PHYSICAL REVIEW 8 VOLUME 46, NUMBER 23 15 DECEMBER 1992-I

Binding energies of excitons and donors in a double quantum well in a magnetic field

J. Cen and K. K. BajajDepartment of Physics, Emory University, Atlanta, Georgia 30322

(Received 22 July 1992)

We study binding energies of both excitons and donors in a symmetric double quantum well in the

presence of a magnetic field applied parallel to the growth direction. We express wave functions as com-binations of Gaussian-type orbitals and subband wave functions of even and odd parities, with variation-

ally determined expansion parameters. By varying the interwell barrier width and well sizes (hence vary-

ing the interwell coupling), we obtain binding energies ranging in character from that for a strongly cou-

pled double well to that for a system of two isolated single wells. The behavior of the binding energies as

function of the interwell coupling, well sizes, and the magnetic field, as well as the donor position is con-sistently described with our formalism with naturally included subband mixing effects. The magneticfield leads to stronger confinement of the wave functions and enhances binding energies.

I. INTRODUCTION

Double quantum-well structures have attracted a gooddeal of attention, both experimentally and theoretical-ly. ' A double quantum well (DQW) is a semiconductorstructure in which two single quantum wells are separat-ed by only a thin potential barrier, across which electronsand holes from one we11 can tunnel into the other. As insingle quantum wells, the electrons and holes confined ina DQW can form excitons due to their mutual Coulombattraction. The electro-optical properties of such exci-tons promise applications in high-speed spatial-lightmodulators and switches. One advantage that a DQWstructure offers over the single quantum wells is theenhanced exciton electro-optic response.

A magnetic field applied parallel to the growth direc-tion has an additional confining effect on electrons andholes in the quantum wells, and is expected to modify ex-citon binding energies in the DQW. ' ' Together witheffects of the confinement and interwell coupling (throughtunneling across the potential barrier) provided by aDQW, we have an interesting physical system in whichthese competing factors influence those exciton charac-teristics determining the exciton electro-optical proper-ties of the DQW. Although several authors have done aconsiderable amount of work on the properties of exci-tons in DQW's, ' ' to our knowledge the eff'ects of amagnetic field on DQW's have not been theoreticallystudied. In addition, there are some conflicting resultsobtained by these various authors as to how interwell

coupling would qualitatively affect exciton binding ener-

gies in a DQW in the weak and strong interwell couplinglimits. A qualitative and quantitative study is desired togain knowledge of these aspects and to clear up ambigui-ties about the role played by interwell coupling in

affecting exciton binding energies in a double quantumwell.

In the presence of a hydrogenic donor impurity, theCoulomb interaction of the donor with an electron in theDQW leads to the formation of hydrogenic bound statesand the removal of the reflection symmetry. Optical

transitions involving donor states in quantum-well struc-tures have attracted interest for some years. ' ' Thebinding energies of hydrogenic donors in quantum-wellstructures have been calculated by various au-thors. ' ' ' Because a symmetric DQW with a varyingcenter potential barrier thickness (or height) can range incharacter from a double-width well to a pair of decoupledsingle wells, a potentially rich spectrum of physics canarise from the competing effects of quantum confinementand tunneling in this structure. Such a DQW structurewith wide-ranging characteristics, however, also demandsa more careful approach for one to describe consistentlythe physical processes taking place in it. Electron wavefunction tends to pile up around a positively chargeddonor atom due to the Coulomb attraction. Dependingon the donor location, electron wave function in the sym-metric quantum-well structure can be highly asymmetri-cal. A single-subband treatment using a definite paritysubband wave function can be inadequate in such in-stances to describe the electron charge distributionaround the donor, and can lead to a lower estimate of thebinding energy even for thin barrier thicknesses. In-clusion of such mixing plays an important role in correct-ly determining donor binding energies in a symmetricquantum-well structure.

Subband mixing is ignored in the few treatments in thepresent literature dealing with donor binding energies insymmetric DQW structures. Chen and Zhou consideredthe problem of donor binding energy in a symmetric dou-ble quantum mell with the donor atom located within thecenter barrier. Product of the lowest (even-parity) sub-

band wave function and a variational hydrogenic wavefunction was used to obtain the donor binding energiesfor various symmetric DQW structures. These authors,however, are unable to recover the corresponding single-well limits when the center potential barrier becomesvery thick.

It is now clear that such a single-subband approachcannot recover the single-well limit as barrier thicknessincreases. The basic reason is simply as follows. Tunnel-

ing across the potential barrier becomes increasingly less

46 15 280 1992 The American Physical Society

SIXDING ENERarES OF SXCITONS AXD DONORS IN A. . . I5 281

probable as the barrier thickness increases. Not able topenetrate the potential barrier, an electron will be essen-tially confined in one single well. Ignoring subband mix-ing, however unrealistically, forces the electron wavefunction to spread throughout DQW structure, whichreduces the probability of finding an electron in the vicin-ity of the donor, and therefore reduces the donor bindingenergy.

