We present a theoretical analysis of the optical matrix element between the electron and hole ground states in InAs/GaAs quantum dots (QDs) modeled with a truncated pyramidal shape. We use an eight-band k center dot p Hamiltonian to calculate the QD electronic structure, including strain and piezoelectric effects. The ground state optical matrix element is very sensitive to variations in both the QD size and shape. For all shapes, the matrix element initially increases with increasing dot height, as the electron and hole wave functions become more localized in k space. Depending on the QD aspect ratio and on the degree of pyramidal truncation, the matrix element then reaches a maximum for some dot shapes at intermediate size beyond which it decreases abruptly in larger dots, where piezoelectric effects lead to a marked reduction in electron-hole overlap. (c) 2005 American Institute of Physics. (DOI:10.1063/1.2130378

A. D. Andreev

Bimberg D.

E. P. O’Reilly

Applied Physics Letters

Andreev, A. D.

O\u27Reilly, Eoin P.

Irish Universities

Optical matrix element in InAs/GaAs quantum dots: Dependence on quantum dot parameters

O'Reilly, Eoin P.

Cork Open Research Archive

Title Optical matrix element in InAs/GaAs quantum dots: Dependence onquantum dot parametersAuthor(s) Andreev, A. D.; O'Reilly, Eoin P.Publication date 2005Original citation Andreev, A. D. and O’Reilly, E. P. (2005) 'Optical matrix element inInAs/GaAs quantum dots: Dependence on quantum dot parameters',Applied Physics Letters, 87(21), pp. 213106. doi: 10.1063/1.2130378Type of publication Article (peer-reviewed)Link to publisher'sversionhttp://aip.scitation.org/doi/abs/10.1063/1.2130378http://dx.doi.org/10.1063/1.2130378Access to the full text of the published version may require asubscription.Rights © 2005 American Institute of Physics.This article may bedownloaded for personal use only. Any other use requires priorpermission of the author and AIP Publishing. The following articleappeared in Andreev, A. D. and O’Reilly, E. P. (2005) 'Opticalmatrix element in InAsGaAs quantum dots: Dependence onquantum dot parameters', Applied Physics Letters, 87(21), pp.213106 and may be found athttp://aip.scitation.org/doi/abs/10.1063/1.2130378Item downloadedfromhttp://hdl.handle.net/10468/4391Downloaded on 2018-08-23T18:26:46ZOptical matrix element in  quantum dots: Dependence on quantum dotparametersA. D. AndreevE. P. O’ReillyCitation: Appl. Phys. Lett. 87, 213106 (2005); doi: 10.1063/1.2130378View online: http://dx.doi.org/10.1063/1.2130378View Table of Contents: http://aip.scitation.org/toc/apl/87/21Published by the American Institute of PhysicsOptical matrix element in InAs/GaAs quantum dots: Dependence onquantum dot parametersA. D. AndreevDepartment of Physics, University of Surrey, Guildford, GU2 7XH, United KingdomE. P. O’ReillyTyndall National Institute, Lee Maltings, Cork, IrelandReceived 13 May 2005; accepted 3 October 2005; published online 15 November 2005We present a theoretical analysis of the optical matrix element between the electron and hole groundstates in InAs/GaAs quantum dots QDs modeled with a truncated pyramidal shape. We use aneight-band k·p Hamiltonian to calculate the QD electronic structure, including strain andpiezoelectric effects. The ground state optical matrix element is very sensitive to variations in boththe QD size and shape. For all shapes, the matrix element initially increases with increasing dotheight, as the electron and hole wave functions become more localized in k space. Depending on theQD aspect ratio and on the degree of pyramidal truncation, the matrix element then reaches amaximum for some dot shapes at intermediate size beyond which it decreases abruptly in largerdots, where piezoelectric effects lead to a marked reduction in electron-hole overlap. © 2005American Institute of Physics. DOI: 10.1063/1.2130378Semiconductor quantum dots QDs have been widelystudied because of their unique fundamental properties andalso because they are considered to