PHYSICAL REVIEW B 15 JUNE 1998-IIVOLUME 57, NUMBER 24

Off-center D2 centers in a quantum well in the presence of a perpendicular magnetic field:Angular-momentum transitions and magnetic evaporation

C. Riva,* V. A. Schweigert,† and F. M. Peeters‡

Departement Natuurkunde, Universiteit Antwerpen (UIA), Universiteitsplein 1, B-2610 Antwerpen, Belgium~Received 19 February 1998!

We investigate the effect of the position of the donor in the quantum well on the energy spectrum and theoscillator strength of theD2 system in the presence of a perpendicular magnetic field. As a function of themagnetic field, we find that whenD2 centers are placed sufficiently off-center, they undergo singlet-triplettransitions which are similar to those found in many-electron parabolic quantum dots. The main difference isthat the number of such transitions depend on the position of the donor, and only a finite number of suchsinglet-triplet transitions are found as a function of the strength of the magnetic field. For sufficiently largemagnetic fields the two-electron system becomes unbound. For the near centerD2 system, no singlet-triplettransition and no unbinding ofD2 is found with increasing magnetic field. A magnetic field vs donor positionphase diagram is presented that depends on the width of the quantum well.@S0163-1829~98!05924-4#

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I. INTRODUCTION

In multilayer and quantum-well structures, suchGaAs/AlxGa12xAs, electrons bound to donor impurities sitated in the barrier tend to migrate into the well, due tofavorable potential gap. There they are trapped by the imrity donors, such as Si, that are naturally or artificiapresent in the material. The trapping of one electron bdonor does not completely screen the charge of the doitself, thus bounded states of negative charged donorspossible in principle and in practice.1

A great deal of attention has been given in recent yearthe formation and stability of negative donor centers in seconductors. Those systems, indeed, being the simplest mbody system, represent an interesting occasion to studyelectron-electron interactions in solids.

In previous experimental and theoretical studies, thependence of the binding energy ofD2 on the magnetic-fieldstrength and on the dimension of the quantum well hbeen investigated. While a great part of these works consthe on-centerD2 problem,2–4 i.e., when the impurity donoris at the center of the well, the study of the off-center prolem, i.e., when the impurity donor is displaced from the ceter of the well, and the barrierD2 problem, i.e., when thedonor is in the barrier, are much less investigated. Ontheoretical side, Zhu and Xu5 studied the spin-singletL50and spin-tripletL521 states for a quasi-two-dimension~2D! D2, while Fox and Larsen6 studied the barrierD2 inwhich the electrons are moving in a perfect 2D plane. Tdependence of the properties of aD2 system on the positionof the donor with respect to the center was partly invegated in Ref. 7. The authors considered the problem odouble quantum well in which one of the two wells hosts,its center, the donor, while the other contains the electroOn the experimental side, we point out the work of Jiaet al.,8 in which experimental evidence of an off-centerD2

system was presented. All these studies on off-centerbarrier D2 show spin-singlet–spin-triplet transitions of thground state with increasing strength of the magnetic fie

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But the situation studied in previous works differs from treal problem of the off-centerD2. The work of Zhu and Xuis most close to the real experimental situation, but they stied only the first two states of theD2 system. Such singlettriplet transitions have also been observed in electron stems confined in quantum dots, and are known asmagicmagnetic numberground-state transitions.9 In quantum dots,the electrons are held together by a parabolic or hard wconfinement potential, which for theD2 problem is replacedby the Coulomb potential of the donor impurity. Thusseems that the appearance of singlet-triplet transitionscharacteristic feature of confined electronic systems, anthis paper we will shed more light on the condition undwhich such transitions appear in theD2 system. TheD2

problem has the added flexibility that the singlet-triplet trasition can be influenced by changing the position of thenor with respect to the center of the quantum well. It is evpossible that for certain donor positions there is no singtriplet transition at all.