In this paper we develop a formalism to calculate thebinding energies of excitons and donors in DQW struc-tures, with effects of subband mixing included, in thepresence of a magnetic field directed along the growthaxis. The formalism is applied to a GaAs-A1„Ga, „AsDQW for various physical parameters. The preliminaryresults in the case of the exciton binding energies werefirst presented at the 1991 March Meeting of the Ameri-can Physical Society. In Sec. II, we describe this for-malisrn, in which we solve for electron and hole wavefunctions in the double-well potential profile; take mixingof electron and hole wave functions of neighboring sub-bands into account; express the electron-hole andelectron-donor internal-state wave function in terms ofGaussian-type orbitals and determine expansion parame-ters and exciton binding energies variationally. In Sec.III we show that our formalism correctly describes exci-ton and donor binding energies for all interwell couplingstrengths, and discuss the binding energies as a functionof the quantum confinement, the magnetic field, and theinterwell coupling.

mass, m, is the effective electron mass, and y, and y2 arethe Kohn-Luttinger band parameters, the + sign corre-sponds to the heavy-hole exciton, and the —sign to thelight-hole exciton. We then scale all lengths in the exci-ton Bohr radius az =Kpfi /Pe, and energies in the exci-ton Rydberg R =e /2Kpaz, to obtain the dirnensionlessform of the Hamiltonian

pH,'(z, ) = — + V, (z, ),77l Qz

(3a)

pH/, (z/, ) —— + V„(z/, ),

mh azh(3b)

H exC} a a2

P~P ~P p ~yp

2

&p'+z'

p + +H,'(z, )+Ht, (z/, )P~p ~P p ~q

2 +yL, + py' 2 (2)2+ 2

where p= +(x,—x„) +(y, —y„) is the in-plane dis-tance between a pair of electron and hole, z =z, —zz, L,is the z component of the angular rnomenturn, and y isthe first Landau level expressed in R, y=eAB!2pcR.The Hamiltonian H above is grouped into three terms,namely the electron part H,', the hole part H&, and theexciton part H,„,H =H,'+ H& +H,„,where

II. FORMALISM 2

+yL, + p (3c)We consider a DQW consisting of two identical GaAs

layers sandwiched between two semi-infiniteAl„Ga1 „As slabs, with a thin layer of Al .Ga, As be-tween them. A uniform magnetic field B is applied per-pendicular to the layers (in the growth direction).

The Hamiltonian of the electron-hole system is

H =H, —ikey+ —A —H~ iAV+ —Ae ec C

e+ V, (z, ) + V/, (z/, )—Kp~r r

where V, (z) and V/, (z) are, respectively, the potentialprofiles for the electrons and holes, A=(BXr)/2 is thevector potential of the magnetic field B, Kp is the dielec-tric constant of the layers (assumed to be uniform here),r, and rl, are the electron and hole positions. The elec-tron Harniltonian H, is adequately described by aneffective-mass approximation, using parabolic bands.The hole Hamiltonian H& is the 4X4 Kohn-LuttingerHamiltonian. To gain physical insight with a tractablemodel, we assume parabolic hole bands in the x-y planeand in the z direction and retain only the diagonal termsin H&, thereby ignoring coupling between the heavy-holeand light-hole bands. Following standard procedure toseparate the constant center-of-mass motion of anelectron-hole pair in the x -y plane, we define thereduced mass of an electron-hole pair p with/M '=m, '+(y, y~)mo ', where mo is the free-electron

and m/, is the heavy- (light-) hole mass defined in

mh '=(yi+2yz)mo '. The Hamiltonian in Eq. (2)reduces to that for an electron bound to an immobiledonor if we replace r& with position of the donor rd andset V/, (z/, ), y„and yz equal to zero (which correspondsto infinite hole masses).

Wave function f(r„r/, ) of the electron-hole system issolved from the Schrodinger equation

(4)

where E is the total energy. We write g(r„r/, ) in the fol-lowing form to express the explicit dependence on z„z&and on the relative distance r =r, —r&.

q(r„r/, )=/t(r) y ~/, F,"(z, ) y b/Fh(z/, ),k=1 1=1

where P(r) is the wave function describing the internalstate of an exciton, F,"(z,) is the kth electron subbandwave function, and F/', (z„) the 1th hole subband wavefunction, aI, and bl are the expansion coefficients to bedetermined, both F, (z, ) and F/', (z/, ) are normalized. Inthe calculation of the donor binding energiesIFh(zz ) I ~5(zh —zd). Equation (5) is a good approxima-tion to the exciton wave function, as long as thedifference between the subband levels that are included inthe summation and those that are not is larger than theexciton binding energies. The two wave functions in the zdirection are determined by the following two equations:

15 282 J. CEN AND K. K. BAJAJ

H,'F,"(z, ) =E,"F,"(z, ),

HbF/ (zb ) E—bF/, (z„),

(6a)

(6b)

V. , lzl&z,V(z)=.0, z, &lzl&z,

vb Izl &zi,in which E, and Eh are the electron and hole subbandenergies.