be promising candidatesfor a new generation of semiconductor laser.1 One of themain advantages initially proposed for QDs was the atomic-like zero-dimensional density of states, which should pro-vide a relatively large optical gain at a reduced carrierdensity.2 In real QD structures the gain is, however, signifi-cantly reduced due both to inhomogeneous broadening of theatomic-like density of states and also because of a reducedoptical matrix element. This can even prohibit lasing fromthe ground state in some QD structures.3–5 Previous theoret-ical studies6,7 suggested that the low gain arises because thebuilt-in piezoelectric field leads to the ground state holewave function being elongated, with the matrix element thenreduced due to lower overlap between the electron and holewave functions. We show here that this is not the only reasonfor low optical matrix element. The QDs in Refs. 6 and 7were assumed to be pyramidal with a base to height aspectratio of 2. In real structures the typical QD shape is markedlydifferent, strongly influencing the electron and hole wavefunctions,8 and thereby modifying the optical matrix ele-ment. Previous theoretical studies of InAs/GaAs QDs eitherdo not contain calculations of the optical matrix element8,9 orconsider only one specific QD shape.10The aim of this letter is to identify the factors determin-ing the ground state optical matrix element in QDs, and itsdependence on QD shape and size. In a bulk semiconductorthe optical matrix element decreases for transitions awayfrom the Brillouin zone center, and also depends on the wavevector direction with respect to the light polarization. Thisfundamental property of semiconductors, as well as the pi-ezoelectric effect, causes the matrix element in QDs to bevery sensitive to variations in the QD geometrical param-eters, leading to a lower optical matrix element in QDs com-pared to the maximum “ideal” band edge value of a bulksemiconductor.To calculate the electronic structure of the QDs we use aplane-wave expansion method, assuming a periodic array ofwidely separated QDs.11. The wave function of each electronor hole is described by a linear combination of bulk statesr = k,Ck,k, , 1where k ,expikr is an eigenstate of the bulk 88 k·p Hamiltonian given in Ref. 12, and  denotes elec-tron, heavy-, light-, or spin-split-off holes including spin.The electron and hole energies in the QDs and the coeffi-cients Ck, are found by diagonalizing a large matrix, whosematrix elements depend on the Fourier transform of the QDshape, strain, and piezoelectric field distribution.8,11,13 For agiven light polarization, the optical matrix element Meh be-tween an electron and hole state in a QD is given byMeh = k,,Ck,e *Ck,h Mk , 2where Mk is the optical matrix element between twobulk states  and  at wave vector k, and Ck,eand Ck,harethe coefficients defined by Eq. 1 describing the electronand hole QD states. Since the coefficients in Eq. 1 arenormalized, k,Ck,e,h2=1, we immediately conclude fromEq. 2 that the squared optical matrix element in a QD isalways smaller than that at k=0 in a bulk or quantum wellmaterial.We consider InAs QDs with a truncated pyramidalshape.8 The QD size and shape can be described by threeindependent parameters; namely the QD height h, the aspectratio =bb /h, and the truncation degree =bt /bb the QDbottom and top are bbbb and btbt squares, respectively;for a full pyramid we have =0, while =1 for a QD box.For the QDs under consideration the ground state matrixelement is much larger for light polarized in the QD x-yplane than it is for light polarized along the growth directionz. We therefore set the light polarization vector along the xaxis. Keeping in mind the main application of the presentstudy for gain calculations, we consider the quantity Mtot2,which is the modulus squared of the optical matrix elementAPPLIED PHYSICS LETTERS 87, 213106 20050003-6951/2005/8721/213106/3/$22.50 © 2005 American Institute of Physics87, 213106-1for the electron-hole ground state