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Physica E 19 (2003) 332–335

www.elsevier.com/locate/physe

Geometrical e�ects on shallow donor impuritiesin quantum wires

E. Kasapoglua ;∗, H. Saria, I. Sokmenb

aDepartment of Physics, Cumhuriyet University, 58140 Sivas, TurkeybDepartment of Physics, Dokuz Eyl!ul University, Izmir, Turkey

Received 31 January 2003; received in revised form 2 May 2003; accepted 4 June 2003

Abstract

We have studied theoretically the impurity binding energy for wires of di�erent shapes (V-shaped quantum wire (V-QWR)and rectangular wire) with a variational procedure without using any coordinate transformation. The e�ective potentialfor V-QWR used in this work consists of a square well potential in the z-direction and full graded well potential in thex-direction. Our results are in good agreement with previous theoretical results, found by the coordinate transformationmethod. Furthermore, it is shown that the impurity binding energy in quantum wires is sensitive to the geometrical e�ects.? 2003 Elsevier B.V. All rights reserved.

PACS: 73. 20. Dx; 73. 20. Hb

Keywords: V-shaped quantum wires; Impurity binding energy

1. Introduction

In the past few years, it has become possible to con-=ne the carriers in one (quantum wells), two (quantumwell wires) or all three dimensions (quantum dots).These structures enable many new phenomena to bediscovered and provide potential device applicationsin future laser technology and the optical modulationtechnique. The presence of impurities in these struc-tures contributes to additional responses. Because ofthe quantum con=nement in two directions, the bind-ing energies of impurity states are greatly enhanced inwires compared with those in quasi-two-dimensional(2D) quantum wells [1–3]. V-shaped quantum wires

∗Corresponding author. Tel.: +90-346-2101010-1937; fax: +90-346-219-1186.

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

(V-QWRs) are one of the typical quantum wire struc-tures. To fully understand the optical and electronicproperties of V-QWRs, a description of the impurityproperties would be important.The calculation of shallow impurities in square and

circular quantum wires is currently well established[4–7]. However, the variational technique employedfor the calculation is numerically intensive if the wireis not of circular shape. Due to the unusual form ofthe potential pro=le in V-QWRs, the calculation ofshallow impurity states may become impractical if themethod of calculation of the carrier ground state is alsonumerically intensive. In order to explain and explorethe physical mechanism in these quantum wire struc-tures, a number of theoretical methods were proposed:Sa’ar et al. [8] proposed a local envelope states expan-sion, Pescetelli et al. [9] used a tight-binding approachfor T- and V-shaped quantum wires, and Amman

1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.doi:10.1016/S1386-9477(03)00382-5

E. Kasapoglu et al. / Physica E 19 (2003) 332–335 333

et al. [10] used a quasi-factorization scheme. How-ever, in general, the 2D e�ective mass SchrIodingerequation has been calculated numerically using eitherplane-wave expansion [11–15] or by adapting =niteelement methods [16]. Recently, Creci and Weber[17] proposed an e�ective potential method whichconsiderably eases the calculation of energy levelsand wave functions in V-groove quantum wires, andthey reported on the calculation of shallow impuritybinding energies in single V-QWRs [18]. Deng etal. presented calculated donor [19] and acceptor [20]binding energies in single V-QWRs at some speci=cimpurity positions using a coordinate transformation.In this work, we calculate the binding ener-

gies for shallow donor impurities in V-grooveGaAs=Ga1−xAlxAs quantum wire using a varia-tional technique. The carrier ground states are calcu-lated by an e�ective potential method without usingany coordinate transformation, where the e�ectivepotential pro=le consists of a square quantum wellin the z-direction and full graded quantum well inthe x-direction. One important aspect of electronicband structure engineering is the realization of gradedheterostructures, in which the composition is variedcontinuously in space. Electronic and optoelectronicdevices that exploit these e�ects include, to date:graded-base heterostructure bipolar transistors, whichpromote the egress of carriers through the base;graded separate con=nement heterostructure laseractive regions, which not only con=ne light to thequantum wells but may also promote transport withinthe active region and increase device bandwidth[21,22].

2. Theory

In the e�ective mass approximation, the Hamil-tonian for a shallow-donor impurity in a V-shapedGaAs=Ga1−xAlxAs quantum wire is

H =− ˝2m∗

d2

dx2+ V (x)− h2

2m∗d2

dz2

+V (z)− ˝22m∗

d2

dy2 − e2

�0→r; (1)

where m∗ is the electron e�ective mass, �0 is thestatic dielectric constant,

→r is the distance between

Fig. 1. (A) Total e�ective potential as a function of the coordinatesfor V-QWR. (B) Total e�ective potential which is obtained byusing graded well potential in both directions.

