Excitonic properties in (111)B-grown (In,Ga)As/GaAs piezoelectric multiple quantum wells

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PHYSICAL REVIEW B 15 DECEMBER 1997-IVOLUME 56, NUMBER 23

Excitonic properties in „111…B-grown „In,Ga…As/GaAs piezoelectric multiple quantum wells

P. Ballet,* P. Disseix, J. Leymarie, A. Vasson, and A-M. VassonLaboratoire des Sciences et Mate´riaux pour l’Electronique, et d’Automatique, Unite´ Mixte de Recherche No. 6602 du Centre National

la Recherche Scientifique, Universite´ Blaise Pascal Clermont-Ferrand II, 63177 Aubie`re Cedex, France

R. GreyDepartment of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 4DU, United Kingdom

~Received 21 July 1997!

The excitonic properties in two~111!B-grown In0.15Ga0.85As multiple quantum wellp- i -n diodes, with 7and 14 quantum wells, respectively, are investigated by thermally detected optical absorption~TDOA! and byelectroreflectance~ER! as a function of applied bias, the latter modifying the electric-field distribution in theheterostructure. The line shapes of the ER signals are analyzed by means of a multilayer model enabling theenergies and the oscillator strengths of excitons to be deduced while the direct measurements of the energypositions of the TDOA peaks provide an accurate determination of the excitonic transition energies at zero-voltage applied bias. The excitonic characteristics are calculated by using a variational approach with atwo-parameter trial function. The piezoelectric field in the strained InxGa12xAs layers is determined by in-cluding the excitonic contribution. The theoretical oscillator strengths are compared to those obtained from ERexperiments for several excitonic transitions; all the physical trends are well reproduced but it appears that aquantitative agreement cannot be found without taking into account the in-plane valence-band mixing. A studyis also presented for the optimization of optoelectronic devices by means of a figure of merit that combines theoscillator strength of the fundamental excitonic transition and the ability for such devices to produce the largestenergy shift for a 1-V additional applied bias.@S0163-1829~97!04447-0#

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I. INTRODUCTION

The growth of semiconducting strained layers on highdex planes has enabled a further field of investigation toexplored in the physics of heterostructures due to the pence of a strong internal piezoelectric field. This piezoeltric field results from the displacement of the anion and cion sublattices induced by the off-diagonal terms in tdeformation tensor and is consequently absent in the casgrowth on conventional~100! substrates. The~111! orienta-tion offers the possibility to produce the largest piezoelecfield, which can easily exceed 100 kV cm21;1 in addition, thecritical thickness is increased with respect to the~100!growth and thus enables thicker multiple quantum-wstructures to be grown.2

The effects of such a strong piezoelectric field on the bstructure, leading to large energy shifts and to optical nlinearity, are used to design electro-optic devices.3–5 The op-timization of their performances imposes the knowledgethe piezoelectric field effects on the excitonic propertiessuch structures. The most important consequence of thewell electric field is the reduction of the electron and howave-function overlap by pushing back the carriers toopposite sides of the well. This phenomenon induces ladecreases in the exciton binding energy and oscillastrength of the fundamental exciton. It is enhanced whenwell is thick and can even lead to the formation of a quatype-II configuration.6

Another important effect of the internal electric fieldbreaking of the symmetry along the growth direction. In thcase, transitions forbidden from the classical selection ruare now allowed and the associated oscillator strengths

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be larger than that of the fundamental exciton if the in-wfield is sufficiently strong~typically 100 kV cm21!.

Since the electro-optic devices mentioned above usequantum-confined Stark effect~QCSE!, which produces alarge energy shift of the band edge, their efficiency couldproportional to the magnitude of the in-well electric fielHowever, the larger the piezoelectric field the larger thein the oscillator strength for the fundamental excitonic trasition that dominates the band edge.

In this work, in order to detect the different excitontransitions, we have carried out thermally detected optabsorption~TDOA! and electroreflectance~ER! experimentson two InxGa12xAs/GaAs multiple-quantum-wellp- i -n di-odes grown on~111!B substrates. The piezoelectric fielddetermined by compensating the in-well field by means obias voltage. The effects of the piezoelectric field on texcitonic properties are investigated through a variatiomethod. The oscillator strengths for the fundamental asome excited transitions, determined from our experimeand calculated theoretically, are compared within the raof the reverse applied bias investigated. Finally, in orderdetermine quantitatively the capability of piezoelectric herostructures to produce the largest energy shifts combwith the largest oscillator strength, we present a figuremerit. This characterizes the type of structure investigahere; it is equivalent to that defined by Rodriguez-Gironand Rees for all optical non-linear devices.7

