PHYSICAL REVIEW B VOLUME 44, NUMBER 19 15 NOVEMBER 1991-I

Binding energies of excitons in type-II GaAs-A1As quantum-well structuresin the presence of a magnetic field

XuDong ZhangCenter for Solid State Electronics Research, Department ofElectrical Engineering, Arizona State University, Tempe, Arizona 85287

K. K. BajajDepartment ofPhysics, Emory University, Atlanta, Georgia 30322

(Received 8 April 1991)

We report a calculation of the binding energies of both the heavy-hole and the light-hole excitons intype-II GaAs-A1As quantum wells as a function of the size of the A1As layer (or the GaAs layer) in thepresence of a magnetic field applied parallel to the direction of growth. We follow a variational ap-proach and assume that the electrons and holes are confined in infinite potential barriers. For a given setof values of GaAs- and A1As-layer thicknesses, the binding energies of excitons are found to increasewith increasing magnetic field.

INTRODUCTION

There has been a great deal of interest in the study ofthe properties of excitons in semiconductor superlatticesand quantum-well (QW) structures during the past 15years. Most of this work, however, has been done in theso-called type-I quantum wells where the electron and thehole are confined spatially in the same region. For nar-rower well sizes in a GaAs-A1As system [typicallyGaAs-layer thickness & 30 A (Refs. 1 —4)] the band-edgeconfiguration at the GaAs-A1As heterojunction may be-come staggered or type-II. In this case an electron and ahole are confined in spatially separate wells (see Fig. 1)and the binding energy of the exciton will be less than itscorresponding value in type-I quantum wells. This effectwas noticed by Bastard et a/. while studying the proper-ties of excitons in type-II, InAs-GaSb quantum wells.Duggan and Ralph have calculated the binding energiesof excitons in type-II GaAs-A1As quantum wells using avariational approach assuming infinite potential barriersfor both the electrons and holes. The values they calcu-lated are comparable with those in type-I quantum wells.This is because the longitudinal electron mass at the Xminimum of A1As is larger compared to its value in the Iminimum of GaAs, thus compensating for the reductionin the Coulomb interaction. Matsuura and Shinozukahave studied the same problem following a variationalapproach using a different trial wave function and find re-sults essentially similar to those obtained by Duggan andRalph. Salmassi and Bauer have studied the exchangeinteraction in type-II QW's and have also calculated thebinding energies of excitons following a variational ap-proach assuming finite potential barriers. They find thatdue to the nonzero overlap between the electron and thehole wave functions, the values of their binding energiesare about 10—15 %%uo larger than those obtained usinginfinite barriers. Recently Degani and Farias have cal-culated the exciton binding energies in type-II quantumwells in the presence of a static electric field using a varia-

THEORY

The band-edge configuration for the type-II GaAs-AlAs system and the corresponding coordination systemare shown in Fig. 1. The holes are confined in the GaAslayer and the electrons reside in the indirect-gap A1Aslayer. Within the framework of an effective-mass approx-imation, the Hamiltonian of an exciton system in this

AIAs GaAs AIAs

Conduction

Band

Valence

Band

-Lh Le

FIG. 1. Schematic energy-band diagram of type-II GaAs-AlAs quantum-well structure.

tional approach.In this paper we report a calculation of the binding en-

ergies of both the heavy-hole and the light-hole excitonsin type-II GaAs-AlAs quantum wells as a function of thesize of the A1As layer (or GaAs) layer in the presence of amagnetic field. We assume that the magnetic field is ap-plied parallel to the direction of growth [001] of thequantum-well structure. We use a variational approachand assume that the potential barriers are infinite.

10 913 1991 The American Physical Society

BRIEF REPORTS

+ 1

2(m, ),

+ 1

2(m, ),

e—iAV +—A

e—iA'V, +—A,

2

(2)

Here (m, )I and (m, ), are the longitudinal and transversemasses of the electron in the X minimum of AlAs and Hhis defined by the well-known Kohn-Luttinger Hamiltoni-an. ' The relative coordinate is r=r, —r&, where r, andr& are the positions of the electron and hole, respectively.The confinement potentials V, (z, ) and Vh (zh) are as-sumed to be infinite.