the carrier and the impurity site, V (x) and V (z)are the =nite con=nement potentials in the x- andz-direction, respectively. V (z) is a square well poten-tial with height V0 and width Lz. V (x) is a full graded(i.e. V-shaped) potential pro=le. The full gradedpotential pro=le is obtained by linearly changing thealuminium concentration−x from zero to 0.3 in theGa1−xAlxAs layer. The functional form of the fullgraded con=nement potential, V (x) is

V (x) =

V0; x¡− Lx=2;

2V0

Lx|x|; |x|¡Lx=2;

V0 x¿Lx=2:

(2)

When we combined the potential pro=les of bothdirections (x; z), the e�ective potential of the V-QWRis obtained, given in Fig. 1. Note that the form of thispotential resembles the e�ective potential, which is

334 E. Kasapoglu et al. / Physica E 19 (2003) 332–335

obtained after the coordinate transformation methodused by Creci and Weber [17] and Deng et al. [19,20].The impurity binding energy in V-QWRs is calcu-

lated by a traditional variational method. The follow-ing trial wave function for ground impurity state inV-QWRs is given by

�(r) = (x) (z)’(y; �); (3)

where the wave function in the y-direction ’(y; �) ischosen to be a Gaussian-type orbital function [23]:

’(y; �) =1√�

(2�

)1=4

e−y2=�2 (4)

in which � is a variational parameter. With the choiceof ’(y; �), the degrees of freedom are limited toone dimension along the axis of the wire. (z) isexactly obtained from the SchrIodinger equation in thez-direction, and (x) is a linear combination of Airyfunctions.The ground impurity binding energy is obtained as

follows:

Eb = Ex + Ez −min�

〈�(r)|H |�(r)〉; (5)

where Ex and Ez are the ground-state energies of elec-tron obtained from the SchrIodinger equation in the x-and z-direction without the impurity, respectively.

3. Results and discussions

The values of the physical parameters used inour calculations can be determined by the following*k = 0 values in Ga1−xAlxAs [24]: m∗ = (0:067 +0:083x)m0 (where m0 is the free electron mass),V e0 = 0:65 LE�

g (x), LE�g (x)= 1:247x (eV) where the

di�erence between the Ga1−xAlxAs and GaAs bandgaps at the � point is given by Ref. [24]; �0 = 12:6is the dielectric constant. The di�erence in dielec-tric constant between the well material and barriermaterial is neglected. The aluminium concentrationx = xmax is 0.3.Fig. 2 shows the variation of the binding en-

ergy of a donor impurity located in the centreof the wire as a function of wire dimensions forthree di�erent quantum wire shapes. Curve 1 inFig. 2 shows the variation in impurity bindingenergy with the dimension of V-QWR: while a gradedwell potential V (x) is used for the x-direction, the

0 40 80 120 160 200 240 28010

15

20

25

30

35

L ( )

E

( m

eV )

b

L = L = L x z

1- VQWR

2- GQWR

3- RQWR

Fig. 2. Variations of the impurity binding energy as a function ofwire dimension for three di�erent quantum wire shapes. Curve 1shows the variation in impurity binding energy with the dimensionof VQWR: while a graded well potential V (x) is used for thex-direction, the potential V (z) along the z-direction is a simplesquare well potential. Curve 2 shows the results of variation inimpurity binding energy with the dimension of the wire, whichis obtained by using graded well potential (V-shaped) in bothdirections. And curve 3 shows the results of a rectangular quantumwire.

potential V (z) along the z-direction is a simple squarewell potential. Curve 2 shows the results of variationin impurity binding energy with the dimension of thewire, which is obtained by using graded well poten-tial in both directions. And curve 3 shows the resultsof rectangular quantum wire. As seen in this =gure,when the dimension of QWRs increases the impuritybinding energy for all curves increases until it reachesa maximum value, and then decreases. This behaviouris related to the change of the electron con=nementin QWRs. When the dimension of wire decreases, thecon=nement of electrons is strengthened, and there-fore the impurity binding energy increases. When thedimension of QWRs is reduced to a small limitedvalue, most of the electronic wave function beginsto leak out of the well and therefore the impuritybinding energy decreases. As seen in these results,for small wire dimensions (206L6 50 MA) impuritybinding energy for a rectangular wire is greater thanV-QWRs (curves 1 and 2), since the wave functionof the electron becomes strongly localized in thesquare well. As the wire dimension increases, the

E. Kasapoglu et al. / Physica E 19 (2003) 332–335 335

Table 1Some calculated binding energy values for V-QWR

L ( MA) EB (meV)

100 ∼= 17:5a ∼= 24b

150 ∼= 15a ∼= 20b

200 ∼= 12:5a ∼= 17:5b

300 ∼= 15b

aParameters corresponding to the results of Ref. [19].bParameters corresponding to our results.

