Shallow donors in the triple-graded quantum well under the hydrostatic pressure and external fields

ARTICLE IN PRESS

Physica B 373 (2006) 280–283 www.elsevier.com/locate/physb

Shallow donors in the triple-graded quantum well under the hydrostatic pressure and external fields E. Kasapoglua,, H. Saria, I. So¨kmenb a

Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey b Department of Physics, Dokuz Eylu¨l University, I˙zmir, Turkey

Received 14 October 2005; received in revised form 17 November 2005; accepted 24 November 2005

Abstract The binding energy of the donor in the triple-graded GaAs-(Ga,Al)As quantum well under the hydrostatic pressure, electric and magnetic fields are calculated by using a variational approach. The results have been obtained in the presence of hydrostatic pressure, magnetic and electric fields applied along the growth direction as a function of the impurity position without consideration of a mass mismatch or dielectric mismatch. r 2005 Elsevier B.V. All rights reserved. PACS: 71.55.Eq; 73.61.Ey Keywords: Triple quantum wells; Hydrogenic impurities; Graded well; Hydrostatic pressure

1. Introduction Currently noticeable interest is focused on the electronic and optical properties of double and triple quantum wells (DQW’s and TQW’s) and their potential application to optical devices [1–3]. One of the most interesting phenomena in quantum well structures is anticrossing of eigenenergies and localization or delocalization of the wavefunctions [4]. The electronic states in multi-quantum well structures are known to be easily modified by applying an external electric field, resulting in coupling and decoupling of the wavefunctions. These effects cause a significiant change in the optical properties Among them Wannier–Stark localization [5–7] in superlattices and quantized level mixing in multi-quantum well structures have been investigated [8]. Impurities in semiconducting heterostructures are known to promote a number of qualitative changes in electronic and optical properties which may be properly controlled by an adequate choice of the sample geometry and external fields. Therefore, hydrogenic impurity states, Corresponding author.

E-mail address: [email protected] (E. Kasapoglu). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.155

binding energy and impurity related optical spectra have been calculated in a large number of different semiconductor heterostructures [9–18]. The hydrostatic pressure effects on the electronic and impurity states in low-dimensional heterostructures such as QWs, QWWs and QDs have been studied in several theoretical works [19–24]. Elabsy [19] has calculated the effects of the hydrostatic pressure on the binding energy of donor impurities in Qws, finding that the binding energy increases with increasing hydrostatic pressure for a certain well thickness and temperature. Oyoko et al. [24] have studied the effects of an unaxial stress on the binding energy of shallow impurities in parallelpiped-shaped GaAs/GaAlAs QDs. They have found that the binding energy increases almost linearly with applied stress and diminishes with the size of the structure. In this paper, we considered the combined effects of hydrostatic pressure, electric field and magnetic field on the binding energy of shallow donor impurity in a coupled GaAs-(Ga,Al)As triple-graded quantum well (TGQW) structure with finite potential-energy barriers, without consideration of a mass mismatch or dielectric mismatch.

ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 373 (2006) 280–283

281

2. Theory 300

The triple-graded quantum well (TGQW) heterostructure is formed by a central GaAs quantum well coupled through thin barriers to graded wells of on both sides. The schematic representation of TGQW structure is given in Fig. 1. The Hamiltonian of a donor impurity located in the TGQW, and in the presence of hydrostatic pressure, magnetic and electric fields may be written as
 _2 q 1 q 1 q2 H¼
þ þ 2me qr2 r qr r2 qF2 _2 d2 e_B e2 L
 þ þ B2 r2 z 2me ðPÞ dz2 2me ðPÞc 8me c2 e2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , þ V ðz; PÞ þ jejFz
eðPÞ r2 þ ðz
zi Þ2

Lb

Lo L 200

V/2

100

0 -c -100

-1.5

-b

-1

-0.5

-a

0

0

a

b

0.5

c

1

1.5

Fig. 1. The schematic representation of the TGQW.

ð1Þ

where the hydrostatic pressure is applied only to the zdirection. me ðPÞ is the effective mass of the electron in GaAs as a function of the hydrostatic pressure [25–27], e(P) is the pressure dependence of static dielectric constant [28], e is the elementary charge, F is the electric field, and   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 is the distance between the electron and impurity in the (x–y) plane. The confinement potential V(z, P) of TGQW structure is given by 8 V ðPÞ; zo
c; > > > V ðPÞ > >
ðz þ bÞ;
cozo
b; > 2LðPÞ > > > >
bozo
a; > < V ðPÞ;
aozoa; (2) V ðz; PÞ ¼ 0; > > > V ðPÞ; aozob; > > > > V ðPÞ > ðz
bÞ; bozoc; > 2LðPÞ > > : V ðPÞ; z4c;

Using the variational method, it is possible to associate a trial wave function, which is an approximated eigenfunction of the Hamiltonian described in Eq. (1). The groundstate wave function of the impurity is given by cðrÞ ¼ cðzÞ Fðr; aÞ,

(7)

where c(z) is the first subbands wave function of the electron, which is exactly obtained from the one-dimensional Schro¨dinger equation in the z-direction and the wave function in the (x–y) plane is chosen to be the wave function of the ground state of a two-dimensional hydrogen-like atom [9,10]:
 1 2 1=2 Fðr; aÞ ¼ exp ð
r=aÞ (8) a p in which a is a variational parameter. The ground-state impurity energy is evaluated  by minimizing  the expectation value of the Hamiltonian, cðrÞjHjcðrÞ with respect to a. The ground-state donor binding energy is given by [31,32]   E B ¼ E z þ g
cðrÞjHjcðrÞ , (9)

where jaj ¼ Lo ðPÞ=2, jbj ¼ Lo ðPÞ=2 þ Lb ðPÞ, jcj ¼ Lo ðPÞ=2 þLb ðPÞ þ LðPÞ, L(P) is the graded quantum well’s width, Lb ðPÞ is the barrier’s width, Lo ðPÞ is the central quantum well’s width. The pressure dependence of the well and barrier widths is obtained from Refs. [19–21]. The barrier height is given by [29]

where Ez is the ground state energy of the electron obtained from the Schro¨ndinger equation in the z-direction without the impurity, and g ¼ e_B=2mn c is the first Landau level.