Due to reduction in symmetry along the direction ofgrowth and the presence of energy-band discontinuities,the degeneracy of the valence band is removed, leading tothe formation of heavy- and light-hole excitons. Weneglect the contributions of the o6'-diagonal terms of theKohn-Luttinger Hamiltonain. ' With this assumption,the Hamiltonian of an exciton associated with either theheavy hole or the light band can be expressed as"

structure can be expressed as

H=H, ( —ifiV+(elc) A) H—h(iAV+(elc) A)2

+ V, (z, )+ Vh (zh) .EpT

Here A is the vector potential associated with themagnetic field, ep is the dielectic constant of the GaAs-A1As system, taken to be 12.3 (average value of thedielectric constants of GaAs and A1As), H, is the con-duction electron Hamiltonian, and H&, is the valence-band Hamiltonian. The explicit expression for H, is

'2

H, = 1

2(m, ),

The parameter y is a dimensionless measure of the mag-netic field and is defined as

eA'8y+=

2p+cA ~

where the efI'ective rydberg

e p+4

2/2

In Eq. (3) L, is the z component of the angular momen-tum operator (in units of A). Finally it should be notedthat the exciton Hamiltonian [Eq. (3)] is in dimensionlessform where the energies have been expressed in terms of%+ and the lengths in terms of effective Bohr radius;

epAa+=p+e

We now calculate the energy of the ground state of theHamiltonain described by Eq. (3) following a variationalapproach. We use the following trial wave function inwhich the one-dimensional square-well functions are ex-plicitly factored,

0'=f, (z, )fh(zh )G(p, z, g),where

&zf, (z, ) = Ae 'sin

e(1 la)

hole band and the lower sign to the light-hole band. Weuse a cylindrical gauge and defined vector potential A as

A= —,'(Bxr) .

H —— — p +1a a 1a'p ap ap p

p+ a(m, ), Bz,'

pz„fh(zh ) =Be "sin (11b)

p+ a 22 +rL. +4l—r—p +V,.(z, )+Vh. (zh) .

m+ azg2

(3)

1 1+ (ri+r2)(m, ), mo

(4)

Here m+ is the heavy- (+) or light - (—) hole mass along

the z direction, and p+ is the reduced mass correspondingto the heavy- (+) or light- ( —) hole bands in the planeperpendicular to the z axis. Both p+ and m+ are ex-pressed in terms of Kohn-Luttinger parameters' y &

and

y2 asE=(q iHiq ) l(q ie) (13)

and finally the binding energy of the exciton is given as

and G(p, z, g) describes the internal motion of the exci-ton. For G(p, z, g) we choose the following form:

G(p, z, g) =Ce (12)

Here A, 8, and C are normalization constants and a, 13,

5, and k are the variational parameters to be determined.The parameters a and P refiect the pileup of the electronand hole wave functions at the interface due to Coulombinteraction. The eigenvalue of the Hamiltonian is deter-mined by minimizing

and Eb =E,+Eh+(r+ E)%+, — (14)1 (r, +2rz)

mp(5) where

2

where mp is the free-electron mass.In these equations, the upper sign refers to the heavy- 2(m, )i L,

(15a)

BRIEF REPORTS 10 915

'2m

2(m~ }( L~

and y+ is the energy of the first Landau level.

RESULTS AND DISCUSSION

(15b)

We have calculated the values of the binding energies(E~) of the heavy-hole and light-hole exciton as a func-

tion of A1As-layer thickness (or GaAs-layer thickness) forseveral values of magnetic field using infinite potentialbarriers. The values of the various physical parameterspertaining to the GaAs-AlAs system used in our calcula-tions are (m, )&

= 1. lmo, (m, ), =O. 19mo, y, =6.93,ye=2. 15, ' co= 12.3. The values of the heavy-hole massm+ and the light-hole mass m, obtained using thevalues of y, and y2 are m+ =0.38mo, m =0.11mo.