In the present paper we study the properties of thecenterD2 as a function of the position of the donor in thwell, and as function of the quantum-well width in thquasi-2D approximation. In Sec. II, we present our moand explain how we obtain the wave function and energythe differentD0 andD2 levels. Next, in Sec. III, we presenand discuss the energy spectral behavior for quantum wof widths 200 and 100 Å. Then we compare the results oftwo calculations in order to have a better understandingthe reasons that underlie the different behaviors of theenergy spectra. Next, in Sec. IV, we evaluate and studydependence of the oscillator strength and of the transienergies on the magnetic field and on the position ofdonor with respect to the center of the well. In Sec. V, wuse our model to explain the cyclotron-resonance experimof Jianget al.8 Our conclusions are presented in Sec. VI.

II. MODEL

The properties of the off-centerD2 in a finite-heightquantum well under the influence of a perpendicular m

15 392 © 1998 The American Physical Society

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57 15 393OFF-CENTERD2 CENTERS IN A QUANTUM WELL IN . . .

netic field will be treated in the present paper. In the framwork of the effective-mass approximation, the Hamiltoniof the D2 system is given by

HD25H1D0

1H2D0

1Vee~ urW12rW2u!, ~1!

where HiD0

is the Hamiltonian for thei th one-electronD0

system andVee is the electron-electron repulsive Coulominteraction. Using cylindrical coordinates and the effectBohr radiusaB5\2e0 /m* e2 and the effective RydbergRy5e2/2eaB as units of length and energy, respectively, t

neutral donor HamiltonianHiD0

and the electron-electroCoulomb potential assume the forms

HiD0

52¹21g

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1

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222

urW i2zu1VQW~z!, ~2!

Vee~ urW12rW2u!52

urW12rW2u, ~3!

where the vector potential is taken in the symmetric gaAW 5rW3BW /2. The magnetic field is expressed in the dimesionless quantityg5\vc/2Ry , with vc5eB/m* c the cyclo-tron frequency;z is the position of the donor along thezaxis, as measured from the center of the well, andVQW(z) isthe confining potential due to the quantum well of widthW.For GaAs/AlxGa12xAs with x50.25, we tooke512.5, andobtainedaB598.7 Å, Ry55.83 meV, andg50.148B(T).We took the mass of the electron equal in the well and inbarrier, namely,m* 50.067m0, and the height of the barrieis given byV050.63(1.155x10.37x2) eV.

The strong confinement along thez axis allows us to ne-glect the correlation induced by the Coulomb interactionthez direction; thus we can write the wave functions forD2

as

C~rW1 ,rW2!5c~rW 1,rW 2! f 1~z1! f 1~z2!, ~4!

with f 1(zi) the 1D ground state wave function for the eletron confined in a quantum well of heightV0.10

The two-electron functionc(rW 1,rW 2! expresses the correlation between the two electrons, and is obtained by dianalizing the Hamiltonian~1! in which the electron-electronVee and the electron-donorVed Coulomb interaction are replaced by their average along thez axis,

Vee~ urW 12rW 2u!5E dz1E dz2u f 1~z1!u2u f 1~z2!u2

32

A~rW 12rW 2!22~z12z2!2~5!

and

Ved~rW !5E dzu f 1~z!u22

Ar21~z2z!2, ~6!

respectively. In a previous work,11 it was shown that in thecase of hard wall confinement Eq.~5! can be replaced by thexpression

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A2pleurW 12rW 2u2/4l2

K0S urW 12rW 2u2

4l2 D ,

~7!

wherel>0.2W andK0(x) is the modified Bessel function othe third kind. In the present paper we use the same expsion for a finite height quantum well in whichl is deter-mined by fitting Eq.~7! to Eq. ~5!. A comparison betweenthe potential~5! that was evaluated numerically and the aproximate expression~7! is shown in Fig. 1~a! for a quantumwell of width W5200 Å where the fitting parameter wafound to bel50.607aB . On the other hand, no simple anlytic approximation of Eq.~6! could be found. This is shownin Fig. 1~b! for an off-center donor withz50.7aB andW5200 Å, where we compare Eq.~6!, which we fitted topotential~7! with l50.92aB ~solid curve! and the screenedCoulomb potential 1/Ar21l2 with l50.803aB ~dashedcurve!. None of the two fits give a good approximation

FIG. 1. Comparison between the numerical evaluation andanalytical fitting of the average in-plane potentials for a quantwell of width W5200 Å. In ~a!, thee-e potential~5! is fitted to Eq.~7! ~dashed curve!. In ~b!, the e-d potential~6! is fitted to Eq.~7!~solid curve! and to 1/Ar21l2 ~dashed curve!.

