Available online at www.sciencedirect.com

Physica E 16 (2003) 237–243

www.elsevier.com/locate/physe

Exciton absorption in quantum-well wiresunder the electric %eld

E. Kasapoglua ;∗, H. Saria, M. Bursala, I. S.okmenb

aPhysics Department, Cumhuriyet University, 58140 Sivas, TurkeybPhysics Department, Dokuz Eyl ul University, Izmir, Turkey

Received 12 July 2002; received in revised form 30 September 2002; accepted 8 October 2002

Abstract

The binding energy of excitons, and interband optical absorption in quantum-well wires of GaAs surrounded by Ga1−xAlxAsis calculated in e9ective-mass approximation, using a variational approach. Results obtained show that the exciton bindingenergies, and optical absorption depend on the sizes of the wire and strength of the electric %eld. The additional con%nementof particles in quantum well wire o9ers greater variety of electric %eld dependence in comparison to bulk materials andquantum well structures.? 2002 Elsevier Science B.V. All rights reserved.

PACS: 71.35.−y; 73.21.Hb

Keywords: Quantum well-wires; Electric %eld; Exciton absorption

1. Introduction

There is a great deal of interest in quasi-one-dimen-sional structures, whose physical properties have moreadvantages in comparision to those observed in three-and two-dimensional structures [1–12]. The reductionof the dimension leads to new physics and to phe-nomena with potential optoelectronic device appli-cations. One- and zero-dimensional devices are theresults of the natural evolution of growth technology;controlled growth has allowed us to easily obtain 2Dmaterials and devices [13,14]. Due to the increasein the number of con%nement directions, which re-sults in the con%nement of carriers to regions that are

∗ Corresponding author.E-mail address: ekasap@cumhuriyet.edu.tr (E. Kasapoglu).

smaller than the characteristic size in bulk materi-als, one-dimensional systems exhibit a rich varietyof enhanced optical properties relative to those ofthree-dimensional systems [15–19]. The important andunusual nature of the subband structure and the elec-tronic properties for a one-dimensional system hasbeen summarized by Li et al. [20]. The progress incrystal growth and the fabrication techniques to ob-tain low-dimensional systems has inspired numerousstudies of optical and transport properties, electronicstructure and excitonic binding [21–29]. As most ofthe calculations have by far been performed for theexcitonic binding in the e9ective-mass approxima-tion within the variational approach, the theoreticalinvestigations of the excitonic binding in quantumwires involve quite similar subjects as in the caseof quantum wells. Thien calculated the ground statebinding energy of an exciton in a quantum-well wire

1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S1386 -9477(02)00671 -9

238 E. Kasapoglu et al. / Physica E 16 (2003) 237–243

subjected to an external electric %eld perpendicular tothe wire axis for the case of the parabolic con%nementpotential [29].The optoelectronic devices depend upon the inter-

actions of photons or electromagnetic %elds with semi-conductors. It is therefore essential to understand theelectroabsorption in semiconductors. Recently, opticalmeasurements on quantum-well wires have been madeand important excitonic features have been observedin the spectra [16, and references therein]. The ad-ditional exciton con%nement in band-gap-engineeredstructures gives rise to an increased oscillator strengthfor the lowest-energy exciton transition. The interbandoptical absorption involving valance and conductanceband states is, of course, important for optical devicessuch as lasers and detectors. Consequently, excitonsplay an important role in the optical absorption andthus the symmetry dependence of the excitons andoptical absorption under external electric %eld is in-teresting. In the case of the excitonic absorption thee9ects of several factors on excitonic spectra havebeen investigated. In this study we are particularlyinterested in external %eld and the symmetry of theQWW for the enhancement excitonic binding and op-tical absorption. In this paper, we calculate the groundstate exciton binding energy in a %nite quantum-wellwire of GaAs surrounded by Ga1−xAlxAs under theelectric %eld in the e9ective mass approximation, us-ing variational approach. We also calculated the linearabsorption coeIcient of the ground state exciton fordi9erent electric %eld values as a function of the wiredimensions.

