centre-of-mass quantization of excitons in ga as quantum boxes

8/13/2019 Centre-Of-mass Quantization of Excitons in Ga as Quantum Boxes

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Semicand. Sci. Technal. 8 1993) 67M74. rinted in the UK

Centre-of-mass quantization of excitons inGaAs quantum boxes

S Jazir i tx G Bastards and R Bennaceurt7 Laboratoire de Physique de la Matiere Condensbe, Facult6 des Sciences deTunis, Campus Universitaire, Belvedkre.1060 Tunis , Tunisia7021, Bizerte, Tunisia9Laboratoire de Ph y s iq u e de la Matikre Condensee d e I'Ecole NormaleSup rieure, 4 rue Lhomond, F-75005 Paris, FranceReceived 4 December 1992, accepted for publication 20 January 1993

Dkpartment de Physique de L'Ecole Normale Super i eu re d e Bizerte, Jarzouna

Abstract. By using a variational-perturbative method, w e have studied the excitonsconfined in wide quantum boxes, i.e. the size quantization of t h e exciton centre-of-mass. The exciton energies are significantly shifted to h igh energy. The oscillatorstrength is enhanced with decreasing box size.

1. IntroductionRecently nanofabncation technology has attracted greatinterest, and it has become possible to confine carriers inall three spatial dimensions, thereby creating quantumboxes and quantum dots [l-131. The optical propertiesof confined systems have been shown to be stronglyinfluencedby the carrier confinement, leadingto a signifi-cant enhancement of the excitonic binding energy andoscillator strength, as in two-dimensional structures(quantum we QW)14] and in one-dimensional systems(quantum well wire, QW) [l5] and probably in zero-dimensional systems (quantum box, QB) [S, 16JTo our knowledge the transitions between the differ-ent dimensional systems are not well known. The transi-tion from 3D bulk-like excitons to 2D quantum wellexcitons is characterized by the quantization of thecentre-of-mass motion of the exciton [17-191. Recently[ZO] the transition from the 2D quantum well regime tothe ID quantum well wire exciton regime wasalso shownto be dominated by the quantization of the centre-of-mass motion, Here, we shall study the excitons in a largequantum box the volume of which is significantly largerthan U:, where a, is the exciton Bohr radius. Thiscorresponds to the dimensions of the boxes currentlybeing fabricated. In these wide boxes, the Coulombinteraction correlates the relative motion of electron-hole pair while the confinement influences the centre-of-mass motion of the pair. We shall show that the excitonlevels follow the centre-of-mass quantization like a 'sin-gle particle' and not the separate electron-hole quantiza-tion. We present the theoretical framework in section 2,the results for exciton binding energies and oscillatorstrengths in a quantum box in section 3 and our conclu-sions in 4.0268-1242/93/050670 + 05 07.50 @ 1993 IOP Publ i shing Ltd

2. Determinationof energy levels and wavefunctionsWe consider a rectangular GaAs quantum box Lx,,,Lz) embedded in Ga,-Al,As barriers, with the condi-tion Lx > L, > L , Lx,,, L, are the well widths alongthe x, y and z directions respectively). The Hamiltonianor the exciton in this quantum box, within the effective-mass approximation, is given by

- -EmLi E)z: + V,(Xi,y;, 2;

3)O(x) is the step function, E , is the bulk bandgap energyand is the dielectric constant for GaAs; the distortiondue to the image potential effect is neglected. K=c is thepotential of the finite well in one direction determined bythe energy discontinuity between well and barrier mater-ials. In the GaAs/Ga,- As system this discontinuitycan be expressed as AEG = 1.247~ev) at 300K andx 5 0.45,wherex represents the mole fraction ofAl in thealloy. In our calculationx is'equal to 0.32. This discontin-uity is then split between conduction and valence bands.A splitting ratio of 0.65/0.35 was used in this work. Theeffectivemass in the directions (x, y) is m,,,,nd inLi s theeffective mass in the perpendicular direction 2). At the


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Quantization of excitons in GaAs q u an tu m boxesband edge, the effective masses are mi/,= m,, = 0.665m0;the heavy hole has miihb= mo/(yl + y Z , mlhh =mo/ y,- y,), and for the light hole m,,lh=mo/ yl - 2),mLlh= mo/ y , + 2y,), where m is the free electron massand y = 6.85 and y 2 = 2.1 are the Luttinger parameterslor vans . n t nc inierrdces oetween w c u s m u amcrs tilediscontinuity in effective masses was taken into account.Using the centre-of-mass and relative coordinates (R, r )of the electron-hole. pair with the corresponding totalmasses M / / , M , ) and reduced masses (,u/,, pi), theiiamiiionian caRDe iew.riiien ag

c n . . .,....*.-c ..__...- .... 1. ^ _ _ I L 7

+ V ( R ,P) + E , (4)v R = v e ( r e ) f hcUh) 5)

H = Hl(r) + H 2 ( R )+ V ( R , . (6)H,(r) represents the relative motion of the electron-holepair and characterizes the Coulomb correlation betweenthe two particles. As a first simplifying step we canapproximate the reduced masses in H , ( r ) by

