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Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots
Author's Accepted Manuscript
Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots E. Sadeghi, M. Moradi LM, P. Zamani
www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(13)00306-8 http://dx.doi.org/10.1016/j.physe.2013.08.032 PHYSE11371
To appear in:
Physica E
Received date: 23 July 2013 Revised date: 28 August 2013 Accepted date: 28 August 2013 Cite this article as: E. Sadeghi, M. Moradi LM, P. Zamani, Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots, Physica E, http://dx.doi.org/10.1016/j.physe.2013.08.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots E. Sadeghi 1 , M. Moradi LM, P. Zamani Department of Physics,Yasouj University, Yasouj, Iran 75914-353 Abstract The present study seeks to scrutinize the effect of polarization charges on the electronic properties of double ellipsoidal quantum dots. In this regard, the effective-mass approximation within a variational scheme is used and the binding energy of hydrogenic impurity located at the center of ellipsoidal quantum dot (EQD) is calculated for GaAs/GaAlAs/AlAs structure. The effect of surface polarization charges due to impurity and self-polarization charges on the binding energy is considered. The results showed that the binding energy depends not only on the thickness of the intermediate layer but also on the ellipticity constant.
Keywords: Bessel function, Binding energy, Ellipsoidal quantum dot, Polarization charges
1
Introduction
The study of low-dimensional systems has been the interesting subject of investigation recent years [1]. Because of quantum size effects in these structures, the physical properties of confined quantum systems strictly depend on external shape of the system under consideration. In theoretical works, it is customary to assume a spherical shape for the quantum dot (SQD). Since deformation of spherical shape during quantum dot growth is unavoidable, other shapes of quantum dot (QD) are probably achieved. The ellipsoidal shape may be a better representation of the actual problems [2, 3]. Impurities have important effects on physical characteristic in low-dimensional semiconductor systems. The impurity changes the energy levels of the materials and therefore, the electronic and optical properties are affected [4]. A significant number of studies have been carried out on calculation methods of impurity states in nanostructures [5-10]. In many of investigations devoted to calculation of binding energy in nanostructures, the dielectric constant mismatch between different regions of a system was not considered. When the dielectric constant changes abruptly across the heterojunction, the discontinuity of the square quantum well model implies an infinite internal electric field at the heterojunctions, which is not physically possible. In reality, the parameters such as dielectric constant and effective mass change over a few monolayers for a perfect microscopic interface [11]. There are also a number works done on the polarization charges in nanostructures. The effect of dielectric mismatch 1 Corresponding author: +98 741 224 2167
e-mail:
[email protected], Phone:
1
+98 741 222 3048, Fax:
on binding energy in quantum well and quantum wire is investigated in [12-14]. Coulomb interaction in thin semiconductor and semimetal films was considered by Keldysh [15]. Boichuk and et al. [16-19] calculated the effect of charge polarization on the energy of electrons of bivalent impurity and optical properties in spherical quantum dot. The effects of electric field and dielectric mismatch on physical properties in quantum dot are also calculated by Niculescu and Cristea in [20]. The effect of dielectric constant and magnetic field on binding energy in cylindrical quantum wire is considered by Karki and et al. in [21]. The electronics properties of quantum structures confined in other nanostructures have attracted many researchers for their experimental applications. Study of impurity binding energy of a multilayered spherical GaAs/(Ga, Al)As quantum dot by Aktas and Boz [22], three dielectric layer model for the interface between a spherical quantum dot and the surrounding matrix by Deng [23], the binding energy of an impurity located at the center of multilayered quantum dot by Boz et al. [24] and the effect of dielectric-constant mismatch and magnetic field in multilayered SQD by Manaselyan and Kirakosyan [25] are a few examples of some interesting investigations in this field. In this paper the binding energy of hydrogenic impurity located at the center of a GaAs ellipsoid with GaAlAs coating in the environment AlAs surrounding the system is calculated. In this regard, the interaction of electron with the polarization charges ( which are caused by the impurity ion) and also selfinteraction potential ( due to the interaction of electron and its induce polarization charges) are taken into account. The outline of the paper is as follows: in Section 2 the Schr¨ odinger equation solved for a hydrogenic impurity in an ellipsoidal quantum dot confined in other EQD, held at a finite potential, and the ground state energy and wave function are calculated by applying the boundary conditions at the interfaces, then using the Ritz variational method, the impurity binding energy is calculated. Section 3 contains our results and discussions. Finally, the conclusions are presented in section 4.
