Physica B 315 (2002) 261–266

Hydrogenic impurities in graded GaAs–(Ga,Al)Asquantum-well wires in an electric field

E. Kasapoglua,*, H. Saria, I. S .okmenb

a Department of Physics, Cumhuriyet University, 58140 Sivas, Turkeyb Physics Department, Dokuz Eyl .ul University, ’Izmir, Turkey

Received 7 September 2001; received in revised form 16 November 2001

Abstract

The electric field dependence of polarizability and binding energy of shallow-donor impurities in graded quantum-

well wires is calculated by a variational method and in the effective-mass approximation. We have considered a finite

confinement model and the results are compared with that of infinite confinement potential. Our calculations have

revealed the dependence of the impurity binding and polarizability on the field direction in the graded quantum-well

wire. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 68.65+g; 71.55.�i; 71. 55.Eq

Keywords: Shallow donors; Quantum-well wires; Electric field; Polarizability; Graded well

1. Introduction

The behavior of a hydrogenic impurity centerlocated in a quantum-well structure has been atopic of considerable interest [1–4]. The interest insuch systems stems mainly from the differencesbetween the behavior of shallow-donor impuritiesconfined to a quantum well and the equivalentbulk system. For instance, the confinement of thecarriers in the low-dimensional systems is respon-sible for the appearance of quantum phenomena,which cannot be observed in bulk semiconductors.A new impulse is due to Bastard who hascalculated the binding energy of the ground state

of a hydrogenic donor associated with the lowestelectron subband level, as a function of GaAsquantum-well size and the position of the impurityion. The original study of Bastard [1] was followedby several other calculations. In these studies,variational techniques are used and essentially thesame results are obtained. Until now, almost allstudies on the binding energy of hydrogenimpurities in low-dimensional semiconductorstructures have exclusively been limited to theeffect of confinement potentials: the square [5–7],or parabolic [8–11]. For quantum wells the bindingenergy of hydrogen impurities was investigated ingreat detail [1,12,13].

Interesting physical properties on low-dimen-sional systems appear when an electric field isapplied. The importance of studying the influence

*Corresponding author.

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

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 4 6 9 - 6

of an electric field on the binding energy, the statedensity, the polarizability have theoretical as wellas technological relevance. The effect of an electricfield, and the geometric form of the system on thebinding energies of shallow-donor impurities inGaAs quantum-well wires was presented byseveral authors [14–16], considering an infiniteconfinement potential and using a variationalscheme. Duque et al. investigated the effect of anapplied electric field on the binding energy, andpolarizabilities of shallow-donor impurities im-planted in rectangular cross-section GaAs–GaA-lAs quantum-well wires [17], and in cylindricalGaAs low-dimensional systems [18]. They ob-tained, as a general feature, that the polarizabilityincreases almost linearly for all applied electricfields, and the results qualitatively agree with othertheoretical calculations [19–22]. Osorio et al. [23]presented the electric field direction dependence ofthe binding energy and the polarizability of donorimpurity in a GaAs–GaAlAs quantum wire.

In the present work, we investigated the effect ofan applied electric field on the binding energies,and polarizability of shallow-donor impurities inquantum-well wires with different finite confine-ment potentials (square and graded). To the bestof our knowledge, theoretical and experimentalresults of the binding energy and polarizability ingraded quantum-well wires have not been re-ported. By using a graded confinement potentialwe can investigate the influence of the electric fielddirection, and the symmetry of the system on thebinding energy and polarizability in low-dimen-sional structures.

2. Theory

In the framework of the effective-mass approx-imation, the Hamiltonian for a hydrogen-donorimpurity in a GaAs quantum-well wire in anelectric field, F ; applied perpendicular to the axisof the wire and along the x-direction is