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15 394 57C. RIVA, V. A. SCHWEIGERT, AND F. M. PEETERS

Eq. ~6! in the small-r region. Therefore, in the Hamiltoniawe retain the numerical expression for Eq.~6!.

Using a finite-difference technique, as explained in R12, the Schro¨dinger equation associated with Hamiltonia~2! was numerically solved on a nonuniform grid inrW space,and the eigenvalues and eigenvectorsRn,l(r)eil f for D0

were found for different values ofz and an arbitrarymagnetic-field strength. The eigenfunctions forD2 can thenbe constructed as a linear combination of theD0 wave func-tions. Due to the rotational symmetry in therW plane ofHamiltonian~2!, thez component of the orbital angular momentumL is a good quantum number for those functionand therefore theD2 wave functions are taken as

cL~rW 1 ,rW 2!5 (k51

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l 5 l m

8 Cknl Rn,~L1 l !/2~r1!

3Rk,~L2 l !/2~r2!ei [ l ~f12f2!1L~f11f2!]/2, ~8!

where(8 indicates the summation is only over even~odd!values of the indexl whenL is even~odd!.

III. ENERGY SPECTRUM

First we solve our model for an off-center donor inGaAs/Al0.3Ga0.7As quantum well with widthW5200 Å('2aB) and height of the potential barrierV050.23 eV. Thedependence of the energy on the position of the donor wrespect to the center of the well is investigated numerica

The binding energy of theD2 state, with az componentof the orbital angular momentum equal toL, is defined as

Ebn~D2,L !5E0~D0,0!1E~e,0!2En~D2,L !, ~9!

whereE0(D0,0) is the energy of the ground state of theD0

in the well;E(e,0)5g is the energy of a free electron in thN50 Landau level; andEn(D2,L) is thenth energy level ofD2, with L the z component of the orbital angular mometum.

The results of our numerical calculation are plotted in F2 for W5200 Å. The binding energies of the firstL50 state,a spin singlet, and of the stateL521, a spin triplet, areplotted against the magnetic field for different positionsz ofthe donor with respect to the center of the well.

We note, first, that the binding energy decreases whendonor center is displaced from the center of the quanwell. The reason for this is that the electron-donor interactdecreases with increasingz. This is because, due to thstrong confinement along the growth axis of the well, telectrons tend, even in the case of an off-center donortem, to be localized in the center of the quantum well,though the donor is displaced a distancez from the center.

A second feature to be noted is that the magnetic fidependence of the binding energy changes qualitatively wincreasing z. For a sufficiently largez, we find thatEb

n(D2,L) has a maximum as a function ofg. The bindingenergy starts to decrease after this maximum, and for sciently largeg it can even become negative, indicatingunbinding of theD2 state.

Third, in the absence of a magnetic field the ground sof D2 is, regardless of the position of the donor, the sp

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singlet state. When increasing the magnetic field, the grostate for a well-centerD2, i.e., z50, remains the singleone. In contrast, the ground state of the off-centerD2 withz.0.45aB shows a transition to a spin-triplet state for larenough magnetic fields. The magnetic field at whichsinglet-triplet transition occurs depends on the position ofdonor as it appears from Fig. 2. This dependence willstudied further below, where it is found that the magnefield at which the transition occurs decreases with increasz.