2
Theory
In the effective mass approximation, the Hamiltonian of an electron in each region of double ellipsoidal quantum dots in the presence of hydrogenic donor impurity can be written as Hi =
−¯ h2 ∆R + Ui (R) + Vi (R) + Wi 2m∗i
(i = 1, 2, 3)
(1)
where m∗i , Ui (R), Vi (R), and Wi are the position-dependent effective mass, the confinement potential , the Coulomb potential and the self-polarization potential, respectively. For an EQD with circular cross-section in X, Y-plane the ellipticity constant is c (2) β =1− a
2
where a and c are the semi-axes of the ellipsoid. It is useful to change the variables as follows [26]: X→
ax ay cz , Y → , Z → , r0 = (a2 c)1/3 r0 r0 r0
(3)
This change transforms the ellipsoid into a sphere of radius r0 , with the same volume. After these coordinate transformation the Hamiltonian can be represented as Hi = H0i + H1i + W (r) + ∆U (r) (4) where H0i =
p2 + Ui (r) + Ai . 2m∗i
(5)
Ai is the effect of the polarization charges induced on the quantum dot surface due to impurity ion, and is achieved by electrostatic image method A1 =
−e2 (ε1 − ε2 ) −e2 (ε2 − ε3 ) + A2 , A2 = , A3 = 0 r01 ε1 ε2 r02 ε2 ε3
(6)
where r01 and r02 are the radius of double sphere and εi is the dielectric constant of different regions of dot. The confining potential Ui (r) can be written as U1 = 0 r ≤ r01 U2 r01 < r < r02 U (r) = (7) U3 r > r02 The last term on the right-hand-side of Eq. (4), the non-spherical part of confinement potential, can be neglected [26]. For small ellipticity, β ≪ 1, the eigenvalues of the system can be determined using the variational method with H1i as e2 βe2 β H1i = − (ˆ p2 − 3pˆz 2 ) (8) − (1 − 3 cos2 θ) + εi r 3εi r 3m∗i The electron will also induce polarization charges at the boundary. W (r) in Eq.(4) is the electron self-polarization potential which originates from the interaction between the electron and its self-image [18]. The W (r) is the solution of Poisson equation and can be written in terms of Green’s function (G) [19] W (r) = −
2πe2 ∆G ε(r)
(9)
where ε(r) =
r − r01 ε1 + ε2 + ε3 {1 − (γ1 tanh( ) 2 L1 r − r01 r − r02 r − r02 +γ2 tanh( ) tanh( ) − γ2 tanh( ))} L1 L2 L2
3
(10)
Li is the thickness of transition layer between regions i and i+1, and γi = The wave function of H0i can be written as ψnl (r, θ) = Rnl (r) Pl (θ)
εi Σn ε . i=1 i
(11)
where Rnl (r) and Pl (θ) are the radial and angular parts of wave function. The radial part of Schr¨ odinger equation is exactly solvable and the solutions are Bessel functions Ri (r) = N1i Jl (Ci r) + N2i Nl (Ci r),
√
(12) 2m∗ i h ¯2
where N1i and N2i are normalization constants, and Ci = (E 0 − Ui − Ai ). The angular part of the wave function is √ √ (13) Pl (θ) = B1l sin( l(l + 1)θ) + B2l cos( l(l + 1)θ). To calculate the eigenvalues of unperturbed part of the Hamiltonian, E0 , we use the continuity of wave function and its derivative at the boundary surface. The ground state energy for H is obtained by variational method E = min λ
< ϕ|H|ϕ > < ϕ|ϕ >
(14)
where the wave function, ϕ = ψ0 (r, θ) e−λr , and λ is the variational parameter. The binding energy is given as Eb = E0 − E
(15)
where E0 is energy of system without impurity.