V ðz; PÞ ¼ Qc DE Gg ðx; PÞ,

3. Results and discussion

(3)

where Qc( ¼ 0.6) is the conduction band offset parameter [29], DE Gg ðx; PÞ is the band gap difference between QW and the barrier matrix at the G-point as a function of pressure, which for an aluminum fraction x ( ¼ 0,3) is given by DE Gg ðx; PÞ ¼ DE Gg ðxÞ þ PDðxÞ,

(4)

where DE Gg ðxÞ ¼ ð1:155x þ 0:37x2 Þ eV

(5)

is the variation of the energy gap difference and the pressure coefficient of the band gap D(x) is given by [30] DðxÞ ¼ ½
ð1:3  10
3 Þx eV=kbar:

(6)

We have chosen the following parameters in the numerical calculations: outer graded quantum well’s width ˚ the barrier’s width Lb ¼ 25 A. ˚ L ¼ 100 A, As known, both electron mass and dielectric constant mismatches in the barrier and wells are important especially in the small dimensions of the GaAs=Ga1
x Alx As QWs. For instance; as in Ref. [33], while the mass mismatch decreases the exciton binding energy approximately by 1 meV, the dielectric constant mismatch increases it approximately by 1 meV for a well width of 50 A˚. Thus, it has been observed that the combined effect of mass and the dielectric constant mismatches does not alter the

ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 373 (2006) 280–283

exciton binding energy in Ref. [33]. Therefore, we have considered both the electron mass and dielectric constant as constants through the heterostructure. We have also used the same method in our previous studies [34–36]. The variation of the ground state binding energy of a hydrogenic donor impurity as a function of the normalized impurity position in the TGQW under the hydrostatic pressure, electric field and magnetic field which are applied to the z-direction is given in Fig. 2 for the central well width ˚ The insets in these Lo ¼ 25 A˚ and in Fig. 3 for Lo ¼ 50 A. figures show the spatial distribution of the ground state wave function of the electron in the TGQW in the absence of the pressure, electric and magnetic field for Lo ¼ 25 A˚ ˚ respectively. As seen in these figures, impurity and 50 A, binding energy as a function of the position behaves like a map of the spatial distribution of the ground state wave function of the electron. For the central well width ˚ the probability of finding of the electron in Lo ¼ 25 A, the outer graded quantum wells is higher than in the central quantum well. So, the binding energy for the impurity located in the central well is smaller than in the graded quantum wells. When we increase the central well width, the probability of finding of the electron in the central well becomes larger than in the outer graded quantum wells and thus, the binding energy increases for the impurity located in the central well. As the hydrostatic pressure increases the well width and dielectric constant decrease, the effective 20 ----

F=0

250

F = 30 kV/cm

200 150

16

100 50 0

-1.5

-1

-0.5

0

0.5

1

1.5

B = 10 T

Eb(meV)

12

2 1 8

2 1 B=0

4 1 2

P=0 P = 30 kbar Lo = 25

1

2

0 -2

-1

0 ~ zi

1

2

Fig. 2. The variation of the ground state binding energy of a hydrogenic donor impurity as a function of the normalized impurity position in the TGQW under the hydrostatic pressure, electric field and magnetic field ˚ The which are applied to the z-direction for central well width Lo ¼ 25 A. inset show the potential profile for system and amplitude of normalized subband wave function of electron,jcð~zÞj2 , versus the normalized position,~z, in the absence of the hydrostatic pressure, electric field and magnetic field.

20

250

F=0 F = 30 kV/cm

----

200 150 100

16

2

B = 10 T

50

2

0

-1.5

-1

-0.5

0

0.5

1

1.5

1 1 12 Eb(meV)

282

2

2

1

8

1 B=0 4 1 P=0 2 P = 30 kbar Lo = 50 0 -2

-1

0 ~z i

1

2

Fig. 3. The variation of the ground state binding energy of a hydrogenic donor impurity as a function of the normalized impurity position in the TGQW under the hydrostatic pressure, electric field and magnetic field ˚ The which are applied to the z-direction for central well width Lo ¼ 50 A. inset show the potential profile for system and amplitude of normalized subband wave function of electron,jcð~zÞj2 , versus the normalized position,~z, in the absence of the hydrostatic pressure, electric field and magnetic field.

mass of electron increases [19–21], leading to more confinement in the well in the z-direction of the impurity electron and so the impurity binding energy increases for all impurity positions. When the electric field is applied, since the electron shifts to the left side of the well in the z-direction with the effect of electric field the probability of finding the electron in the left side of the well increases, while it decreases in the right side of the well so, we observe that for impurity position close to the left side of the well the binding energy is higher than in right side. As the magnetic field increases, the wave function of the electron is squeezed due to the extra confinement of the magnetic field. This squeezing leads to an increase in the binding energy, as we see in Figs. (2) and (3). The results obtained, show that the expected enhancement in the impurity binding energy induced by the magnetic field is much more pronounced in the regions where the electron distribution is more localized. 4. Conclusion As a result, we have found that the binding energy of hydrogenic donor impurity strongly depends on the geometry of the TQW system, hydrostatic pressure and external fields. We have considered both the electron mass and dielectric constant as constants through the

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