Superlattices and Microstructures 45 (2009) 618–623

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Superlattices and Microstructures

journal homepage: www.elsevier.com/locate/superlattices

Binding energies of donor impurities in modulation-dopedGaAs/AlxGa1−xAs double quantum wells under an electricfieldE. Kasapoglu a,∗, F. Ungan a, H. Sari a, I. Sökmen ba Cumhuriyet University, Physics Department, 58140 Sivas, Turkeyb Dokuz Eylül University, Physics Department, 35160 Izmir, Turkey

a r t i c l e i n f o

Article history:Received 17 January 2009Received in revised form12 February 2009Accepted 18 February 2009Available online 26 March 2009

Keywords:Double quantum wellImpurity binding energyModulation doped double quantum wells

a b s t r a c t

In this study, we have investigated theoretically the bind-ing energies of shallow donor impurities in modulation-dopedGaAs/Al0.33Ga0.67As double quantumwells (DQWs) under an elec-tric field which is applied along the growth direction for differentdoping concentrations as a function of the impurity position. Theelectronic structure of modulation-doped DQWs under an electricfield has been investigated by using a self-consistent calculation inthe effective-mass approximation. The results obtained show thatthe carrier density and the depth of the quantumwells in semicon-ductors may be tuned by changing the doping concentration, theelectric field and the structure parameters such as thewell and bar-rier widths. This tunability gives a possibility of use in many elec-tronic and optical devices.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Modulation-doped structures have been widely used in high-speed electronic devices. Thecharged carriers can be controlled by appropriately doping the barriers without the need for opticalpumping, which also has the desirable effect of having the states below the Fermi level filled at lowtemperatures. Thus, the electron (or hole) gas in quantum wells is an ideal quantum liquid systemwhich can be used to investigate many-body interactions [1].Double quantum wells (DQWs) have been intensively investigated because of their potential

applications in advanced opto-electronic devices. Recently, the evolution of the growth techniques

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

0749-6036/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.spmi.2009.02.011

E. Kasapoglu et al. / Superlattices and Microstructures 45 (2009) 618–623 619

Fig. 1. A schematic representation of a DQW system. Ldm and Ldp are doped Al0.33Ga0.67As thicknesses, Ls1 and Ls2 are undopedAl0.33Ga0.67As thicknesses, Lw1 and Lw2 are GaAs quantum well widths, Lbis the barrier width and LD = ZL + ZR is the totalthickness of the DQW structure. The barrier layers have the same thickness (Ls1 = Ls2 = 10 Å and Ldm = Ldp) and the samedoping concentration (Nd1 = Nd2).

like molecular beam epitaxy (MBE) and metal–organic chemical vapour deposition (MOCVD)combined with the use of the modulation doping technique made it possible to achieve a new two-dimensional system at the semiconductor heterojunction interface between GaAs and AlxGa1−xAs[2–5]. Since the ionized impurity scattering is greatly reduced by separating the electrons from theirparent donors and the Coulomb scattering is reduced by the screening effects due to the extremelyhigh density of the two-dimensional electron gas (2DEG), high electron mobilities can be obtainedin these structures. Under equilibrium conditions, electrons in the donor levels of AlxGa1−xAs aretransferred to the GaAs layer, leading to considerable band bending. Modulation doping of coupleddouble quantum well (CDQW) structures creates two parallel 2DEG layers [6–10]. This additionalelectronic degree of freedom in the growthdirection canbe controlled by varying the barrier thickness,external gate voltages and external fields.In this paper, the effects of the electric field and doping concentration on the binding energy

of shallow donor impurities in modulation-doped GaAs/Al0.33Ga0.67As DQWs with different welland barrier widths have been investigated. An electric field is applied along the growth direction(z-direction). The electronic structure of modulation-doped DQW system has been investigated byusing a self-consistent calculation in the effective-mass approximation.