II. EXPERIMENTAL DETAILS

In0.15Ga0.85As/GaAs multiple quantum wellp- i -n diodeshave been grown by molecular beam epitaxy on the B fac

15 202 © 1997 The American Physical Society

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56 15 203EXCITONIC PROPERTIES IN~111!B-GROWN . . .

n1 ~111! GaAs substrates tilted by 2° towards (211) for anenhanced crystal quality.8 The n1 and surfacep1 layers ofthe diodes consist of 3-mm GaAs, Si and Be doped, respetively; the nominal dopant concentrations, confirmedC(V) measurements are 231018 cm23. Wells of 100 Åwidth are incorporated centrally in the intrinsic GaAs regiand are separated by 150-Å GaAs barriers. Details concing growth conditions are available in Ref. 9.

The two samples investigated here differ only by tnumber of wellsN, namely, 7 in sample 1 and 14 in samp2. The length~7350 Å! of the intrinsic region is the same fothe two samples leading to an equivalent built-in elecfield. In order to allow for the potential drop across theodes, the electric fields are shared between the wellsbarriers as described by Pablaet al.10 This sharing of thepiezoelectric field induces a larger in-well field in the samcontaining the smallest number of wells leading to a stronQCSE. In this way, the excitonic transition energiessample 1 are expected to be smaller than those in samp

TDOA is a nonconventional technique based on thetection of the temperature rise of the sample caused by pnon emission due to nonradiative de-excitation occurringter optical absorption. This heating is detected at liquhelium temperatures~typically 0.35 K! by a germaniumresistor thermometer. This method enables temperavariations much less than 1023 K to be detected and it isthus very appropriate for studying weak absorptions occring in the high electric field regime. The monochromalight source is a halogen lamp followed by a HR 640 JobYvon monochromator, which provides a weak excitatipower that prevents any optical screening of the in-well eltric field. Details concerning the experimental setupavailable in Refs. 11 and 12.

ER experiments are performed at 4 K on sample pieces inthe form of 400-mm-diameter annular diodes with a 200-mmoptical access. The modulation source is a square alternvoltage of amplitude 0.5 V. In order to balance the in-wfield, we superimpose a bias dc voltage. The excitatsource is a halogen lamp; the optical screening is limitedthe addition of a filter that stops photons of energy larthan that of the barrier gap. The modulated signal is analythrough the HR640 monochromator and detected withliquid-nitrogen-cooled germanium detector using a standlock-in amplification.

III. EXPERIMENTAL RESULTS

The TDOA spectra of the two samples are displayedFig. 1. In order to make the analysis easier we have silated the monotonic increase, with photon energy, ofTDOA signal by a polynomial curve and subsequently elimnated it by a simple subtraction. This increase of the TDOsignal is attributed to the impurity absorption front of thGaAs barriers.

For the two samples investigated, four transitions, invoing the first (e1) and second (e2) levels of electrons and thfirst (hh1), second (hh2), and third (hh3) levels of heavyholes are detected. For sample 2, the wide structure obseat higher energy corresponds to the superposition of sevtransitions such ase2hh2, e1hh4,..., which are close in energies and in oscillator strengths. It is worth noting that t

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intensities of the transitions forbidden from the classicallection rules may be of the same order of magnitude as thof the allowed transitions; this is the signature of the prence of a large in-well electric field.

As expected, the transition energies of sample 1, conting only 7 wells, are redshifted by a few meV with respectthose of sample 2. This fact confirms that the electric fieldlarger in the wells of sample 1 and offers the possibilitytailor this in-well field by a simple modification of the number of wells in such heterostructures.

Figure 2 shows the evolution of the ER spectra of sam1 when the applied bias is varied from 0 to 12 V. The insindicates how this bias influences the band structure; theplied electric field balances progressively the in-well fieuntil reaching the flat well conditions and then reversesThe spectra are noisy because of the reduced optical acimposed by the processing of the samples but excitonictures associated with GaAs and with confined states upe2hh2 are well defined. One can follow the position and itensity changes of the fundamental exciton signal from 1.512 V; the maximum in the transition energy indicates thatflat band configuration in the wells has been reached. Tsituation occurs for voltages near to 10 V. In this case,lack of in-well electric field leads to the vanishing of thforbidden transitions. At lower voltages, the fundamental aexcited transitions can be detected together. For a 3-Vplied bias, their intensities are of the same order; forsmaller applied bias, corresponding to larger in-well fie

FIG. 1. TDOA spectra of sample 1 (N57) ~a! and sample 2(N514) ~b!. To facilitate the analysis, a baseline~dashed! corre-sponding to the impurity absorption front of the GaAs barriers hbeen removed. The result of the subtraction is shown after ampcation at the bottom of each part of the figure.

