Exciton Confinement in InAs InP Quantum Wires and Quantum Wells in the Presence of a Magnetic Field

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Exciton confinement in InAs InP quantum wires and quantum wells in the presence

of a magnetic field

Y. Sidor,1 B. Partoens,1 F. M. Peeters,1 J. Maes,2 M. Hayne,3 D. Fuster,4 Y. Gonzlez,4 L. Gonzlez,4 and V. V. Moshchalkov21Departement Fysica, Universiteit Antwerpen (CGB), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium

2INPAC-Institute for Nanoscale Physics and Chemistry, Pulsed Field Group,

K.U. Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium3Department of Physics, Lancaster University, Lancaster LAI 4YB, United Kingdom

4Instituto de Microelectrnica de Madrid (CNM-CSIC), Isaac Newton 8, Tres Cantos, Madrid 28760, Spain

Received 24 July 2007; revised manuscript received 19 September 2007; published 19 November 2007

The charge confinement in InAs / InP based quantum wells and self-assembled quantum wires is investigatedtheoretically and experimentally through the study of the exciton diamagnetic shift. The numerical calculationsare performed within the single-band effective mass approximation, including band nonparabolicity and straineffects. The exciton diamagnetic shift is obtained for quantum wires and quantum wells incorporating localwidth fluctuations, as well as the electron-hole Coulomb interaction energy. Both electrons and holesbut to alesser extentshow a substantial penetration into the InP barrier. A detailed comparison is made between thetheoretical and experimental data on the magnetic field dependence of the exciton diamagnetic shift. Ourtheoretical analysis shows that excitons in the InAs / InP quantum well are trapped by local well widthfluctuations.

DOI:10.1103/PhysRevB.76.195320 PACS numbers: 73.21.Fg, 73.21.Hb, 75.75.a, 78.55.m

I. INTRODUCTION

The optical properties of quantum nanostructures havebeen intensively studied to achieve device applications andbecause they provide the confinement for electrons and holesin such systems. The confinement of the electron and thehole is responsible for the properties of an exciton in suchsystems. Confinement of the particles depends on the sizeand shape of the semiconductor nanostructure. It can also becontrolled through a selection of structure and barrier mate-rials to obtain various band offsets. Moreover, the application

of an external magnetic field can give important informationabout the carrier confinement.The magnetoexcitons in a quantum wire QWR have

been theoretically investigated by a number of authors,13 aswas discussed in Ref.4.An exciton in a quantum well QWin the presence of an external magnetic field has been inten-sively studied over the past decades. For the QW exciton inthe presence of an in-plane magnetic field, only a small num-ber of studies have been published5,6 due to the increasedcomplexity as a consequence of the breaking of the conser-vation of the total pseudomomentum.7 Most of the investiga-tions on magnetoexcitons were restricted to the case whenthe magnetic field is applied along the growth directionsee,for example, Refs. 810. However, all the mentioned studiesof the exciton in a perpendicular magnetic field did not in-clude local QW width fluctuations, where excitons as wellas trions and biexcitons can be trapped.1114 Such fluctua-tions can induce an additional weak lateral confinement,which leads to a confinement of the particles in all threedimensions; i.e., a shallow quantum structure is formed.

One of the promising fabrication methods made it pos-sible to create self-organized nanostructures, formed duringthe initial stage of heteroepitaxial growth in lattice-mismatched systems. Deposition of InAs on InP by molecu-lar beam epitaxy can lead to a QW or self-assembled quan-

tum wires and dots, depending on the growth conditions.Self-assembled InAs / InP quantum wires and dots are prom-ising candidates for optical applications at telecommunica-tion wavelengths of 1.3 and 1.55m,1518 among otherthings because of the enhanced charge confinement as com-pared to QW. In this paper, we theoretically and experimen-tally investigate the charge confinement in InAs / InP QWand self-assembled InAs / InP QWRs. Both are designed toemit photoluminescence PL around 1.3 m at room tem-perature. We report temperature, laser power, and pulsedmagnetic field dependence of the PL spectrum. We perform

the calculations within the single-band effective mass ap-proximation, but including strain effects and conductionband nonparabolicity. We study the exciton energy shift ofQWR and QW with a local thickness fluctuation in magneticfields up to 50 T and compare theoretical and experimentaldata.