3
Results and discussion
In this study, the numerical calculations are carried out for a QD consisting of a GaAs ellipsoid coated by a layer of Ga1−x Alx As, embedded in the dielectric medium AlAs. The material parameters used in the calculations are [25]: m∗i = (0.067 + 0.083x)m0 , εi = 13.18 − 3.12x, x = 0.4 and the barrier potential Ui = Qc ∆Eg where ∆Eg = (1.155x + 0.37x2 ), the energy is expressed in terms of the effective Rydberg constant, Ry∗ = m∗3 e4 /2¯h2 ε23 . Figure 1 shows the variation of the impurity binding energy as a function of semi-axis of ellipsoidal quantum dot, a1 , for various values of ellipticity constant,β, for a fixed coating semi-axis a2 = 100˚ A. As it is seen, the binding energy increases, reaches a maximum value and then decreases as the a1 increases. As the β increases, the maxima of the curves shift to the dot center. For a fixed value of a1 , the binding energy increases with β, so that for spherical quantum dot β = 0, it is minimum. This is because, with increases the β, the volume of QD decreases and the Coulomb interaction increases. The results are similar with work done by Manaselyan and Kirakosyan for SQD [25]. Dependence of the impurity binding energy on the semi-axis of ellipsoid, a1 , with β = 0.15 for various value of dielectric constant is presented in Fig.2. The 4
binding energy while the regions 1 and 3 are the same (ε1 = ε3 < ε2 ) is greater than when the dielectric constant of environment is less than the well region, (ε1 < ε2 , ε3 ). This is because in the small dimension of the system (r01 < r02 ), the Coulomb interaction spreads in the surrounding environment, and therefore the dielectric constant changes of the environment have a considerable effect on the binding energy. To investigate the effect of polarization charges due to dielectric mismatch on the carrier energy, the polarization energy Epol is calculated and shown in Fig.3 for different β. Epol defines as the difference between the total energy and the energy without induced charges at the interface of heterostructure, (Ai = 0, W (r) = 0) which will show the effect of induced charges at the separation boundary of the media on the energy of the system. Because the dielectric permeability of the well region is greater than the barrier, the W (r) is a positive function and therefore, the energies of carriers are increased by the surface polarization. As it could be seen, the energy Epol decreases as the a1 increases. This is because the effect of surface (charges) potential on electrons decreases with increases in the semi-axis a1 . The behavior of binding energy versus the thickness of intermediate layer for a1 = 30˚ A and β = 0.1 and 0.2 is shown in Fig.4. There is a slight variation in the binding energy and it decreases as the layer thickness increases. This is because with the enhancement of L, the polarization charges increase and the electron energy is enhanced, thus the binding energy decreases. The results are similar to work done by Boichuk and et. al. for spherical quantum dot [17] and Niculescu and Cristea for nanodots [20] .
4
Conclusion
In this paper the binding energy of hydrogenic impurity located at the center of double ellipsoidal quantum dot with different dielectric constant and rectangular potential well model using an appropriate coordinate transformation is calculated. The effects of polarization charges due to impurity on binding energy are considered. The results show that the binding energy increases with the enhancement of ellipticity constant and it decreases when the dimension of quantum dot increases.
References [1] D. Bimberg, Semiconductor, 33 No: 9 (1999) 951-955 [2] G. Cantele, D. Nino, and G. Iadonisi, J. Phys.: Condens. Matter 12 (2000) 9019-9037 [3] E. Sadeghi, and A. Avazpour, Physica B 406 (2) (2011) 241-244 [4] Z. Xiao, J. Zhu, F. He, Superlattices and Microstructures, 19 (2) (1996) 137-149 5