Fox and Larsen6 investigated the ideal 2D problem, neglecting the finite extension of the electron wave functionthe z direction, i.e., f 1(z)5d(z), and calculated theD2

spectrum for a donor out of the plane in the limit of higmagnetic fields, and found an infinite number of singltriplet transitions. The situation for a quasi-2D off-centerD2

is quite different. In this case, in contrast to the 2D case,extension of the electron wave function in thez direction istaken into account, together with the finite height of the brier. Let us investigate more deeply the behavior of theergy spectrum of such a system, with, e.g.,z50.7aB . Theresults for the binding energy of the different levels, i.different angular momentum states, are shown in Fig. 3~a!,for the case of a quantum well of widthW5200 Å. Note thatdifferent transitions occur at higher magnetic fields. Tground state exhibits a singlet-triplet transition atg51.5 anda triplet-singlet transition atg516.1. For g.22.7, whichcorresponds toB.154 T, theD2 ground state unbinds, i.eD2 magnetically evaporates.

While for the ideal 2D system an infinite numbersinglet-triplet transitions are found, for a quasi-2D systeonly a finite number of such transitions are possible asclearly visible from Figs. 3~a! and 3~b!. The critical g ’s atwhich the singlet↔ triplet transitions occur depend on thposition of the donor@see Fig. 3~b!#. The g2z phase dia-gram for the ground state of a quantum well of widW5200 Å is given in Fig. 4. We found that forz,0.45aB

FIG. 2. The magnetic-field dependence of theL50 spin-singletbinding energy~solid curves! and theL521 spin-triplet bindingenergy~dotted curves! for a GaAs/Al0.3Ga0.7As quantum well withwidth W5200 Å52.02aB are shown for different positions,z ofthe donor with respect to the center of the well.z is in units ofaB .For increasing magnetic field there is a crossing between the ssinglet and -triplet states whenz.0.45aB .

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57 15 395OFF-CENTERD2 CENTERS IN A QUANTUM WELL IN . . .

FIG. 3. In ~a!, the binding energies for different values of thezcomponent of the angular momentum are shown for aD2 with thedonor placed atz50.7ab'70 Å from the center of the quantumwell. In ~b!, the binding energies for a barrierD2 are displayed,with z51.4aB'140 Å.

FIG. 4. Phase diagram for a quantum well of widthW5200 Å.The curves show the magnetic fields at which the singlet-triptransitions occur for a given position of the donor, as well asfield at which theD2 system evaporates.

the ground state is a singlet for all magnetic fields,0.45aB,z,0.65aB only one singlet-triplet transition~seeFig. 4! is possible, and forz.0.65aB there are two suchtransitions. Increasingz further, such that the donor is in thbarrier~i.e., z.1.01aB), the number of singlet-triplet transition does not increase as illustrated in Fig. 3~b! for D2 withz51.4aB .

The physical origin of the singlet-triplet transitions is rlated to the decrease of the electron-donor attraction withdisplacement of the donor from the center of the well whcompared to the constant electron-electron repulsion.corresponding electron-donor and electron-electron in-plpotentials are shown in Fig. 5 for two values ofz. For smallvalues of z ~e.g., z50 in Fig. 5!, the attractive singleelectron-donor potential is larger than the electron-electpotential, and consequently theD2 system ‘‘prefers’’ a con-figuration in which the two electrons are as close as possto the donor in order to enhance the binding energy, i.e.,L50 state is favored. Whenz is sufficiently large~e.g.,z50.7aB in Fig. 5! the repulsive electron-electron interation dominates the attractive single-donor potential at smdistances, andD2 can have bound states only when the twelectrons are sufficiently apart to render the repulsive inelectron interaction lower than or of the same order asattractive electron-donor potential. For small magnetic fiethis can still be realized in theL50 state. Increasing themagnetic field brings the electrons closer tor50, which willalso increase the electron-electron repulsive energy. Forficiently small z this can still be compensated for by thattractive electron-donor energy. Forz sufficiently large, theelectron-electron repulsive energy increases faster thenelectron-donor energy with increasingB. TheD2 system candecrease its energy in this case by placing the electronsther apart, which is achieved by placing the electronshigher L states. Similar singlet-triplet transitions have rcently been found in quantum-dot systems.9,12,13 Thequantum-dot system is an extreme case in whichelectron-donor potential is replaced by the confinementtential which is usually taken of a quadratic form, i.e.,Ved→v2r2.

