Physica B 315 (2002) 240–246

Spectra of excitons in quantum dots under a magnetic field

Wenfang Xie

Department of Physics, Guangzhou University, Guangzhou 510405, People’s Republic of China

Received 7 August 2001; received in revised form 26 November 2001

Abstract

An interacting system of two-layer spatially separated electrons and holes quantum dots is considered in the presence

of external magnetic fields. The energies of the low-lying states of the exciton in two-layer quantum dots are calculated

for the different projections of the total angular momentum as a function of the applied magnetic field in the effective-

mass-approximation by using the method of few-body physics. We also calculate the binding energies of the ground and

some excited states of excitons in a disk-like quantum dot for different values of the magnetic field. r 2002 Elsevier

Science B.V. All rights reserved.

PACS: 73.20.Dx; 71.35.Gg; 78.66.Li

Keywords: Exciton; Quantum dot; Energy spectrum

1. Introduction

The optical properties of semiconductor quan-tum dots (QDs) are of interest both because oftheir potential for application in optoelectronicdevices and because of the insight they provideinto the nature of confined electrons and holes.Excitons dominate the optical properties of thesestructures, typically producing sharp absorptionand luminescence lines. Two factors are respon-sible for the properties of the exciton in a QD. Thefirst is confinement of the electron and the hole bythe QD. The second is the Coulomb interactionbetween the electron and the hole. Confinementcan be controlled through the size and shape of theQD as well as by the selection of structure andbarrier materials to produce various band offsets.

Confinement localizes the electron and hole,enhancing exciton-binding energies. The Coulombinteraction is controlled by the static dielectricconstant of the QD material and produces excitonbinding. Both factors significantly influence theenergy of a confined exciton.

Many theoretical studies have been devoted toexciton states in microcrystals [1–21]. Most ofthem are related to variational studies of theexcitonic ground state in spherical dots or tocalculations in some limiting cases [1–9,18,19].Typically, infinite barriers are considered. A fullnumerical analysis of this problem is carried outby Hu and co-workers [10,11], expanding theexcitonic wave functions in terms of solutions ofthe single-particle Schr .odinger equations. Thereare also papers studying nonspherical dots, e.g.,boxes [12], square flat plates [13], and cylindricalQDs [14,15]. Effects such as dielectric confinementE-mail address: gzxiewf@public.guangzhou.gd.cn (W. Xie).

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 1 4 - 8

and electron–hole exchange interaction on exci-tonic states in semiconductor QDs are also studied[16]. Recently, Marfn et al. performed variationalcalculation of the ground state energy of excitonsconfined in spherical QDs with a finite-heightpotential wall [20].

In contrast to the various studies on the excitonstates in a spherical QD, theoretical results relatedto the quantum confinement effect on excitonstates in a disk-like QD are still rare. However, inexperimentally realized QDs, the motion in the z-direction is always frozen out into the lowestsubband. Since the corresponding extent of thewave function is much less than that in the xy-plane, we can treat the QDs in the two-dimen-sional limit of thin disks. What about the quantumconfinement effect and character of exciton-bind-ing energy in such QDs? Xie calculated theenergies of the low-lying states of an exciton ofa disk-like QD with a parabolic potential by usingthe method of few-body physics in the absence ofa magnetic field [21].

On the other hand, external perturbation ofa system, such as the application of an electric ormagnetic field, can provide valuable informationabout the exciton. The purpose of this paper isto present model calculations of excitons ina structure consisting of two strictly parallelQDs, with the electron being confined in one QDand the hole being confined in another paralleldot. Recently, the properties of two-dimensionalsystems with spatially separated electrons andholes in a strong magnetic field have attractedconsiderable interest [22–26]. Hence, in the presentwork we will use the method of few-body physicsto study the low-lying states of the exciton systemwith an electron and a hole in two-layer spatiallyseparated QDs as a function of magnetic field andthe dot radius.