[5] G. Bastard, Phys. Rev. B 24 (1981) 4714-4722 [6] A. J. Peter, Physica E 28 (2005) 225-229 [7] C. Bose, C. K. Sarkar, Solid State Electron. 42 (1998) 1661-1668 [8] C. Bose, C. K. Sarkar, Physica B 253 (1998) 238-241 [9] J. L. Zhu, J. J. Xiong, B. L. Gu, Phys. Rev. B 41 (1990) 6001-6007 [10] V. Ranjan, V. A. Singh, J. Appl. Phys. 89 (2001) 6415-6422 [11] L. Li Tsung and J. Kuhn Kelin, Phys. Rev. B B47 (1993) 12760-12770 [12] S. Fraizzoli, F. Bassani, R. Buczko, Phys. Rev. B 41 (1990) 5096-5103 [13] M. M. Aghasyan, A. A. Kirakosyan, Physica E 8 (2000) 281-289 [14] E. Sadeghi, F. Vahdatnejad, and M. Moradi LM., Superlattices and Microstructures 58 (2013) 165-170 [15] L. V. Keldysh, JETP Lett. 29 (1979) 658-661 [16] V. I. Boichuk, I. V. Bilynsky, Phys. Stat. Solidi (b) 174 (2) (1992) 463-470 [17] V. I. Boichuk, I. V. Bilynsky, R. Ya. Leshko, condensed Matter Physics 11, No 4(56) (2008) 653-661 [18] V. I. Boichuk, I. V. Bilynsky, R. Ya. Leshko, L. M. Turyanska, Physica E 44 (2011) 476-482 [19] V. I. Boichuk, R. Yu. Kubai, Physics of the Solid State 43 (2) (2001) 235241 [20] E. C. Niculescu, M. Cristea, Eur. Phys. J. B 84 (1) (2011) 59-67 [21] H. D. Karki,S. Elagoz, R. Amca, P. Baser, K. Atasever, Physica E 42 (2010) 1351-1355 [22] S.Aktas,and F.K.Boz,Physica E 40 (2008) 753-758 [23] S.Deng, Computer Physics Communication 181 (2010)787-799 [24] F.K.Boz, S.Aktas, A.Bilekkaya, S.E.Okan, Applied Surface Science 256 (2010) 3832 [25] Aram Kh. Manaselyan, and Albert A. Kirakosyan, Physica E 22 (2004) 825-832 [26] K. G. Dvoyan, E. M. Kazaryan, L. S. Petrosyan, Physica E 28 (2005) 333-338
6
captions: Fig1: The variation of impurity binding energy in terms of the semi-axis of the ellipsoidal quantum dot for several β with ε1 = 13.18, ε2 = 11.932, ε3 = 10.06. Fig2: The variation of binding energy versus semi-axis of the ellipsoidal quantum dot for different dielectric constants. Fig3: The variation of polarization energy (Epol ) versus semi-axis of the ellipsoidal quantum dot for different β. Fig4: The variation of binding energy in terms of transition layer thickness L1 = L2 = L with a1 = 30(˚ A).
7
3.5
3.0
β=0.2
*
Binding Energy (R y)
=0.1 =0.05 =0.0
2.5
2.0 a2=100 A
1.5
1.0 20
40
60
80
100
a1 (A) Figure 1: The variation of impurity binding energy in terms of the semi-axis of the ellipsoidal quantum dot for different β with ε1 = 13.18, ε2 = 11.932, ε3 = 10.06.
8
ε1=13.18, ε2=11.932, ε3=10.06
10
ε1=ε3=13.18, ε2=11.932 ε1=13.18, ε2=ε3=11.932
8
*
Binding Energy (R y)
β=0.15
6
4
2
20
40
60
80
100
a1 (A)
Figure 2: The variation of binding energy versus semi-axis of the ellipsoidal quantum dot for different dielectric constants.
9
0.24
0.20
β=0.0
β=0.2
β=0.05 β=0.1
*
Epol (R y)
0.16
0.12
0.08
0.04 20
40
60
80
100
a1 (A)
Figure 3: The variation of polarization energy (Epol ) versus semi-axis of the ellipsoidal quantum dot for different β.
10
3.25 β=0.2
3.15 a1=30 A, a2=100 A L1=L2
*
Binding Energy (R y )
3.20
3.10
3.05
3.00
β=0.1
2.95 4
5
6
7
8
L (A)
Figure 4: The variation of binding energy in terms of transition layer thickness L1 = L2 = L with a1 = 30(˚ A).
11
Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots E. Sadeghi, M. Moradi LM, P. Zamani
www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(13)00306-8 http://dx.doi.org/10.1016/j.physe.2013.08.032 PHYSE11371
To appear in:
Physica E
Received date: 23 July 2013 Revised date: 28 August 2013 Accepted date: 28 August 2013 Cite this article as: E. Sadeghi, M. Moradi LM, P. Zamani, Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots, Physica E, http://dx.doi.org/10.1016/j.physe.2013.08.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of dielectric mismatch on impurity binding energy in double ellipsoidal quantum dots E. Sadeghi 1 , M. Moradi LM, P. Zamani Department of Physics,Yasouj University, Yasouj, Iran 75914-353 Abstract The present study seeks to scrutinize the effect of polarization charges on the electronic properties of double ellipsoidal quantum dots. In this regard, the effective-mass approximation within a variational scheme is used and the binding energy of hydrogenic impurity located at the center of ellipsoidal quantum dot (EQD) is calculated for GaAs/GaAlAs/AlAs structure. The effect of surface polarization charges due to impurity and self-polarization charges on the binding energy is considered. The results showed that the binding energy depends not only on the thickness of the intermediate layer but also on the ellipticity constant.
