DOI: 10.1007/s00339-003-2161-1

Appl. Phys. A 78, 1053–1058 (2004)

Materials Science & ProcessingApplied Physics A

e. kasapoglu1,�

h. sari1

i. sokmen2

The electric field dependence of a donorimpurity in graded GaAs quantum wires1 Cumhuriyet University, Physics Department, 58140 Sivas, Turkey2 Dokuz Eylül University, Physics Department, Izmir, Turkey

Received: 2 October 2002/Accepted: 4 March 2003Published online: 2 September 2003 • © Springer-Verlag 2003

ABSTRACT The effect of the electric field on the binding en-ergy of the ground state of a shallow donor impurity in a gradedGaAs quantum-well wire (GQWW) was investigated. The elec-tric field was applied parallel to the symmetry axes of the wire.Within the effective mass approximation, we calculated thebinding energy of the donor impurity by a variational methodas a function of the wire dimension, applied electric field, anddonor impurity position. We show that changes in the donorbinding energy in GQWWs strongly depend not only on thequantum confinement, but also on the direction of the electricfield and on the impurity position. We also compared our resultswith those for the square quantum-well wire (SQWW). The re-sults we obtained describe the behavior of impurities in bothsquare and graded quantum wires.

PACS 68.65.-k; 71.55.-i; 71.55.Eq

1 Introduction

Low dimensional heterostructure systems have re-ceived a great deal of attention because of their intrinsicallyinteresting physical properties and their technological appli-cations in electronic and optical devices. The confinementof carriers in these structures is responsible for the appear-ance of quantum phenomena that cannot be observed in bulksemiconductors. This yields unique properties, such as the en-hanced quantum confined Stark effect, which lead to potentialapplications in optical devices [1–3].

There have been a number of studies of the physical prop-erties of shallow donors in quantum wires [4–16]. For im-purity states in quantum wires, the effects of several fac-tors on impurity spectra have been investigated: the shapeof the wire’s cross-section, the transverse dimensions of thewire, the height of the potential barriers, the position of thecenter with respect to the axis of the wire, and the exter-nal applied electric and magnetic fields [17–31]. Most cal-culations have been performed for a hydrogenic impurity inthe effective-mass approximation and within the variationalapproach. Electric fields have become an interesting probe

� Fax: +90-346/219-1186, E-mail: ekasap@cumhuriyet.edu.tr

for studying the physical properties of low-dimensional sys-tems, both from theoretical and technological points of view.Bin and You-Tong [22], using variational and perturbationalmethods, have studied the impurity energy levels in a cylin-drical quantum-well wire in a weak magnetic field. Monteset al. [17] have calculated the binding energy of a shallow-donor impurity in a rectangular cross-sectional area of a GaAsquantum-well wire, in an electric field applied perpendicularto the one of the interfaces and assuming an infinite con-finement potential. They also studied the binding energies ofthe ground and first few excited impurity states of quantum-well wires in electric fields [19]. They found that the impuritybinding energy depends strongly on not only quantum con-finement, but also the applied electric field and the distributionof impurities inside the quantum-well wires. Weber et al. [16]have studied the binding energy as well as the density ofshallow impurities in rectangular cross-sectional area GaAs–(Ga, Al)As QWWs. Duque et al. [23] have studied bindingenergies and polarizabilities in QWWs. Szwacka [18], usinga simple Gaussian-like trial envelope functions and a variablegauge of the vector potential, investigated the ground and 2p−states of the D0 center in a rectangular quantum wire.

In this study we investigate the binding energy of a shallowdonor impurity in a graded quantum-well wire in an appliedexternal electric field, as a function of the wire dimensions andthe impurity position. In our calculations we consider finiteconfinement potentials.