II. THEORETICAL MODEL

In our calculations, we model the quantum wire as a two-dimensional2Dquantum box with height h and widthw, asshown in Fig. 1a, and we follow our earlier theoreticalapproach of Ref. 4. We identify the crystallographic axes

110, 001, and 110 of the InAs/ InP self-assembled

QWR4,15,16 with the x, y, and z axes of the wire, and the zaxis of the QW with the growth direction of the well seeFig.1. For the narrow quantum well of widthh, we includea local circular well width fluctuation of 1 MLmonolayerwith radius R , as illustrated in Fig.1b.

The single-band effective mass theory is used to calculatethe exciton states in both QW and QWR. In our previouswork on InAs / InP self-assembled QWR,4 we presented thebasic equations of the QWR in the presence of parallel andperpendicular magnetic fields. In the present work, we ex-

PHYSICAL REVIEW B 76, 1953202007

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tend that work to QW in the presence of parallel and perpen-dicular magnetic fields and when a local well width fluctua-tion is present. The components of magnetic field B are

perpendicular and parallel to the plane motion: B =B+ B,where B z and Bz. Without going into a detailed de-scription of the basic equations for the QW in the presence ofperpendicular and parallel magnetic fields, three points haveto be stressed.

1The effect of strain on the electron and the hole con-finement potentials was included19,20 by calculating the hy-drostatic strain in the conduction and valence bands of theQW, and the shear separation potential in the valence band ofthe QW. The heavy hole is found to be the ground state in theInAs / InP QW due to the shear separation potential. We in-clude the effect of band nonparabolicity21 in our calculationsby using an energy dependent electron mass

mnonp.e=me1 + E, 1

where me and mnonp.e is the bulk and nonparabolic electronmass, respectively,is the nonparabolicity parameter, whichwe assume to be22,23 =1.4 eV1, and Eis the energy of theelectron obtained by solving the single-particle Schrdingerequation using the electron bulk mass.

2In a parallel magnetic fieldBz, we introduce thecenter of mass c.m. R =mere + mhrh /Mand relative mo-tion coordinates r = re rhto describe the exciton state in theQW, where m emhdenotes the electronholemass, r erhis the electron hole coordinates in the xy plane, and M=me + mh is the total mass of the exciton. In the case of aparallel magnetic field, we compare the numerical results fordifferent well widths, but we do not include the local QWwidth fluctuation see the dark region in Fig. 1b. This ismotivated by the fact that the magnetic field mainly influ-ences the extent of the exciton wave function in the zdirec-tion, while a local well width fluctuation will trap the excitonand shift its energy slightly, which will have a very weakmagnetic field dependence. This leads to the followingHamiltonian for the exciton:

H=

2

2r

2

2

2MR

2 +ieB

c ze

me+

zh

mhry + ieBMc

zezhRy +e 2B

2

2c2ze2

me+

zh2

mh ze

2

2mezeze

zh

2

2mhzhzh+Veze+Vhzh

e2

r2 +zezh2,

2

where = memh /Mdenotes the exciton reduced mass in thexy plane, p =iris the relative mass momentum in the x yplane, P=iR is the exciton c.m. momentum in the xyplane, Veze Vhzh the confinement potential of the elec-tron hole along the growth direction of the QW, e is thefree-electron charge, and is the dielectric constant. Further,the total in-plane pseudomomentum =K is an exact in-tegral of motion,7 which implies that the slow motion of thec.m. is decoupled from the fast relative motion in Eq. 2.Therefore, the exciton wave function can be written as