Keywords: Bessel function, Binding energy, Ellipsoidal quantum dot, Polarization charges
1
Introduction
The study of low-dimensional systems has been the interesting subject of investigation recent years [1]. Because of quantum size effects in these structures, the physical properties of confined quantum systems strictly depend on external shape of the system under consideration. In theoretical works, it is customary to assume a spherical shape for the quantum dot (SQD). Since deformation of spherical shape during quantum dot growth is unavoidable, other shapes of quantum dot (QD) are probably achieved. The ellipsoidal shape may be a better representation of the actual problems [2, 3]. Impurities have important effects on physical characteristic in low-dimensional semiconductor systems. The impurity changes the energy levels of the materials and therefore, the electronic and optical properties are affected [4]. A significant number of studies have been carried out on calculation methods of impurity states in nanostructures [5-10]. In many of investigations devoted to calculation of binding energy in nanostructures, the dielectric constant mismatch between different regions of a system was not considered. When the dielectric constant changes abruptly across the heterojunction, the discontinuity of the square quantum well model implies an infinite internal electric field at the heterojunctions, which is not physically possible. In reality, the parameters such as dielectric constant and effective mass change over a few monolayers for a perfect microscopic interface [11]. There are also a number works done on the polarization charges in nanostructures. The effect of dielectric mismatch 1 Corresponding author: +98 741 224 2167
e-mail:
[email protected], Phone:
1
+98 741 222 3048, Fax:
on binding energy in quantum well and quantum wire is investigated in [12-14]. Coulomb interaction in thin semiconductor and semimetal films was considered by Keldysh [15]. Boichuk and et al. [16-19] calculated the effect of charge polarization on the energy of electrons of bivalent impurity and optical properties in spherical quantum dot. The effects of electric field and dielectric mismatch on physical properties in quantum dot are also calculated by Niculescu and Cristea in [20]. The effect of dielectric constant and magnetic field on binding energy in cylindrical quantum wire is considered by Karki and et al. in [21]. The electronics properties of quantum structures confined in other nanostructures have attracted many researchers for their experimental applications. Study of impurity binding energy of a multilayered spherical GaAs/(Ga, Al)As quantum dot by Aktas and Boz [22], three dielectric layer model for the interface between a spherical quantum dot and the surrounding matrix by Deng [23], the binding energy of an impurity located at the center of multilayered quantum dot by Boz et al. [24] and the effect of dielectric-constant mismatch and magnetic field in multilayered SQD by Manaselyan and Kirakosyan [25] are a few examples of some interesting investigations in this field. In this paper the binding energy of hydrogenic impurity located at the center of a GaAs ellipsoid with GaAlAs coating in the environment AlAs surrounding the system is calculated. In this regard, the interaction of electron with the polarization charges ( which are caused by the impurity ion) and also selfinteraction potential ( due to the interaction of electron and its induce polarization charges) are taken into account. The outline of the paper is as follows: in Section 2 the Schr¨ odinger equation solved for a hydrogenic impurity in an ellipsoidal quantum dot confined in other EQD, held at a finite potential, and the ground state energy and wave function are calculated by applying the boundary conditions at the interfaces, then using the Ritz variational method, the impurity binding energy is calculated. Section 3 contains our results and discussions. Finally, the conclusions are presented in section 4.