We first solve for the subband envelope functionsF,(z, ) and Fb (zb). Since H,', and Hb have the same form,we can solve the general double-well problem for the fol-lowing profile V(z);

where z& =Lb/2, z2=z&+L, LI, is the width of thecenter potential barrier, L is the width of a single quan-tum well, Vb is the center barrier height, V is outer po-tential wall height. We write F(z} in the following formto reflect the symmetry of V(z):

T

D exp[ —P2(z —z2)], z &z2

A sin[a(z —z, }]+Bcos[a(z —z, }], z, &z &z2

F(z}= 2Cy+(P, z) lzl &z,

+ A sin[a(z+z, })+Bcos[a(z+z, )] —z2 &z & —z,+D exp[Pi(z+z2)], z & —zz,

where + indicates the wave function F(z) as being ofeven parity, —as being of odd parity, a, P„and P2 arethe parameters determined by the subband

energy E„, a= +m „E Ip, 13,=Qm l Vb El lp, —

P2=+m (V E)lp, (—cr=e, h), and

coshP, z (+ )

sinhp, z ( —), Vb & EcosP,z (+ )

sinp, z (—}, Vb &E .

By requiring the continuity of F (z) and BF(z)IBz at in-terfaces z =+z& and z =+z2, we obtain the secular equa-tion for the subband energy E,[h],

a~.(p... )

13~—a y+(P, z, ) sinaL

z]

aq, (P,z, )+a +P2X+(P,z, ) cosaL =0 . (10)z]

After the subband energy is obtained, coefficients A, B,C, and D for the wave function F(z) are then determinedby the continuity and normalization conditions. Thewave function of the lowest subband is of even parity,that of the second subband is of odd parity.

Next we express the exciton internal-state wave func-tion P(r) in terms of Gaussian-type orbitals and use avariational calculation to determine the expansion pa-rameters and the exciton binding energy.

eim ~f(' 2n

P(p, p;z)= p 'e ~~ g c,exp[ —a, (p'+z }]&2m

(m =0,+1,+2, . . . ),where P is the variational parameter, c are the expansioncoefFicients, a- are sets of constants. ' m is the azimu-thal quantum number; for the 1s state, m =0; for the 2p+

states, m =+1; and so on. For excitons in the doublequantuin well, P is varied to adjust these Gaussian-typebasis functions to minimize the total energy E.

For DQW's consisting of narrow wells with strong in-

terwell couplings (for center barriers of small widths orlow heights), effects of the coupling between neighboringsubbands on exciton binding energies are shown to besmall; therefore it is sufficient to assume the exciton to beassociated with a single electron subband and a singlehole subband. ' In general, however, the single-subbanddescription of an exciton in a double quantum well isinadequate and can lead to qualitatively misleading re-sults. When two single quantum wells are separated by apotential barrier, the wave functions in these wells arescrambled to form a "bonding" (even-parity) and an "an-tibonding" (odd-parity) combination (total) wave func-tion. If the barrier is thin and the wells are narrow, thesingle-well wave functions are strongly modified by thepresence of the neighboring well because of the tunnelingof the electron (hole) across the potential barrier. As aresult, the bonding and antibonding total wave functionshave significantly different subband levels. In otherwords, subband levels in such a thin-barrier, narrow-wellDQW are nondegenerate. When the barrier is thicker orthe wells are wider, single-well wave functions are essen-tially confined to one single well and are therefore dimin-

ishingly affected by the presence of its neighboring we11.

Bonding and antibonding combinations would yield simi-lar subband levels, with one slightly lower and one slight-

ly higher than the isolated single-well subband levels. Allsubband levels are almost doubly degenerate. A con-sistent description of exciton in DQW structures shouldtherefore include pairs of subband levels to properly ac-count for contributions to exciton binding energies fromboth the even-parity and odd-parity subband wave func-tions. In the case of an electron bound to the donor, theelectron system no longer has a definite symmetry, andneither the even- nor the odd-parity wave function alonecan be the true wave function. We take a linear combina-

BINDING ENERGIES OF EXCITONS AND DONORS IN A. . . 15 283

tion of the wave functions of different subbands as the en-velope function.