R,r,ze,zh

expiK Rr,ze,zh. 3Because of the strong confinement along the z direction, weare allowed to separate the lateral motion in the xy planefrom the zmotion. Further, we assume that the confinementinteraction is larger than the Coulomb i.e., excitoninterac-tion, which allows us to use the adiabatic approximation

r,ze,zh=ezehzhexp r2, 4

where eze hzh is the electron hole wave function,which is the solution of Eq. 3 when neglecting the Cou-lomb interaction. The wave function for the relative motionof the exciton is approximated by a Gaussian, where is a

variational parameter. In the final step, we average the ki-netic part and the Coulomb interaction part in the excitonHamiltonian see Eq. 3 using the above wave function,minimize the total exciton energy with respect to , andobtain the exciton energy.

3For the perpendicular magnetic field direction B z,the local well width fluctuations will influence the diamag-netic shift much more, the reason being that the exciton willbe trapped in such local well width fluctuations. For such aweakly confined exciton, there will be a competition betweenthe magnetic confinement and the localization due to wellwidth fluctuations. A local increase of the well width resultsin a lower zero point energy for the electron and the hole,which can be viewed as an attractive potential well in theplane of the QW.1114 For example, for h =3 ML and a localwell with an increase of 1 ML, the difference in zero pointenergy for the electronholeis Ve =4050meV. The cor-responding results for h =4 ML is Ve =39 36meV. Nor-mally, one would expect a larger difference in zero pointenergy for the electron than for the hole, which is the caseforh =4 ML. However, for the very narrow QW, the electronground state energy is close to the conduction band offsetcausing VeVh, as we found for h =3 ML. We use amean field theory in the Hartree approximation to describeour system. Due to the axial symmetry of the problem, the

x [110]

y [001]

z [110]

h

w

(a)

h

z(b)

R

1 MLR

h h + 1 ML

FIG. 1. Schematic drawing ofaour model of rectangular self-assembled InAs / InP quantum wire with height h and width w , andb our model of two-dimensional InAs /InP quantum well withwidth h. The quantum well width fluctuation of 1 ML and radius Ris modeled by the gray area.

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wave function of the particles can be written as ,z ,=,zeil. The electron and the heavy-hole states in theQW are obtained by solving the following 2D Schrdingerequations:

ze 22mezeze

2

2mee2 + 1

e

e+ 2

2me

le2

e

l e

2

eB

mec

+ e2

B2e

2

8mec2 +Vee,ze+Uef fe,zeee,ze

=Eeee,ze , 5

zh 22mhzhzh

2

2mhh2 + 1

h

h+ 2

2mh

lh2

h

+l h

2

eB

mhc

+e 2B

2h

2

8mhc2 +Vhh,zh+Uef fh,zhhh,zh

=Ehhh,zh, 6

where eh =xeh2 +yeh2 , Veheh ,zehis the confinementpotential of the electronhole, which takes into account theQW width fluctuation, and Uef feh ,zeh is the effectiveHartree potential felt by the electron hole,

Uef feh,zeh= e 2

hedhe dzhe

dhe hehe,zhe2

e2 + h2 2ehcose h+zezh2.

7

To solve self-consistently the single-particle differentialequationsEqs.5and 6, we have to evaluate a threefold

Hartree integral, which is given by Eq. 7. The integrationover the angle was performed analytically, and the partwhich contains the and z dependence was performed nu-merically using the logarithmically weighted method.24 Inthe final step, we calculate the exciton energy as follows:

Eex=Ee+Eh+e 2

ee,ze

2hh,zh2

eh2 +zezh2, 8

where the last term also equalsEe+EhEe Eh /2, with thesingle-particle energiesEeandEhobtained from Eqs.5and6 by neglecting the Hartree potential. Note that this termdefines the Coulomb interaction energy, but its value has theopposite signthe Coulomb interaction energy is negative.

The input parameters used in our simulations for theInAs / InP material are taken from Ref.4,except for the non-parabolicity parameter .