2. Calculation method

Consider a double-layer QD with a spatiallyseparated electron and a hole. We assume that theelectron and hole layers are infinitesimally thin,and the layers are separated by a distance d in thez-direction along which an external magnetic field

B is applied. With the effective-mass approxima-tion, the Hamiltonian is given by

H ¼1

2mne

ð~ppe þe

c~AAeÞ

2 þ1

2mn

eo2e0r

2e

þ1

2mnh

~pph �e

c~AAh

� �2

þ1

2mn

ho2h0r

2h

�e2

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij~rre �~rrhj

2 þ d2p ð1Þ

mne ; mn

h are the effective electron and hole masses;~AAe; ~AAh are the vector potentials in electron andhole location, respectively; ~ppe; ~pph; ~rre and ~rrh are,respectively, the momenta and positions of theelectron and the hole; e is the dielectric constant ofthe medium that the electron and the hole aremoving in; oe0 and oh0 are, respectively, theconfinement frequencies of the electron and thehole. With the symmetric gauge for the magneticfield ~AA ¼ ðB=2Þð�y;x; 0Þ; the Hamiltonian thenreads as

H ¼p2e

2mne

þ1

2mn

eo2er

2e þ

p2h

2mnh

þ1

2mn

ho2er

2h

þoce

2cez �

och

2chz þ

1

2mn

hðo2h � o2

eÞr2h

�e2

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij~rre �~rrhj

2 þ d2p ; ð2Þ

where oe ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

e0 þ o2ce=4

q; oh ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

h0 þ o2ch=4

q;

oce ¼ eB=ðcmne Þ and och ¼ eB=ðcmn

hÞ stand for thecyclotron frequencies of the electron and the hole,respectively. cez and chz are the orbital angularmomenta along the z-direction of the electron andthe hole.

Introducing, as usual, the center of masscoordinate and the relative coordinate of theelectron–hole pair defined by

~RR ¼mn

e~rre þ mnh~rrh

mne þ mn

h

; ð3Þ

~rr ¼~rre �~rrh; ð4Þ

and then substituting into Eq. (2), we obtain

H ¼ H0 þ Veff

W. Xie / Physica B 315 (2002) 240–246 241

with

H0 ¼p2

R

2Mþ

1

2Mo2

eR2 þp2

r

2mþ

1

2mo2

er2

þoce

2cez �

och

2chz;

Veff ¼1

2mn

hðo2h � o2

eÞr2h �

e2

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ d2

p ; ð5Þ

where the reduced mass m and that of the center-of-mass M are given by

1

m¼

1

mne

þ1

mnh

; ð6Þ

M ¼ mn

e þ mn

h: ð7Þ

Owing to the cylindrical symmetry of theproblem, the exciton wave function can be labeledby the total orbital angular momentum L: Toobtain the eigenenergies and eigenstates, H isdiagonalized in model space spanned by transla-tional invariant harmonic product states

F½K� ¼ ½joe

n1c1ðrÞjoe

n2c2ðRÞ�L; ð8Þ

where jonc is a 2D harmonic oscillator state with a

frequency o and an energy ð2n þ jcj þ 1Þ_o [27],and [K] denotes the set quantum numbersðn1; c1; n2; c2Þ in brevity, c1 þ c2 ¼ L is the totalorbital angular momentum. In practice, o servesas a variational parameter around oe to minimizethe eigenenergies. The matrix elements of H arethen given by the following expressions:

/F½K�jH0jF½K 0 �S

¼ 2ðn1 þ n2Þ þ jc1j þ jc2j þ 2½ �_oe þoce

2cez

n

�och

2chz

od½K �½K 0 �; ð9Þ

/F½K�jVeff jF½K 0�S

¼ �U In1n0

1dc1c01dn2n0

2dc2c02 þ

1

2mn

hðo2h � o2

eÞ

X

½K 00 �½K 000 �

B½K �½K 00 �B½K 0 �½K 000 �UIIn001n0001dc001c0001

dn002n0002dc002c0002 ð10Þ

with

U In;n0 ¼

ZN

0

RncðrÞe2

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ d2

p Rn0cðrÞr dr; ð11Þ

U IIn;n0 ¼

ZN

0

RncðrÞr2Rn0cðrÞr dr; ð12Þ

B½K �½K 0 � ¼Z

F½K �ð~RR;~rrÞF½K 0 �ð~RR0;~rr0Þ d~RR d~rr; ð13Þ

where Rnc is the radial part of two-dimensionalharmonic oscillator function, B½K �½K 0 � is the trans-formation bracket of two-dimensional harmonicproduct states with two different sets of relativecoordinates for three-body systems, which allowsus to reduce the otherwise multi-integral intosingle-integral. Nonvanishing B½K �½K 0 � occurs onlywhen both the states F½K �ð~RR;~rrÞ and F½K 0 �ð~RR0;~rr0Þhave exactly the same eigenenergy and eigen-angular momentum. An analytical expression forB½K �½K 0 � has already been derived in Ref. [28]. Theset of canonical coordinates f~RR0;~rr0g is defined by~RR0 ¼~rre; and ~rr0 ¼ �~rrh: The accuracy of solutionsdepends on how large the model space is. Thedimension of the model space is constrained by0p2ðn1 þ n2Þ þ jc1j þ jc2jp20: If N is increased by2, the ratio of the difference in energy is less than0:01%:

3. Numerical results

In this section we apply the method developedabove to discuss the low-lying state energies andthe binding energy of an exciton in QDs. In thefollowing, the units of energy and length will betaken as meV and nm, respectively. We take theelectrons and holes to exist in isotropic bands withmasses mn

e ¼ 0:067me ðme is the bare mass of singleelectron) and mn

h ¼ 0:099me in both the QDs andin the barrier materials, and we take the back-ground dielectric constant to be e ¼ 12:5 in eachmaterial, which are relevant to GaAs. For thisdiscussion we have defined the dot radius R ¼ffiffiffiffiffiffiffiffiffiffiffiffi

_=mop

as the characteristic length associatedwith the lateral confining potential, and confininglengths Re ¼ Rh ¼ R ¼ 45 nm yield confining

W. Xie / Physica B 315 (2002) 240–246242

energies _oe0 ¼ 0:277 meV; _oh0 ¼ 0:188 meV; forelectrons and holes.

We took R ¼ 45 nm and two different values ofd; i.e., a disk-like QD, to calculate the energy

spectrum of the low-lying states ðLp3Þ of anexciton as a function of the magnetic field(Figs. 1). From Fig. 1a, we find that the L ¼ 1excitons are especially interesting as these may beexcited in infrared spectroscopy as well as in two-photon spectroscopy. Both these phenomena haverecently received some experimental attention. Forthe L ¼ 0 state, we observe that the energy of theexciton increases in absolute value for the strengthof magnetic fields up to Bmin ¼ 2:1 T; where itreaches a minimum value. It is obvious that thisminimum value position is dependent on theangular momentum of the exciton, e.g., for L ¼1 and 2 states they, respectively, appear at Bmin ¼1:7 and 1:5 T: It is the competition between thesingle particle energy, Zeeman energy, and Cou-lomb interaction energy that finally determines theenergies of low-lying states of an exciton in QDs.The stronger the confinement in QDs, the higher isthe single particle energy. On the other hand, wehave known that the orbit radius of the electronand the hole in a QD is proportional to the dot sizeand the quantum number of angular momentum.When the strength of the magnetic field increases,the dot size will decrease, the confinement willincrease, the spatial overlap between an electronand a hole is increased, leading to an increase inthe Coulomb binding energy. In addition, whendot size decreases, the rotational inertia of excitonsreduces and the rotational energy of excitonsincreases. Hence, the exciton energies are propor-tional to the total orbital momentum L and theminimum value position decreases with increasingL: For BoBmin and with increasing B the excitonenergy decreases due to the fact that the increase ofthe Coulomb interaction energy is higher than thatof the single particle energy. At B ¼ Bmin they areequal. For B > Bmin and with increasing B theexciton energy increases with B; which is due to thefact that the increase of the Coulomb interactionenergy is less than that of the single particleenergy; in a small-size QD, the increase in thesingle energy becomes predominant and cannot becompensated for by the increase of electron–holeinteraction. From Fig. 1b, we note that theenergies are higher than that in Fig. 1a and theminimum Bmin position decreases. This physicalorigin is such that the Coulomb binding energy