In what follows, we calculate the binding energies ofthe 1s exciton state associated with the first two electronand hole subbands and binding energies of the 1s and 2pdonor states associated with the first two electron sub-bands. In a system with thin barrier and narrow wells,the separation between the adjacent subband levels arelarge compared with the expected exciton binding ener-gies and there is little intersubband coupling, i.e.,a, b, =1 and akb&~0(k +1 )2). As Lb —+ ~, these sub-band levels become degenerate and coupling betweenthem becomes important, i.e., all ak and bI ~ould playcomparable roles. In the absence of the Coulomb interac-tion and without mixing of the even-parity and odd-parity subbands, the total energy E is just the sum of firstelectron and hole subband energies E,'", Ez" and theLandau-level energy y. The Coulomb interaction be-tween the electron and hole lowers the total energy andleads to the formation of the exciton. The binding energyof the lowest-lying exciton E~ is defined in

Ez =E,'"+E&"+y —E. Similarly, binding energies ofthe 1s and 2p donor states is Ez =E,"'+y —E. Bindingenergies of the 2p+ state can be obtained by simply add-ing 2y to the 2p state values.

The total energy E is obtained by expressing theSchrodinger equation (4) as an eigensystem problem withthe nonorthogonal Gaussian-type orbitals as basis func-tions, which is then solved by a generalized Rayleigh quo-tient iteration method. By choosing the appropriate ei-genvalue A, and minimizing it as a function of the varia-tional parameter, we obtain the total energy E of a givenexciton/donor state and its binding energy Ez.

III. RESULTS AND DISCUSSION

A. Kxciton binding energies

We have calculated the binding energies of the heavy-hole exciton and the light-hole exciton as functions of themagnetic field, the well width L, and the center barrierthickness LI, of a symmetric GaAs-Al Ga& As doublequantum well. The values of physical parameters per-taining to GaAs used in our calculations arem, =0.067mo, so=12.5, y) =7.36, y2=2. 57. Thevalues for the heavy-hole (J, =+—', ) exciton are

0

mI, =0 45mo p=004mo a 165 A R =3 49 eVthose for the light-hole (J, =+—,') exciton are

m& =0.08mo, p=0.05mo, a&=131 A, R =4.39 meV.We use an empirical formula AEs =(1.36+0.22x)x (eV)to determine the band-gap discontinuity, with 60% ofhE contributing to the conduction-band discontinuityhE, and 40% to the valence-band discontinuity hE„.Mole fraction x =0.3 is used for all Al concentrations.Differences between other material parameters of GaAsand those of Al Ga, As are not included in the calcula-tions (see Fig. 1).

In Fig. 2, we compare the binding energies of theheavy-hole exciton in a DQW calculated by us, withthose obtained by Kamizato and Matsuura (KM hereaf-ter)' with and without subband mixing, and those byDignam and Sipe (DS hereafter)' with subband mixing,as a function of the barrier thickness Lb. It is evidentthat the two-subband treatment by DS underestimatesthe binding energy in the strongly interwell coupling lim-it (Lz ~0.2as) and cannot reconcile with the fact that, atLb=0, the DQW is simply a single well of width 2L

AN&a 1-.As c'.Anh»r. tinn-band edge

GaAs

Veb

GaAs

L

Ve

AMsal-xAs Anal-xAsW

p Zl Z2

Vel

I I~V~ valent. e-band edge

FIG. 1. Schematic band diagram of a symmetric GaAs-Al„Ga, As double quantum well and the applied magnetic field B in thegrowth direction.

15 284 J. CEN AND K. K. BAJAJ

On the other hand, the two-subband DS result in theweak interwell coupling limit approaches that obtainedby KM without subband mixing, which cannot recoverthe single-well result at large barrier thicknesses either.It appears that although the DS two-subband treatmentworks at the intermediate interwell coupling strengths, itoverestimates the strength of the interwell coupling inboth the strong and weak interwell coupling limits. Ourresult agrees with that of KM, including subband mixingat both strong and weak interwell coupling limits. It isalso evident that our formalism gives higher binding en-ergies for all interwell coupling strengths. Furthermoreour formalism correctly describes exciton binding ener-gies when the additional confining effect of the magneticfield is also included.

In Fig. 3(a), we show variation of the binding energyEz of the heavy-hole exciton as a function of well widthsL„ for several different combinations of the barrier thick-ness Lb and the magnetic field B. The results of E~ forLb =0 have been compared with those of Greene and Ba-jaj for exciton binding energies in a single quantum wellin a magnetic field, ' based on the expansion of the exci-ton wave function into Gaussian basis orbitals. These forzero magnetic field (B =0}and Lb «a~ have been com-pared with the results obtained by KM for the excitonbinding energies in a symmetric double quantum well,with material parameters roughly corresponding to thoseof heavy-hole excitons in GaAs-Al Ga, As quantumwells. ' The agreement in both cases is excellent, as ex-pected.

For Lb=0, as well width L decreases, electron andhole wave functions first become compressed in the nar-

25-'

20--

)15-

X

10--

50 100

L (A)

150 2()0

rowing wells and the exciton binding energy E~ climbsup due to the decreasing average distance between theelectron and the hole, which is mainly determined by thewell size L and barrier thickness Lb in a given magneticfield, until E~ reaches a maximum. As L further de-creases, subband energies are pushed up and leakage ofthe wave functions into the barrier regions becomessignificant, and E~ begins to fall off rather rapidly as theexciton assumes more of a 3D-like nature.