III. EXPERIMENTAL SETUP

The samples were grown by solid-source molecular beamepitaxy MBE. For the QW sample, 4 ML of InAs weredeposited1 ML is 3 , followed without interruption by a20 nm thick InP barrier, both at 485 C. The fast growth rateof 1 ML /s and the absence of interruption between InAs and

InP growth inhibit the kinetics, and thus no wires wereformed. This is confirmed by the reflection high-energy elec-tron diffractionRHEEDsignal, which showed a 24 pat-tern during InAs growth, corresponding to a 2D layer. Forthe QWRs sample, 1.7 ML of InAs was deposited at0.1 ML /s and 515 C in a pulsed dynamic way pulsed Insequence: 1 s on and 2 s off. The deposition of InAs wasinterrupted immediately after the change from 2D to three-dimensional growth was observed in the RHEED pattern.Then, the sample was kept 1 min under As pressure andfinally covered by a 20 nm thick InP layer grown at 1 ML /s.PL measurements were performed in a He bath cryostat at4.2 K and in a He flow cryostat 20200 K. Excitation ofthe samples was achieved by a solid-state laser operating at

532 nm via a 200 m core optical fiber. The laser power wascontrolled between 0.6 and 600 mW by neutral density filterscorresponding to power densities at the sample from0.06 to 60 W cm2. The luminescence, collected by six sur-rounding fibers, was analyzed in a 0.25 m spectrometer witha liquid-nitrogen cooled InGaAs detector. A magnetic field of50 T is obtained by pulsed fields. During a 25 ms magneticfield pulse, up to 23 spectra were taken at various fields witha photon integration time of 0.2 ms each.

IV. COMPARISON OF VERTICAL CONFINEMENT IN

QUANTUM WIRES AND WELLS

The QWR structure shows three distinct PL peakssee thethick dashedbluecurve in Fig.2. From our previous the-oretical study of such quantum wires,4 we know that theycorrespond to a wire height of 46 ML and a width of180 . The temperature dependence of these three PL linesis similar to that reported by Fuster et al.25 on a similarsample, revealing a unipolar electron escape toward the bar-rier at room temperature. We also note that in our sample, theintensity of the low-energy peak originating from the 6 MLthick wires increases up to 80 K, evidencing the occurrenceof phonon assisted tunneling or recapture of carriers that

FIG. 2. Color online Photoluminescence spectra of theInAs /InP quantum well and wires at 20 K. The thick fullredanddashedbluecurves correspond to the PL energies of the well andwires, respectively. The thin full and dashed lines are Gaussian fitsto the spectrum.

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have thermally escaped from thinner wires.25

In Fig.3,we compare the theoretical results of the excitondiamagnetic shift E=EB E0 in the InAs / InP QWR,which are denoted by four curves corresponding to the wireheighth =3 6 ML and widthw =180 , with the experimen-tal data, which are denoted by three different symbols. Thenumerical calculations for the electron parabolic effectivemass and the electron nonparabolic effective mass are shownsee Figs.3aand3b. Note that the experimental data inFig. 3 differ slightly from those reported in our previousworks4,16 due to different growth conditions of the QWR.However, the theoretical results for the electron parabolic

effective mass, when height hequals 35 ML, are the sameas those given in Ref. 4. The theoretical results for the bulkparabolic mass of the electron in Fig. 3a only slightlydiffer from those in Fig.3b. However, in the case of para-bolic electron effective mass, we obtained a better agreementbetween the theoretical result and experimental data of thesame heighth, as compared to the case of nonparabolic elec-tron effective mass. The best agreement in the whole B re-gion is found when h =6 ML compare the dashed-dotgreencurve with the full trianglesgreenin Fig.3a. Forthe height of 4 and 5 ML, we see a fair agreement in Fig.3a between the experimental points and the theoreticalcurves. At 45 T, a discrepancy of 1.3 meV for h =4 ML and1.0 meV forh =5 ML is observed. The reason for these smalldeviations is probably due to the fact that we assume a rect-angular shape of the QWR in our theoretical model, whichmay not be the exact experimental shape.16 Comparing thetheoretical and experimental PL energies for different heightsof the wire at zero magnetic field in Fig. 3b, we see adifference of about 24 meV for the wire height h =6 ML anda smaller difference for h=45 ML. When we compare thecomputed PL energies with the experimental PL energies forthe parabolic mass of the electron in Fig.3a,we found thatthe 5 ML computed PL energy EPL5 ML=986 meV is closerto the experimental one EPLexpt5 ML =987 meV, as com-

