Shallow donor impurities in different shaped double quantum wells under the hydrostatic pressure and applied electric field

ARTICLE IN PRESS

Physica B 362 (2005) 56–61 www.elsevier.com/locate/physb

Shallow donor impurities in different shaped double quantum wells under the hydrostatic pressure and applied electric field E. Kasapoglua,, H. Saria, I. Sokmenb a

Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey b Department of Physics, Dokuz Eylu¨l University, I˙zmir, Turkey Received 4 January 2005; accepted 27 January 2005

Abstract The combined electric field and hydrostatic pressure effects on the binding energy of the donor impurity in double triangle quantum well (DTQW), double graded (DGQW) and double square (DSQW) GaAs–(Ga,Al)As quantum wells are calculated by using a variational technique within the effective-mass approximation. The results have been obtained in the presence of an electric field applied along the growth direction as a function of hydrostatic pressure, the impurity position, barrier width and the geometric shape of the double quantum wells. r 2005 Elsevier B.V. All rights reserved. PACS: 71.55.Eq; 71.55.
i Keywords: Hydrogenic impurities; Double graded well; Double triangle well; Hydrostatic pressure

1. Introduction A double-quantum well (DQW) structure incorporating a two-dimensional electron gas (2DEG) in each quantum well (QW) is a unique system from the viewpoint of both device applications and fundamental physics. One feature making this system particularly interesting is the ability to manipulate the electron wave functions using Corresponding author. Tel.: +90 3462191010;

fax: +90 3462191186. E-mail address: [email protected] (E. Kasapoglu).

gates. That is, when the gate biases are turned in such a way that the eigenstates of the two QWs are aligned in energy, the electronic states initially localized in the individual QWs are quantum mechanically coupled, thus forming delocalized states extending over the two QWs. In recent years, a considerable amount of work has been devoted to the study of artificial lowdimensional systems, such as semiconductor heterostructures due to their interesting basic physical properties and the possible technological applications as in photo-detector and optoelectronic devices. Among the various systems under current

0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.475

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investigation, the QWs have attained considerable theoretical experimental attention. The QWs are commonly taken to be symmetric, but the asymmetric QWs give new tunable properties, which are very important for device applications [1–6]. Within the last few years, there has been great interest in the electronic properties in the asymmetric QW structures because they are the ideal systems for the study of terahertz electromagnetic radiation from semiconductor heterostructures [7–14]. This interest is mainly due to the possible applications of quantum devices high-speed electronics and to the generation and detection of terahertz bandwidth signals. In order to get sufficient tuning range, different QW structures such as stepped QW and asymmetric coupled double QW have been tried [15–17]. Various QW structures such as graded-gap, two-step, and coupled asymmetric QW have been investigated in an effort to enhance the electric-field-induced changes [8 and references therein]. In these structures, the Stark shifts, the changes in oscillator strengths and absorption coefficients were predicted theoretically and confirmed experimentally to be larger than changes that occur in conventional square potential QWs [8]. Studies of the effect of hydrostatic pressure have proven to be invaluable in the context of the optical properties of semiconductors and their heterostructures [18]. For a given structure, the difference in energy between the type-I and type-II transitions can be tuned with external hydrostatic pressure in a continuous and reversible manner. Theoretical works related to the effects of hydrostatic pressure on shallow-donor impurity states in single and double GaAs–(Ga,Al)As QWs have been reported [19–22]. These authors have considered the G2X crossover and, as general feature, they have found a linear dependence on the binding energy in the direct gap regime under the applied pressure, while in the indirect gap regime the energy grows with the pressure until reaching a maximum and then it decreases. Additionally, they have shown a redshift in the shallow donor related optical absorption spectra associated with the pressure dependence of the semiconductor band gap. For the single QW