2. Theory

A schematic representation of the DQW system is given in Fig. 1. The origin of the z-axis is taken atthe centre of the structure. The system consists of two GaAswells withwidths Lw1 and Lw2, separatedby an undoped AlxGa1−xAs barrier with width Lb, surrounded by two AlxGa1−xAs barrier layers eachside. Each of these two barrier layers consists of an undoped spacer layer with widths Ls1 and Ls2. Thedepletion lengths Ldm and Ldp are determined self-consistently in our calculations.In the effective mass approximation, the Hamiltonian for a shallow-donor impurity of a DQW

system under an electric field is given by

H = −h̄2

2m∗E∇2+ V (z)−

e2

εEr, (1)

wherem∗ is the electron effectivemass, ε is the static dielectric constant, Er is the distance between thecarrier and the donor impurity site (r =

√ρ2 + (z − zi)2) and ρ (=

√x2 + y2) is the distance between

the electron and impurity in the (x–y) plane, e is the electron charge, z and zi are the coordinates ofthe electron and impurity along the structure, respectively and V (z) = VH(z)+Vconf(z)+ eFz, F is thestrength of the electric field, VH(z) is the effective Hartree potential and Vconf(z) is the confinementpotential in the z-direction of the DQW. The functional form of the confinement potential is

Vconf(z) =

0, z < −(Lb/2+ Lw1)−V o, −(Lb/2+ Lw1) < z < −Lb /20, −Lb/2 < z < Lb/2−V o, Lb/2 < z < Lb/2+ Lw20, z > Lb/2+ Lw2.

(2)

620 E. Kasapoglu et al. / Superlattices and Microstructures 45 (2009) 618–623

For the solution of Schrödinger’s equation:(−h̄2

2m∗d2

dz2+ V (z)

)ψi(z) = Eiψi(z). (3)

The Hartree potential is determined from the Poisson equation:

d2VH(z)dz2

= −4πe2

ε[N(z)− Nd(z)] (4)

with

N(z) =nd∑i=1

ni |ψi(z)|2 (5)

where Nd(z) is the total density of ionized dopants, ψi(z) is the wave function, which is obtainedfrom Eq. (1), nd is the number of filled states, ni is the temperature-dependent (or zero-temperature)number of electrons per unit area in the ith sub-band, given by Eq. (6a) (or Eq. (6b)):

ni =m∗kBTπ h̄2

ln(1+ exp[(EF − Ei)/kBT ]) (6a)

ni =m∗kBTπ h̄2

(EF − Ei); (6b)

i is the sub-band index, kB is the Boltzmann constant and EF is the Fermi energy; at the low-temperature limit (T → 0), the Fermi energy can be taken as the donor level, which is supposedto lie at an energy EF = 0.070 eV below the conduction band of the GaAs layers [11]. All donors areassumed to be ionized, i.e.

Ldm × Nd1 + Ldp × Nd2 =nd∑i=1

ni (7)

where Nd1 is the doping concentration for thickness Ldm and Nd2 is the doping concentration forthickness Ldp. The potential profile, density profile, sub-band energies and sub-band populations areobtained from the self-consistent solution of equations (3)–(7).We choose the trial wave function as a product of the three-dimensional wave function in the

Coulomb potential with the ground state wave function of the DQW:

Ψ (ρ, z) = N exp

−√ρ2λ2+(z − zi)2

β2

ψi(z) (8)

where N is the normalization constant, and λ and β are variational parameters. The ground stateimpurity energy is evaluated by minimizing the expectation value of the Hamiltonian in Eq. (1) withrespect to λ and β .The ground state donor binding energy is calculated as

Eb = Ei −minλ,β〈Ψ |H |Ψ 〉 (9)

where Ei is the ground-state energy of an electron obtained from Eq. (2) without the impurity.

