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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices

Donor impurity states in a GaAs square tangent quantum dot Zhongmin Zhang a, Kangxian Guo a,⇑, Sen Mou b, Bo Xiao a, Yingchu Zhou a a b

Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China Dipartimento di Fisica, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy

a r t i c l e

i n f o

Article history: Received 30 January 2015 Received in revised form 26 March 2015 Accepted 27 March 2015 Available online 3 April 2015 Keywords: Impurity Quantum dot Pressure

a b s t r a c t Based on the effective-mass approximation, the impurity binding energy in a square tangent quantum dot is calculated variationally. The impurity binding energy has been calculated as a function of d; U 0 , pressure and the impurity position. It is found that d; U 0 , pressure and the impurity position have great effects on the impurity binding energy. It is worth to note that the effect of pressure should be taken into consideration in the experimental study of semiconductor nanostructures. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Thanks to the advances in semiconductor nanofabrication technology in recent years, it is possible to produce semiconductor nanostructures which can be classiﬁed as quantum wells, quantum dots, quantum wires, quantum rings and superlattices [1–5]. It is well known that the reduced dimensionality can lead to the presence of many new physical effects [6–11]. As a result, many theoretical and experimental studies have been concentrated on understanding physical properties of these semiconductor nanostructures [12–17]. As we know, impurity plays an important role in the photoelectron devices. Thus, many attentions are focused on the studies of impurity states in semiconductor nanostructures. In 2005, Peter studied the effect of hydrostatic pressure on binding energy of impurity states in spherical quantum dots [18]. His results showed that the impurity binding energy is strongly affected by the dot size and the

⇑ Corresponding author. E-mail address: [email protected] (K. Guo). http://dx.doi.org/10.1016/j.spmi.2015.03.053 0749-6036/Ó 2015 Elsevier Ltd. All rights reserved.

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hydrostatic pressure. In 2008, Xia et al. investigated hydrostatic pressure effects on the impurity states in InAs/GaAs coupled quantum dots [19]. In their investigations they found that the impurity binding energy is the largest as the impurity is placed at the dot center and the impurity binding energy has a minimum value when the interdot barrier width is increased. In 2014, Moussaouy et al. reported the role of hydrostatic pressure and temperature on bound polaron in semiconductor quantum dot [20]. Their results told us that the hydrostatic pressure, temperature and polaronic correction are main factors to the variation of the impurity binding energy. This paper is aimed at studying the impurity binding energy in a GaAs square tangent quantum dot and organized as follows. In Section 2, the theoretical model is presented and the impurity binding energy is obtained by variational method. In Section 3, numerical results and some discussions are displayed. In Section 4, a brief conclusion is exhibited. 2. Theory In our paper, we consider an impurity conﬁned by the system. The effective-mass Hamiltonian of the system can be written as

b ¼H b0 H

e2 ; 4pe0 eðPÞr

ð1Þ

with

b0 ¼ H

" # h2 1 @ @ 1 @2 @2 þ Vðq; zÞ; þ þ q @q 2m ðPÞ q @ q q2 @/2 @z2

where e means the electron charge, P denotes the pressure, r ¼

ð2Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q2 þ ðz zi Þ2 is electron–impurity

distance, zi is the position of the impurity, e0 is vacuum permittivity. The effective relative dielectric constant eðPÞ and the effective mass of electron m ðPÞ are functions of P. For eðPÞ [21],

h

i

eðPÞ ¼ 12:74 exp ð1:67 kbar1 P 6:73Þ 103 ;

ð3Þ

and for m ðPÞ [21],

m ðPÞ ¼ 1 þ

15020 1

1519 þ 10:7 kbar P

þ

1

7510 1

1519 þ 10:7 kbar P þ 341

m0 ;

ð4Þ

where m0 is the free electron mass. According to Eqs. (3) and (4), we can ﬁnd that P has great inﬂuences on eðPÞ and m ðPÞ, which can be seen in Figs. 1 and 2, respectively. Vðq; zÞ is the sum of the radial and z-direction conﬁnement of the system which has the following form,

( Vðq; zÞ ¼ VðqÞ þ VðzÞ ¼

U 0 tan2 ðpz=dÞ jzj 6 d=2; q 6 R 1

otherwise:

ð5Þ

in the equation above, VðzÞ is the bound potential which is related to U 0 and d. In Figs. 3 and 4, we plot the bound potential VðzÞ as a function of the direction z for different values of d and U 0 . In order to calculate the impurity binding energy, the trial wave function may be chosen as [17]

U ¼ NWðq; /; zÞekr ;

ð6Þ

where N is the normalization constant and k is the variational parameter, Wðq; /; zÞ ¼ f ðqÞhðzÞeim/ is the electron wave function. The radial function f ðqÞ can be gotten by using the 0-order Bessel function b pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ J 0 , and the z-direction wave function hðzÞ can be written as hðzÞ ¼ cos pdz , where b ¼ 12 ð1 þ 1 þ 4U 1 Þ

2

and U 1 ¼ 2m hðPÞd 2 2 p

U0

[22].

Z. Zhang et al. / Superlattices and Microstructures 83 (2015) 439–446

Fig. 1. The effective relative dielectric constant eðPÞ as a function of the pressure P.

Fig. 2. The effective mass of electron m ðPÞ as a function of P.

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Z. Zhang et al. / Superlattices and Microstructures 83 (2015) 439–446

Fig. 3. The potential VðzÞ as a function of z, with d ¼ 20 nm, for two different values of U 0 ; U 0 ¼ 15 meV and U 0 ¼ 25 meV.