2 Theory

In the framework of the effective-mass approxima-tion, the Hamiltonian for a hydrogen donor impurity in a GaAsquantum-well wire in an electric field, F, applied perpendicu-lar to the axis of the wire and along the x-direction is

H = −h2∇2

2 m∗ + V(x)+ V(z)+|e| Fx − e2

εr, (1)

where m∗ is the electronic effective mass, ε is the static di-electric constant, r is the distance between the carrier andthe impurity site, and V(x) and V(z) are the finite confine-ment potentials in the x- and z-directions, respectively. V(z)is a square well with height V0 and length Lz . For the con-finement potential V(x), square and graded profiles, structuresnamed in what follows as a square quantum-well wire SQWW

1054 Applied Physics A – Materials Science & Processing

and a graded quantum-well wire GQWW, were considered tostudy the influence of the geometric and induced-field con-finement mechanisms on the binding energy. It is assumed thatthe functional form of the graded confinement potential V(x)

is

V(x) ={|e| R(x + Lx

2 ), |x| ≤ Lx/2V0, otherwise,

(2)

where Lx is the length of the graded quantum well. The gradedpotential profile is obtained by linearly changing the Al con-centration x from zero up to 0.15 in a thin layer of Ga1−x AlxAsthat is sandwiched between two thick layers of a Ga0.7Al0.3Asalloy. This profile represents the situation in which the exter-nal electric field is applied in the x-direction with a magnitudeR in the well region. We used this method in our previous stud-ies [32]. Therefore, the electric field R needed to obtain thegraded potential profile should be given by

|e| RLx = V0

2. (3)

Using the variational method, it is possible to associate a trialwave function, which is an approximated eigenfunction of theHamiltonian described in (1). The ground state wave functionof the impurity is given, for example, by

ψ(r) = ψ(x)ψ(z)φ(y, λ), (4)

where the trial-wave function φ(y, λ) in the y-direction,a function describing the binding between the electron and thedonor, is taken to be of hydrogenic form, i.e. a Gaussian-typeorbital function [33, 34],

φ(y, λ) = 1√λ

(2

π

)1/4

e−y2/λ2, (5)

in which λ is a variational parameter. With this choice ofφ(y, λ), the degrees of freedom are limited to one dimen-sion along the axis of wire, which makes it possible for usto proceed further with the problem analytically. The samekind of solution has been successfully used before in differenttypes of heterostructure calculations [33–35]. Here ψ(x) isa linear combination of Airy functions, and ψ(z) is exactly ob-tained from the Schrödinger equation in the z-direction. Theground state impurity energy is evaluated by minimizing theexpectation value of the Hamiltonian, 〈ψ(r) |H |ψ(r)〉, withrespect to λ.

The ground state donor binding energy is given by [6, 17,36]

EB = Ex + Ez −minλ

〈ψ(r) |H | ψ(r)〉 , (6)

where Ex and Ez are the ground state energies of the elec-tron obtained from the Schrödinger equation in the x- andz-directions, respectively, without the impurity.

3 Results

For numerical calculations, we took V0 = 228 meV,m∗ = 0.0665 m0 (where m0 is the free electron mass), and ε =12.58. These parameters are suitable for GaAs/Ga1−xAlxAs

heterostructures with an Al concentration of x ∼= 0.3 [17, 37].The position of the donor impurity is given as (xi, 0, 0) and thebinding energy is calculated in Rydberg units.

In Fig. 1a we display the ground state binding energy ofthe donor impurity at the center of the GQWW as a functionof the electric field for several different wire dimensions. Asseen in this figure, for a finite confinement potential the donorbinding energy increases as the wire dimensions are reduced.

FIGURE 1 a Binding energy of a donor impurity localized at the center ofa GQWW as a function of the electric field applied in the ±x-direction forLz = 50 Å and different Lx values. The dashed line indicates the result forthe SQWW with Lx = 50 Å. b Binding energy of a donor impurity localizedat the center of a GQWW as a function of the electric field applied in the ±x-direction for Lz = 200 Å and different Lx values. The dashed line indicatesthe result for the SQWW with Lx = 200 Å

KASAPOGLU et al. The electric field dependence of a donor impurity in graded GaAs quantum wires 1055

