Optical properties of (In,Ga)As capped InAs quantum dots grown on [11k] substrates
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Optical properties of (In,Ga)As capped InAs quantum dots grown on [11k] substrates
V. Mlinar ∗ and F. M. Peeters † Departement Fysica,Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Dated: November 29, 2018)Using three-dimensional k · p calculation including strain and piezoelectricity, we showed that thesize of the quantum dot (QD) in the growth direction determines the influence of the (In,Ga)Ascapping layer on the optical properties of [11k] grown InAs QDs, where k=1,2,3. For flat dots,increase of In concentration in the capping layer leads to a decrease of the transition energy, asis the case of [001] grown QDs, whereas for large dots an increase of the In concentration in thecapping layer is followed by an increase of the transition energy up to a critical concentration of In ,after which the optical transition energy starts to decrease. PACS numbers: 73.21.La, 71.35.Ji, 78.20.Ls, 71.70.Gm
Manipulation of quantum dots’ (QDs) properties isdriven by current and potential applications, rangingfrom QD lasers, and photodetectors to single polarizedphoton sources. Growth conditions, such as growth tem-perature, substrate orientation, or capping procedures,determine the QD electronic and optical properties. Inorder to produce good quality QD structures with highdensities and low size dispersion, or to control lateral andvertical ordering of QDs in QD lattices, growth on highindex surfaces has been put forward.
Furthermore,to achieve long wavelength emission, e.g. larger than1.3 µ m in a QD laser diode, or alternatively user-defineddetection wavelength, e.g. for quantum dot infraredphotodetectors, QD-in-a-well (DWELL) structures wereintroduced. Namely, optical properties of a QD aretuned by size and chemical composition of QW layer,where the QD is embedded in.In a widely investigated DWELL system, [001] grownInAs QD embedded in In x Ga − x As QW, variation of the In concentration in the capping layer as well as the thick-ness of the layer influence the hydrostatic component ofthe strain tensor and consequently the transition ener-gies: Increase of the In concentration in the QW leadsto a decrease of the transition energy. What is happen-ing in the case of QDs grown on [11k] substrates, wherek=1,2,3? How does the (In,Ga)As capping layer influ-ence the optical properties of the InAs QDs grown on[11k] substrates? In this Letter we answer these questionsand provide a guideline for the variation of the transitionenergy of [11k] grown QDs, as function of the cappinglayers thickness and chemical profile, and for differentdot composition.Prior to understanding how the capping layer influ-ences the transition energies of [11k] grown InAs QDs,one has to know the effect on the transition energies ofQD growth on [11k] substrates (k=1,2,3). The originof the variation of the transition energy with the sub-strate orientation can be traced back to the competitionof several effects: (i) hydrostatic component of the straintensor is responsible for a shift of the conduction bandupwards and the valence bands downwards, (ii) biaxialcomponent of the strain tensor influences the degree ofthe valence band mixing, and (iii) variation of the hole effective mass with the substrate orientation, which cansignificantly alter the effects of the size quantization inthe QD. Actually, we have shown that the QD size inthe growth direction determine which of the three abovementioned effects will be the dominant one, regardlesson the dot shape. Therefore, we consider here two modellens-shaped QDs with different height: L1 QD with ra-dius R=9.04nm and height h=4.52nm, and L2 QD withradius R=9.04nm and height h=9.04nm. The thicknessof the capping layer is assumed to be the same as theheight of the dots, whereas the In concentration in thecapping layer is varied from 0 to 30%.A model QD, as it enters our calculations, is con-structed on a three-dimensional (3D) rectangular gridwith a grid step equal to the lattice constant of GaAs,and is shown in Fig. 1(a). In our full 3D model, thestrain distribution is calculated using continuum elastic-ity and the single particle states are obtained from aneight-band k · p theory including strain and piezoelec-tricity. In order to properly take into account the effectof the different substrate orientation, the coordinate sys-tem is rotated in a way that the Cartesian coordinate z ′ coincides with the growth direction [Figs. 1(b)]. Thegeneral [11k] coordinate system ( x ′ , y ′ , z ′ ) is related tothe conventional [001] system ( x, y, z ) through a transfor-mation matrix U=U( φ, θ ). The angles φ and θ representthe azimuthal and polar angles, respectively, of the [11k]direction relative to the [001] coordinate system. Transi-tion energies are calculated taking into account the directCoulomb interaction.What will happen when both effects, QD growth onhigh index surfaces and capping, are present? Transitionenergies of L1 and L2 QD, extracted from our numericalcalculations, as they vary with substrate orientation and In concentration in the capping layer, are shown in Figs.1(c) and (d), respectively. On can see that for both, [001]grown L1 and [001] grown L2 QD, an increase of the In concentration in the capping layer leads to a decrease ofthe transition energy. On the other hand, our findings onthe transition energies versus In concentration for [11k]grown QD, depend heavily on the dot size in the growthdirection. Let us first discuss the case of the L1 QD.The variation of the transition energy with the In con- FIG. 1: (color online) (a) Model InAs QD. (b) Transformationof the general [11k] coordinate system to the conventional[001] coordinate system. Transition energies of L1 (c) andL2 (d) QDs as they vary with the In concentration in thecapping layer for different substrate orientations. centration does not depend qualitatively on the substrateorientation, i.e. with increase of In concentration thetransition energy decreases, as was the case for [001]grown QDs [black line in Fig. 1(a)]. This is a simpleconsequence of the variation of the hydrostatic compo-nent of the strain tensor with the substrate orientationand In concentration in the capping layer, as shown inFig. 