AlxGa1−xAs double quantum wells under an electric field

Superlattices and Microstructures 45 (2009) 618–623

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Binding energies of donor impurities in modulation-doped GaAs/Alx Ga1−x As double quantum wells under an electric field E. Kasapoglu a,∗ , F. Ungan a , H. Sari a , I. Sökmen b a

Cumhuriyet University, Physics Department, 58140 Sivas, Turkey

b

Dokuz Eylül University, Physics Department, 35160 Izmir, Turkey

article

info

Article history: Received 17 January 2009 Received in revised form 12 February 2009 Accepted 18 February 2009 Available online 26 March 2009 Keywords: Double quantum well Impurity binding energy Modulation doped double quantum wells

a b s t r a c t In this study, we have investigated theoretically the binding energies of shallow donor impurities in modulation-doped GaAs/Al0.33 Ga0.67 As double quantum wells (DQWs) under an electric field which is applied along the growth direction for different doping concentrations as a function of the impurity position. The electronic structure of modulation-doped DQWs under an electric field has been investigated by using a self-consistent calculation in the effective-mass approximation. The results obtained show that the carrier density and the depth of the quantum wells in semiconductors may be tuned by changing the doping concentration, the electric field and the structure parameters such as the well and barrier widths. This tunability gives a possibility of use in many electronic and optical devices. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Modulation-doped structures have been widely used in high-speed electronic devices. The charged carriers can be controlled by appropriately doping the barriers without the need for optical pumping, which also has the desirable effect of having the states below the Fermi level filled at low temperatures. Thus, the electron (or hole) gas in quantum wells is an ideal quantum liquid system which 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: [email protected] (E. Kasapoglu).

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

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Fig. 1. A schematic representation of a DQW system. Ldm and Ldp are doped Al0.33 Ga0.67 As thicknesses, Ls1 and Ls2 are undoped Al0.33 Ga0.67 As thicknesses, Lw 1 and Lw 2 are GaAs quantum well widths, Lb is the barrier width and LD = ZL + ZR is the total thickness of the DQW structure. The barrier layers have the same thickness (Ls1 = Ls2 = 10 Å and Ldm = Ldp ) and the same doping 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 twodimensional system at the semiconductor heterojunction interface between GaAs and Alx Ga1−x As [2–5]. Since the ionized impurity scattering is greatly reduced by separating the electrons from their parent donors and the Coulomb scattering is reduced by the screening effects due to the extremely high density of the two-dimensional electron gas (2DEG), high electron mobilities can be obtained in these structures. Under equilibrium conditions, electrons in the donor levels of Alx Ga1−x As are transferred to the GaAs layer, leading to considerable band bending. Modulation doping of coupled double quantum well (CDQW) structures creates two parallel 2DEG layers [6–10]. This additional electronic degree of freedom in the growth direction can be 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/Al0.33 Ga0.67 As DQWs with different well and 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 by using 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 at the centre of the structure. The system consists of two GaAs wells with widths Lw1 and Lw2 , separated by an undoped Alx Ga1−x As barrier with width Lb , surrounded by two Alx Ga1−x As barrier layers each side. Each of these two barrier layers consists of an undoped spacer layer with widths Ls1 and Ls2 . The depletion 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 =−

h¯ 2 2m∗

2 E 2 + V (z ) − e , ∇ εEr

(1)

where m∗ is the electron effective mass, εpis the static dielectric constant, p Er is the distance between the carrier and the donor impurity site (r = ρ 2 + (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 of the electron and impurity along the structure, respectively and V (z ) = VH (z ) + Vconf (z ) + eFz, F is the strength of the electric field, VH (z ) is the effective Hartree potential and Vconf (z ) is the confinement potential in the z-direction of the DQW. The functional form of the confinement potential is

 0,   −V o ,  Vconf (z ) = 0,    −V o , 0,

z < −(Lb /2 + Lw1 ) −(Lb /2 + Lw1 ) < z < −Lb /2 −Lb /2 < z < Lb /2 Lb /2 < z < Lb /2 + Lw2 z > Lb /2 + Lw2 .

(2)

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For the solution of Schrödinger’s equation: h¯ 2

 −

d2

 + V (z ) ψi (z ) = Ei ψi (z ).