In Fig. 6, the pair-correlation functiond(r2urW 12rW 2u)&

te

FIG. 5. The in-plane electron-donor potential for the well-cenD2 and for the off-centerD2 are compared to the electron-electropotential.

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15 396 57C. RIVA, V. A. SCHWEIGERT, AND F. M. PEETERS

is shown for the spin-singletL50 @Fig. 6~a!# and spin-tripletL521 @Fig. 6~b!# states for different values of the magnefield for an on-center~i.e., z50) D2 system and an off-center~i.e.,z50.7aB) D2 system. The magnetic field behaior of the two states is essentially the same for bothcenter and off-centerD2 systems. The magnetic field tendto localize the wave function more with increasing magnefield. For theL50 state, the pair-correlation function becomes more and more peaked atr50, which means that theelectrons are closer and closer to each other with increamagnetic fields. ForL521, the peak of the correlationfunction is shifted towardr50 with increasingB, and, thus,the magnetic field localizes the electrons further. The effof the electron-electron repulsion can be seen in the shapthe correlation function itself. For the off-center system tpair-correlation function is broader than the one for the cter D2 system, even for increasing magnetic fields, and tthe electrons tend to repel each other more, which is a csequence of the diminished electron-donor interaction.

When the dimension of the well is reduced, the localiztion of the electrons in the center of the well is increased.example, if we neglect the penetration of the electrons inbarrier, the width of thef 1(z) is equal to L. Thus theelectron-electron repulsion increases and, at the same

FIG. 6. The pair correlation function of theD2. In ~a!, thecorrelation function of the spin-singletL50 state is presented fodifferent values of the magnetic field both for the off-center~dottedcurves! and centerD2 ~solid curves!. In ~b!, the same plot as in~a!is made, but now for the spin-tripletL521 state.

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for the off-center case, the electron spends more timefrom the position where the donor is located. Thus we expthat systems with a smaller well width will show more spisinglet to spin-triplet transitions with increasing magnefield, and that these transitions will occur at smaller field

Indeed, for aW5100 Å quantum well, withz50.7aB ,we observe~see Fig. 7! as many as four transitions beforD2 evaporates at a magnetic field ofB'81 T ~i.e., g'12.0). The full phase diagram for those transitionsshown in Fig. 8. The well width dependence of the singltriplet transitions and of the evaporation magnetic fieldshown in Fig. 9 forz50.7aB . Notice that the critical mag-netic field for the same transitions, e.g., for the spin-singL50 to spin-tripletL521 states, decreases with decreaswell width. At the same time, the number of transitions icreases. But the evaporation magnetic field first decreaand then, forW,140 Å, increases again. An explanationthis feature is that other transitions are allowed for small wwidth, and this ensures stability ofD2 up to higher magneticfields.

The increase of the number of singlet-triplet transitionwith decreasing dimensions of the quantum well, explathe larger number of transitions found by Fox and Larsen6 inthe ideal 2D system with respect to the smaller numfound in the present study of realistic quasi-2D systems.

IV. CYCLOTRON-RESONANCE TRANSITIONS

The oscillator strength for cyclotron transitions, in thpresent units, is defined as

Fi , f5~Ef2Ei !U^C i u(j 51

21

2r je

6 if j uC f&U2

~10!

whereEf andEi are, respectively, the final- and initial-staenergies, andc f and c i are, respectively, the final- aninitial-state wave functions. The6 sign in Eq.~10! refers tocircular left/right polarization of the light. Note that the peturbation induced by the electric field is spin independeand thus the initial and final states conserve the total si.e., they are both spin-triplet or both spin-singlet stat

FIG. 7. The binding energies for aD2 with z50.7aB in a 100-Åquantum well, for different values ofL. Four transitions occur before theD2 system evaporates, which is atg513.5.

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57 15 397OFF-CENTERD2 CENTERS IN A QUANTUM WELL IN . . .

Equation~10! leads to the selection rulesDL561, while noselection rule is present for the quantum numbern.