For LbWO, the binding energy is lower for small Land higher for large L, in comparison to that in theDQW with Lb=0. For narrow wells, the electron andhole wave functions spread throughout the DQW struc-

25 ~(

(b)

/ g+ A

20--

)15—

(~F

10 j

6

0 50 100 150 200 250 300 350

Lb (A)

5 t

0 50

!

100 150 200

L (A)FIG. 2. Comparison of binding energies of the heavy-hole ex-

citon, calculated with and without subband mixing, as a func-tion of the barrier thickness Lb. The well width is fixed atL =0.6a~. The solid line ( } is our result with subbandmixing included; the dotted line (- -) is by Dignam and Sipe(Ref. 16) with subband mixing; the short-dashed line (

———) is

by Kamizato and Matsuura (Ref. 14) without subband mixing;the long-dashed line (——) is by Kamizato and Matsuura withsubband mixing. The material parameters are as in Ref. 14.

FIG. 3. (a) The binding energy of the heavy-hole exciton as afunction of the well width L„,. (b) The binding energy of thelight-hole exciton as a function of the well width L„„with mag-netic field B and barrier thickness Lb as the two other parame-ters. Material parameters are noted in text. The solid lines

( } stand for Lb =0 (corresponding to that in a single quan-

tum well of width 2L„,); the dotted lines (. . . -) stand forL~ =25 A.

46 BINDING ENERGIES OF EXCITONS AND DONORS IN A. . . 15 285

ture, and the presence of the barrier merely increases theaverage distance between the electron and hole, leadingto a lower binding energy. As the wells become wider,however, the wave functions become more and moreconfined in one single well due to the presence of the bar-rier, and the average distance between the electron andhole decreases, leading to a higher binding energy. At awell width L', the Es curves in the DQW with L„WOwill cross over with that in the DQW with Lb =0. Sincea magnetic field provides an extra confinement of thewave function in the quantum well, such a crossover will

occur at a smaller L' at higher field strengths. Also no-tice that a shoulder develops in the binding-energycurves. As Lb increases, this shoulder will become moreevident and appears at smaller well widths L . At thelimit Lb ~~, it merges with the maximum that is caused

by the leakage of lowest subband wave functions into thebarrier regions. This shoulder is attributed to the mixingof wave functions of the odd-parity second subbands withthose of the even-parity first subbands. For small Lb,E'" and E' ' are significantly different, and wave func-tions of the second subbands are more spread out due totheir higher energies. The binding energy influenced bythe second subbands would reach its maximum at thelarge well width L . As Lb increases, the second sub-band lowers down and eventually becomes degeneratewith the first subband, and the maximum in E~ caused byit coincides with that of the first subband.

In Fig. 3(b), we show values of the binding energy Esof the light-hole exciton as a function of L . Qualitative-

ly E~ behaves similar to the binding-energy of theheavy-hole exciton. However, it is larger and reaches themaximum for larger L compared to that of the heavy-hole exciton. "' Also, the values of light-hole excitonbinding energy are higher than those obtained by Greeneand Bajaj, who used 85—15 % conduction-valence bandoffsets in their calculations, ' as light holes are now moreseverely confined in the quantum wells by higher poten-tial barriers. The binding energies of the heavy-hole exci-ton associated with the lowest subband are not as sensi-tive to the change of band offsets used in the calculations,since the heavier longitudinal mass results in strongerconfinement of the heavy-hole wave function in the quan-tum wells. However, for excitons associated with highersubbands, higher valence-band offsets are expected toaffect binding energies more significantly for both thelight-hole and heavy-hole excitons. In all instances, thepresence of a magnetic field in the growth direction leadsto higher exciton binding energies. Our results on theheavy-hole exciton fit rather well with those measured byPerry et al. ,

' when appropriate material parameters areused in the calculations.

In Fig. 4(a), we show the binding energies of theheavy-hole exciton as functions of barrier thickness LI,for several different values of (L,y). Similar results forthe light-hole exciton are displayed in Fig. 4(b). AtLb =0, results for single quantum wells of width 2L arerecovered, as we have noted earlier. The average dis-tance between the pair of electron and hole forming theexciton increases as Lb increases from zero, and as a re-

suit the binding energy Ez drops down first. For smallbarrier thicknesses, a significant portion of wave func-tions is present in the barrier regions. However, thisleakage decreases sharply as well size increases. There-fore for wider wells the rate at which the binding energydrops down as Lb increases is higher, as it is easier toseparate the wave function in the two neighboring wellswhen the center barrier size Lb increases.