Fig. 1. The energy spectrum of the four lowest exciton states in

QDs as a function of the external magnetic field is plotted for

R ¼ 45 nm (a) d ¼ 0; (b) d ¼ 1:0 nm: The levels are labeled by

the total angular momentum L:

W. Xie / Physica B 315 (2002) 240–246 243

diminishes with increasing d so that Bmin takes asmaller value.

The exciton-binding energy Eb is defined by

Eb ¼ EX � EðeÞ � EðhÞ; ð14Þ

where EX is the energy of an exciton, EðeÞ and EðhÞare the energies of an electron and a hole in theQD in the presence of external magnetic fields,respectively. In Fig. 2 we present the bindingenergy of the L ¼ 0 (Fig. 2a), L ¼ 1 (Fig. 2b) and

Fig. 2. Binding energy of an exciton in QDs as a function of the dot radius R which is plotted for different values of the magnetic field

with d ¼ 0 (a) L ¼ 0; (b) L ¼ 1; (c) L ¼ 2:

W. Xie / Physica B 315 (2002) 240–246244

L ¼ 2 (Fig. 2c) states as a function of the QDradius and for different values of the appliedmagnetic field. From Fig. 2a, we observe that inthe absence of magnetic field, the binding energydecreases with increasing R: For Ba0 and forsmall values of the dot radius, Ro20 nm; thebinding energy of the ground state decreases withincreasing R: For R > 20 nm the effect of themagnetic field begins to be apparent, the bindingenergy being approximately constant due to thefact that the single particle energy and theinteraction energy are also constants for theseranges of the radius. At zero magnetic field, it isinteresting to calculate the binding energy of theground state of excitons and find that it tends to4:208 meV at a much larger dot radius R: Thisresult agrees with those of Refs. [29,30] inquantum wells and is larger than that ofRef. [31] in the bulk semiconductors. In Figs. 2band c we can see that the binding energy of the twostates increases with the magnetic field. Also, it isseen that in the absence of magnetic field, thebinding energy of the two states increases as theradius decreases and reaches a maximum beforethe characteristic radius Rc: The characteristicradius raises a larger value with increasing L: ForQD radius R > Rc; the binding energy increaseswith decreasing R and for RoRc; the bindingenergy decreases with decreasing R: The physicalorigin is such that the increase in the rotationenergy becomes predominant and cannot becompensated for by the increase of electron–holeinteraction for a smaller dot radius in the La0case. In the presence of magnetic field, we find thatat the beginning, the binding energies increase fastwith increasing R and then they tend to someapproximately constant energies due to the effectof the magnetic fields. On the other hand, fromFigs. 2b and c, we find that when dot radius tendsto 10 nm; the binding energies drop sharply tozero. This physical origin is such that the Coulombinteraction increase cannot compensate for thesharp increase of the kinetic energies of theelectron and the hole in an excited state case whendot radius is nearly equal to 10 nm:

The calculated results have shown that theenergy and the binding energy of low-lying statesof an exciton in QDs are strongly dependent on the

confinement, i.e., the strength of magnetic fieldand the QD radius. This will be useful forunderstanding the electronic properties in quasi-zero-dimensional systems and for designing somedevices in the future.

Acknowledgements

This work is supported by the National NaturalScience Foundation of China under Grant No.19975013 and by the Backbone Teacher Founda-tion of the Universities in China.

References

[1] Al.L. Efros, A.L. Efros, Fiz. Sov. Phys. Semicond. 16

(1982) 772.

[2] L.E. Brus, J. Chem. Phys. 80 (1984) 4403.