transition summed overdegenerate spin states. Figure 1 shows the calculated varia-tion of Mtot2 with dot height for five different shapes. Mtot2initially increases with dot height, in four of the five casesreaching a peak value beyond which it decreases rapidly. Wealso note that in each case the value of Mtot2 is noticeablysmaller than the maximum value of unity which might beexpected for an “ideal” QD. The main aim of this letter is toidentify the causes of this behavior.It can be seen from Eq. 2 that the magnitude of theoptical matrix element is determined by two factors: i thedependence of the bulk matrix element M on k, and iihow the coefficients Ck,eand Ck,hoverlap in k space witheach other and with Mk. This overlap is very sensitive tothe QD size and shape. The hole wave function consists ofmainly two components: a heavy-hole HH contribution anda light-hole LH contribution each is represented by a sum-mation in Eq. 1 over the corresponding set of bulk states.For the hole ground state, the admixture of LH bulk states istypically about 10%–20%, depending on the QD geometry.We therefore concentrate our analysis on the electron-HHcomponent. The dependence of the bulk matrix elementM2 on k for the electron-heavy-hole transition is illus-trated in Figs. 2a and 2b The optical matrix element iscalculated following the method developed in Ref. 14. Forthe electron-heavy-hole bulk states, the maximum matrix el-ement is achieved for a given k when k is perpendicular tothe light polarization vector e=ex see Fig. 2b. In addition,the bulk matrix element ME-HH2 decreases quite rapidly ask increases Fig. 2a. For example when ky=0.46 nm−1which corresponds to a distance in real space of 2 /k13.7 nm, the InAs bulk matrix element summed over spinstates is only 0.5 of its maximum value at k=0. This reduc-tion is due predominantly to band-mixing effects, and so isbest described in a k·p model that includes both the conduc-tion and valence bands, along with corrections due to otherbands.14 The effect of decreasing matrix element with thein-plane momentum k	 is well-known in quantum wellstructures.15 For example, the calculated optical matrix ele-ment for E-HH transitions in an InGaAs quantum well de-creased by more than a factor of three when k	 increasedfrom 0 to 1 nm−1.16 The significant decrease of ME-HH inFig. 2a at relatively large wave vectors occurs when thekinetic energy of the electron approaches the band gap en-ergy. This effect is therefore considerably more pronouncedhere than would be the case for bulk materials with largerband gap. The described properties of the E-HH bulk matrixelement have important consequences for the matrix elementin QDs. As the QD height decreases, the characteristic size ofthe electron and hole wave function in k space becomeslarger compare Figs. 2c and 2d with Figs. 2e and 2fand the overlap in k space between Ck,e, Ck,h, and Mkbecomes smaller: this can be clearly seen by comparing Fig.2b with Figs. 2c–2f. The terms with larger k and con-sequently a smaller average M in Eq. 2 play a moreimportant role in small QDs, leading to a reduction in theQD optical matrix element as the QD height decreases. Thus,the trend of increasing Mtot2 with h is a direct consequenceof the k dependence of ME-HH2 in a bulk semiconductor. Wediscuss later how this trend is interrupted for some QDshapes by piezoelectric effects, which lead to a sharp de-crease in Mtot2 beyond a critical value of h, see Fig. 1.It can be seen in Fig. 2b that the unstrained bulk matrixelement ME-HH2 has a strong angular dependence. ThisFIG. 