H ¼ �_2r2

2m� þ V ðxÞ þ V ðzÞ þ jejFx �e2

er; ð1Þ

where m� is the electronic effective mass, e is thestatic dielectric constant, r is the distance between

the carrier and the impurity site, V ðxÞ and V ðzÞ arethe finite confinement potentials in the x- and z-direction, respectively. V ðzÞ is a square well withheight V0 and length Lz: For the confinementpotential V ðxÞ; square and graded profiles, struc-tures named in what follows square quantum-wellwire (SQWW) and graded quantum-well wire(GQWW), were considered to study the influenceof the geometric and induced-field confinementmechanisms on the binding energy and polariz-ability. The graded potential profile is schemati-cally given in Fig. 1. By changing the Alconcentration x in Ga1�xAlxAs one obtains alinearly changing conduction-band profile asshown in Fig. 1. This linear variation representsthe situation when external electric field is appliedin the x-direction with a value R in the well region.We used this method in our previous study [24].Therefore, the electric field R needed to obtain thesame profile should be given by

jejRLx ¼V0

2; ð2Þ

where Lx is the length of the graded quantum well.The functional form of the graded confinementpotential V ðxÞ is

V ðxÞ ¼jejR x þ

Lx

2

� �; jxjpLx=2;

V0 otherwise:

8><>: ð3Þ

Using the variational method, it is possible toassociate a trial wave function, which is anapproximated eigenfunction of the Hamiltoniandescribed in Eq. (1). The ground-state wavefunction of the impurity is given by

cðrÞ ¼ cðxÞcðzÞfðy; lÞ; ð4Þ

where the wave function in the y-direction fðy; lÞis chosen to be a Gaussian-type orbital function[25–27].

fðy; lÞ ¼1ffiffiffil

p 2

p

� �1=4

e�y2=l2

ð5Þ

in which l is a variational parameter. With thechoice of fðy; lÞ; the degrees of freedom arelimited to one dimension along the axis of wire,which makes it possible for us to proceed furtherwith the problem analytically. Here cðxÞ is a linear

E. Kasapoglu et al. / Physica B 315 (2002) 261–266262

combination of Airy functions, and cðzÞ isexactly obtained from the Schr .odinger equationin the z-direction. The ground-state impurityenergy is evaluated by minimizing the expectationvalue of the Hamiltonian, /cðrÞjH jcðrÞS withrespect to l:

The ground-state donor binding energy is givenby [14,28,29]

EB ¼ Ex þ Ez � minl

/cðrÞjH jcðrÞS; ð6Þ

where Ex and Ez are the ground-state energies ofthe electron obtained from the Schr .odingerequation in the x- and z-direction without theimpurity, respectively.

The polarizability is calculated by means of[17,20]

aP ¼ �e

F/cjxjcSFa0 �/cjxjcSF¼0

: ð7Þ

3. Result and discussion

The values of the physical parameters used inour calculations are m� ¼ 0:0665 m0 (where m0 isthe free electron mass), e ¼ 12:58 (static dielectricconstant is assuming to be same every here),V0 ¼ 228 meV. These parameters are suitable inGaAs/Ga1�xAlx heterostructures with an Al con-centration of xD0:3 [14, 30]. We have assumed theconduction-band discontinuity to be 56% of thetotal band gap difference between GaAs andGa1�xAlxAs, and the donor is located at thecenter of the wire. The donor binding energy iscalculated in Rydberg units.

In Fig. 2, we present the variation of the bindingenergy of a hydrogenic impurity as a function ofthe applied electric field for the SQWW andGQWW with different dimensions. In order tocompare our results with that of Duque et al. [17],we choose the dimensions Lz ¼ 650 (A, considering

x-direction

-L/2 L/2

Vo

AsAlGamaxxmaxx1_ Ga1_xAlxAs

x-direction

X

maxX

-L/2 L/2

(A)

(B)

AsAlGamaxxmaxx1_

Fig. 1. (A) The schematic representation of the conduction band in the graded Ga1�xAlxAs structure; (B) the variation of the

aluminium concentration x in the x-direction which is used in the structure.

E. Kasapoglu et al. / Physica B 315 (2002) 261–266 263

500 and 200 (A for the Lx dimension. As seen inFig. 2 the binding energy in the SQWW is largerthan that of the GQWW, since for zero electricfield the donor electron is mostly confined to theleft side of the graded quantum well whereas in thesquare well the electron moves freely in the wholewell region. Thus, in the SQWW the probability ofthe finding the donor electron and the impurity ionin the same plane is larger than for the GQWW. Inthe SQWW, for the smaller dimensions, where thegeometric confinement is predominant, the varia-tion of the binding energy is very small in thewhole range of electric fields. On the other hand,for the wider SQWW, Lx ¼ 500 (A, the variation islarge especially for small electric field values.Qualitatively, we see that, these results give thesame behavior as found by Duque et al. We shouldpoint out, as expected, that our results are smallerthan those of Duque et al. [17], since we considerthe finite potential confinements in the x- and z-directions. In their model the donor electron isconfined in a quantum box. The comparisonshows that for Lz ¼ 650 (A, Lx ¼ 500 (A, F ¼ 0;their result for the binding energy is nearly equalto the 1.5 Ry. Our value in this case is 1.2 Ry. Inthe finite potential model, the penetration of the