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15 204 56P. BALLET et al.

the magnitudes of the forbidden transitions become stronthan that of the fundamental transition. However, for tlargest in-well fields~without applied bias!, all transitionsassociated with the wells disappear. This shows that for vstrong electric fields, all transition probabilities are weaThis vanishing of the excitonic features of the wells canalso related to the strong GaAs excitonic signal, which incates that many carriers recombine in the barrier regwhere the electric field is weaker.

Another effect of increasing the bias is the increase ofbarrier field. From the top part of the inset in Fig. 2 it canseen that, for a strong barrier electric field, the potential dcontinuity forms a tip through which the carriers may escaIn this case, the states are no longer quasibound but becresonant. This phenomenon is responsible for the dampinthe excitonic features and leads to their complete vanishfor applied voltages larger than 12 V.

In order to have a precise knowledge of the energy ptions of all excitonic transitions in the ER spectra, we haused a model that is based on a calculation of the refleccoefficientr and on the determination of its variations wielectrical modulation. The reflection coefficient is written agebraically by considering the different reflections and tramissions at the interfaces between successive layers offerent optical indices taking into account the differedephasing terms provided when the light goes throughlayers of the heterostructure. The reflectanceR can be ex-pressed as the square modulus of the reflection coeffic

FIG. 2. ER spectra of sample 1 for different values of applbias (Vapp) between 0 and 12 V. Spectra are shifted for clarity. Tinset shows the potential profiles of one of the wells for differevalues of applied voltage.

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and taken as a function of the real and imaginary parts ofdielectric functione1 ande2 in the form

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The modulation voltage in the ER experiments modifithe electric field inside the sample and thus also modifiesreflectance by mean of the dielectric function. In the caseheterostructures, the change in reflection can be expressterms of the first derivative of the dielectric function leadinto the well-known expression13

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It can be written algebraically by one derivative of threflectance with respect toe1 ande2 . Finally, the dielectricfunction is modeled by a standard damped Lorentzian oslator including three adjustable parameters:Eexc, which isthe energy of the oscillator,g, a phenomenological dampinparameter, andA, the integrated intensity. The latter can brelated to the oscillator strength by matching the absorpcoefficient obtained from classical and semiclassitheories.14

A change in the electric fieldDF induced by a change inthe modulated voltageDV leads to a modification in thedielectric function given by

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The three terms in square brackets correspond to the dient mechanisms of modulation and give the shape ofspectra.

Modulation spectroscopy is a relevant technique for ostudy because the magnitudes of the excitonic signalstained depend upon the modulation efficiencies.15 Since themodulation spectra in the case of heterostructures are ofderivative type and since the dominating modulation mecnism involves the excitonic energyEexc,

13 the magnitudecan be expressed as the productf osc(]Eexc/]F) wheref osc isthe oscillator strength of the excitonic transition. In this wait is expected that no transitions could be detected forbias corresponding to the exact flat well because the t]Eexc/]F vanishes. This can be used to locate the flat baconfiguration with a high precision leading to an accurdetermination of the in-well field strength. Unfortunately, whave never found such a flat spectrum. This is due to thethat the band structure is not perfectly symmetric becausthe presence of a strong barrier electric field. In addition,such a bias, the damping of the excitonic features increaleading to the contribution of the second modulation te]g/]F.

IV. THEORY

We have considered a free-standing heterostructurethat the strain is entirely accommodated by the quantwells.16 The strain tensor has been determined from themalism described by Yanget al. giving analytical expres-sions of the deformation tensor for an arbitrarily orientsubstrate.17 We have applied this formalism to the case o

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TABLE I. Parameters used to calculate the transition energies.

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GaAs 5.6533a 11.88a 5.38a 5.99a 28.16b 25.4a 0.0667a 6.85a 9.29a 0.341a 12.50a

aReference 18.bReference 19.cReference 20.