pared to the calculated and experimental PL energies for theheight of 4 ML, EPL4 ML =1056 meV and EPLexpt4 ML=1034 meV, and the height of 6 ML, EPL6 ML =929 meVand EPLexpt6 ML =945 meV.

When the magnetic field is applied along the 110direc-tion, the confinement properties in the direction of the wireheight are investigated. The diamagnetic shift in the range ofBbetween 0 and 50 T increases with decreasing wire heightfor both theoretical and experimental results shown in Fig.3.This indicates a larger exciton wave function extent forsmaller wires. To illustrate this, we calculated the wave func-tion radii in the InAs / InP QWR theoretically. In TableI, we

present a list of the wavefunction radii for the electron andheavy hole along the 001 direction of the QWR with h=46 ML, where we define the particle radius as the averagequadratic distance along the 001 direction of the wire gr=y2 /2. Note that the electron extent gre decreases withincreasing well width, while the opposite behavior is foundfor the hole extent grhh. The unusual behavior for the elec-tron extent is a consequence of the electron wave function

TABLE I. Wave function radii for the electrongre, the heavyhole grhh, and the exciton grexc along the growth direction ofthe InAs/ InP quantum wellzdirectionwith width of 35 ML andthe InAs / InP quantum wire 001 direction with height of 46ML.

gre

grhh

grexc

QW 3 ML 9.0 2.4 9.3

QW 4 ML 7.5 2.5 7.9

QW 5 ML 6.9 2.7 7.4

QWR 4 ML 7.7 2.6 8.1

QWR 5 ML 7.0 2.7 7.5

QWR 6 ML 6.7 3.0 7.3

FIG. 3. Color onlineThe exciton diamagnetic shift as a function of magnetic field B 110for InAs / InP quantum wires. The dashedblue, fullblack, dottedred, and short dash-dot greencurves correspond to our numerical results with parabolic mass of the electronaand when the effect of band nonparabolicity is included bfor different heights of the wire. The full squares black, circlesred, andtriangles greenrepresent the experimental data for the wire height 4, 5, and 6 ML, respectively, and width equal to 180 . The value of theexperimental and the calculated PL energies in the absence of the magnetic field are given in parentheses.


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spillover effect. As a result, also the exciton size increaseswith decreasing well widthsee the data for grexc in TableI.

In Fig. 4, we examine the exciton diamagnetic shift forthe InAs / InP quantum well when the magnetic field is ap-plied perpendicular to the z direction. The PL experimentsshow that the QW structure clearly has a main peak, high-energy HE peak, and a second PL peak, low-energy LEpeak, about 20 meV belowsee the fullredcurves Fig.2.We compare the theoretical results for the QW with widths

equal to 3, 4, and 5 ML, with the experimental data of theHE peak denoted by the full circles the diamagnetic shift ofthe LE peak was not measured. Band nonparabolicity has asmall effect on the resultscompare the curves with open andclosed symbols in Fig.4. The effect of band nonparabolicity

is to increase the electron effective mass. The exciton dia-magnetic shift is inversely proportional to the reduced massof the electron and heavy hole; therefore, the exciton dia-magnetic shift when including the nonparabolic electronmass is smaller than the one with the bulk electron mass. UptoB =30 T, the calculated diamagnetic shift energieswhenthe effect of the nonparabolicity is includedfor the QW withwidth of 4 ML are in close agreement with the experimental