[3] A.I.E. Kimov, A.L. Efros, A.A. Onushchenko, Solid State

Commun. 56 (1985) 921.

[4] H.M. Schmidt, H. Weller, Chem. Phys. Lett. 129 (1986)

615.

[5] S.V. Nair, S. Sinha, K.C. Rustagi, Phys. Rev. B 35 (1987)

4098.

[6] Y. Kayanuma, Phys. Rev. B 38 (1988) 9797.

[7] J.B. Xia, Phys. Rev. B 40 (1989) 8500.

[8] Y. Kayanuma, H. Momiji, Phys. Rev. B 41 (1990) 10261.

[9] D.B. Tran Thoai, Y.Z. Hu, S.W. Koch, Phys. Rev. B 42

(1990) 11261.

[10] Y.Z. Hu, S.W. Koch, M. Lindberg, N. Peyghambarian,

E.L. Pollock, F.F. Abraham, Phys. Rev. Lett. 64 (1990)

1805.

[11] Y.Z. Hu, M. Lindberg, S.W. Koch, Phys. Rev. B 42 (1990)

1713.

[12] R. Del Sole, A. D’Andrea, in: H. Haug, L. Banyai (Eds.),

Optical Switching in Low-Dimensional Systems, Plenum,

New York, 1989, p. 289.

[13] G.W. Bryant, Phys. Rev. Lett. 59 (1987) 1140;

G.W. Bryant, Phys. Rev. B 37 (1988) 8763;

G.W. Bryant, Phys. Rev. B 41 (1990) 1243.

[14] Y. Kayanuma, Phys. Rev. B 44 (1991) 13085.

[15] S. Le Goff, S. St!eb!e, Phys. Rev. B 47 (1993) 1383.

[16] T. Takagahara, Phys. Rev. B 47 (1993) 4569.

[17] S.W. Koch, Y.Z. Hu, B. Fluegel, N. Peyghambarian,

J. Crystal Growth 117 (1992) 592.

[18] A.L. Ekimov, F. Hache, M.C. Schanne-Klein, D. Ricard,

C. Flytzanis, I.A. Kudryavtsev, T.V. Yazeva, A.V.

Rodina, Al.L. Efros, J. Opt. Soc. Am. B 10 (1993) 100.

[19] L.M. Ramaniah, S.V. Nair, Phys. Rev. B 47 (1993) 7132.

[20] J.L. Marfn, R. Riera, S.A. Cruz, J. Phys.: Condens. Matter

10 (1998) 1349.

[21] W.F. Xie, Physica B 279 (2000) 253.

W. Xie / Physica B 315 (2002) 240–246 245

[22] Yu.E. Lozovik, A.M. Ruvinsky, Phys. Lett. A 227 (1997)

271.

[23] Yu.E. Lozovik, A.M. Ruvinsky, Phys. Scripta 58 (1998)

90.

[24] L.V. Butov, A. Zrenner, G. Abstreiter, A.V. Petinova,

K. Eberl, Phys. Rev. B 52 (1995) 12153.

[25] A.B. Dzyubenko, A.L. Yublonskli, Phys. Rev. B 53 (1996)

16355.

[26] M. Bayer, V.B. Timofeev, F. Faller, T. Gutbrod,

A. Forchel, Phys. Rev. B 54 (1996) 8799.

[27] X. Wenfang, R. Wenying, Commun. Theor. Phys. 29

(1998) 531.

[28] W.Y. Ruan, J. Math. Phys. 37 (1996) 3760.

[29] M. de Dios-Leyva, A. Bruno-Alfonso, A. Matos-Abiague,

L.E. Oliveira, J. Phys.: Condens. Matter 9 (1997) 8477.

[30] B. Gerlach, J. Wusthoff, M.O. Dzero, M.A. Smondyrev,

Phys. Rev. B 58 (1998) 10568.

[31] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors,

Springer, Berlin, 1995, p. 271.

W. Xie / Physica B 315 (2002) 240–246246