57

case under applied electric field they have found that the electron polarizability decreases as the impurity moves in the opposite direction to the electric field, where the electron cloud is more confined by the electric field and pressure effects. The combined effect of an applied electric field and hydrostatic pressure modifies the photoluminescence line shape and the donor and excitonic related pressure coefficient in low-dimensional semiconductor heterostructures. These effects could be used to tune the output of optoelectronic devices. In this work, we calculate the binding energies for shallow donor impurities in the symmetric double graded, triangle and square GaAs/ Ga1
xAlxAs QWs under hydrostatic pressure and the parallel-applied electric field to the growth direction using a variational technique. The pressure effects will be considered in the direct gap regime for the barrier material. The charge image effects have not been considered. These potential profiles (DGQW and DTQW) for P ¼ 0 and Pa0; and amplitude of normalized subband wave function of electron, jcð~zÞj2 ; versus the normalized position, z~ ¼ L=a0 ; are given in Figs. 1 and 2 for cases (a) F ¼ 0 and (b) F ¼ 50 kV=cm; respectively. Graded potential profile is obtained by changing linearly from zero to 0.3 the aluminium concentration, x, in the Ga1
x Alx As layer. As known, one important aspect of electronic band structure engineering is the realization of graded heterostructures, in which the composition is varied continuously in space. Electronic and optoelectronic devices which exploit these effects include, to date; graded-base heterostructure bipolar transistors which promote the egress of carriers through the base; graded separate confinement heterostructure laser active regions which not only confine light to the QWs, but may also promote transport within the active region and increase device bandwidth.

2. Theory In this study, we focus our attention a double graded QW (DGQW), double triangle QW

ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 362 (2005) 56–61

58

300

400 F=0 300 200

P≠0

Lb

P=0

(a)

200 P=0

150

L

V/2

P≠0

F=0

250

100

V 100

50 0

0 -b -100 400

-2

-a -1

0 0

a

-50

b 1

2

-2

-1

0

1

2

-1

0

1

2

F≠0

(b)

F≠0

(a)

-100 400

300

300

200

200

P=0 P≠0

P=0 100

100

P≠0

0

0 -100

-2

-1

0

1

-100

2

(b) -2

Fig. 1. The potential profile of DGQW for P ¼ 0 and Pa0 and amplitude of normalized subband wave function of electron, jcð~zÞj2 ; versus the normalized position, z~ ¼ L=a0 ; for cases (a) F ¼ 0 and (b) F a0:

Fig. 2. The potential profile of DTQW for P ¼ 0 and Pa0 and amplitude of normalized subband wave function of electron, jcð~zÞj2 ; versus the normalized position, z~ ¼ L=a0 ; for cases (a) F ¼ 0 and (b) F a0:

(DTQW) and double square QW (DSQW) structures, consisting of two QW separated by a thin potential barrier, subject to an electric field applied parallel to the growth direction (chosen as the z-direction). In the effective mass approximation, the Hamiltonian for a shallow-donor impurity under the electric field is

DTQW and DSQW are, respectively, 8 V ðPÞ; zo
b; > > > > V ðPÞ > > > >
2LðPÞ z;
bozo
a; > < V ðz; PÞDGQW ¼ V ðPÞ;
aozoa; > > V ðPÞ > > > z; aozob; > > 2LðPÞ > > : V ðPÞ; z4b;

H¼


2 _2 ~2 þ V ðz; PÞ
e þ eFz, r 2m ðPÞ ðPÞ~ r

(1)

where m is the electron effective mass as a function of P, ðPÞ is the static dielectric constant as a function of pressure, ~ r is the distance between the carrier and the donor impurity site qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (r ¼ r2 þ ðz
zi Þ2 ) and rð¼ x2 þ y2 Þ is the distance between the electron and impurity in the ðx
yÞ plane, F is the electric field strength, V ðz; PÞ are the finite confinement potentials in the zdirection. The functional forms of the DGQW,

(2a)

V ðz; PÞDTQW

8 V ðPÞ; zo
b; > > > V ðPÞ > > > z;
bozo
a;
> > > < LðPÞ ¼ V ðPÞ;
aozoa; > > > V ðPÞ > > z; aozob > > LðPÞ > > : V ðPÞ; z4b; (2b)

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V ðz; PÞDSQW

8 V ðPÞ; zo
b; > > > > > > < 0;
bozo
a; ¼ V ðPÞ;
aozoa; > > > 0; aozob; > > > : V ðPÞ; z4b;

(2c)