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pure @111# direction and to that of a tilt of 2° towards th@21 1# direction and found a piezoelectric field differenless than 1% between the two cases. Therefore, we haveconsidered the case of a pure~111!B growth axis to investi-gate theoretically the electronic properties of such system

The parameters used for the calculation of the transienergies are listed in Table I. The effective masses ofheavy holes in the growth direction are determined fromLuttinger-Kohn Hamiltonian@mhh'5m0 /(g122g3)#.21 Thelight holes play no role in our spectroscopy results becathey are resonant with the GaAs barriers for the indium coposition investigated. The effective masses for electronsthose related to the~100! growth axis assuming that the efect of the growth direction on the band structure is maiconfined to the valence band.22,23The effects of the strain onthe energy gap are taken into account;24 the unstrained wellgap is deduced from the Goetz relation.25 All the alloy pa-rameters are deduced from those of the binary constituby using Vegard’s law.

To calculate eigenenergies and the associated wave ftions, we have used a staircase approximation withintransfer matrix formalism, which is very appropriate to stuirregular potentials such those provided by electric field26

The potential is discretized in stairs; the length of one statypically the monolayer thickness and its height represethe potential variation induced by the electric field.

In order to evaluate the effects of strong internal elecfields on the excitonic properties, the transfer matrix formism has been combined with a variational method.27,28 Thetotal Hamiltonian of the excitonHexc can be expressed assum of three contributions:

Hexc5He1Hh1Heh .

He andHh are the classical Hamiltonians of the electron athe hole in the presence of a longitudinal electric field aHeh contains the kinetic energy and Coulomb terms such

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where r is the distance between the two carriers,r is theradial component in the cylindrical coordinates,m representsthe reduced exciton effective mass in the layer plane ande isthe relative dielectric constant.

The hole in-plane effective mass is determined byglecting the longitudinal component of the wave-vectorthe Luttinger Hamiltonian. This approximation leads to texpressionmhhi5m0 /(g11g3); the implications of this ap-

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proximation are discussed in the next section. The dielecconstant in the InxGa12xAs layers is determined by a lineainterpolation~see Table I!.

The excitonic trial wave function contains two variationparametersa andl and takes the form:

Fexc~ze ,zh ,r!5we~ze!wh~zh!expS 2Ar21a~ze2zh!2

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ze , zh , we , andwh are the coordinates of the electron ahole along the quantization direction and their envelofunctions, respectively. The parameterl is a measure of theexciton Bohr radius anda stands for the exciton dimensionality. The limiting two-dimensional~2D, a50! and three-dimensional~3D, a51! excitons in an InxGa12xAs/GaAsquantum well, in the presence of longitudinal internal eletric fields, have been studied frequently,29,30 but Andreet al.have shown thata takes intermediate values in the caseII-VI compounds.6

The binding energy is calculated numerically by minimiing the total energy of the exciton. The oscillator strengththen deduced from the expression:

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The variations with the well width have been calculatedan In0.15Ga0.85As quantum well, with 120 kV cm21 in-wellfield (Fw) and 40 kV cm21 barrier opposite field (Fb) andare plotted in Fig. 3 for the fundamental and first excitexcitons. The lowest limit for thicknesses is 30 Å becauthe electrons become no longer localized in the well butcape through the potential tips induced by the barrier fieThe results are compared to these for a fundamental excin an equivalent structure without any electric field. One csee that the binding energy and oscillator strength of excitstrongly decrease with well width because of the enhancarrier separation leading to a quasi-type-II configuratiFor thicknesses larger than 150 Å, thee1hh1 and e1hh2wave-function overlap reaches a zero value. With increaswell width, the dimensionality parametera first follows theevolution of that in a square quantum well in the regiwhere the confinement effects are predominant; the elecfield perturbation is not seen by the carriers that are strondelocalized for small well widths. Note that, in this case, tmajor effect of an electric field in a quantum structure, whiis to allow transitions forbidden from classical selectirules, vanishes. When the thickness increases above 10the electrons and holes are pushed towards opposite sidthe quantum well leading to a drastic decrease of the C

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lombic interaction between the two carriers. This induce2D character for the exciton and explains the continuouscrease ofa. The e1hh2 exciton exhibits the same behavioas these corresponding toe1hh1, but due to the larger extension of the wave function of the second level of hole, toverlap remains and begins to decrease for larger quanwell thicknesses than fore1hh1.

V. RESULTS AND DISCUSSION

The experimental transition energies of the fundameexciton in samples 1 and 2 and the second and third excexcitons in sample 2, obtained from both TDOA and ERplotted as a function of applied bias in Fig. 4. There is a goagreement between the results from the two spectrosctechniques; the slight difference between the TDOA dataextrapolations of the energy values deduced from ERzero bias are attributed to inhomogeneities in the quant

FIG. 3. Variations with well width of the oscillator strength~a!,binding energy~b!, and exciton dimensionality~c! obtained with thetwo variational parameter excitonic trial function. Solid lines corspond toe1hh1 ~thick lines! and e1hh2 ~thin lines! excitons in apiezoelectric quantum well in the intrinsic region of ap- i -n diodeand the dashed lines represent the results calculated for a pesquare well with equivalent characteristics. The electric field limthe study to well width larger than 30 Å; under this limit the staare no longer bound.