2. Theory

2.1. Exciton binding energy

We consider a rectangular quantum-well wire whichconsists of GaAs surrounded by Ga1−xAlxAs as shownschematically in Fig. 1. The axis of wire is y. Elec-tric %eld F is applied along the z-direction. Usually,the microstructures designed for Stark e9ect experi-ments are insulating. The e9ective mass approxima-tion is used in constructing the Hamiltonian for theinteracting electron–hole pair that forms the exciton.The Hamiltonian of the single electron–hole system

y

z

x

Lx

F

Fig. 1. The schematic representation of the quantum-well wire.

is given by

H =p2ye

2me+

p2yh

2mh+ H e

x (xe) + H hx (xh)

+H ez (ze) + H h

z (zh) + Vcoul(r); (1)

where H ex (xe) and H h

x (xh) are the one-dimensionalHamiltonian for electrons and holes in the x-direction:

H ex (xe) = p2

xe=2me + V (xe);

H hx (xh) = p2

xh=2mh + V (xh); (2)

H ez (ze) and H h

z (zh) are the one-dimensional Hamilto-nian for electrons and holes in the z-direction;

H ez (ze) = p2

ze=2me + V (ze)− eFze;

H hz (zh) = p2

zh=2mh + V (zh) + eFzh ; (3)

where V (xe(h)) and V (ze(h)) are the con%nement poten-tials for electrons and holes in the x and z-directions,respectively, Vcoul(r) is the Coulomb potential be-tween electrons and holes, including an e9ectivedielectric constant for the system:

VCoul =− e2

0 |re − rh| ; (4)

where re and rh are the electron and hole positions. Itis convenient to introduce the center of mass coordi-nate R= (mere +mh rh)=(me +mh), the relative coor-dinate (r = re − rh) and by scaling all lengths in the

E. Kasapoglu et al. / Physica E 16 (2003) 237–243 239

exciton Bohr radius (aB = 0˝2=�e2), and energies inthe exciton Rydberg (Ry = �e4=2 20˝2), we can writethe dimensionless Hamiltonian of the exciton as

H ex =− �me

929x2e

+ V (xe)− �mh⊥

929xh

+ V (xh)

− �me

929z2e

+ V (ze)− F ze − �mh⊥

929z2h

+ V (zh)

+F zh − 929y 2 − 2√

(xe − xh)2 + y 2 + (ze − zh)2;

(5)

where F is the dimensionless electric %eld: F =eFaB=Ry; me, is the electron conduction-band e9ec-tive mass which is isotropic, mh⊥ is the heavy-holevalance-band e9ective mass perpendicular to the axisof the wire, � is the heavy-hole exciton reduced massin the free direction. In terms of the Luttinger para-meters, the expressions for reduced mass in the freedimension and the hole mass are

1�=

1me

+1

mh‖;

1mh⊥

=(�1 − 2�2)

m0;

wherem0 is the free-electron mass, �1=7:36; �2=2:57,and m−1

h‖= (�1 + �2)m−1

0 is the heavy-hole e9ectivemass parallel to the axis of the wire.The Schr.odinger equation of the excitonic struc-

ture is

H exc�(xe; xh ; ze; zh ; y; �) = E�(xe; xh ; ze; zh ; y; �); (6)

where E is the total energy and y = y e − y h. Forthe Schr.odinger equation, we take the following trialwave function [2,17,30,31]

�(xe; xh ; ze; zh ; y; �)

=�e(xe)�h(xh)�e(ze)�h(zh)gt(y; �) (7)

where wave function in the y-direction gt(y; �) ischosen to be Gaussian-type orbital function

gt(y; �) =1√�

(2�

)1=4

e−y 2=�2 (8)

in which � is a dimensionless variational parameter.�e(xe) and �h(xh) are the wave functions of elec-tron and hole which are exactly obtained from theSchr.odinger equation in the x-direction. The wave

functions �e(ze) and �h(zh) represent the motion ofthe electron and hole in the z-direction and are Airyfunctions. Our experience with variational calcula-tions of the excitonic binding energy in quantumwells and quantum wires suggests that very simpleGaussian-type function gives quite accurate results inthe case of moderate and strong electric %elds [30].The ground state exciton binding energy is given by