2(a). The hydrostatic component of the strain ten-sor of [11k] grown QDs reduces with the increase of the In concentration, as in the case of [001] grown QDs. Thesubstrate orientation only determines the degree of theinfluence of the capping layer on the hydrostatic strain,and consequently on the transition energy. Namely, for[111] grown QDs, transition energies decrease slower withthe increase of the In concentration, whereas same de-pendence for [113] grown QDs is similar to the referencecase of [001] grown QDs.However, for the L2 QD surprising results are obtained:with the increase of the In concentration in the cappinglayer the transition energies of [11k] grown L2 QD in-crease, exactly the opposite to the dependence for [001]grown L2 QD. After the concentration of In in the cap-ping layer reaches some critical value, ∼ ∼ ∼ In concentration in the cappinglayer [it starts to follow the pattern of [001] grown L2QD]. What is the origin of such a behavior? Dependenceof the hydrostatic strain on the substrate orientation and In concentration in the capping layer is shown in Fig.2(b) demonstrating that the increase of the In concen-tration leads to a decrease of the hydrostatic strain, asin the case of [001] grown QDs. We single out the most FIG. 2: (color online) Hydrostatic component of the straintensor as it varies with the substrate orientation and In con-centration in the capping layer of L1 QD (a) and L2 QD (b).FIG. 3: (color online) Variation of the electron and hole en-ergy levels of [001] (short dash) and [111] (dash-dot-dot) L2QD with In concentration in the capping layer. Insets showthe variation of the biaxial component of the strain tensorwith In concentration of [001] and [111] grown L2 QD. FIG. 4: (color online) ∆E trans =E trans (x)-E trans (0), wherex is In concentration in the capping layer, of L2 QDs, ver-sus In concentration in the capping layer when (a) the dotcomposition is varied from pure InAs to In . Ga . As andIn . Ga . As; (b) the thickness of the capping layer, d,is varied from d=h, where h is the dot height, to d=h/3,d=2h/3, and d=4h/3. pronounced case, [111] grown L2 QD, and show in Fig. 3calculated electron and hole ground state for x=0% andx=30% In concentration in capping layer. As a compar-ison, we show the results for [001] grown L2 QD as well.Clearly, the variation of the hole ground state, which ismost strongly influenced by the strain, with the cappingleads to an increase of the transition energy. It is causedby the increase of the biaxial component of the strain, asshown in the inset of Fig. 3. Actually, the competitionbetween the increased biaxial component of the straintensor, responsible for the decrease of the valence band mixing, and the decrease of the hydrostatic strain withincrease of the In concentration determine the transitionenergy. Note also that even for [001] grown L1 QD, bi-axial strain is increased, but the hydrostatic strain has adominant influence on the transition energy.At the end we address how the above conclusions areaffected by the variation in the dot composition and thethickness of the capping layer, since those are the un-certainties to be expected in the experiment. For thatpurpose we modified the L2 QD composition from pureInAs to In . Ga . As and In . Ga . As, and thicknessof the capping layer, d, from d=h, where h is the L2 dotheight, to d=h/3, d=2h/3, and d=4h/3. Our results areshown in Figs. 4(a) and (b). An increase of the transitionenergy with increase of the In concentration in the cap-ping layer is observed regardless of the variation of thedot composition or capping layer thickness. Note thatthe critical In concentration after which the transitionenergy vs. In concentration dependence starts to followthe expected is reduced for L2 QD with 50%Ga in thedot, as it can be seen in Fig. 4(a). For example, for [111]grown pure InAs QD critical In concentration is 35%,whereas for In . Ga . As QD critical In concentration inthe capping layer is 20%.In conclusion, our 3D k · p calculation includingstrain and piezoelectricity showed that the QD size inthe growth direction determines the influence of the(In,Ga)As capping layer on the optical properties of [11k]grown InAs QDs, where k=1,2,3. For flat dots, an in-crease of In concentration in the capping layer leads toa decrease of the transition energy, as it is the case of[001] grown QDs, whereas the large dots exhibit an op-posite behavior i.e. increase of the transition energy withincrease of In concentration up to a critical In concentra-tion after which the transition energies start to decrease.We have shown that our conclusions were not sensitive onthe dot composition and thickness of the capping layer,therefore possible to verify experimentally.This work was supported by the Belgian Science Policy(IAP), and the European Union Network of Excellence:SANDiE. ∗ Electronic address: [email protected] † Electronic address: [email protected] V. Shchukin, N. N. Ledentsov, and D. Bimberg,
Epitaxyof Nanostructures, Nanoscience and Technology (Springer,New York, 2003). M. Schmidbauer, Sh. Seydmohamadi, D. Grigoriev, Zh. M.Wang, Yu. I. Mazur, P. Sch¨afer, M. Hanke, R. K¨ohler, andG. J. Salamo, Phys. Rev. Lett. , 66108 (2006). P. Caroff, C. Paranhoen, C. Platz, H. Folliot, N. Bertru,C. Labbe, R. Piron, E. Homeyer, A. Le Corre, and S.Loualiche, Appl. Phys. Lett. , 243107 (2005). C. C¸ elebi, J. M. Ulloa, P. M. Koenraad, A. Simon andA. Letoublon, and N. Bertru, Appl. Phys. Lett. , 23119(2006). M. C. Xu, Y. Temko, T. Suzuki, and K. Jacobi, Phys. Rev.B , 75314 (2005). C. H. Lin, W.-W. Pai, F. Y. Chang, and H. H. Lin, Appl.Phys. Lett , 63102 (2007). Q. Gong, P. Offermans, R. N¨otzel, P. M. Koenraad, and J.H. Wolter, Appl. Phys. Lett , 5697 (2004). V. Mlinar and F.M. Peeters, Appl. Phys. Lett. , 261910(2006). V. Mlinar, M. Tadi´c, B. Partoens, and F.M. Peeters, Phys.Rev. B , 205305 (2005). R. H. Henderson and E. Towe, J. Appl. Phys.78