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well thicknesses.~The pieces of the samples used for TDOare different from these processed for ER.! It can be seen thathere is an inversion of the curvature of the ER energysmall voltages ('2 V). This is due to screening effects; alow bias voltages, the in-well field is stronger and therefothe in-well screening efficiency is a maximum, becomisufficiently large to increase the excitonic energies by reding the QCSE.

The best fit to the experimental data is plotted in Figand concentrates on the fundamental exciton becausepossible to locate the bias and energy positions of theband in the latter case and because the curvature of theperimental points for thee1hh2 ande1hh3 excitonic energiescannot be well reproduced with any set of fitting parameteThe energy differences betweene1hh2-e1hh1 ande1hh3-e1hh1 transitions are too large. In addition, thmaxima of the theoretical curves occur at lower voltagese1hh2 ande1hh3 than fore1hh1. These two behaviors suggethat indium surface segregation may occur; its effects arincrease the well width narrowing the sublevels, and to blshift the excitonic energies enabling the in-well electric fieto be increased in our model to push the theoretical curvethe excited excitons to larger bias voltages. This point is nunder investigation.

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FIG. 4. Experimental excitonic transition energies deduced frTDOA ~squares! and from fits to ER spectra~circles! for sample 1~filled symbols! and sample 2~hollow symbols!. The lines are fromtheoretical calculations with wells of 95-Å width, indium compostion is x50.143 for sample 1 andx50.144 for sample 2; the pi-ezoelectric discontinuity is taken to be 165 kV cm21. The experi-mental and theoretical results fore1hh2 ande1hh3 for sample 1 arenot plotted for clarity; they exhibit the same behaviors as thosesample 2.

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tal exciton in both samples. The parameters used to fitexperimental data are close to the nominal values giventhe growth sequences. In agreement with the fits to ER stra, the width of the wellsLw is estimated to be 95 Å for thetwo samples. The indium compositionx is then found to be0.143 for sample 1 and 0.144 for sample 2. These two vables,x andLw , are evaluated simultaneously by considing the energies of the flat band configuration, which canlocated unambiguously from the top of the experimencurves plotted in Fig. 4. Thus we assume that the barrier fidoes not influence the exciton energy,7 therefore the pair(x,Lw) is determined by calculations for a perfect squawell. The third adjustable parameter is the piezoelectric fiin the InxGa12xAs layers; this electric field arises from thproduct of a piezoelectric constant and a term includingdisplacements of atoms under a field of deformationsduced by the lattice mismatch. Note that if one assumesthe piezoelectric constant of the alloy can be calculated bsimple linear interpolation, the piezoelectric field is therefodetermined from the knowledge of the indium compositiand is not to be taken as an adjustable parameter. In factpiezoelectric field calculated by this formalism220 kV cm21 for x50.144, which is inconsistent with ouexperimental results. This value must be drastically reduto 165 kV cm21 for a correct description of the experimentexcitonic transition energies to be made. The uncertaintthe determination of the piezoelectric field from the latter5% and is mainly due to the lack of precision in the estimtion of the exact bias for reaching the flat band configuratiSuch a discrepancy between theoretical and experimevalues of the piezoelectric discontinuity has been alreadyported in the literature but its origin is still unclear.15,31–33

Several authors attribute this disagreement to an approximknowledge of the piezoelectric constants of the binconstituents;32,33 Tober and Bahder suggest that it may arfrom charge accumulation at the interfaces screening thewell electric field.15 Other physical effects may be mentionesuch as indium surface segregation whose effects on bstructure are strongly enhanced by the presence of lbuilt-in electric fields. Note that the paper of Shenet al. re-ports an experimental determination of the value of theezoelectric field in good agreement with the theoretiprediction.34 However, the uncertainty given correspondsthe discrepancy between our results and the theory.