As Lb further increases, coupling between the well di-minishes, and the binding energies will eventually climbup and approach the isolated single-well values Es(L ).The E~ curves will bottom out at barrier thickness Lband then rise up. Again for wider wells, this minimum in

E& will occur at smaller barrier thickness Lb. Noticealso that as the magnetic field increases, the exciton wave

25

(a)

20-8=200 kG~ ~ ~ ~ ~ ~

~ ~ ~ ~

15- 8=100 kG.

B=50 kG~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

10—~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~

8=0~ ~ ~ ~ ~ ~ ~

25

25 50

Q (A)

75 100

~ ~

20--

~ ~M ~

(b)

kG

15-Pl

10—.~ .

~ ~

8=100 kG—~ ~ ~

~ ~~ ~ ~ ~

8=50 kG~ ~ ~

~ ~

=0

25 50

tb (A)

75 100

FIG. 4. (a) The binding energy of the heavy-hole exciton as afunction of the barrier thickness Lb. (b) The binding energy ofthe light-hole exciton as a function of the barrier thickness Lb.The other two parameters are the magnetic field 8 and the well

width L . Solid lines ( ) stand for L =100 A; dotted lines0

(- - -) for L =10 A.

15 286 J. CEN AND K. K. BAJAJ 46

function spread is reduced, and as a result the interweHcoupling decreases faster as the barrier thickness in-creases. The minimum of the binding energy occurs atsmaller Lb.

It is worth pointing out that so fax only our approach,to our knowledge, has produced consistent results for allbarrier sizes. Although we have calculated only the bind-ing energies of excitons associated with the first electronand hole subbands, our formalism here can be applied toexcitons associated with other subbands.

B. Donor binding energies

We have also calculated the binding energies of the 1sand 2p+ donor states in symmetric double GaAs-

35-

30--B=200 kG

{a)

25--

15--

10--

50 100 150

I

200

LR (A)

(b)B=100 kG ',

I r/

I /'B=50kG'. ' ~ 8=200kG, /. ---.

I I 8 0( r

20 —g'~' ', '~

'~,'I I

~II

I

30--

10--

I

0 I

0 50 100 150

L (A')

200

FIG. 5. The binding energy of the 1s donor state E~ as a0

function of well size L„„for a fixed barrier thickness Lb =50 A,for several values of magnetic field B (kG). (a) The solid lines

( ) stand for a well-center donor, the dotted lines (. - . ~ )

for a barrier-center donor. (b) The long-dashed lines (——)

stand for a barrier-edge donor, the short-dashed lines (————)

for an outer-well-edge donor.

Al Ga, As quantum-well structures in a magnetic field.0

The effective Bohr radius a~ =99.5 A, and effective Ryd-berg R =5.79 meV. Using only the lowest subband inour calculations, we easily recover for the 1s donor statein the case of zero magnetic field the results of Chen andZou, who used the even-parity lowest subband in theircalculations of donor binding energies in a symmetricdouble quantum well.

In Fig. 5, we show the binding energy of the 1s state asa function of well size L . Four different cases are con-sidered: (l) a barrier-center donor, (2) a barrier-edge(inner-well-edge) donor, (3) a well-center donor, and (4)an outer-well-edge donor. The center barrier thickness is

0fixed at L& =50 A. The well size is varied from L =10to 200 A. All four donor locations yield similar bindingenergies at small well widths. This is due to the fact that,since the electron wave function is spread throughout anarrow-well DQW structure, the probability of finding anelectron in the vicinity of a randomly distributed donor isalmost independent of the donor location. As well sizeincreases, the binding energy of the barrier-center donormonotonically decreases, as the wave function becomesmore and more concentrated inside the quantum well, in-

creasing the average electron-donor distance. For thebarrier-edge donor, however, it is possible that there ex-ists a favorable probability of an electron being in the vi-

cinity of the donor at some small L„, since the electronwave function tends to pile up around a positive charge.A maximum in the binding energy exists at some smallwell sizes. As well size further increases, the elctronwave function is increasingly confined inside the well, andthe binding energy then decreases due to the larger aver-age electron-donor distance. For the well-center donor,the confinement of the electron wave function inside thewell at increasing L strongly favors a larger binding en-

ergy. As well size further increases, the spread of thewave function in the well and surrounding regionsreduces the probability of the electron staying close tothe donor, and the binding energy decreases. For theouter-well-edge donor, the increasing confinement of theelectron wave function to the inside of the well as L„, in-

creases leads to higher binding energies at small well

sizes, but eventually leads to a larger average electron-donor distance and hence to lower binding energies. No-tice that the maximum binding energies of the donor atthe outer-well edge are higher than those at the well

center. This is attributed to a stronger mixing of theeven- and off-parity DQW wave functions in the vicinityof the donor.