Fig. 4. The potential VðzÞ as a function of z, with U 0 ¼ 15 meV, for two different values of d; d ¼ 20 nm and d ¼ 25 nm.

Z. Zhang et al. / Superlattices and Microstructures 83 (2015) 439–446

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The impurity energy can be obtained by minimizing [23]

E ¼ min k

b Ui hUj Hj : hUjUi

ð7Þ

Then the impurity binding energy can be deﬁned as follows [24]:

Eb ¼ E0 E;

ð8Þ

where E0 means the ground state energy of Eq. (2).

3. Results and discussion In Fig. 5, the impurity binding energy Eb is shown as a function of d for three different U 0 values: U 0 ¼ 10 meV; U 0 ¼ 20 meV and U 0 ¼ 30 meV, when the impurity is located at the center of the system and P ¼ 0 kbar. From the ﬁgure, we can see that the impurity binding energy Eb decreases with the enhancement of d. This behavior is in agreement with the fact that the mean relative electron–impurity distance increases and the electron wave function is more spread with the increasing d. Moreover, we can also ﬁnd that at each value of d, the impurity binding energy Eb increases as U 0 enhances. The origin of this characteristic is that the quantum conﬁnement becomes stronger and the probability of wave function appearing around the impurity increases with the augment of U 0 . In Fig. 6, the variation of the impurity binding energy Eb with d is depicted for three different pressure values: P ¼ 0 kbar; P ¼ 30 kbar and P ¼ 60 kbar, here we set U 0 ¼ 20 meV and consider the impurity is located at the center of the system. It is clearly found that the impurity binding energy Eb diminishes as d increases. The physical reason of this feature is similar to that of Fig. 5. Additionally, it is also revealed that Eb shows an increase as P increases. The reason of this trait can be provided as follows. When pressure P increases, eðPÞ decreases and m ðPÞ increases, which leads

Fig. 5. The impurity binding energy Eb as a function of d, with P ¼ 0 kbar and zi ¼ 0, for three different values of U 0 ; U 0 ¼ 10 meV; U 0 ¼ 20 meV and U 0 ¼ 30 meV.

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Z. Zhang et al. / Superlattices and Microstructures 83 (2015) 439–446

Fig. 6. The impurity binding energy Eb as a function of d, with U 0 ¼ 20 meV and zi ¼ 0, for three different values of P; P ¼ 0 kbar; P ¼ 30 kbar and P ¼ 60 kbar.

Fig. 7. The impurity binding energy Eb as a function of P, with d ¼ 10 nm and U 0 ¼ 20 meV, for three different values of the impurity position zi ; zi ¼ 0; zi ¼ d=4 and zi ¼ d=3.

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445

Fig. 8. The impurity binding energy Eb as a function of zi , with d ¼ 10 nm and U 0 ¼ 10 meV, for three different values of P; P ¼ 0 kbar; P ¼ 30 kbar and P ¼ 60 kbar.

to the intensive quantum conﬁnement. As a consequence, the wave function is more strongly conﬁned inside the system. In Fig. 7, we plot the impurity binding energy Eb as a function of P for three different impurity position values, zi ¼ 0; zi ¼ d=4 and zi ¼ d=3, with d ¼ 20 nm and U 0 ¼ 20 meV. The results show that the impurity binding energy Eb has a nearly linear augment when P enhances for any impurity position. The reason of this trait is clearly made in Fig. 6. What calls for special attention is that the smaller the value of impurity position, the lager the curve slope. It can be explained as that the impurity binding energy Eb is more susceptible to the pressure P when the impurity is placed at the center of the quantum dot. Moreover, it can be found that the impurity binding energy Eb has an obvious decrease when the impurity position value increases. This is due to the truth that the probability of electron wave function inside the quantum dot is greater than that near the edge of the quantum dot. In Fig. 8, the impurity binding energy Eb is investigated as a function of impurity position zi for different values of pressure P. From the ﬁgure, we can mainly ﬁnd three traits. The ﬁrst one is that the impurity binding energy Eb has a maximum value when the impurity is placed at the center of the quantum dot. This is because the electron wave function mainly exists in the periphery of impurity. The second one is that the curve of the impurity binding energy Eb has a nice symmetry. The reason of this feature is located in the fact that the potential has a symmetry which can be seen from Figs. 3 and 4. The third one is that the impurity binding energy Eb increases with the increment of pressure P, as expected. This characteristic can be explained as that the existence of pressure brings in an extra conﬁnement and makes the electron wave function locate around the impurity. 4. Conclusion In short, the impurity binding energy has been calculated as a function of d; U 0 , pressure P and the impurity position zi . Our calculative results show that the impurity binding energy Eb decreases with the enhancement of d and increases as U 0 enhances. Moreover, the impurity binding energy Eb also strongly affected by the pressure P. When pressure P increases, the impurity binding energy Eb

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enhances. In addition, the impurity position plays a primordial role in the change of the impurity binding energy. From our calculation, we conclude that, in experimental work of electron properties of semiconductor nanostructures, we should take consideration of the effect of pressure. At last, we hope that our results will stimulate further experimental activity in semiconductor nanostructures and make a helpful contribution to the application of pressure transducer. Acknowledgments This Work is supported by the National Natural Science Foundation of China (under Grant Nos. 61178003, 61475039), Guangdong Provincial Department of Science and Technology (under Grant Nos. 2012A080304010, S2012010010115, 2012A080304005) and Guangzhou Municipal Department of Education (under Grant No. 12A005S). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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