2
Theory
In the effective mass approximation, the Hamiltonian of an electron in each region of double ellipsoidal quantum dots in the presence of hydrogenic donor impurity can be written as Hi =
−¯ h2 ∆R + Ui (R) + Vi (R) + Wi 2m∗i
(i = 1, 2, 3)
(1)
where m∗i , Ui (R), Vi (R), and Wi are the position-dependent effective mass, the confinement potential , the Coulomb potential and the self-polarization potential, respectively. For an EQD with circular cross-section in X, Y-plane the ellipticity constant is c (2) β =1− a
2
where a and c are the semi-axes of the ellipsoid. It is useful to change the variables as follows [26]: X→
ax ay cz , Y → , Z → , r0 = (a2 c)1/3 r0 r0 r0
(3)
This change transforms the ellipsoid into a sphere of radius r0 , with the same volume. After these coordinate transformation the Hamiltonian can be represented as Hi = H0i + H1i + W (r) + ∆U (r) (4) where H0i =
p2 + Ui (r) + Ai . 2m∗i
(5)
Ai is the effect of the polarization charges induced on the quantum dot surface due to impurity ion, and is achieved by electrostatic image method A1 =
−e2 (ε1 − ε2 ) −e2 (ε2 − ε3 ) + A2 , A2 = , A3 = 0 r01 ε1 ε2 r02 ε2 ε3
(6)
where r01 and r02 are the radius of double sphere and εi is the dielectric constant of different regions of dot. The confining potential Ui (r) can be written as U1 = 0 r ≤ r01 U2 r01 < r < r02 U (r) = (7) U3 r > r02 The last term on the right-hand-side of Eq. (4), the non-spherical part of confinement potential, can be neglected [26]. For small ellipticity, β ≪ 1, the eigenvalues of the system can be determined using the variational method with H1i as e2 βe2 β H1i = − (ˆ p2 − 3pˆz 2 ) (8) − (1 − 3 cos2 θ) + εi r 3εi r 3m∗i The electron will also induce polarization charges at the boundary. W (r) in Eq.(4) is the electron self-polarization potential which originates from the interaction between the electron and its self-image [18]. The W (r) is the solution of Poisson equation and can be written in terms of Green’s function (G) [19] W (r) = −
2πe2 ∆G ε(r)
(9)
where ε(r) =
r − r01 ε1 + ε2 + ε3 {1 − (γ1 tanh( ) 2 L1 r − r01 r − r02 r − r02 +γ2 tanh( ) tanh( ) − γ2 tanh( ))} L1 L2 L2
3
(10)
Li is the thickness of transition layer between regions i and i+1, and γi = The wave function of H0i can be written as ψnl (r, θ) = Rnl (r) Pl (θ)
εi Σn ε . i=1 i
(11)
where Rnl (r) and Pl (θ) are the radial and angular parts of wave function. The radial part of Schr¨ odinger equation is exactly solvable and the solutions are Bessel functions Ri (r) = N1i Jl (Ci r) + N2i Nl (Ci r),
√
(12) 2m∗ i h ¯2
where N1i and N2i are normalization constants, and Ci = (E 0 − Ui − Ai ). The angular part of the wave function is √ √ (13) Pl (θ) = B1l sin( l(l + 1)θ) + B2l cos( l(l + 1)θ). To calculate the eigenvalues of unperturbed part of the Hamiltonian, E0 , we use the continuity of wave function and its derivative at the boundary surface. The ground state energy for H is obtained by variational method E = min λ
< ϕ|H|ϕ > < ϕ|ϕ >
(14)
where the wave function, ϕ = ψ0 (r, θ) e−λr , and λ is the variational parameter. The binding energy is given as Eb = E0 − E
(15)
where E0 is energy of system without impurity.
3
Results and discussion
In this study, the numerical calculations are carried out for a QD consisting of a GaAs ellipsoid coated by a layer of Ga1−x Alx As, embedded in the dielectric medium AlAs. The material parameters used in the calculations are [25]: m∗i = (0.067 + 0.083x)m0 , εi = 13.18 − 3.12x, x = 0.4 and the barrier potential Ui = Qc ∆Eg where ∆Eg = (1.155x + 0.37x2 ), the energy is expressed in terms of the effective Rydberg constant, Ry∗ = m∗3 e4 /2¯h2 ε23 . Figure 1 shows the variation of the impurity binding energy as a function of semi-axis of ellipsoidal quantum dot, a1 , for various values of ellipticity constant,β, for a fixed coating semi-axis a2 = 100˚ A. As it is seen, the binding energy increases, reaches a maximum value and then decreases as the a1 increases. As the β increases, the maxima of the curves shift to the dot center. For a fixed value of a1 , the binding energy increases with β, so that for spherical quantum dot β = 0, it is minimum. This is because, with increases the β, the volume of QD decreases and the Coulomb interaction increases. The results are similar with work done by Manaselyan and Kirakosyan for SQD [25]. Dependence of the impurity binding energy on the semi-axis of ellipsoid, a1 , with β = 0.15 for various value of dielectric constant is presented in Fig.2. The 4