EB = Eex + Eh

x + Eez + Eh

z −min�

〈�|Hexc|�〉

=− 1

�2+

2

�

√2�

∫ +∞

−∞|�e(xe)|2 dxe

×∫ +∞

−∞|�h(xh)|2 dxh

∫ +∞

−∞|�e(ze)|2 dze

×∫ +∞

−∞|�h(zh)|2 dzh

×∫ +∞

−∞

e−2y 2=�2√(xe − xh)2 + (y)2 + (ze − zh)2

dy;

(9)

where Eex; E

hx and Ee

z ; Ehz are the lowest electron and

hole subband dimensionless energies in the x andz-directions, respectively.

2.2. Exciton absorption

The general expression for the linear absorptioncoeIcient in the dipole approximation is givenby [32]

�(!) =1V

4�2e2

ncm20!

×∑if

|〈i| P|f〉|2 ( f − i − ˝!); (10)

where ! is the angular frequency of the incident pho-ton, V is the volume of the sample, n is the refrac-tion index of the material, is the polarization vectorof the incident radiation, c is the speed of the light,P is many-electron momentum operator, |i〉, and |f〉are initial and %nal state of system, respectively. ForHH1 → E1 excitonic transitions in a QWW the dis-persion relations of HH1 and E1 are taken as parabolic

240 E. Kasapoglu et al. / Physica E 16 (2003) 237–243

upon ky, and the one electron wave functions are

F (h)ky; v = Uhh(r)�h(zh)�h(xh)

1√Ly

eikyvy;

F (h)ky; c = Uc(r)�e(ze)�e(xe)

1√Ly

eikycy; (11)

where Uhh and Uc are the heavy hole and electronperiodic parts of the Bloch function at the zone centerof the host materials. The initial and %nal states areSlater determinants [33]. The initial state |i〉 is uniqueand corresponds to the N electrons that occupy theHH1 subband. For the %nal state |f〉, we use a methodthat is parallel to the Bastard’s developments for ex-citons in a two-dimensional quantum well [33]. The%nal state in the form of a wave packet summed overSlater determinants for N − 1 electrons in the valanceband and one electron in the conduction subband. Thuswe obtain for QWW

|〈i| p|f〉|2 = |〈Uhh| p|Uc〉|2|�h(zh)|�e(ze)〉|2

×|〈�h(xh)|�e(xe)〉|2

× kyc−kyv ;0Ly|gt(y = 0)|2; (12)

where �’s are the normalized eigenfunctions, Ly isthe length of the wire (free dimension), and Ky =ky;c−ky;v is the di9erence between the conduction andvalance band electron wave vectors. The Kronecker function dictates that, in the exciton absorption, theelectron momentum in the y-direction is conserved.The appearance of the term |gt(y=0)|2 in absorptioncoeIcient means that, only excitons with non-zeroamplitude at y = 0 can absorb the light.By examining the matrix element square for light

polarized along various orientations, we see that thez-polarized light has no coupling to the HH states.From the symmetry we see that, the allowed opti-cal transitions between the HH1 and E1 subbands aregiven by

|〈Uv|Px|Uc〉|2 = |〈Uv|Py|Uc〉|2 = m0

4Ep; (13)

where the quantity Ep is the Kane matrix element.Thus, exciton absorption coeIcient in Eq. (10) isreduced to

�(!) =1

LxLz�

√2�e2Ep

ncm0!

×|〈�i(xh)|�f (xe)〉|2|〈�i(zh)|�f (ze〉|2

⊗ '=2(Ee

1x+Eh1x+Ee

1z+Eh1z+Eg−EB−˝!)2+('=2)2

;

(14)

where ' is the broadening parameter obtainedby replacing the function in Eq. (10) by aGaussian-shaped function, ˝! is photon energy re-quired to excite the system from its ground state.