For moderate Lx values (∼= 200 Å), the effect of the electricfield applied in the +x-direction (+F) on the binding energyis appreciable, whereas for very small and large Lx values, thedonor binding energy is not very sensitive to the applied field+F. For the +F case, as general feature, the donor bindingenergy decreases as the field increases, since the probabilitydensity of the donor electron decreases around the impurityposition due to the action of the applied electric field. For theelectric field applied in the −x-direction, the −F case, thefield dependence of the donor binding energy is quite differentfrom the +F case. This difference is especially clearly seenfor Lx = 500 Å and Lz = 50 Å. In this case, when −F is in-creased, the binding energy increases up to a maximum andthen begins to decrease. We can explain this behavior as fol-lows: since the wavefunctions in the GQWWs are asymmetricfor F = 0, the donor electron is mostly confined to the left sideof the well, and the graded quantum well becomes flatter ifthe electric field is applied in −x-direction, when the negativefield is slightly increased, the donor electron becomes closeto the donor impurity ion, resulting in an increase in the bind-ing energy. The maximum in the binding energy occurs closeto 20 kV/cm for Lx = 500 Å, Lz = 50 Å, and xi = 0, whereasfor small Lx values the position of the maximum peak is ob-served at large field values. For example, for Lx = 200 Å themaximum in the binding energy occurs nearly at 60 kV/cm.However after a critical field value the donor binding energybegins to decrease, since at a critical negative field value thegraded structure is completely leveled off and beyond thisfield the donor electron moves towards the right. This givesa reduction in the probability of the finding the donor elec-tron and impurity in the same plane. From our calculations, wefound that the electric field dependence of the donor bindingenergy is more pronounced at large wire dimensions. It shouldalso be noted that, in conventional symmetrical structures,the wavefunctions have a symmetric character and the donorbinding energy does not depend on the direction of the electricfield. In Fig. 1b the on-center donor (xi = 0) binding energy isgiven as a function of the electric field for Lz = 200 Å and sev-eral Lx values. By comparing this figure with Fig. 1a, we seethat, due to the weak electron confinement in the z-direction,the donor binding energy is small with respect to the previ-ous case, and the electric field dependence is almost the same.That is, by increasing −F the donor binding energy increasesup to a maximum and then begins to decrease. This behavioris explained in detail for Lx = 500 Å and Lz = 50 Å in Fig. 1a.As a general feature, we conclude that for large Lx values thedonor binding energy is not very sensitive to the length of theLz side, whereas for small Lx values increasing the length ofthe Lz side causes the binding energy to decrease for all im-purity positions. In Fig. 1a and b we also present the variationof the binding energy of a hydrogenic impurity for a squarequantum-well wire (SQWW) with different dimensions andimpurity positions. For the SQWW, in order to compare ourresults with those of Montes et al. [17, 19], we chose thedimensions Lx = Lz = 50 and 200 Å in Fig. 1a and b, respec-tively. As seen in this figures the binding energy in the SQWWis larger than that of the GQWW, since at zero electric field thedonor electron is mostly confined to the left side of the gradedquantum well, whereas the electron moves freely in the wholewell region in the square well. Thus, in the GQWW the prob-

ability of finding the donor electron and the impurity ion inthe same plane is smaller than for the SQWW. Qualitatively,we see that the electric field dependence of the donor bindingenergy gives the same behavior as found by Montes et al. Weshould point out that, as expected, our results are smaller thanthose of Montes et al. [17, 19], since they considered infiniteconfinement potentials. In an actual QWW, the donor electronwavefunctions have evanescent tails penetrating into the bar-rier region. This penetration weakens the Coulomb interactionbetween the donor electron and impurity ion. The comparisonshows that, as the wire dimensions increase, the difference be-tween the binding energy of the finite and infinite confinementpotentials decreases, since at large enough wire dimensions,the band offsets act on the donor electron as infinite potentialbarriers, and the donor electron wavefunctions are identicalin both models. For example, from the paper of Montes etal. [19], we estimate the donor binding energy for the SQWWobtained by them to be ∼ 40 meV for Lx = Lz = 50 Å andxi = 0. Our value in this case is ∼ 28 meV. On the other hand,for the wire with Lx = Lz = 200 Å and xi = −Lx/2, the donorbinding energy obtained by Montes et al. [17] is ∼ 8 meV, andour value in this case is 7.75 meV.