1 2 1 1 1- = _ _ +--,P 3 P// 3 PlThe Hamiltonian HI@)becomes

(7)

and the solutions for H,(r ) are the well known hydro-genic wavefunctions.The Hm.i tonian li12 11) + V R, describes thecentre-of-mass motion, like a single-particle systemconlined in a box with barriers characterized by thepotential V(R, r). To solve this Hamiltonian we intro-duce a parameter 1 by adding and subtracting thepotential AV@): in order to be able to split this Hamil-

tonian into two terms [Zl]:H L R ) +W (9)H L R , 1.) + H A R , F, Awith

1 \hZ 2H2(R, 1.) = - -2M,, (X 5)-?- E)~ , zz + AV(R)

where1 Y ( R )= 1AE& X 2--( :and

H,(R, r , A = V ( R , ) - V( R) . (12)

The Hamiltonian H,(R, A) is the unperturbed part whichcan he solved exactly.H,(R, r A is the perturbation termand the parameter A is a variational parameter. Thismethod combines the variational method with the per-turbation one and its validity can be assessed immedi-the quasidecoupling approximation [22] to solve theHamiltonian H2(R, I.), which is valid because we con-sidera arge quantum boxwith the condition L, > L y>Lz , the calculation is processed from the motion along

2 axis io motion x .+e obtain theenvelope function and the energy level for the centre-of-mass (CM) motion along the z axis by the followingequations:

aieiy ,Decause creates own smdd p a r ~ e i e r ,ising

with the potential given byV, Z, A = lAEG@ 2 -- = VZO Z 2 -- 14)?) 7By the same calculation as that in a conventional quan-tum well [14] the wavefunction@, a s a simple trigo-nometric function in the well and exponential outside thewell in the case of the bound states. At the interfacebetween the weU and the barrier, the discontinuity ineffective masses is taken into account and the energy levelE, is determined by

In a second step, we obtain the envelope functiona, ?)and the energy ievei E , along the y axis by tine sameprocedure, with the harrier potential given by

Finally we obtain the envelope function X) and theenergy level E,,, in the x direction with the barrierpotential.

x o x2 .?)We thus obtain the total wavefunctionsof the CM in thequantum box asY,,,(X, x 2, . = W Z , )@, E: A)@, X, I.). 18)

(1%They correspond to the energies

% M N ( ~ ) = EL@) + EM @ )+ EN@).These solutions, which still depend on the parameter A,are for the unperturbed part of the CM Hamiltonian. To

671


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S Jaziri et a/obtain the actual CM energies and wavefunctions, wedetermine the best choice of 1 s the one which ensuresthe fastest convergence of the perturbation series. Underthese circumstances we shall write

ELMN= E h N ( 4+ A E W " 4 (20)where AELMN(A)is approximated by the first-orderenergy matrix correction calculated by perturbationmethodAE & (4 =

X.~,,,,,,~~ R,, A)lH3 R,r, L ) ) I X ~ ~ ~ , L M N ( R ,, A (211where

x ~ ~ ~ . L M N ( R >, 4 = Rni ( r )YX ~ Y L M N ( RFinally, the energy levels of the exciton in a quantum boxare given by

= Cp dr W L M ~ ( R I. (22)

RYE n i m , ~ ~ ~EG - E ~ M N &,,,,LMN (23)nwhere R , is the effective Rydberg of the 3D exciton.The best solution, which no longer depends explicitlyon the parameter A, is determined by minimization of thetotal energy value by the principle of minimal sensitivity[21,23]. This condition is satisfied by requiring

with respect to L. If AEnlm,LMdA)s approximated byAE,$A,LMN(%)of equation (21), the condition (24) can besatisfied by requiringAEL;A,LMdl) = 0 (25)

yielding the L value.In order to calculate the exciton ground state relativeto the heavy-hole exciton E-HH) and light-hole (E-LH),we numerically evaluate the first-order perturbation en-ergy; the optimum condition requires AE\i lll(J.) = 0and thus determines the best variational parameter Lo.Hence the exciton ground state determined by the pertur-bative variational criterion is

E E , - R y +E11l(do). (26)We also calculate the oscillator strength for the excitonground state using the normalized wavefunction given by(22); the oscillator strength for the ground state is [24]

where C is a constant of proportionality which includesthe electric dipole matrix element and E is the excitonenergy in the box given by (26). The oscillator strength isproportional to the probability amplitude of finding anelectron and a hole at the same site, as is well known inthe case of the bulk [26] corresponding to Y = 0 and672

R = re= r,. According to equation (27) the oscillatorstrength for a 3D bulk-like exciton is

where E , is the exciton energy given by E., = E , - R,in the bulk,2 is the sample volume and ]91s(0)12s equalto 1/naz.Substituting C in equation (28) we find