1. Variation with dot height, h, of the modulus squared of the opticalmatrix element Mtot2 for the ground state electron-hole transition inInAs/GaAs QDs of truncated pyramidal shape, summed over degeneratespin states. The value of Mtot2 is given in units of P02 where P0 is theinterband momentum matrix element, P0=  /m0spxx, see Ref. 11. Thelight polarization vector is taken along the x direction.FIG. 2. a Modulus squared of the optical matrix element for electron-heavy hole transitions ME-HH2 in bulk unstrained InAs summed over de-generate spin states; units are the same as in Fig. 1, the light polarizationvector is along the x direction. b Contour plot of ME-HH2 in the kx-ky planefor bulk InAs; c–f Contour plot of the wave function coefficients Ck,e 2and Ck,h 2 in the kx-ky plane for QDs with =5, =0.75, and h=3 nm c,d and h=9 nm e,f. Five grades of grey color correspond to the values of0%–20%; 20%–40%; 40%–60%; 60%–80% and 80%–100% of the maxi-mum value, respectively.213106-2 A. D. Andreev and E. P. O’Reilly Appl. Phys. Lett. 87, 213106 2005arises due to the change in the relative contribution of thevalence x, y, and z states to the unstrained bulk HH bandalong different directions. Interband optical transitions withex polarization require valence states with x character. Inthe unstrained bulk material, the HH state has no x charac-ter along the kx direction: therefore giving zero matrix ele-ment, see Fig. 2b. The anisotropy is reduced in quantumwell and in biaxially strained structures. The reduction insymmetry splits the degeneracy of the HH and LH states atthe valence band maximum, thereby reducing the ability tomix z-like character from the LH states into the uppermostHH band. The relative contribution to the highest hole stateof the z Bloch components can then provide a quantitativemeasure of the anisotropy effect displayed in Fig. 2b.17When the unstrained bulk HH band is averaged over all di-rections, the valence states have equal contributions of about33% from each of x, y, and z. The anisotropy observedin Fig. 2b is eliminated for a bulk material which is biaxi-ally strained in the z direction, with the valence z compo-nent then making nearly zero contribution to the uppermostvalence states. For the various QDs we considered the situ-ation is between these two extreme cases and the z termsaccount for 7%–20% of the ground state hole wave function.This is a significant contribution to the overall reduction inmatrix element, although not as large as might be expectedbased on the bulk matrix elements presented in Fig. 2b.The piezoelectric field has previously been considered asthe main cause of a reduced matrix element in pyramidalQDs.6 Due to this field, the hole wave function becomeselongated and asymmetric in the x-y plane, leading to a de-crease of the electron-hole overlap in real space. This situa-tion is illustrated in Fig. 3a. To demonstrate that the piezo-electric field need not be the prime cause of a reduced matrixelement in medium-sized QDs with h5.5–6.5 nm, we setthe piezoelectric constants to zero in a test calculation on theQD of Fig. 3a. The hole wave function is indeed moresymmetric without the piezoelectric field Fig. 3b, whichvisually increases the overlap between the electron and holewave functions. However, the calculated total matrix elementMtot2 increases only from 0.447 to 0.471 in units of P02, seecaption to Fig. 1. The situation changes, however, when thedot size is increased. Depending on the dot shape, the holewave function either remains nearly symmetric Fig. 3c orbecomes strongly localized in two potential pockets Fig.3d. This localization leads to a steep drop in the matrixelement Mtot2 beyond a critical dot size for some QD shapessee Fig. 1. The piezoelectric field creates two potentialpockets for holes, with the size and depth of these pocketsdetermined by the QD geometry. In InAs QDs we find thatthe piezoelectric potential pockets lose their ability to bind ahole for flat QDs =5, 0.75, and the hole wave func-tion remains almost symmetric even for quite large QDs h=10 nm. The effect of the piezoelectric field on the opticalmatrix element in large QDs then depends on the QD shape.For many shapes this leads to a steep decrease of Mtot2beyond a critical QD height Fig. 1; however, for othershapes with a large aspect ratio  and truncation factor ,the piezoelectric field has little effect on the QD ground statematrix element.In summary, we have analyzed