In Fig. 2a and b we present the binding energy of a donorlocated at the position (−Lx/2, 0, 0) in the GQWW withLz = 50 and 200 Å, respectively, as a function of the appliedelectric field for several values of Lx . By comparing Figs. 1and 2, we see that for small Lx values, the donor electron ismore energetic, since Ex gets to be larger, and it moves freelyin the well region. Thus the binding energy of the donor impu-rity located at the center of the GQWW is larger than that ofthe impurity located at the left side of the well. But for largeenough Lx values, such as Lx = 500 Å, the donor electron ismostly confined to the left side of the well and the binding en-ergy of the donor impurity located at the center of the GQWWis smaller than that of the impurity located at the left side of thewire. For the impurity located at xi = −Lx/2, it is observedthat, for Lx = 500 Å, the binding energy reaches a constantvalue as −F is greater than 30 kV/cm. It should also be notedthat, for xi = −Lx/2, when the electric field is applied in the+x-direction, the binding energy is higher than when the elec-tric field is applied in the −x-direction. This result is due tothe fact that, in the former situation, the probability of find-ing the donor electron around the impurity is higher than inthe later case. For xi = 0 the situation is reversed. It shouldalso be noted that in that case (xi = −Lx/2), especially forLx = 500 Å, the variation of the donor binding energy ver-sus the electric field is quite different with respect to that ofthe on-center donor impurity. In that case, the binding energydecreases as −F increases.

In Fig. 3a and b we show the results for the binding en-ergy of a donor in a GQWW with the impurity on the rightside of the well as a function of the electric field for severalLx values. As seen in Fig. 3a and b the sensitivity of the donorbinding energy to the electric field applied in the +x-directionis not appreciable for all considered Lx values. By comparingFigs. 3a and 2a it is observed that the binding energy is notdegenerate for impurity states corresponding to symmetricpositions of the impurity, due to the asymmetric character ofthe donor impurity wavefunctions in the graded quantum well.Whereas the donor binding energy in conventional symmet-

1056 Applied Physics A – Materials Science & Processing

FIGURE 2 a Binding energy of a donor impurity localized at the left sideof a GQWW as a function of the electric field applied in the ±x-direction forLz = 50 Å and different Lx values. b Binding energy of a donor impurity lo-calized at the left side of a GQWW as a function of the electric field appliedin the ±x-direction for Lz = 200 Å and different Lx values

ric quantum-well wires changes symmetrically with respectto the direction of the electric field, this variation of the bind-ing energy behaves differently in GQWWs, as shown in Figs.2 and 3. By comparing Figs. 2 and 3, it is also observed thatfor zero electric field, the binding energy is not degenerate forimpurity states corresponding to the symmetrical positions ofthe impurity, due to the fact that in the asymmetric GQWWsthe wave functions of the donor impurity are asymmetric forF = 0. The donor binding energy increases slightly with in-creasing −F up to 20 kV/cm. With further increases in the ap-

FIGURE 3 a Binding energy of a donor impurity localized at the right sideof a GQWW as a function of the electric field applied in the ±x-direction forLz = 50 Å and different Lx values. The dashed line indicates the result forthe SQWW with Lx = 50 Å. b Binding energy of a donor impurity localizedat the right side of a GQWW as a function of the electric field applied in the±x-direction for Lz = 200 Å and different Lx values

plied field, the binding energy increases rapidly to ∼ 13 meVfor −F = 60 kV/cm, since at a critical field (20 kV/cm) thegraded quantum well is completely leveled off and beyondthis field, the electronic probability density around the im-purity increases with increasing −F. The binding energy ofa hydrogenic impurity in the SQWW with Lx = 50 Å andxi = Lx/2 is also given in Fig. 3a. By comparing the resultsfor the SQWW with that of a previous study [19] we con-clude that, for the impurity located at the corner of the SQWW,

KASAPOGLU et al. The electric field dependence of a donor impurity in graded GaAs quantum wires 1057

the donor binding energy for small Lx values in the finiteand infinite confinement potentials are nearly the same. ForLx = Lz = 50 Å and xi = −Lx/2, the donor binding energyobtained by Montes et al. [19] is ∼ 21 meV, and our value inthis case is 20.98 meV.