IIn the following we will present the calculated oscillatorstrength of the exciton ground state in the box normal-ized to that of a free exciton in a bulk material withvolume C2 = LxL,Lz. In the large C2 limit f/f., becomesindependent of C2 and converges towards 2 /n6 = 0.53, alimit obtained by assuming an infinite confinement forthe CM coordinates along x , y and z. It is troublesomethat one does not recover lim (f/f,,) = 1. This point hasalready been noticed by Kayanuma [16] (who studiedthe case of cylindrical confinement) and directly resultsfrom the assumed even distribution of the free exciton CMin the volume a, i.e. disregarding any edge effect. Onemay, however, remark that the infiniteL imit is to a largeextent unreachable in actual materials. Indeed, if thebox size is very large, the scattering phenomena willunavoidably limit the coherence of the exciton wavefunction to a volume V,,, thereby preventing theinfinite L limit being meaningful.3. Numerical resultsWe have numerically evaluated the expressions derivedin section 2 and minimized the CM energy levels forheavy-hole exciton (E-HH) and light-hole exciton (E-LH)for several values of the box dimensions. In figure 1 weplot the CM energies of the two quantum states as afunction of the length Lz, keeping L, = 500 . andLx= 1000 8, to show the importance of the CM sizequantization. We see that the energies decrease withincreasing Lz, a feature reminiscent of quantum wellbehaviour (the size quantization for the x and y motionin negligible compared with that of the z motion in figure1 . At large Lz the exciton energies converge to therespective bulk Rydbergs. Note that in this limit themodel of CM quantization provides a lower excitonenergy than the one which assumes an independentconfinement for the electron and the hole (since l/(m, +mh) < /m, + l/mh).It is not easy to find a lower limit onLzwhere the CM quantization model becomes worse thanthe model of independent particle quantization. A quali-tative criterion would be to consider the CM quantizationmeaningful if the CM confinement were smaller than orequal to the energy separation between the 1s and 2sstates, i.e. if

30)


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Quantization of excitons in G a A s qua n t um boxes150

~ - . . . ,

LY=500 .I u=innn

125LY=500 .;::\50 - u=1000%.

25-: -HH0 0 1do 200 400 5000-100 200 400 500

Figure 1. Excitonic CM energies as a functionof L, for t h eground state of heavy-hole excitons E-HH) and light-holeexcitons (E-LH).

c- - - .--: _I_^ 2- -< - . . TA _.._I.I U I a VlIC-UUUCU>IUIldl blLG q U l U ~ t l U l l . ll bUU I a W a ythe effectiveHamiltonian acting on the exciton CM cansafely be taken as scalar, and the effective potential energyis the actual one weighed by the 1s probability density forthe reduced motion. The criterion quantitatively meansLz 2 00 A for GaAs-like parameters. In materials with awider bandgap or more polar materials the CM quantiza-tion model should work down to lower thicknesses thanin GaAs-Ga(A1)As.We show in figure 2 the CM ground state energyversus L, for the (E-HH) nd the (E-LH) excitons forLz = 250 8, and Lx = 2000 A. One finds the same kind of

Figure 2. Excitonic CM energies as a function of L, for h eground state of heavy-hole excitons E-HH)and light-holeexcitons (SLH).

behaviour as in figure 1. The heavy-hole CM s lower inenergy than the light-hole CM because of its heavier massalong the z axis.Figures 3 and 4 show plots of the reduced excitonoscillator strength f/f., versus Lz (at fixed Lx, y) rversus L, (at fixed L,, Lz) . In these plots the change off/f., -with Lz or L, is dominated by that ofI~YlI1(R)d3RIZ hile the CM energy-dependent termdisplays negligible variation. In the strong confinementregime, our calculated increase of f/f,. s less importantthan the ones found by Bryant [SI or Kayanuma [16]who used models of separate confinement for the electronand the hole. In the intermediate regime (Lz 2 200 A ,however, our model of CM quantization leads to a gent1e.rparticle quantization..l-- :-- - ___:It r Le I.-^^ :--I-UCUIUC UI J , er W l U l fiz W61l LUG UllC5 d>>UUUUgSLUpG-

. Ixs2-

LY=500 A .U=2000 A- -LH--- E-HH

Figure 3. The normalized oscillator strength of t h e excitonground state per unit volume as a function of the length Lz.

~

I 7 n E nL J Y A.LX=2000 AE-HH2 ,

--- E-LH

Figure 4. The normalized oscillator strength of t h e excitonground state per unit volume as a function of the length L,.673


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S Jaziri et a /4. ConclusionWe have studied the size quantization of the excitoncentre of mass in quantum boxes by combining varia-tional and perturbation methods. Our calculationsshould work relatively well in wide boxes (with dimen-sions 2 2-3 Bohr radi but are not expected to be betterthan the models which assume independent size quanti-zations for the electron and the hole in narrow boxes.The transition from the CM quantization to the single-particle independent quantizations remains to be morequantitatively studied.

[q Gilliot P, Merle J C, Levy R, Robino M and HonerlageB 1989 Phvs. status Solidi b 153 403

hd...nu.l. Anm. nt.=- R'. ...'~J.. ...

The Laboratoire de Physique de la MatiLre Condendede l'Ecole Normale SupCrieure s Laboratoire associk auCNRS UA 1437) et aux UniversitCs Paris 6 et Paris 7.This work has been partially supported by the Com-mission of the European Communities (Esprit Project6719)

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centre-of-mass quantization of excitons in ga as quantum boxes

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