data. At 45 T, the discrepancy between theory and experi-ment is only 0.7 meV. When comparing the calculated PLenergies at zero magnetic fieldwhen including band nonpa-rabolicity, we found that the 4 ML resultsEPL4 ML =1032 isclosest to the experimental PL energy EPLexpt=1006 meVwe assumed a bulk band gap energy for InAs of Eg=472 meV. This is consistent with the fact that the 4 MLresults also give the best agreement for the exciton diamag-netic shift and with the intended thickness of the QW in theMBE growth. From Fig. 4, we see that the calculated dia-magnetic shift increases with decreasing well width. The dia-magnetic shift is determined by the lateral size of the wavefunction perpendicular to the magnetic field. In the case ofthe parallel magnetic field, it reflects the exciton extent alongthe z direction of the QW. We found a large wave functionspillover in the growth direction of the well.

Figure5 shows the density of the electron when the ef-fect of band nonparabolicity is included and heavy holealong the z direction. Because the QW is very thin and theelectron mass is at least six times smaller along the zdirec-tion than the heavy-hole mass, the largest part of the electrondensity is in the barrier of the QW in contrast to the heavy-hole density, which is mostly situated in the well. The wavefunction radii for the particles along the growth direction ofthe QWgr=z2 /2withh =35 ML are listed in TableI.When compared to the theoretical results of the exciton ex-tent in the QWR with the one in the QW see TableI, we

found that the extent of the electron, heavy hole, and excitonalong the z direction for the QW with h =4 and 5 ML isslightly smaller than the corresponding extent along the001direction for the QWR with h =4 and 5 ML. The elec-tron and the exciton wave function spillover was found in

FIG. 4. Color onlineThe exciton diamagnetic shift as a func-tion of parallel magnetic fieldBzfor InAs /InP quantum wells.The curves with the open and closed symbols correspond to theexciton diamagnetic shift with and without taking into account theeffect of band nonparabolicity for the electron, respectively. Thecurves with the diamonds blue, squares black, and circles redcorrespond to the quantum well with width of 3, 4, and 5 ML,respectively. The full circlesblackcorrespond to the experimentaldata. The value of the experimental and the calculated PL energiesin the absence of the magnetic field are given in parentheses.

FIG. 5. Color onlineThe probability density for the electronaand the heavy hole balong the growth direction z directionin aInAs/ InP quantum well with widths 3, 4, and 5 MLthe dashedblue, fullblack, and dotted redcurves, respectively. The probabilityof the electron and the heavy hole to be inside the quantum well is indicated in parentheses.


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both QW and QWR, but not for the heavy holesee TableI,which proves that the exciton extent is predominantly deter-mined by the electron extent. We also found that along the zdirection of the QW with different h, the effective mass ofthe exciton is different. From TableII, we can see a largerexciton effective mass for the thinner well26 we also givea list of exciton effective masses for the QWR, where forthe QW,

1

= dzedzheze2hzh2 1

meze+

1

mhzh , 9

and for the QWR,

1

= dxedyedxhdyhexe,ye2hxh,yh2

1mexe,ye

+ 1

mhxh,yh . 10

The QW shows a second PL peak about 20 meV belowthe main peak Fig.2. This LE peak can be attributed to amonolayer fluctuation in the QW thickness. The observationof the LE shoulder is very surprising since QW excitons cantypically migrate to regions with the largest well thicknessand so to regions of the lowest confinement energyat he-lium temperature. We assume that the exciton energy statecorresponding to the main PL peak HE peak is a localminimum originating from the predominant thickness of h=4 ML, while the LE state is attributed to 5 ML thicknessfluctuations that are less abundantFigs.7aand7b. Theratio of the intensities of the 4 and 5 ML peaks does notdepend on the laser power at 20 K see Fig.6. Hence, weconclude that the charge carriers in the QW are trapped bythickness fluctuations and that there is no coupling betweenthe optically active 4 and 5 ML fluctuations. Therefore, thesestates are spatially well separated. Indeed, charge carriersthat are in a region of 4 ML well thickness, which is close to5 ML fluctuation, would not significantly be optically activesince such charge carriers would easily migrate toward the 5ML fluctuation. Since there is no coupling at 20 K, the ratioof the peak intensities reflects the ratio of the number ofcorresponding states: There are 3.3 times more optically ac-tive 4 ML states than 5 ML states. When temperature Tin-creases, the 4 ML peakHE peakloses intensity in favor of