In Fig. 6, we show the 2p -state binding energies as afunction of well size L„ for a fixed barrier size L~ =50 A.For the 2p state, donors at all locations have lower

binding energies than those of the 1s state because of thelarger wave-function spread. The barrier-center andbarrier-edge donor binding energies behave in a qualita-tively similar way with their 1s counterparts. The behav-ior of the well-center and outer-well-edge binding ener-

gies of the 2p states, however, is qualitatively differentfrom that of the 1s state for the following reason. In nar-row quantum-well structures, the 2p -state wave func-

tion is highly compressed, much more so than that of the

46 BINDING ENERGIES OF EXCITONS AND DONORS IN A. . . 15 287

ls state. As well size increases, the 2p -state wave func-tion is allowed to relax in the structure, both in thegrowth direction and in the x-y plane. For the well-center donor, such an expansion of the electron wavefunction reduces the probability of finding the electron inits vicinity, therefore its binding energy decreases. Afterthe expansion is largely accomodated by even widerwells, the electron wave function tends to pile up aroundthe donor and the binding energy will first increase toreach a maximum and then behave like that of the 1sstate. For the outer-well-edge donor, the binding energywill initially also decrease. Since it is more sensitivelyaffected by the spatial extent of the electron wave func-tion, it recovers to a maximum at smaller well sizes andthen drops down at a steeper rate as well size further in-creases.

Both 1s- and 2p -state binding energies in a magneticfield behave qualitatively like zero-field counterparts. Inaddition to giving higher binding energies, the strongerquantum confinement by the magnetic field also increasesthe rate at which the binding energies change at increas-ing well sizes.

In Fig. 7, we show binding energies of the 1s state as afunction of barrier thickness Lb for a fixed well sizeL =50 A. As the barrier thickness increases, thebarrier-center donor binding energy monotonically de-creases, as the probability of finding the electron in its vi-cinity can only become more remote. The barrier-edgedonor binding energy initially also decreases when L in-b

creases from zero, as the single-well (of width 2L ) wavefunction begins to separate and the probability of findingthe electron in its vicinity decreases. As Lb further in-creases, however, the DQW wave function gradually be-

40

30--

(a)

B=200 kG

100 kG

=50 kG

B=O

10--~ ~

~ ~ ~ ~ ~ ~ ~ ~ P ~

~ ~ ~~ \ ~ ~

~ ~ ~~ ~ ~ ~ ~ ~ ~ ~

~ ~

~ ~ ~ 0 ~ ~ ~ P~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~

~ ~ ~~ ~ ~ ~ ~ ~

~ ~ ~

~ ~ ~ ~ ~ ~ ~~ ~ ~ ~

~ ~~ ~ ~ ~ ~P

~ ~ ~ ~ ~P

~~

~ ~~ ~ ~

~~

~ ~~ ~ 0 ~

~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ P ~ ~~ P ~ P ~ ~ ~ ~

50 100 150

comes an efFectively single-well wave function (with sub-band mixing), and the binding energy quickly recovers itsvalue in a single quantum well of width L . It should bepointed out that a single-subband treatment will alwaysassume comparable portions of the electron wave func-tion in both wells, no matter how thick the barrier hasbecome, and will therefore always underestimate donorbinding energies in the symmetric DQW structure. Thewell-center donor starts out with roughly the same bind-ing energies as the barrier-center and barrier-edge donorsat small barrier thicknesses, since the binding energy isnot very sensitive to changes in location for a donor notclose to the surrounding potential walls in a single quan-tum well (size -2L ). This situation quickly changes as

16 40

12--

B=200 kG(b)

B=200 kGr

3R r

B=100 kG~ ~

GPE ~

~ ~

B=50 kGPy~r ' ~ ~r ~ ~ ~

P aP ~ ~ ~ ~ r ~ ~~

~ ~ ~ ~ ~ o~ P ~ ~ ~

4 B=O

~ ~peirg

P ~ ~ P ~ P

~ s ~ ~ P~ ~ ~ ~ ~ ~ 0 ~

20-

10--

Prrr

P r P

r

r

PP P P

B=O

P P P P P

B=50 kG

PPPPPP ~~ PPPPPP PPPPPP

50 100

L (A)

150 200 50

Q (A)

100 150

FIG. 6. The binding energy of the 2p donor state E as afunction of well size L, for a fixed barrier thickness Lb =50 A 7

for several values of magnetic field B (kG). The solid lines( ) stand for the well-center donor, the dotted lines (. ~ - ~ )for the barrier-center donor, the long-dashed lines (——) forthe barrier-edge donor, and the short-dashed lines (————) forthe outer-well-edge donor.

FIG. 7. The binding energy of the 1s donor state Ez as afunction of barrier thickness Lb, for a fixed well size L =50 AW 7

for several values of magnetic field B (kG). (a) The solid lines( ) stand for the well-center donor and the dotted lines(. - . .) for the barrier-center donor. (b) The long-dashed lines(——) stand for the barrier-edge donor and the short-dashedlines (————) for the outer-well-edge donor.