V ðz; PÞ ¼

gap (in eV) for a GaAs QW at the G-point with the hydrostatic pressure in units of kbar, which in turn is expressed [24–27] as E Gg ðPÞ ¼ 1:425 þ 1:26 10
2 P
3:77 10
5 P2 . (9)

where L is the well’s width (L ¼ L1 ¼ L2 ), Lb is the barrier width, jbj ¼ LðPÞ þ Lb2ðPÞ ; a ¼ Lb ðPÞ=2 and V ðz; PÞ Qc DE Gg ðx; PÞ

59

(3)

In the presence of the static dielectric constant can be expressed as [28] ðPÞ ¼ ð0ÞedP ,

(10)

is the barrier height and Qc ð¼ 0:6Þ is the conduction band offset parameter[23]. DE Gg ðx; PÞ is the band gap difference between QW and the barrier matrix at the G-point as a function of P; which for an aluminum molar fraction xð¼ 0:3Þ is given by

where d ¼
1:73 10
3 kbar
1 and ð0Þ ¼ 13:18 is the value of the dielectric constant of the QW at atmospheric hydrostatic pressure (static dielectric constant is assumed to be same GaAs and GaAlAs, charge image effects have not been considered). The following trial wave function of the ground impurity state is given by

DE Gg ðx; PÞ ¼ DE Gg ðxÞ þ PDðxÞ,

(4)

CðrÞ ¼ cðzÞjðr; lÞ,

DE Gg ðxÞ ¼ ð1:155x þ 0:37x2 ÞeV

(5)

is the variation of the energy gap difference and DðxÞ is the pressure coefficient of the band gap, given by DðxÞ ¼ ½
ð1:3 10
3 ÞxeV=kbar

(6)

In Eqs. (2)

(11)

where the wave function, cðzÞ; is exactly obtained from the Schro¨dinger equation in the z-direction, the wave function in the ðx
yÞ plane, jðr; lÞ; is chosen to be the wave function of the ground state of a two-dimensional hydrogen-like atom [29,30]:

 1 2 1=2
r=l jðr; lÞ ¼ e (12) l p

LðPÞ ¼ Lð0Þ½1
ðS 11 þ 2S12 ÞP,

(7a)

in which l is a variational parameter. The ground state binding energy of impurity is obtained as follows:

Lb ðPÞ ¼ Lb ð0Þ½1
ðS11 þ 2S 12 ÞP,

(7b)

E b ¼ E z
minhCjHjCi,

where S 11 ¼ 1:16 10
3 kbar
1 and S12 ¼
3:7

10
4 kbar
1 are the elastic constants of the GaAs [13–15], Lð0Þ is the original width of the confinement potentials for the electron in the z-direction, Lb ð0Þ is the original barrier width. The pressure dependence of the effective mass of the electron in GaAs is determined from expression [24–27] m0 " #, me ðPÞ ¼ (8) 2 1 1 þ E Gp G þ E g ðPÞ E Gg ðPÞ þ D0 where m0 is the free electron mass, E Gp ¼ 7:51 eV; D0 ¼ 0:341 eVand E Gg is the variation of the energy

l

(13)

where E z is the ground-state energy of electron obtained from Schro¨dinger equation in the zdirection without the impurity.

3. Results and discussion The values of the physical parameters used in our calculations are m ¼ 0:0665m0 (where m0 is the free electron mass), V 0 ¼ 228 meV and the distances are given in units of the effective Bohr radius a0 ¼ 0 _2 =m e2 : The variations of the ground state impurity binding energy of DSQW, DGQW and DTQW

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Eb(meV)

12 1 P = 10 kbar 2 3 8

4

10

3

F=0

2

9

1

P = 30 kbar

3 2

8

1

7 6 5 -1.0

P = 10 kbar

-0.5

0.0

z~i

F = 50 kV/cm L (0) = 50 Å Lb(0) = 50Å

0.5

1.0

P = 30 kbar

1 DSQW 2 DGQW 3 DTQW 0 -1.0

-0.5

0.0 ~ zi

0.5

1.0

Fig. 3. The variations of the ground state impurity binding energy of DSQW, DGQW and DTQW structures as a function ( well of the impurity position for barrier width, Lb ¼ 50 A; ( and F ¼ 50 kV=cm: The inset shows the width L ¼ 50 A binding energy of all confinement potentials for F ¼ 0:

structures as a function of the normalized impurity ( position (~zi ¼ z=a0 ) for barrier width, Lb ¼ 50 A; ( well’s width, L ¼ 50 A for two different pressure values is given in Fig. 3 for F ¼ 50 kV=cm and for F ¼ 0 (the inset of the Fig. 3). For F ¼ 0; we observe that the binding energy is degenerate for symmetrical positions with respect to the center barrier of different shaped DQW’s and the binding energy increases with the pressure. The maximum binding energy is observed in DTQW for the impurity located in the barrier center (bc) and than for DGQW and DSQW, respectively due to the geometric confinement. When the electric field is applied, degeneracy is broken. The Coulombic interaction between the electron and a donor impurity located in the bc decreases since the electrons shift to the left side of the wells, and the binding energy of a donor impurity located in the center of left well increases. As seen in this figure, DSQW structure is very sensitive to the electric field with respect to the other structures, and the impurity binding energy for DSQW is greater than the others structures. As the pressure increases the

well’s width and dielectric constant decrease, the effective mass of electron increases, leading to more confinement in the well’s of the impurity electron and so the impurity binding energy increases for all impurity positions. The effects of the coupling between the wells on donor impurity binding energy, the variation of the ground state impurity binding energy of DSQW, DGQW and DTQW structures as a function of the normalized impurity position for ( for two different presbarrier width, Lb ¼ 80 A; sure values is given in Fig. 4. for F ¼ 50 kV=cm and F ¼ 0 (the inset of Fig. 4). When we compare ( with previous the obtained results for Lb ¼ 80 A ( as the coupling between the results ðLb ¼ 80 AÞ; well decreases we see that the impurity binding energy decreases. As known, as the barrier width increases the effective length, Leff ¼ L1 þ Lb þ L2 ; ðL ¼ L1 ¼ L2 Þ increases and the Coulombic interaction between the electron and a donor impurity decreases, for F ¼ 0 the probability of finding of the electrons in both of the wells is the same and so, impurity binding energy is minimum for donor

16

9 F=0

1 2 3

P = 30 kbar

2 3

12

P = 30 kbar

3

8

1

Eb(meV)

1 2 3

Eb(meV)

16

2 3

7

1

2 1 P = 10 kbar

6

Eb(meV)

60

5 -1.5

8

F = 50 kV/cm L (0) = 50 Å Lb(0) = 50 Å

4

1 DSQW 2 DGQW 3 DTQW

-1.0

-0.5

0.0

~ zi

0.5

1.0

1.5

P = 10 kbar

0 -2

-1

0 ~ zi

1

2

Fig. 4. The variations of the ground state impurity binding energy of DSQW, DGQW and DTQW structures as a function ( L ¼ 50 A ( of the impurity position for barrier width, Lb ¼ 80 A; and F ¼ 50 kV=cm: The inset shows the binding energy of all confinement potentials for F ¼ 0:

ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 362 (2005) 56–61

impurity located in the bc. As the electric field increases, the Coulombic interaction between the electron and a donor impurity located in the bc or right well center decreases and so, impurity binding energy decreases while it increases for impurity located left well center, since the electrons shift to the left side of the structure. In summary, we have studied the binding energy of the hydrogenic impurities of different shaped double QWs (graded, triangle or full graded and square QWs) under the electric field. The calculations were performed within the effective-mass approximation and by using a variational method. The dependence of the ground state impurity binding energy on the applied electric field, pressure, geometric shapes of the wells and barrier width was discussed. To the best of our knowledge, this is the first study for hydrogenic donor impurities in double graded and triangle shaped QWs. We expect that this method will be of great help for theoretical studies of the physical properties of graded and triangle shaped QWs. Furthermore, the simplicity of the method might be useful for describing the correct behavior of shallowdonor impurities in double quantum-wells with different shapes in the external fields, and for designing some devices in the future. References [1] W.Q. Chen, T.G. Anderson, Appl. Phys. Lett. 60 (1992) 1591. [2] G. Senders, K.K. Bajaj, J. Appl. Phys. 68 (1990) 5348. [3] M. Morita, K. Goto, T. Suziki, Jpn. Appl. Phys. 29 (1990) L1663. [4] E.J. Austin, M. Jaros, J. Phys. C 19 (1986) 533.

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