the 5 ML peakLE peak, as shown in the inset of Fig.6. So,in the QW, we find a phonon assisted tunneling or the recap-ture of carriers that have thermally escaped from the thinnerregions of the QW see Fig. 7c. For temperatures aboveabout 70 K, the 4 ML peak regains some of its relative in-tensity as the laser power increases see Fig. 6, indicatingthat the thermal redistribution rate is smaller than the rate atwhich the charge carriers are injected at elevated laser power.

TABLE II. The exciton reduced mass m0 is the vacuum elec-tron massalong the z direction of the InAs / InP quantum well andquantum wire with and without including the effect of bandnonparabolicity.

parm0 nonpm0

QW 3 ML 0.042 0.051

QW 4 ML 0.036 0.047QW 5 ML 0.033 0.044

QWR 4 ML 0.037 0.047

QWR 5 ML 0.033 0.044

QWR 6 ML 0.031 0.042

FIG. 6. Color onlineLaser power dependence of the ratio ofthe intensities of the HE and LE peaks of the quantum well at thedifferent temperatures denoted by the different symbols. At 20 K,this ratio remains constant with varying laser power. With increas-ing temperature, the laser power dependence becomes more pro-nounced. The inset shows the temperature dependence of the abso-lute intensities of the HE squares black and LE circles redpeaks.

InP

InAs

(a)(b)

(c)

(d)

Conduction band

Valence band

4 ML 5 ML 5 ML 4 ML

FIG. 7. Color onlineSchematic diagram of the thickness fluc-tuations within the quantum well topand the band structure bot-tom. Most excitons can freely move to the most abundant 4 MLfluctuation at all temperaturesa, and some excitons get trapped bya 5 ML fluctuationb. At moderate temperature excitons are able tomigrate from the 4 ML to the 5 ML statec. At temperatures above70 K, thermal redistribution causes excitons to migrate from the 5ML to the 4 ML state d.


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for R =150 is the most prolongated in comparison to theprevious two radii, as expected, because the radius of thefluctuation increases; i.e., the electron is less localized in theplane.

Let us examine the exciton diamagnetic shift in theInAs / InP QW with the local width fluctuations in the pres-ence of a magnetic field parallel to the growth directionB z. In Figs.10aand10b,we compare the theoreticalresults of the exciton diamagnetic shift for the 4-5 and 3-4QWs with the experimental results of the HE peak see thecurves with the symbols and the full circles black in Fig.

10,respectively. We present the calculations with and with-out inclusion of band nonparabolicity see the curves withthe open and closed symbols in Fig. 10, respectivelywhenthe QWs fluctuation radii are in the range of 25 and 300 for both the 3-4 and 4-5 QWs. Again, we observe a smallerdiamagnetic shift for the case when nonparabolicity for theelectron is included than in the case when it is excluded, asseen previously for the parallel magnetic field see Fig. 4,but now the effect is much larger. Up to 30 T, we found thebest agreement between theory and experiment for the 4-5QW with a well width fluctuation radius R =50 , while afull agreement in the whole B region is achieved for the 3-4QW with the same well width fluctuation radius comparethe curves with the open circles red and experimentalpoints in Figs. 10a and10b. Nevertheless, for the 4-5QW, the computed PL energy is within 15 meV from theexperimental PL energy of the HE peak compare the curvewith the open symbolsredfor R =50 and the experimen-tal value of the PL energy in Fig.8a, while for the 3-4 QWa difference of about 60 meV is found compare the curvewith the open symbolsredfor R =50 and the experimen-tal value of the PL energy in Fig. 8b. Besides, for theradius of 300 , the diamagnetic shift is much larger thanthat for R =50, 37.5, and 25 . This is a consequence of thewave function spillover effect for the electron. In the case of