theoretically the groundstate optical matrix element in QDs. For many dots the kdependence of the bulk optical matrix element contributes toa reduction in matrix element compared to that initially ex-pected in an ideal QD. The magnitude of the matrix elementis very sensitive to variations in QD size and shape. It shouldtherefore be possible to engineer the size and shape of QDswith the aim to optimize the optical gain in QD lasers andoptical amplifiers. In particular, “flat” QDs with a large as-pect ratio have a larger matrix element. We conclude that,depending on the QD geometry, the piezoelectric field mayeither have little effect on the ground state matrix element orelse will markedly reduce it, effectively switching off theground state optical transition in some QDs.This work was supported by SFI Ireland.1D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Hetero-structures Wiley, New York, 1999.2M. Asada, Y. Miyamoto, and Y. Suematsu, IEEE J. Quantum Electron.QE-22, 1915 1986.3M. V. Maximov et al., Electron. Lett. 35, 2038 1999.4L. Harris, A. D. Ashmore, D. J. Mowbray, M. S. Skolnick, M. Hopkinson,G. Hill, and J. Clark, Appl. Phys. Lett. 75, 3512 1999.5G. Park, O. B. Shchekin, S. Csutak, D. L. Huffaker, and D. G. Deppe,Appl. Phys. Lett. 75, 3267 1999.6L. V. Asryan, M. Grundmann, N. N. Ledentsov, O. Stier, R. A. Suris, andD. Bimberg, J. Appl. Phys. 90, 1666 2001.7O. Stier, M. Grundmann, and D. Bimberg, Phys. Rev. B 59, 5688 1999.8J. A. Barker and E. P. O’Reilly, Phys. Rev. B 61, 13840 2000.9C. Pryor, Phys. Rev. B 60, 2869 1999; J. Kim, L.-W. Wang, and A.Zunger, ibid. 57, R9408 1998.10W. Sheng and J.-P. Leburton, Appl. Phys. Lett. 80, 2755 2002; W. Yang,H. Lee, T. J. Johnson, P. C. Sercel, and A. G. Norman, Phys. Rev. B 61,2784 2000.11A. D. Andreev, Proc. SPIE 3284, 151 1998; A. D. Andreev and E. P.O’Reilly, Phys. Rev. B 62, 15851 2000.12T. B. Bahder, Phys. Rev. B 41, 11992 1990; 46, 9913 1992.13A. D. Andreev, J. R. Downes, D. A. Faux, and E. P. O’Reilly, J. Appl.Phys. 84, 297 1999.14F. Szmulowicz, Phys. Rev. B 51, 1613 1995.15S. L. Chuang, Physics of Optoelectronic Devices Wiley, New York,1995, page. 387.16D. Ahn, Prog. Quantum Electron. 21, 249 1997, see Fig. 317E. P. O’Reilly, Semicond. Sci. Technol. 4, 121 1989.FIG. 3. Surfaces of constant probability density equal to 35% of the maxi-mum value for the QD ground electron states left-hand side and groundhole states right-hand side for QDs of the following sizes: a h=5.5 nm,=2.9, =0.25; b the same shape and size as a, but without piezoelectricfield; c h=9 nm; =5, =0.75; and d h=9 nm; =5, =0.5.213106-3 A. D. Andreev and E. P. O’Reilly Appl. Phys. Lett. 87, 213106 2005

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Optical matrix element in InAs∕GaAs quantum dots: Dependence on quantum dot parameters

We present a theoretical analysis of the optical matrix element between the electron and hole ground states in InAs/GaAs quantum dots (QDs) modeled with a truncated pyramidal shape. We use an eight-band k center dot p Hamiltonian to calculate the QD electronic structure, including strain and piezoelectric effects. The ground state optical matrix element is very sensitive to variations in both the QD size and shape. For all shapes, the matrix element initially increases with increasing dot height, as the electron and hole wave functions become more localized in k space. Depending on the QD aspect ratio and on the degree of pyramidal truncation, the matrix element then reaches a maximum for some dot shapes at intermediate size beyond which it decreases abruptly in larger dots, where piezoelectric effects lead to a marked reduction in electron-hole overlap.</p

Andreev, A D

O'Reilly, E P

Surrey Research Insight

Optical Matrix Element in InAs/GaAs Quantum Dots: Dependence on Quantum Dot Parameters

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Optical matrix element in InAs/GaAs quantum dots: Dependence on quantum dot parameters

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