binding energy while the regions 1 and 3 are the same (ε1 = ε3 < ε2 ) is greater than when the dielectric constant of environment is less than the well region, (ε1 < ε2 , ε3 ). This is because in the small dimension of the system (r01 < r02 ), the Coulomb interaction spreads in the surrounding environment, and therefore the dielectric constant changes of the environment have a considerable effect on the binding energy. To investigate the effect of polarization charges due to dielectric mismatch on the carrier energy, the polarization energy Epol is calculated and shown in Fig.3 for different β. Epol defines as the difference between the total energy and the energy without induced charges at the interface of heterostructure, (Ai = 0, W (r) = 0) which will show the effect of induced charges at the separation boundary of the media on the energy of the system. Because the dielectric permeability of the well region is greater than the barrier, the W (r) is a positive function and therefore, the energies of carriers are increased by the surface polarization. As it could be seen, the energy Epol decreases as the a1 increases. This is because the effect of surface (charges) potential on electrons decreases with increases in the semi-axis a1 . The behavior of binding energy versus the thickness of intermediate layer for a1 = 30˚ A and β = 0.1 and 0.2 is shown in Fig.4. There is a slight variation in the binding energy and it decreases as the layer thickness increases. This is because with the enhancement of L, the polarization charges increase and the electron energy is enhanced, thus the binding energy decreases. The results are similar to work done by Boichuk and et. al. for spherical quantum dot [17] and Niculescu and Cristea for nanodots [20] .
4
Conclusion
In this paper the binding energy of hydrogenic impurity located at the center of double ellipsoidal quantum dot with different dielectric constant and rectangular potential well model using an appropriate coordinate transformation is calculated. The effects of polarization charges due to impurity on binding energy are considered. The results show that the binding energy increases with the enhancement of ellipticity constant and it decreases when the dimension of quantum dot increases.
References [1] D. Bimberg, Semiconductor, 33 No: 9 (1999) 951-955 [2] G. Cantele, D. Nino, and G. Iadonisi, J. Phys.: Condens. Matter 12 (2000) 9019-9037 [3] E. Sadeghi, and A. Avazpour, Physica B 406 (2) (2011) 241-244 [4] Z. Xiao, J. Zhu, F. He, Superlattices and Microstructures, 19 (2) (1996) 137-149 5
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captions: Fig1: The variation of impurity binding energy in terms of the semi-axis of the ellipsoidal quantum dot for several β with ε1 = 13.18, ε2 = 11.932, ε3 = 10.06. Fig2: The variation of binding energy versus semi-axis of the ellipsoidal quantum dot for different dielectric constants. Fig3: The variation of polarization energy (Epol ) versus semi-axis of the ellipsoidal quantum dot for different β. Fig4: The variation of binding energy in terms of transition layer thickness L1 = L2 = L with a1 = 30(˚ A).
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3.5
3.0
β=0.2
*
Binding Energy (R y)
=0.1 =0.05 =0.0
2.5
2.0 a2=100 A
1.5
1.0 20
40
60
80
100
a1 (A) Figure 1: The variation of impurity binding energy in terms of the semi-axis of the ellipsoidal quantum dot for different β with ε1 = 13.18, ε2 = 11.932, ε3 = 10.06.
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ε1=13.18, ε2=11.932, ε3=10.06
10
ε1=ε3=13.18, ε2=11.932 ε1=13.18, ε2=ε3=11.932
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*
Binding Energy (R y)
β=0.15
6
4
2
20
40
60
80
100
a1 (A)
Figure 2: The variation of binding energy versus semi-axis of the ellipsoidal quantum dot for different dielectric constants.
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0.24
0.20
β=0.0
β=0.2
β=0.05 β=0.1
*
Epol (R y)
0.16
0.12
0.08
0.04 20
40
60
80
100
a1 (A)
Figure 3: The variation of polarization energy (Epol ) versus semi-axis of the ellipsoidal quantum dot for different β.
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3.25 β=0.2
3.15 a1=30 A, a2=100 A L1=L2
*
Binding Energy (R y )
3.20
3.10
3.05
3.00
β=0.1
2.95 4
5
6
7
8
L (A)
Figure 4: The variation of binding energy in terms of transition layer thickness L1 = L2 = L with a1 = 30(˚ A).
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