We have studied the oscillator strength for cyclotroresonance transitions from the first singletL50 state—(n,L,S)5(1,0,0)—to the~1,21,0! and ~1,1,0! states in therange 2–15 T. The transition energies and oscillastrengths forz50.7aB are plotted in Fig. 10 against the manetic field, and are compared to the one forz50. We recallthat for a two-electron atom the oscillator strength satisthe sum rule( iFi , f52. We observe that the off-center ancenterD2 have rather similar qualitative magnetic-field dpendences.

The cyclotron-resonance transition from the ground sshould show a discontinuous behavior in the cyclotron trsition energies as a function of the magnetic field atsinglet-triplet transition points. In Fig. 11, the transition eergies for a donor at positionz50.85aB are shown. Thetransition energies for (1,0,0)→(1,1,0), (1,21,1)→(1,0,1),and (1,22,0)→(1,21,0), respectively, are plotted. The solcurve shows the transition energy, which we expect toserve if the system makes a cyclotron-resonance transstarting from the ground state. Thus steps in the cyclotrresonance energy should be observed at those mag

FIG. 8. The phase diagram for a 100-Å-wide quantum wFour transitions are possible for this quantum well.

FIG. 9. The phase diagram for a fixed donor position,z50.7aB , as a function of the well widthW.

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fields at which the singlet-triplet transition takes place.real experiments, as we will see later, transitions only frthe ground state are not always seen in the neighborhoothe critical field, which is due to the fact that in a real eperiment the temperature is nonzero. Indeed, when theoldground state, i.e., the state that before the transition wasground state, and thenew ground state, i.e., the state thafter the transition now is the ground-state, have a comrable binding energy, they can both be thermally populat

V. COMPARISON WITH EXPERIMENT

In this section we present a comparison between ouroretical results and the experimental data reported by Jet al.8 The experiment of Jianget al. was performed on mul-tilayers of GaAs/Al0.3Ga0.7As with a well width of 200 Åand a barrier width of 600 Å. Such a system can be conered as an ensemble of single quantum wells. The wells wnominallyd doped at34 of the distance between the centerthe well and its edge. In the model discussed in this pathis means thatz50.75aB .

A comparison between the theoretical and experimetransition energies is reported in Fig. 12. The observed tr

.

FIG. 10. The transition energies~a! and oscillator strengths~b!for the (1,0,0)→(1,21,0) and (1,0,0)→(1,1,0) transitions. Thevalues for the donor placed atz50.7aB are compared to the valuefor a well centerD2. The dotted line is the free-electron cyclotrotransition energy\vc .

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15 398 57C. RIVA, V. A. SCHWEIGERT, AND F. M. PEETERS

sitions are theD0 (1,0)→(1,1) and D2 singlet (1,0,0)→(1,1,0) and triplet (1,21,1)→(1,0,1) transitions. Our theoretical results are given by three different curves. Note tour results well fit the data at low magnetic fields. The dviations between theory and experiment observed forB.9 Tcan be attributed to band nonparabolicity and polaron effeBoth effects decrease the transition energy,2 but are not takenin account in this paper.

In cyclotron-resonance experiments the integrated abstion intensities can be measured. The integrated absorpintensities are proportional to the oscillator strength timespopulation densities of the levels involved in the transitioTo compare our results with the experimental data, we hto make an assumption on the form of the population densWe assume that only the initial level of the transitionpopulated. Thus for the off-centerD2, the population densityof the level is proportional toeEb /kT, whereEb is the bindingenergy of the initial state. We remark that for the off-cenD2 in this range of magnetic fields the energies of the tripand singlet states are comparable. For the well-centerD2, onthe contrary, we consider only theL50 spin-singlet state tobe populated, i.e., the population density is 1.