2m∗ dz 2

(3)

The Hartree potential is determined from the Poisson equation: d2 VH (z ) dz 2

=−

4 π e2

ε

[N (z ) − Nd (z )]

(4)

with N (z ) =

nd X

ni |ψi (z )|2

(5)

i =1

where Nd (z ) is the total density of ionized dopants, ψi (z ) is the wave function, which is obtained from 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 = ni =

m∗ kB T

π h¯ 2 m∗ kB T

π h¯ 2

ln(1 + exp[(EF − Ei )/kB T ])

(6a)

(EF − Ei );

(6b)

i is the sub-band index, kB is the Boltzmann constant and EF is the Fermi energy; at the lowtemperature limit (T → 0), the Fermi energy can be taken as the donor level, which is supposed to lie at an energy EF = 0.070 eV below the conduction band of the GaAs layers [11]. All donors are assumed to be ionized, i.e. Ldm × Nd1 + Ldp × Nd2 =

nd X

ni

(7)

i =1

where Nd1 is the doping concentration for thickness Ldm and Nd2 is the doping concentration for thickness Ldp . The potential profile, density profile, sub-band energies and sub-band populations are obtained 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:

 s

 2 ρ ( z − zi )2  ψi (z ) Ψ (ρ, z ) = N exp − + λ2 β2

(8)

where N is the normalization constant, and λ and β are variational parameters. The ground state impurity energy is evaluated by minimizing the expectation value of the Hamiltonian in Eq. (1) with respect to λ and β . The ground state donor binding energy is calculated as Eb = Ei − min hΨ | H |Ψ i λ,β

(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 DQWs which have well widths Lw 1 = Lw 2 = 75 Å and barrier widths Lb = 25 Å and Lb = 60 Å for different doping concentrations versus the impurity position and the variations of the confinement potential profile, ground state energy level and the squared wave function belonging to this energy level 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

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

energy as a function of the position behaves like a map of the spatial distribution of the ground state wave function of the electron. The binding energy for donor impurities located in the barrier region is smaller than that for the well regions since the probabilities of finding the electrons in the wells are higher than for the barrier. As the doping concentration increases (see Fig. 2(c) and (b)), the band bending increases due to the increase of the charge density in the doped layer and this gives rise to the formation of deeper quantum wells. Thus, the binding energies of shallow donor impurities increase in these regions since the probabilities of finding electrons in deep regions of the wells increase. As seen in Fig. 2(b), when the barrier width increases the coupling between the wells decreases and so the binding energies for donor impurities located in the well regions increase while they decrease for donor impurities located in the barrier region since the probabilities of finding electrons in the barrier region decrease due to the weak coupling.

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Fig. 3. The variation of the ground state binding energies of impurities in a modulation-doped DQW which has well widths Lw 1 = Lw 2 = 100 Å as a function of the impurity position for different doping concentrations and barrier widths: (a) Lb = 25 Å and (b) Lb = 60 Å.

When the electric field is applied, a clear distortion from the symmetric results presented for F = 0 is achieved, due to the electronic localization induced by the field which pushes the electrons towards the opposite direction to the electric field, and the binding energies of donor impurities located in the left well region increase. Furthermore, this increase in the binding energy becomes more pronounced with 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 Lw 1 = Lw 2 = 100 Å and barrier widths Lb = 25 Å and Lb = 60 Å for different doping concentrations versus the impurity position, respectively. In addition to the explanations mentioned above, the effects of the doping concentration and the electric field depend weakly on the binding energy in the small well widths and this dependence becomes important as the quantum well width is increased. 4. Conclusion In summary, the effects of an electric field which is applied along the growth direction and doping concentration on the binding energies of shallow donor impurities in modulation-doped GaAs/Al0.33 Ga0.67 As DQWs with different well and barrier widths have been investigated. The electronic structure of modulation-doped DQWs under the electric field has been investigated by using 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 the structure parameters such as the well and barrier widths. This tunability gives a possibility of use in many electronic and optical devices. To the best of our knowledge this is the first study for the binding energies of shallow donor impurities in modulation-doped GaAs/Al0.33 Ga0.67 As DQWs, so the results obtained here cannot be compared with any previous results. It is hoped that the present work will stimulate further experimental activities in semiconductor heterostructures. References [1] D. Pines, P. Noziere, The Theory of Quantum Liquids, Benjamin, New York, 1966. [2] R. Dingle, H.L. Störmer, A.C. Gossard, W. Weigmann, Appl. Phys. Lett. 33 (1978) 665. [3] L. Esaki, R. Tsu, IBM Res. Rep. RC-2418 (1969).

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