donor electron into the barriers is appreciable,which gives a reduction of the electron confine-ment and impurity binding energy. In the GQWW,the variation of the binding energy is small for allwire dimensions in the electric field applied in the+x-direction (þF ). But for electric field applied inthe �x-direction (�F ) the variation of the bindingenergy is quite different, especially for Lx ¼ 500 (A.It is found that the binding energy increases as �F

increases, until it reaches a maximum at a certainvalue of �F ; and then decreases as the fieldbecomes larger. We can explain this behavior asfollows; when the electric field is zero, due to theasymmetric character of the graded well potentialthe donor electron is mostly confined to the left-hand side of the well, and by applying the electricfield in the �x-direction the graded well becomesflatter, so that the donor electron shifts progres-sively to the right-hand side of the well. At acritical field �F ¼ 22:8 kV cm�1, the graded struc-ture is completely leveled off. Concurrent with thisshift of the donor electron, the separation betweenthe donor electron and the impurity ion decreases,leading to an increase in the Coulomb interaction.It should be noted that at this critical field valuethe binding energy value approaches that of theSQWW with no electric field. For Lx ¼ 200 (A, thesame behavior is observed, but the field depen-dence of the binding energy is weak with respect tothe previous one. Actually at the critical fieldvalue, for Lx ¼ 200 (A this critical field value is57 kV cm�1, the donor binding energy in theSQWW with no electric field and the GQWWare nearly the same.

In Fig. 3, the variation of the polarizability of ashallow donor in a SQWW is given as a functionof the electric field. As expected, field dependenceof the polarizability is observed for small dimen-sions of the quantum wire, while for largerdimensions the polarizability tends to be constantover the whole field range. We also find that thepolarizability increases as the dimensions of thewire decrease.

In Fig. 4 we present the results of the polariz-ability in the GQWW in +F and �F fields fordifferent dimensions. For small dimensions of theGQWW, the field dependence of the polarizabilityis very weak for both þF and �F field values,

F (kV/ cm )

+F

-F

L = 200Åx

+F

-F L = 500Åx

L = 650z

E

( R

)

By

Fig. 2. Binding energy of a shallow-donor impurity as a

function of the applied electric field in a SQWW (dashed lines)

and GQWW (solid lines).

E. Kasapoglu et al. / Physica B 315 (2002) 261–266264

while for the larger dimension, Lx ¼ 500 (A, for+F values the reduction in the polarizability isalmost linear, but for �F values the polarizabilityincreases with field, until it reaches a maximum atthe critical field value and then begins to decrease

for further large field values. The general shape ofthis curve can be explained using a similarargument as provided above for explaining thevariation of the binding energy in a negativeelectric field. Also we find that for the large wiredimensions a negative electric field affects thepolarizability more than an electric field in theopposite direction. Accordingly, the probability oftunneling of the donor electron can be increased ordecreased depending on the direction of the field.This control over tunneling could be desirable forsome device applications.

4. Summary and conclusion

As a summary, in this work we have studiedfield dependence of the binding energy andpolarizability of an on-center donor impurity inSQWWs and GQWWs. The calculations wereperformed within the effective-mass approxima-tion and by using a variational method. Weconclude that, for quantum-well wires with largedimensions, the applied electric field produces animportant effect on the electronic and opticalproperties, and in GQWWs, the changes in donorbinding energy and polarizability being stronglydependent on the direction of the field. Our resultsqualitatively agree with other theoretical calcula-tions [17]. We obtain, as a general feature, that forstrong geometric confinements the binding energyis more influenced by the quantum confinementthan by the applied electric field. The method usedin this work is capable of describing the correctbehavior of shallow-donor impurities in quantum-well wires with different shapes in an external fieldapplied parallel to the axis of the wire.

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