Since the in-well electric field is responsible for the shiof the transition energies, it is necessary to reproduce exathe curvature in experimental data for an accurate detenation of its value and thus of that of the piezoelectric fiestrength. The variation of the excitonic binding energy takan important place in this curvature. As can be seen frFig. 5, the variation in the binding energy with applied biexceeds 2 meV. Therefore, if excitonic contributions aretaken into account via the binding energy variations wapplied bias, the piezoelectric field will be underestimatFrom a practical point of view, we have plotted the bindienergies obtained from the two variational parameter tfunction and the two limited cases in Fig. 5. It can be sethat the choice of model does not really influence theTherefore, the use of the simplest 2D model (a50) thatbrings numerical results instantaneously appears to be appriated for a fitting procedure. The values of the two var

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tional parameters also plotted as functions of applied bfollow the physical evolution described above. The Bohrdius increases for large in-well fields due to the enhanspatial separation of electron and hole leading to a dradecrease of the Coulombic interaction responsible for thein the value of the dimensionality parameter. Note thatfall of a on the large bias side begins before reaching theband configuration. This can be explained by the large brier field that attracts the carriers towards the outside ofquantum well making their wave functions nonsymmeteven in a flat well and reducing slightly their Coulombinteraction.

All the experimental and theoretical results concernthe excitonic oscillator strength values are plotted in Fig. 6functions of applied bias fore1hh1, e1hh2, ande1hh3 exci-tons. The experimental points deduced from the fits tospectra are plotted with 15% error bars; most of this relatuncertainty is attributable to that in the damping paramewhich influences drastically the evaluation of the experimtal oscillator strength. Theoretical results are also plottedthe two limiting casesa50 anda51 and for the case whera is taken as a variational parameter. The curves obtainethe latter case stand between the others, very close toobtained by using the 3D exciton model (a51). Such adiscrepancy between experimental and theoretical oscillstrengths, which is of the order of a factor of 2, has alreabeen reported in@001#-grown InxGa12xAs/GaAs quantumwells.27 However, Andre´ et al. found good agreement be

FIG. 5. Excitonic binding energy of the exciton in its fundametal state~a! and the variational parametersa andl ~b!, as functionsof the applied bias, witha50 ~2D model!, a51 ~3D model!, anda variable~aD model!.

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tween experimental data and calculations, employingsame variational method, in the case of CdTe/CdxMn12xTequantum wells.6 The only difference lies in the determinationof the hole in-plane effective masses; the valence-band ming was not taken into account in our calculations. The aproximation that consists of the determination of the in-plahole masses by a simple reduction of the Luttinger Hamtonian of the bulk InxGa12xAs to the in-plane components othe wave vector does not appear to be relevant for a quatative description of the oscillator strength of excitonsstrained quantum wells.

However, our model reproduces accurately the physitrends for the variations of the oscillator strengths with aplied bias and therefore with in-well field. The maximum foeach exciton is well defined and is localized at zero in-welectric field for the fundamental exciton and at a larger fiefor e1hh3 than fore1hh2. The fact that the oscillator strengthfor all excitons are weak in the high electric field regimconfirms the experimental observation that all the excitontransitions become undetectable for bias less than 1.5 V, cresponding to in-well electric fields exceeding 100 kV cm21.

This result is of fundamental interest for the designoptoelectronic devices because it shows that increasingin-well field for an enhanced QCSE, usable to provide larenergy shifts, leads to a strong decrease of oscilla

FIG. 6. Oscillator strength as a function of applied bias for the1hh1, e1hh2, ande1hh3 excitonic transitions in samples 1 and 2Circles with error bars are experimental data from ER experimenlines are from variational calculations usinga51 ~dashed lines!,a50 ~thin solid lines!, anda as a variational parameter~thick solidlines!.

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strengths of excitonic transitions for the fundamental excibut also for excitons involving excited levels. It is evidethat the best device combines the largest energy shift wthe largest oscillator strength for an optimized workinTherefore, and to follow the idea of Rodriguez-Gijones aRees who considered a figure of merit for all optical deviccombining screening efficiency and transition probabilit7

we define a figure of merit for optoelectronic devices thathe product of the energy shift caused by the addition of 1applied bias:dEexc/dV and the oscillator strengthf osc of theexciton at the bias considered. Considering that the devare based on the fundamental transition, Fig. 7 showsvariation of the quantityf oscdEexc/dV with applied bias forthe e1hh1 exciton, for different number of wells with 100-Åwidths in the heterostructure and for several quantum-wthicknesses in the case of seven wells. The excitonic tfunction used to calculate this figure of merit correspondsthat for a pure 3D exciton (a51), the differences betweethe casesa51 anda variable for exciton binding energieand oscillator strength being very small, and the model wa51 reduces drastically the computing time.