In Fig. 4a we present the variation of the binding en-ergy of a hydrogenic impurity as a function of the appliedelectric field for the SQWW and GQWW with Lx = 500 Å,Lz = 50 Å, and xi = 0. As expected for the +F case, the donor

FIGURE 4 Binding energy of a donor impurity localized at the center ofa GQWW and a SQWW (dashed line) as a function of the electric fieldapplied in the ±x-direction for Lz = 50 Å and Lx = 500 Å. b Binding en-ergy of a donor impurity localized at the left and right sides of a GQWWand a SQWW (dashed line) as a function of the electric field applied in the±x-direction for Lz = 50 Å and Lx = 500 Å

binding energy in the SQWW decreases rapidly with elec-tric field and approaches that of the GQWW. On the otherhand, due to the asymmetric character of the donor electronwavefunction, the positive electric field (+F) dependence ofthe binding energy in the GQWW is very weak The impuritybinding energy in the GQWW for the electric field appliedin the −x-direction (−F) increases with increasing field andreaches a maximum value, since the probability of findingthe electron around the impurity increases with increasingfield. After a certain −F value (20 kV/cm), the donor elec-tron moves to the right side of the well. Thus the impuritybinding energy begins to decrease with increasing −F, sincethe Coulombic interaction between the electron and donor im-purity weakens. The variation of the donor binding energy inthe SQWW is qualitatively consistent with results obtained byDuque et al. [23], who considered an infinite potential model.It should be noted that when the position of the electronic dis-tribution drops off to 1/2 of its peak value, it is approximatelyone effective Bohr radius from the impurity. The binding en-ergy corresponds approximately to one effective Rydberg. Forexample in the SQWW with Lx = 200 Å and xi = Lx/2, thissituation is observed at F = 40 kV/cm.

In Fig. 4b the variation of the binding energy of a hy-drogenic impurity as a function of the applied electric fieldfor the SQWW with xi = −Lx/2 and for the GQWW withxi = ±Lx/2, Lz = 50 Å, and Lx = 500 Å is given. As canbe seen in this figure, at a critical value of −F, the bind-ing energy of the donor located at the position xi = −Lx/2 isequal to that of the impurity located at the position xi = Lx/2,since, at this critical field, the graded quantum well is com-pletely leveled off and the structure behaves as a symmetricalone. As expected at zero electric field, for the impurity pos-ition xi = −Lx/2 the binding energy difference between theSQWW and GQWW models is larger than that for the impu-rity located at xi = Lx/2. This behavior can be explained witha similar argument to that used above for explaining the vari-ation of the binding energy in the electric field. The resultspresented here show that in the GQWW, especially for largewire dimensions, a negative electric field affects the donorbinding energy more than an electric field in the opposite di-rection. One should also note that for both the GQWW andSQWW cases, the donor binding energy decreases with thenegative electric field, and approaches a constant value of∼ 2 meV. This may be understood as a consequence of thedonor electron confinement.

4 Conclusions

In summary; we have investigated the effect of anelectric field on the binding energy of a shallow donor impu-rity in GQWWs and SQWWs for different impurity positions.The calculations were performed within the effective massapproximation and using a variational method. We concludethat, in GQWWs with large dimensions, the applied electricfield produces an important effect on the binding energy. Wefound that the degeneracy of symmetrically positioned impu-rities is not observed in GQWWs. The changes in the donorbinding energy strongly depend not only on the quantum con-finement, but also on the direction of the electric field andon the impurity position. Thus the direction of the applied

1058 Applied Physics A – Materials Science & Processing

electric field can be used as a tuning parameter for the elec-tronic structure of a shallow donor impurity in a GQWW. Thedirection of the electric field dependence of the donor bind-ing energy should have important consequences for opticalstudies and transport measurements on QWWs. The obtainedresults were compared with those of previous studies. In con-clusion, the method used in this study is capable of describingthe correct behavior of shallow donor impurities in a GQWWand a SQWW in an external field.

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