3. Results and discussion

The variation of the ground state binding energies of impurities in the modulation-doped DQWswhich have well widths Lw1 = Lw2 = 75 Å and barrier widths Lb = 25 Å and Lb = 60 Å fordifferent doping concentrations versus the impurity position and the variations of the confinementpotential profile, ground state energy level and the squared wave function belonging to this energylevel according to doped concentrations are given in Fig. 2(a), (b), (c) and (d), respectively. Solid(dashed) curves correspond to F = 0 (F = 10 kV/cm). As seen in Fig. 2(a) and (b), the impurity binding

E. Kasapoglu et al. / Superlattices and Microstructures 45 (2009) 618–623 621

Fig. 2. The variation of the ground state binding energies of impurities in a modulation-doped DQW which has well widthsLw1 = Lw2 = 75 Å as a function of the impurity position for different doping concentrations and barrier widths: (a) Lb = 25 Åand (b) Lb = 60 Å. The variation of the confinement potential profile, ground state energy level and the squared wave functionbelonging to this energy level for (c) Nd = 1 × 1017 cm−3 and (d) Nd = 10 × 1017 cm−3 . Solid (dashed) curves correspond toF = 0 (F = 10 kV/cm).

energy as a function of the position behaves like a map of the spatial distribution of the ground statewave function of the electron. The binding energy for donor impurities located in the barrier regionis smaller than that for the well regions since the probabilities of finding the electrons in the wellsare higher than for the barrier. As the doping concentration increases (see Fig. 2(c) and (b)), the bandbending increases due to the increase of the charge density in the doped layer and this gives rise to theformation of deeper quantum wells. Thus, the binding energies of shallow donor impurities increasein these regions since the probabilities of finding electrons in deep regions of the wells increase. Asseen in Fig. 2(b), when the barrier width increases the coupling between the wells decreases and sothe binding energies for donor impurities located in the well regions increase while they decrease fordonor impurities located in the barrier region since the probabilities of finding electrons in the barrierregion decrease due to the weak coupling.

622 E. Kasapoglu et al. / Superlattices and Microstructures 45 (2009) 618–623

Fig. 3. The variation of the ground state binding energies of impurities in a modulation-doped DQW which has well widthsLw1 = Lw2 = 100 Å as a function of the impurity position for different doping concentrations and barrier widths: (a) Lb = 25 Åand (b) Lb = 60 Å.

When the electric field is applied, a clear distortion from the symmetric results presented for F = 0is achieved, due to the electronic localization induced by the field which pushes the electrons towardsthe opposite direction to the electric field, and the binding energies of donor impurities located in theleft well region increase. Furthermore, this increase in the binding energy becomes more pronouncedwith the increase of doping concentration.Fig. 3(a) and (b) show the variation of the ground state binding energies of impurities in

modulation-doped DQWs which have well widths Lw1 = Lw2 = 100 Å and barrier widths Lb = 25 Åand Lb = 60 Å for different doping concentrations versus the impurity position, respectively. Inaddition to the explanationsmentioned above, the effects of the doping concentration and the electricfield depend weakly on the binding energy in the small well widths and this dependence becomesimportant as the quantum well width is increased.

4. Conclusion

In summary, the effects of an electric field which is applied along the growth direction anddoping concentration on the binding energies of shallow donor impurities in modulation-dopedGaAs/Al0.33Ga0.67As DQWs with different well and barrier widths have been investigated. Theelectronic structure of modulation-doped DQWs under the electric field has been investigated byusing a self-consistent calculation in the effective-mass approximation.The results obtained show that the carrier density and the depth of quantum wells in

semiconductors may be tuning by changing the doping concentration, the electric field and thestructure parameters such as the well and barrier widths. This tunability gives a possibility of usein many electronic and optical devices. To the best of our knowledge this is the first study for thebinding energies of shallow donor impurities in modulation-doped GaAs/Al0.33Ga0.67As DQWs, so theresults obtained here cannot be comparedwith any previous results. It is hoped that the present workwill stimulate further experimental activities in semiconductor heterostructures.

References

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