From Fig. 7 it is seen that the quantityf oscdE/dV passesthrough a maximum at a given bias and therefore the numof wells and the well width can be adjusted to obtainoptimized working at the command bias. The modificationthe number of wells influences only the peak position b

s,

FIG. 7. Plot of the quantityf oscdE/dV, for the fundamentaltransition, as a figure of merit for optoelectronic devices for 100-thick quantum wells with varying number of wells~a!, and for a7-well heterostructure with different quantum-well thicknesses~b!.The piezoelectric field is taken to be 165 kV cm21.

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does not act on its shape. Furthermore, the well width pla role on the bias value, the damping, and the maxima ofcurve. A device with narrow wells will work within a largenumber of bias values but its efficiency is weaker thanwide-well-based device, which provides a very localized fiure of merit and therefore enables a better working insmaller window of usable bias. In addition, wide well dvices work at voltages near the flat band conditions and tthe working of these devices would not be perturbed bycited excitonic transitions. Note that the number of weplays a role because contributions from all wells add up athus can be used to balance a weak oscillator strength vaThis effect is not taken into account in our analysis asresults are only valid for one of the wells.

VI. CONCLUSION

In this paper we have investigated experimentally atheoretically, within a variational approach, the excitonproperties of piezoelectric multiple-quantum-wellp- i -n het-erostructures grown along the@111# direction. The piezoelec-tric field in the strained layers is determined by includinexcitonic contributions. Its value (165 kV cm21), which is30% lower than the theoretical prediction, is in good agrment with all the studies reported in the literature. We ha

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shown that the quantum-well thickness influences drasticthe excitonic properties of piezoelectric quantum wells.narrow quantum wells, the perturbation caused by the etric field is weak because of the extension of the carrier wafunctions. However, when the well thickness increases,2D excitonic character is magnified for both the fundamenand first excited excitons. Simultaneously their binding engies and oscillator strengths fall rapidly because of the qutype-II configuration due to enhanced electron and hole stial separation. We have also shown experimentally atheoretically how the in-well electric field affects the osclator strengths of excitons in adding bias voltages. This stuconfirms the allowed or forbidden characters of the excitotransitions involving fundamental and excited hole levels ademonstrates that, for sufficiently large electric fields, toscillator strengths for transitions forbidden from classicselection rules become stronger than that for the fundametransition. In the very high electric field regime, the oscillatstrengths for all the excitonic transitions become very weWe have used a figure of merit for optoelectronic deviccombining oscillator strengths and energy shifts in orderoptimize the device performances. It is clear that the kparameter is the well width; wide-well-based devices provbetter working but reduce the usable range of bias.

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*FAX: 33 4 73 40 73 40.Electronic address: [email protected]

1D. L. Smith, Solid State Commun.57, 919 ~1986!.2T. Anan, K. Nishi, and S. Sugou, Appl. Phys. Lett.60, 3159

~1992!.3K. W. Goossen, E. A. Caridi, T. Y. Chang, J. B. Stark, D. A. B

Miller, and R. A. Morgan, Appl. Phys. Lett.56, 715 ~1990!.4I. W. Tao and W. I. Wang, Electron. Lett.28, 705 ~1992!.5A. S. Pabla, J. L. Sanchez-Rojas, J. Woodhead, R. Grey, J. P

David, G. J. Rees, G. Hill, M. A. Pate, P. N. Robson, R. AHoog, T. A. Fisher, A. R. K. Willckox, D. M. Whittaker, M. S.Skolnick, and D. J. Mowbray, Appl. Phys. Lett.63, 752 ~1993!.

6R. Andre, J. Cibert, and Le Si Dang, Phys. Rev. B52, 12 013~1995!.

7P. J. Rodriguez-Girones and G. J. Rees, IEEE Photonics TechLett. 7, 71 ~1995!.

8R. Grey, J. P. R. David, G. Hill, A. S. Pabla, M. A. Pate, G.Rees, P. N. Robson, P. J. Rodriguez-Girones, T. E. SaleWoodhead, T. A. Fisher, R. A. Hogg, D. J. Mowbray, M. SSkolnick, D. M. Whittaker, and A. R. K. Willckox, Microelec-tron. J.26, 811 ~1995!.

9J. P. R. David, R. Grey, G. J. Rees, A. S. Pabla, T. E. SaleWoodhead, J. L. Sanchez-Rojas, M. A. Pate, G. Hill, P. N. Roson, R. A. Hogg, T. A. Fisher, M. S. Skolnick, D. M. WhittakerA. R. K. Willckox, and D. J. Mowbray, J. Electron. Mater.23,975 ~1994!.