the perpendicular magnetic field, the discrepancy betweentheory and experiment for the 4-5 QW is about 5 meV at45 T when R =50 . However, the exciton extent in theplane of the 4-5 QW with the radius of the fluctuation R=50 is 6.7 nm 6.2 nm for the 3-4 QW, which is in rea-sonable agreement with the exciton radius found from ex-periment, 7.7 nm for the 4 and 5 ML QW.

VI. CONCLUSIONS

In summary, we studied theoretically and experimentally

the vertical confinement in a InAs / InP self-assembled QWRand in a InAs / InP QW. The wave function radii of the elec-tron, heavy hole, and exciton along the growth direction ofthe QW and 001 direction of the QWR were calculated.From a comparison with the qualitative measure of the ver-tical exciton extent obtained from the experimental data, weconclude that the exciton wave function spillover effect inthe 3-5 ML QWs and the 4-6 ML QWRs is large.

The exciton diamagnetic shift for QW in the presence ofparallel magnetic field and for QWR in the presence of mag-netic field applied along the 110 direction is calculated,reflecting in both cases the confinement properties of thewire height and the well width. We obtained the best agree-ment between theory and experiment for the QWR withheight ofh =6 ML, and also a fair agreement for QWR withh =4 and 5 ML, when using the parabolic mass of the elec-tron. Deviation from the experiment at B =45 T is found tobe less than 1.3 meV for 4 ML QWR and about 1 meV for 5ML QWR. When comparing the PL energies and the diamag-netic shift in the QW, the best agreement is found for a 4 MLwell when including the conduction band nonparabolicity ef-fect.

Experimental results of the laser power dependence at dif-ferent temperatures for InAs / InP QW indicate that thecharge carriersexcitonsare trapped by the well width fluc-

FIG. 10. Color onlineThe exciton diamagnetic shift as a function of perpendicular magnetic fieldBzfor InAs/ InP quantum well.The curves with the open and closed symbols correspond to the exciton diamagnetic shift with and without taking into account the effect ofband nonparabolicity for the electron. The curves with the diamonds blue, squaresblack, circlesred, and triangles greencorrespondto the quantum well with width of 4 ML, fluctuation width of 5 ML a, and the fluctuation radius of 25, 37.5, 50, and 300 , respectively.In graphb, the same identification of the curves is used as in grapha, but now for the quantum well with width of 3 ML and fluctuation

width of 4 ML. The full circles blackcorrespond to the experimental data.


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tuations. Numerical results of the PL energy for the QW withlocal width fluctuations show that experimentally found HEand LE PL peaks correspond rather to the QW with width of4 ML and fluctuation width of 1 ML, than to the 3 ML wellwith fluctuation width of 1 ML. The best agreement for theHE peak is found for the 4-5 QW when the fluctuation radiusR is between 25 and 50 . The calculated exciton diamag-netic shift in the presence of a perpendicular magnetic field

for both 3-4 and 4-5 QWs with R =37.5 50 , when usingthe nonparabolic effective mass of the electron, fits best theexperimental obtained diamagnetic shift of the HE peak.

ACKNOWLEDGMENTS

This work is supported by the Belgian Science PolicyIAP, the Flemish Science Foundation FWO-Vl, the Euro-pean Commission network of excellence: SANDiE, theSpanish MEC and CAM through Projects No. TEC-2005-05781-C03-01 and No. NAN2004-09109-C04-01, andConsolider-Ingenio 2010 through Projects No. CSD2006-

0019 and No. S-505/ESP/000200. M.H. acknowledges sup-port of the Research Councils, UK. The authors thank A.Matulis for fruitful discussions.

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Documents

Exciton Confinement in InAs InP Quantum Wires and Quantum Wells in the Presence of a Magnetic Field