3. Results and discussion

In the values of the physical parameters used inour calculations are me = 0:067m0; mh = 0:45m0; �=0:04m0 (where m0 is the free electron mass), 0 =12:5 (static dielectric constant is assuming to be sameeverywhere), Ve(xe) = Ve(ze) = 228 meV; Vh(xh) =Vh(zh) = 176 meV. These parameters are suitable inGaAs=Ga1−xA1xAs heterostructures with an Al con-centration of x ∼= 0:3. We have assumed the conduc-tion band discontinuity to be 56% of the total bandgap di9erence between GaAs and Ga1−xAlxAs; Eg =1:424 eV; Ep = 25:7 eV and ' = 1 meV.In Fig. 2 the variation of the exciton binding

energy is given as a function of the symmetric wire

0 100 200 3000

4

8

12

16

20

EB

(m

eV)

Lx = Lz (Å)

F=10kV/cm

F=50kV/cm

F=100kV/cm

F=0

Fig. 2. The variation of the exciton binding energy as a functionof the wire dimensions Lz; Lx , for di9erent electric %eld values.

E. Kasapoglu et al. / Physica E 16 (2003) 237–243 241

dimensions for di9erent electric %eld values. It is seenthat for large Lx and Lz values the binding energytends to its three-dimensional value (∼4 meV). Bydecreasing wire dimensions the probability of %ndingthe electron and hole in the same plane increases andthe binding energy approaches to that of quasi-onedimensional excitonic structure. For further small Lxand Lz values the electron and hole penetrates intothe barriers thus the overlap function gets smaller andbinding energy begins to decrease rapidly. We can seefrom Fig. 2 that, the %eld dependence of the bind-ing energy in narrow wire (L6 100 RA) is very weak,since the geometric con%nement is predominant. Butin the wider quantum well wires, the binding energyis more sensitive to the external electric %eld and theStark e9ect is more appreciable. Actually, for bestmodulation one should use a wider well, optimal wellsizes are of the order of ∼100 RA for modulators [34].Moreover, for zero electric %eld the general featuresof these results are consistent with the results by previ-ous workers [16,25,26]. As expected our result of thebinding energy is smaller than that of Madarasz [16]and Ando [26], especially for small wire dimensions,since they considered in%nite con%nement potentials.In an actual QWW, the electron and hole wave func-tions therefore have evanescent tails penetrating intothe barrier region. This penetration modi%es the sub-band dispersion relations and weakens the Coulombinteraction. The excitonic binding will be inSuencedby the reduction of the Coulomb interaction. For ex-ample, from Madarasz’s paper, for symmetric wire,we estimate the exciton binding energy obtained bythem to be ∼21 meV at 50 RA, and our value in thiscase is 18 meV. On the other hand, for the wire dimen-sions larger than 150 RA the di9erence between our andtheir results becomes nearly, since for large enoughwire dimensions the band o9sets act on the particlesas in%nite potential barriers, and the electron and holewave functions are identical. We should point out thatthese results are in agreement with the cathodolumi-nescence observed by Petro9 et al. in quantum wellwires [25]. They conclude that the value of the ex-citon binding energy is 8–10 meV higher than thatin two-dimensional quantum wells. As seen in Fig. 2for L = 25 RA the calculated exciton binding energyis 19:6 meV which is nearly two times of that in aquantum well with the same width. The general fea-tures of the binding energy are also qualitatively in

0 50 100 150 200 250 3001200

1400

1600

1800

2000

F=0F=10 kV/cm

Lx = Lz (Å)A

bsor

ptio

n pe

ak p

ositi

on (

meV

)

F = 100 kV/cm

F = 50 kV/cm

Fig. 3. The absorption peak position in the symmetric wire as afunction of the wire dimensions for a few electric %eld values.