20—

18— 2O L18—

16—

14—

8

200 kGI

I150 kGI

I100 kGI

16—

14—

I200 kGI

I150 kGI

I100 kGI

10—Iso kGI 12—

I50 kGI

Io kG(

10—

10 kGI

6—I

30I

40I I I I

50 60 70 80AIAs-LAYER THICKNESS (k)

I

90I

100

8—I

30 40I I I I

50 60 70 80AIAs-LAYER THICKNESS (A)

I

90I

100

20— 20—

18— HEAVY HOLES18—

16—

14—

8

10—

I200 kGI

I150 kGI

I100 kGI

I50 kGI

14—

12—

10—

200 kGI

150 kGI

I100 kGI

I50 kGI

6—30

I

40I I I I

50 60 70 80AIAs-LAYER THICKNESS (k)

I

90

Io kG(

I

100

8—I

30 40I I I I

50 60 70 80AlAs- LA YER THICKNESS (k)

I

90

Io kGI

I

100

FIG. 2. Variation of the binding energy (E& ) of a heavy-holeexciton as a function of AlAs-layer thickness for several valuesof the magnetic field for GaAs-layer thickness (a) 20 A and (b)28 A.

FIG. 3. Variation of the binding energy (E&) of a light-holeexciton as a function of AlAs-layer thickness for several valuesof the magnetic field for GaAs-layer thickness (a) 20 A and (b)28 A.

10 916 BRIEF REPORTS

The reduced mass in the x-y plane for the heavy-hole ex-citon is p+ =0.07m 0 and p =0. 1m 0 for the light-holeexciton. The value of p+ is less than that of p due tothe anisotropic nature of the kinetic-energy expressionsin the diagonal terms of the Kohn-Luttinger Hamiltonianof an exciton. '

In Fig. 2(a) we display the variation of the binding en-

ergy E~ of the heavy-hole exciton as a function of theA1As-layer thickness for a given GaAs-layer thickness of20 A for several values of the magnetic field. We findthat for a given value of the magnetic field the value ofE~ increases as the A1As-layer thickness is reduced, asexpected due to greater confinement. And for a givenvalue of A1As-layer thickness the value of E~ increases asthe magnetic field is increased. This is due to the factthat the application of the magnetic field compresses theexcitonic wave function leading to more binding. In Fig.2(b) we display a similar variation for the GaAs-layerthickness of 28 A. Again the behavior of Ez as a func-tion of AlAs-layer thickness and magnetic field is essen-tially similar to that displayed in Fig. 2(a). In Figs. 3(a)and 3(b) we display the variation of Ez for the light-holeexciton as a function of A1As-layer thickness for twodifferent values of GaAs-layer thickness, i.e., 20 and 280

A, respectively, for various values of the magnetic field.Again the behavior of E~ for the light-hole exciton as afunction of layer thickness and magnetic field is essential-ly similar to that for the heavy-hole exciton. It should bepointed out that our values of Ez both for the heavy-holeexciton and for the light-hole exciton for zero magneticfield are larger than those obtained by Duggan andRalph using the same physical parameters. This is dueto the fact that their expression of G(p, z, g) is propor-tional to e l' and thus has only one variational parame-ter.

Recently Hodge et aI. ' have measured the energy of1s —+2p+ transition of a heavy-hole exciton in type-IIGaAs-A1As quantum-well structures in the presence of amagnetic field using photoinduced far-infrared absorptionspectroscopy. Even though we use infinite potential bar-riers in our calculation of Ez and do not calculate the en-

ergy of the 2p+ level it is still possible to compare our re-sults approximately with those of Hodge et a/. ' We findthat the shift in energy of the 1s~2p+ transition withmagnetic field is comparable to what we calculate, as theenergy of the 2p+ level is small at the value of the mag-netic field (-45 kG) used by them. Calculations of theenergy levels of excitons in type-II GaAs-A1As quantumwells using finite barriers in the presence of a magneticfield are in progress.

CONCLUSIONS

We have calculated the binding energies of both theheavy-hole and the light-hole excitons in type-II GaAs-A1As quantum-well structures as a function of the size ofthe AlAs layer (or the GaAs layer) in the presence of amagnetic field applied parallel to the direction of growth.We have used a variational approach and assumedinfinite potential barriers. For a given set of values ofA1As- and GaAs-layer thicknesses, the binding energiesof excitons are found to increase with increasing magnet-ic field.

ACKNOWLEDGMENTS

The authors are thankful to Dr. Spiros Branis for use-ful discussions. This work was supported by the AirForce Oftice of Scientific Research under Grant Nos.AFOSR-91-0056 and AFOSR-90-0118.

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