10A. S. Pabla, J. L. Sanchez-Rojas, J. Woodhead, R. Grey, J. PDavid, G. J. Rees, G. Hill, M. A. Pate, P. N. Robson, R. AHogg, T. A. Fisher, A. R. K. Willckox, D. M. Whittaker, M. S.Skolnick, and D. J. Mowbray, Appl. Phys. Lett.63, 752 ~1993!.

11A-M. Vasson, A. Vasson, J. Leymarie, P. Disseix, P. Boring, aB. Gil, Semicond. Sci. Technol.8, 303 ~1993!.

12D. Boffety, J. Leymarie, A. Vasson, A-M. Vasson, C. A. Bates,M. Chamberlain, J. L. Dunn, M. Henini, and O. H. Hughe

.

. R..

nol.

., J..

, J.b-,

. R..

d

J.,

Semicond. Sci. Technol.8, 1408~1993!.13B. V. Shanabrook, O. J. Glembocki, and W. T. Beard, Phys. Rev

B 35, 2540~1987!.14L. C. Andreani and A. Pasquarello, Phys. Rev. B42, 8928~1990!.15R. L. Tober and T. B. Bahder, Appl. Phys. Lett.63, 2369~1993!.16E. A. Caridi, T. Y. Chang, K. W. Goosen, and L. F. Eastman

Appl. Phys. Lett.56, 659 ~1990!.17K. Yang, T. Anan, and L. J. Schowalter, Appl. Phys. Lett.65,

2789 ~1994!.18Semiconductors Physics of Group IV Elements and III-V Com

pounds, O. Madelung, Landolt-Bo¨rnstein, New Series, GroupIII, Vol. 17, Pt. a ~Springer, New York, 1982!.

19A. R. Goni, K. Strossner, K. Syassen, and M. Cardona, PhysRev. B36, 1581~1987!.

20R. People and K. Sputz, Phys. Rev. B41, 8431~1990!.21J. M. Luttinger, Phys. Rev.102, 1030~1956!.22J. Los, A. Fasolino, and A. Catellani, Phys. Rev. B53, 4630

~1996!.23R. Winkler and A. I. Nesvizhskii, Phys. Rev. B53, 9984~1996!.24T. S. Moise, L. J. Guido, and R. C. Barker, Phys. Rev. B47, 6758

~1993!.25K. H. Goetz, D. Bimberg, H. Ju¨rgensen, J. Selders, A. V. So-

lomonov, G. F. Glinskii, and M. Razeghi, J. Appl. Phys.54,4543 ~1983!.

26W. W. Lui and M. Fukuma, J. Appl. Phys.60, 1555~1986!.27C. Monier, P. Disseix, J. Leymarie, A. Vasson, A-M. Vasson, B

Courboules, C. Deparis, M. Leroux, and J. Massies,Semicon-ductor Heteroepitaxy, edited by B. Gil and R. L. Aulombard~World Scientific, Singapore, 1995!, p. 511.

28P. Disseix, J. Leymarie, A. Vasson, A-M. Vasson, C. Monier, NGrandjean, M. Leroux, and J. Massies, Phys. Rev. B55, 2406~1997!.

29P. Bigenwald, B. Gil, and P. Boring, Phys. Rev. B48, 9122~1993!.

eyx

J. A

lle

G.

.pl.


30P. Ballet, P. Disseix, A. Vasson, A-M. Vasson, and R. GrMicroelectron. J.~to be published! ~special issue on novel indesemiconductor surfaces!.

31R. A. Hogg, T. A. Fisher, A. R. K. Willcox, D. M. Whittaker, M.S. Skolnick, D. J. Mowbray, J. P. R. David, A. S. Pabla, G.Rees, R. Grey, J. Woodhead, J. L. Sanchez-Rojas, G. Hill, MPate, and P. N. Robson, Phys. Rev. B48, 8491~1993!.

32J. L. Sanchez-Rojas, A. Sacedon, F. Gonzalez-Sanz, E. Ca

,

..

ja,

and E. Munoz, Appl. Phys. Lett.65, 2042~1994!.33P. D. Berger, C. Bru, Y. Baltagi, T. Benyattou, M. Berenguer,

Guillot, X. Marcadet, and J. Nagle, Microelectron. J.26, 827~1995!.

34H. Shen, M. Dutta, W. Chang, R. Moerkirk, D. M. Kim, K. WChung, P. P. Ruden, M. I. Nathan, and M. A. Stroscio, ApPhys. Lett.60, 2400~1992!.

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