wave function of the electron becomes strongly local-ized in the graded well and impurity binding energyfor the wires which have graded well potentials asthe con=nement potentials is greater than rectangularwire. Note that due to the more spatial con=nementin the x-direction, our results for impurity bindingenergy in V-QWR are larger than the results inRef. [19]. Some of the results are given in Table 1.Especially for large wire dimensions, we =nd, at 2Lvalues, the results of impurity binding energy for acertain L value which are obtained by Ref. [19]. Inthe theoretical studies for V-QWRs, two parabolasare used to describe the upper and bottom boundariesof V-QWRs and it was found that after the coordi-nate transformation, the con=ning potential due tothe V-shaped boundaries is a harmonic-like potential[17–20]. The advantage of the present potential pro=lein the x-direction is that the SchrIodinger equation has arather simple form and it can be exactly solvable with-out using coordinate transformation. Furthermore, thepresent con=nement potential has the exact V shapeand gives an additional con=nement for the donor elec-tron, thus the binding energy of the donor impurity isgreater than that of di�erent geometrical con=nementpotentials. For example, especially for large L val-ues, as expected, curve 2 in Fig. 2 which is obtainedby using graded well potential (V-shaped) in bothdirections is greater than other curves.In summary, we have studied theoretically the

impurity binding energy in V-QWR with a variationalprocedure without using any coordinate transforma-tion method. Furthermore, to observe the geometrice�ect of the wire on impurity binding energy weused the wires of di�erent shapes. Our results showthat the structural con=nement is very e�ective andthe impurity wave functions become strongly local-ized in the V-QWR when compared with results in

Ref. [19]. We expect that this method will be ofgreat help for theoretical studies of the physicalproperties of V-groove quantum wires. Furthermore,the simplicity of the method provides an interestinginsight into the main physical properties of carriers inV-groove quantum wires.

References

[1] N.P. Montenegro, L.E. Oliveira, Solid State Commun. 76(1990) 275.

[2] G. Weber, L.E. Oliveira, Mater. Sci. Fonon. 65–66 (1990)135.

[3] A. Latge, N.P. Montenegro, L.E. Oliveira, Phys. Rev. B 45(1992) 6742.

[4] G. Weber, P.A. Schulz, L.E. Oliveria, Phys. Rev. B 38 (1998)2179.

[5] A. Latge, L.E. Oliveria, J. Appl. Phys. 77 (1995) 1328.[6] A. Latge, M. de Dios-Leyva, L.E. Oliveira, Phys. Rev. B 49

(1994) 10450.[7] P. Villamil, N. Porros-Montenegro, J.C. Granada, Phys. Rev.

B 59 (1999) 1605.[8] A. Sa’ar, S. Calderon, A. Givant, O. Ben-Shalom, E. Kapon,

C. Caneau, Phys. Rev. B 54 (1996) 2675.[9] S. Pescetelli, A. Di Carlo, P. Lugli, Phys. Rev. B 56 (1997)

1668.[10] C. Amman, M.A. Depurtius, U. Bockelmann, B. Deveaud,

Phys. Rev. B 55 (1997) 2420.[11] R. Rinaldi, et al., Phys. Rev. Lett. 73 (1994) 2899.[12] R. Rinaldi, P.V. Giugno, R. Cingoloni, F. Rossi, E. Molinari,

U. Marti, F.K. Reinhart, Phys. Rev. B 53 (1996) 13710.[13] C. Kiener, L. Rota, J.M. Freyland, K. Turner, A.C.

Maciel, J.F. Ryan, U. Marti, D. Martin, F. Morier-Gemond,F.K. Reinhart, Appl. Phys. Lett. 67 (1995) 2851.

[14] K. Chang, J.B. Xia, Phys. Rev. B 58 (1998) 2031.[15] Y. Fu, M. Willander, X.Q. Liu, W. Lu, S.C. Shen, H.H.

Tan, C. Jagadish, J. Zou, D.J.H. Cockayne, J. Appl. Phys.89 (2001) 2351.

[16] T. Inoshita, H. Sakaki, J. Appl. Phys. 79 (1996) 269.[17] G. Creci, G. Weber, Semicond. Sci. Technol. 14 (1999) 690.[18] G. Weber, de A.M. Paula, Phys. Rev. B 63 (2001) 113307.[19] Z.Y. Deng, X. Chen, T. Ohji, T. Kobayashi, Phys. Rev. B

61 (2000) 15905.[20] Z.Y. Deng, X. Chen, T. Ohji, J. Phys.: Condens. Matter 12

(2000) 3019.[21] S. Marcinkevicius, U. Olin, J. Wallin, G.L.K. Strebel, in:

Conference on Lasers and Electro-Optics, 1995, Baltimore,p. 280.

[22] G.W. Yih, K. Gioney, J. Bowers, M. Rodwell, P. Silvestre,P. Thiagrajan, G. Robinson, J. Lightwave Technol. 13 (7)(1995) 1490.

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

[24] H. Kalt, Optical Properties of III–IV Semiconductors,Springer, Berlin, 1996, p. 177.