

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approach the ground state energies of an isolated QD in acontinuous way from above or below the respective energy.

To analyze this behavior in more detail, Fig. 4 showsisosurfaces of the probability densities of the electron (e1)and hole (h1) ground states as a function of slt. The blue andred isosurfaces correspond to 10% and 50% of the maximumvalue, respectively. As expected from the single-particleenergies, ftot breaks the symmetry between the two QDs,and bonding and anti-bonding states are not formed.Furthermore, Fig. 4 shows that the built-potential can leadto a ground state switching for both electrons and holes. Forspacer layer thicknesses of slt� 2 nm, e1 is localized at thetop of the upper dot, while h1 is localized at the bottom of thelower QD. This behavior is therefore similar to the behaviorone would expect in two coupled QWs. The situationchanges for slt> 2 nm. In this case e1 is localized at the top ofthe lower QDwhile h1 is localized at the bottom of the upperdot. This ground state switching explains therefore the kinkin the energy spectrum shown in Fig. 3. The reason for thisground state switching can be understood by looking at thetotal built-in potential ftot above and below an isolated QD.Figure 5(a) shows ftot for a line-scan through the center of anisolated lens-shaped In0.25Ga0.75N/GaNQD along the c-axis(z-direction). Outside the QD ftot returns to zero quickly andchanges sign a few nanometer away from the dot. Thisbehavior affects therefore the built-in potential in a QDM.Figure 5(b) and (c) show the same line-scan as in Fig. 5(a),but this time for the In0.25Ga0.75N QDM with slt� 1 nmand slt� 4.1 nm, respectively. The behavior of ftot inFig. 5(b) and (c) can be understood when superimposingbuilt-in potentials of two isolated, identical lens-shapedIn0.25Ga0.75N/GaN QDs with their base centered at z¼ 0 andz� 4.1 nm Fig. 5(b) or z¼ 0 and z� 7.2 nm Fig. 5(c). Incase of the smaller slt Fig. 5(b), the magnitude of ftot in theupper (lower) QD is reduced at the bottom (top) only, andalmost unchanged at the top (bottom) compared to anisolated QD. Therefore, the single-particle electron wavefunctions could be expected to be localized at the top of theupper QDwhile the hole states are expected to be localized atthe bottom of the lower QD. This is exactly the result we

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Figure 5 Total built-in potentialftot shown for a line-scan throughthe center of the lens-shapedQDs. (a) displaysftot in an isolatedQD;(b) shows ftot in a QDMwith a spacer layer thickness of slt� 1 nm.The total built-in potential of theQDMwith a spacer layer thicknessof slt� 4.1 nm is shown in (c).

obtain from our TB analysis (cf. Fig. 4). For slt> 2 nm, thechange in sign in ftot in an isolated QD becomes important,as it can be seen from Fig. 5(c). Here, ftot is slightly reducedin magnitude at the top (bottom) of the upper (lower) QDwhile almost unchanged at the bottom (top) of the lower(upper) QD. In this case the electron wave functions areexpected to be localized at the top of the lowerQDwhile holestates are expected to be localized at the bottom of the lowerQD. Again, this is in accordance with our TB results(cf. Fig. 4).

From this analysis one might expect an even furtherreduction of the oscillator strength in a QDM compared to anisolated QD, since the wave function are localized ondifferent QDs, leading to a large spatial separation ofthe electron and holewave functions.However, herewe havemade the assumption of two identical QDs. During thegrowth process of a stack of QDs, the geometrical propertiesof the dots are probably changed from layer to layer.Furthermore, the indium content is probably also different inthe different QDs, leading therefore to different confinementenergies aswell as strain fields. Thiswill then alsomodify thebuilt-in potential. Consequently, further studies are requiredto analyze in detail how the change in composition and ingeometry affects the built-in field in QDMs compared to anisolated QD and how this influences the electronic andoptical properties of these systems.

4 Conclusion The electronic structure of In0.25Ga0.75NQDMs, made up of two identical lens-shaped QDs, has beenstudied as a function of the slt between the two dots by meansof a TB model, taking strain and built-in fields into account.The impact of these contributions was studied in detail,showing that the molecular-like description of bonding andanti-bonding states breaks down for both electrons and holes,even though we assume two identical QDs. Due to the high

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effective hole mass and the lack of inversion of the systemunder consideration, the strain field on its own is alreadysufficient to prevent the formation of bonding and anti-bonding hole states. In addition to this effect, the electron andhole levels are significantly modified by the presence of thebuilt-in field. When taking the built-in field into account, wefind a ground state switching for electrons and holes. Forspacer layer thicknesses slt� 2 nm the electron ground state islocalized at the top of the upperQDwhile the holeground stateis localized near the bottom of the lower QD. However, whengoing to larger barrier thicknesses (slt> 2 nm), the electronground state wave function is localized near the top of thelower QD while the hole ground state wave function islocalized near the bottom of the upper QD. This ground stateswitching follows from the behavior of the built-in field in anisolated QD, where along the c-axis the sign of the potentialchanges a few nanometer away from the QD.

Acknowledgements The authors acknowledge financialsupport from Science Foundation Ireland.

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