agreement with photoluminescence spectra on a seriesof 50 RA scale T-shaped, GaAs QWWs [21]. In theirpaper [21] Someya et al. conclude that the binding en-ergy of one-dimensional excitons in T-shaped QWWsis to be 17 meV. For the same wire dimensions, ourbinding energy value is ∼16 meV.In Fig. 3, we present the calculated absorption peak

position in the symmetric wire as a function of thewire dimensions for a few electric %eld values. Asseen in this %gure for all electric %eld values the ab-sorption peak position decreases as wire dimensionsincrease, since for increasing wire dimensions theenergy of the ground states of the electron and holedecrease monotonically, and this behavior causes areduction in the e9ective band gap of the system. Forthe narrow wire (Lx=Lz6 50 RA) where the geometriccon%nement is strong enough, the absorption peakposition is independent from the electric %eld. Forfurther wire dimensions, the Stark e9ect becomesdominant and a decrement in the absorption peak po-sition (resonant photon energy) is observed, since inthe wider range wire the perturbative term eFz in Eq.(3) becomes more e9ective on the structure. It is clearthat the tuning range in the resonance incident photonenergy becomes larger in the wider wire. This givesan additional degree of freedom in opto-electronicdevice applications. The electric %eld has a strong

242 E. Kasapoglu et al. / Physica E 16 (2003) 237–243

0 50 100 150 200 250 3000

400000

800000

1200000

1600000

Lx=Lz=(Å)

F = 0

F = 100 kV/cm

1430 1440 1450 1460 1470 14800

20000

40000

60000

80000

F=0

F=10 kV/cm

F=50 kV/cm

Photon Energy (meV)

Lx=Lz=120 Å

F=100 kV/cm

α (c

m)-1

α (c

m)-1

Fig. 4. The absorption coeIcient in the symmetric wire as a function of the wire dimensions for a few electric %eld values. The insetshows the variation of the absorption coeIcient as a function of the incident photon energy for di9erent %eld values.

inSuence on the optical absorption behavior for widerquantum well wire. The wire dimension dependenceof the absorption coeIcients at exciton resonance isqualitatively in agreement with those given by Andoet al. [26]. In the text of their paper, it is pointed thatthe position of the lower-energy peak decreases aswire dimensions increase.In Fig. 4, the optical absorption coeIcient is given

as a function of the wire dimensions for di9erent elec-tric %eld values. As seen in this %gure, especially fornarrow wire dimension, the magnitude of the absorp-tion coeIcient rapidly decreases as wire dimensionsincrease and it is independent from the electric %eld.This decrement is a consequence of the decrease ofthe interaction between the electrons and holes in thesame plane, since their wave functions are con%nedto a larger region in (x–z) plane. For large wire di-mensions the magnitude of the absorption coeIcientbegins to be sensitive to the electric %eld but the reduc-tion in the magnitude becomes nearly constant withincreasing wire dimensions. The %eld sensitivity of theabsorption coeIcient in this case can be explained as,

by applying the electric %eld, the electron and hole lo-calizes at the opposite sides of the wire and the over-lap between the electron and hole decreases rapidlyin comparison to those observed in the narrow wire,thus as wire dimensions increase the magnitude of theabsorption coeIcient decreases with %eld. In orderto show this behavior more clearly, the variation ofthe absorption coeIcient as a function of the incidentphoton energy for several %eld values is given in theinset.As a summary, we have investigated mainly, the

e9ect of the wire dimensions and the electric %eld ap-plied parallel to the one direction (z) of the wire onthe exciton binding energy and the optical interbandabsorption. The e9ect of the electric %eld on the ex-citonic absorption and binding energy is appreciablein quantum well wire with large enough dimensions.But, when the dimension of the wire in the directionparallel to the electric %eld is small enough, the ex-citonic binding energy and optical absorption is inde-pendent of the electric %eld in the range of electric%elds we considered. The additional con%nement of

E. Kasapoglu et al. / Physica E 16 (2003) 237–243 243

particles in quantum well wire o9ers greater variety ofelectric %eld dependence in comparison to bulk ma-terials and quantum well structures. The obtained re-sults were compared to those of previous studies, ourresults were observed to qualitatively agree with othercalculations. We can conclude, in general, that the de-gree of con%nement does enhance the magnitude ofthe linear optical absorption and the excitonic bindingenergy in the low dimensional heterostructures.

References

[1] H. Sakaki, Japan J. Appl. Phys. 19 L (1980) 735.[2] R.O. Klepher, F.L. Madarasz, F. Szmulowichz, Phys. Rev. B

51 (1995) 4633.[3] P.M. Petro9, A.C. Gossard, R.A. Logan, W. Wiegmann, Appl.