The results for the relative integrated intensities ofsinglet transition, as evaluated in our calculation and theperimental results, are plotted in Fig. 13, and are in goagreement, both for the well-center and off-centerD2. Thetemperature in the experiment wasT54.3 K. Note that thedifferent magnetic-field dependence for the centerD2 ~i.e.,the increase withB) and for the off-centerD2 ~i.e., thedecrease withB) is correctly described. The errors bars fthe off-center intensities are rather large. A slight discrancy is observed at certain values of the magnetic fieldthe off-centerD2, but we observe that moving the donorour model slightly closer to the center of the well, i.e.,z50.7aB , the relative integrated intensity changes from tsolid curve to the dotted curve in Fig. 13, and now matcthe experimental data in the magnetic-field region in whthere was not such good agreement before. Thus the appdiscrepancy in the integrated intensity, withz50.75aB , in

FIG. 11. The cyclotron-resonance transition energies for a doat z50.85aB are shown by the dashed curves for the first thlowest states. The solid curve represents the expected transenergy from the ground state as a function of the magnetic fielzero temperature.

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the range 5–14 T, is explained by considering a small disbution of donors around the point of intendedd doping.

VI. SUMMARY AND CONCLUSION

We presented a theoretical study of the off-centerD2,where special attention was paid to the dependence ofbinding energy on the well width and donor position. Wfound that the magnetic field induces spin-singlet to sptriplet transitions in the ground state of the off-centerD2.The number of these transitions dependsbothon the positionof the donor and on the width of the well. In contrast to tideal 2D system and to quantum dots, only a finite numbetransitions are found. If the donor is near the center ofquantum well, no such singlet-triplet transitions occur. Whsuch singlet-triplet transitions occur, we find that at a suciently large magnetic field theD2 system becomes unbounand consequently one observes a magnetic evaporation o

oreionat

FIG. 12. The experimental data of Jianget al. ~Ref. 8! for thecyclotron-resonance transition energy~symbols! are compared toour theoretical results~curves!, for theD0 and the singlet and tripleD2. The donor is atz50.75aB and the well width isW5200 Å.

FIG. 13. Comparison of the relative integrate absorption intsity between the experimentally measured~symbols! and thepresent theoretical results~curves!. The donor position is atz50.75aB . The dotted curve takes into account a displacementhe donor from the position at which the well is nominallyd doped,i.e., z50.7aB .

t

nr

n

he

ehd

thethe

t po-nsif-r ofag-eti-

mionn

f-

57 15 399OFF-CENTERD2 CENTERS IN A QUANTUM WELL IN . . .

D2 system. We also calculated the oscillator strength foroff-center D2 as function of the magnetic field, and compared it to the results for a centerD2. We restrain ourselvesto a study of the optical transitions (1,0,0)→(1,21,0) and(1,0,0)→(1,1,0), and we observed that the off-center acenterD2 have similar magnetic behavior. Our results weused to explain the experimental results recently reportedJianget al. for the cyclotron-resonance transition energy athe absorption intensity of the off-centerD2 system for mag-netic fields up to 15 T.

In conclusion, theD2 center is a natural quantum-dosystem which is confined by the Coulomb potential of timpurity and consequently is more closely related to ratomic systems. A remarkable feature of theD2 centers inquantum wells is the controllability of the effective confinment potential, which is Platzmann-like for a donor in tcenter of the well and screened Coulomb-like when the

he-

debyd

teal

-eo-

nor is placed far away from the quantum-well center. Inlatter case the potential is parabolic near the center ofquantum-well plane, and thus resembles the confinementential of quantum dots. In this case singlet-triplet transitioare found as a function of the magnetic field. A crucial dference from the quantum dots is that only a finite numbesuch transitions occur, and that for sufficiently large mnetic fields theD2 system becomes unbound, i.e., magncally evaporates.

ACKNOWLEDGMENTS

Part of this work was supported by the EC prograINTAS-93-1495-ext, the Flemish Science Foundat~FWO-Vl!, and the ‘‘Interuniversity Poles of AttractioProgram-Belgian State, Prime Minister’s Office-Federal Ofice for Scientific, Technical and Cultural Affairs.’’

i

v

s.

ev.

v.

*Electronic address: riva@uia.ua.ac.be†Permanent address: Theoretical Applied Mechanics, RussAcademy of Science, Novosibirsk 630090, Russia.

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