15 288 J. CEN AND K. K. BAJAJ 46

16+——B=200 kG

12—

B=100 k

4---1 B=O

B=50 kG

0+—

50 100 150

FIG. 8. The binding energy of the 2p donor state E~ as afunction of barrier thickness Lb, for a fixed well size L =50 A,for several values of magnetic field B (kG). The solid lines

( ) stand for the well-center donor, the dotted lines (- . . -)

for the barrier-center donor, the long-dashed lines (——) forthe barrier-edge donor, and the short-dahsed lines (————) forthe outer-well-edge donor.

Lb increases and the wave function in the double-widthwell begins to separate into single-well wave functions.The well-center donor is in a favorable position to have agreater probability of finding the electron in its vicinity;its binding energy quickly recovers the single-well limit.Again, only inclusion of subband mixing allows one tocorrectly describe the realistic situation. The outer-well-edge donor always benefits from evolution of the DQWwave function into the single-well wave function. Itsbinding energy monotonically increases with increasingL&, and eventually flattens out as the average electron-donor distance becomes a constant for a fixed well size.

In Fig. 8, we display the binding energies of the 2pstate as a function of LI, . The 2p donor state, due tocompression of its large natural wave-function spread inthe DQW structure, behaves quite differently with in-creasing barrier thickness Lb. All donor locations startout with roughly the same binding energies, because thehighly spread out electron wave function is not sensitiveto the donor location in the DQW of L&=0. Thebarrier-center donor binding energy monotonically de-creases as Lb increases, for the same reason as in the caseof the 1s state. In contrast to their 1s counterparts,donors outside the barrier first see their binding energiesdecrease with increasing barrier thicknesses. Thereason is as follows. The 2p -state wave function is

highly compressed in the DQW of total width2L +L~ (100 A+L& here), so the wave function will

first expand to occupy the extra space provided by the in-creasing Lb, thereby reducing the probability of findingthe electron in the vicinity of the donor; and the binding

energies will first decrease. After this initial expansionstage, the DQW wave function begins to pile up aroundthe donor and gradually becomes effectively a single-wellwave function, and the binding energies then quickly ap-proach the single-well limit. As expected, extraconfinement by the magnetic field compresses the wavefunction in the structure or around a donor, and there-fore increases binding energies and increases the pace atwhich the binding energies change with increasing bar-rier thickness.

IV. SUMMARY AND CONCLUSIONS

In summary, we have developed a formalism to calcu-late the binding energies of excitons and donors in a sym-metric double quantum well in the presence of a magneticfield applied parallel to the growth axis. The extra quan-tum confinement by the magnetic field increases the bind-ing energies. Effects of interwell (intersubband) couplingon the light-hole and heavy-hole exciton binding energiesin the double quantum well are consistently included inour calculation. In the limit of thick potential barriers,the even-parity and odd-parity subband wave functionshave degenerate energy levels; mixing of electron andhole subband wave functions strongly modifies the exci-tonic wave function and consequently lets one recover re-sults for excitons in decoupled single quantum wells. Wehave shown that ignoring such subband mixing is a goodapproximation only for narrow wells and thin-barrierdouble quantum-well structures, and that such a single-subband approach can lead to qualitatively misleadingconclusions when applied in wide wells or thick barrierDQW's.

The presence of the donor atom breaks the reflectionsymmetry along the growth axis, and leads to strong mix-

ing of electron subband wave functions of even and oddparities. We have shown that inclusion of such mixing iscrucial for a correct description of the donor binding en-

ergies in DQW structures with wide-ranging characteris-tics, and that a single-subband treatment would underes-timate the binding energy in a symmetric double quan-tum well except in the thin-barrier and narrow-wellstructures. Using our formalism, we have correctly de-scribed the donor binding energies in DQW structureswith Lb ranging from zero to several effective Bohr radii,and recovered the corresponding single-well limits in

both extremes. Effects of quantum confinement provided

by the magnetic field and the potential wells, and that oftunneling across the center potential barrier on the exci-ton binding energies, are discussed.

We have used the first two single electron and hole sub-

bands in calculations of the exciton binding energies, andour results cover most cases one would encounter in ex-

periments. While we have not included differences in theeffective masses and dielectric constants across theGaAs-Al„Ga& As interfaces in our calculations, theyhave been shown by various authors to lead only to small

increases in the binding energies, and would not alter theconclusions here. After the original manuscripts weresubmitted, Ranganathan et al. also reported their work

BINDING ENERGIES OF EXCITONS AND DONORS IN A. . . 15 289

on the donor binding energies in double quantum wells,with the inclusion of subband mixing effects. Our gen-eralized formalism, when applied to the case of donorbinding energies for the specific parameters, obtained re-sults which are in agreement with the results of their cal-culation and experiment.

ACKNOWLEDGMENTS

We wish to thank Dr. S. M. Lee for fruitful discus-sions. This work was supported by the Air Force Officeof Scientific Research under Grant Nos. AFOSR-91-0056and AFOSR-90-0118.

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