Phys. Lett. 41 (1982) 635.[4] H. Temkin, G.J. Dolan, M.B. Panish, S.N.G. Chu, Appl. Phys.

Lett. 50 (1987) 413.[5] M. Tanaka, H. Sakaki, Japan J. Appl. Phys. 27 L (1988)

2025.[6] K. Hiruma, T. Katsuyama, K. Ogawa, T.M. Koguchi, H.

Kakibayashi, G.P. Morgan, Appl. Phys. Lett. 59 (1991) 431.[7] G.P.Morgan, K. Ogawa, K. Hiruma, H. Kakibayashi,

T. Katsuyama, Solid, State Commun. 80 (1991) 235.[8] H. Weman, M. Potemski, M.E. Lazzouni, M.S. Miller,

J.L. Merz, Phys. Rev. B 53 (1996) 6959.[9] T. Szwacka, J. Phys. Condens. Matter 8 (1996) 10521.[10] M.H. Degani, O. Hipolito, Phys. Rev. B 35 (1987) 9345.[11] A.D’ Andrea, R. Del Sole, Phys. Rev. B 46 (1962) 2363.[12] Y. Takagaki, K. Ploog, Phys. Rev. B 51 (1995) 7017.[13] J.N. Randall, M.A. Reed, G.A. Frazier, J. Vacsci. Technol.

B 7 (1989) 1398.[14] R. Bate, G. Frazier, W. Frensly, M. Reed, TI Technical J. 6,

no. 4 (1989) 13.

[15] C. Weisbuch, B. Vinter, in: C.S.F. Thomson (Ed.), QuantumSemiconductor Structures, Academic Press, New York,1991.

[16] F.L. Madarasz, F. Szmulowichz, F.K. Hopkins, D.L. Dorsey,Phys. Rev. B 49 (1994) 13528.

[17] T. Fukui, H. Saito, Appl. Phys. Lett. 50 (1987) 824.[18] D.S. Chemla, D.A.B. Miller, S. Schmitt-Rink, in: H.

Haug (Ed.), Optical Nonlinearities and Instabilities inSemiconductors, Academic Press, New York, 1988, pp. 83–120 and 325–359, and references therein.

[19] P. Dawson, J. Lumin. 53 (1992) 293.[20] P.Q. Li, S.D. Sarma, Phys. Rev. B 43 (1991) 11768.[21] T. Someya, H. Akiyama, H. Sakaki, Phys. Rev. Lett. 76

(1996) 2965.[22] R. Rinaldi, et al., Phys. Rev. Lett. 73 (1994) 2899.[23] T. Someya, H. Akiyama, H. Sakaki, Phys. Rev. Lett. 74

(1995) 3664.[24] D. Gershoni, H. Temkin, G.J. Dolan, J. Dunsmuir, S.N.G.

Chu, M.B. Panish, Appl. Phys. Lett. 53 (1988) 955.[25] P.M. Petro9, A.C. Gossard, R.L. Logan, W. Wiegmann, Appl.

Phys. Lett. 41 (1982) 645.[26] H. Ando, S. Nojima, H. Kanbe, J. Appl. Phys. 74 (1993)

6383.[27] F. Rossi, G. Goldoni, E. Molinari, Phys. Rev. Lett. 78 (1997)

3527.[28] A. Lorenzoni, L.C. Andreani, Semicond. Sci. Technol. 14

(1999) 1169.[29] C.H. Thien, Commun. Phys. 7 (3) (1997) 25.[30] E. Kasapoglu, H. Sari, I. S.okmen, J. Appl. Phys. 88 (2000) 5.[31] A. Balandin, S. Bandyopadhyay, Superlattices Microstruct.

19 (1996) 2.[32] F. Bassani, G. Pastori Parravicini, Electronic States and

Optical Transitions in Solids, Pergamon, New York, 1975,pp. 152–154.

[33] G. Bastard, Wave Mechanics Applied to SemiconductorHeterostructures, Halsted, New York, 1988.

[34] J. Singh, in: W. Stephen (Ed.), Semiconductors Opto-electronics, McGraw-Hill, New York, 1995, p. 269.