- Email: [email protected]
External electric field, hydrostatic pressure and temperature effects on the binding energy of an off-center hydrogenic impurity confined in a spherical Gaussian quantum dot
Physica E 44 (2012) 1562–1566
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
External electric field, hydrostatic pressure and temperature effects on the binding energy of an off-center hydrogenic impurity confined in a spherical Gaussian quantum dot G. Rezaei n, S.F. Taghizadeh, A.A. Enshaeian Department of Physics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 March 2012 Accepted 26 March 2012 Available online 1 April 2012
Based on the effective-mass approximation within a matrix diagonalization scheme, simultaneous effects of external electric field, hydrostatic pressure and temperature on the binding energy of an offcenter hydrogenic donor confined by a spherical Gaussian potential have been calculated. The binding energy dependencies on the dot radius, the potential depth, the impurity position, the electric field strength, the hydrostatic pressure and the temperature are reported. We found that, not only the internal parameters (such as: the dot radius, the potential depth and the impurity position) but also the external perturbations (such as: electric field strength, hydrostatic pressure and temperature) have a great influence on the impurity binding energy. & 2012 Elsevier B.V. All rights reserved.
1. Introduction Hydrogenic impurity states in semiconductor nano-structures are a subject of interest due to possible technological applications in electronic and optoelectronic devices associated with these systems. Impurity states in these structures are very essential as they govern the thermal, optical and electrical properties. After Bastard’s pioneering work on the donor impurity in a semiconductor quantum well [1], the study of impurities in semiconductor quantum dots (QDs) has also been of great interest to researchers. Thus, a great number of theoretical works has been devoted to the understanding of confined impurity states in low-dimensional semiconductor QDs with different shapes and various confinement potentials [2–13]. Furthermore, external electric field, hydrostatic pressure and temperature effects on the electronic structure of the semiconductor QDs have also attracted much attention due to practical and fundamental points of view. There are several reports on the electric field, hydrostatic pressure and temperature dependence of binding energy of hydrogenic impurities in semiconductor QDs [14–33]. For example: Duque et al. [14] have used the effective mass and parabolic band approximations and a variational scheme to calculate the combined effects of intense laser radiation, hydrostatic pressure, and applied electric field on shallowdonor impurity confined in cylindrical-shaped single and double
n
Corresponding author. Tel./fax: þ 98 741 222 3048. E-mail addresses: [email protected], [email protected] (G. Rezaei).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.03.028
GaAs2Ga1x Alx As QD. Using the direct diagonalization method and by means of the Luttinger–Kohn effective mass equation, Rezaei and Doostimotlagh [15], have investigated the effects of electric field, hydrostatic pressure and conduction band nonparabolicity on the binding energies of the lower-lying states and the diamagnetic susceptibility of an on-center hydrogenic impurity confined in a typical GaAs2Alx Ga1x As spherical QD. Simultaneous effects of pressure and magnetic field on the donor states in a parabolic quantum dot was studied by John Peter [16]. Gerardin Jayam and Navaneethakrishnan have calculated the effects of electric field and hydrostatic pressure on the donor binding energies in a spherical quantum dot with parabolic confinement potential [17]. Perez-Merchancano et al. [18] have studied the influences of hydrostatic-pressure on the donor binding energy in GaAs–(Ga, Al)As quantum dots . Based on the variational method and the effective-mass approximation, combined effects of hydrostatic pressure and temperature on the ground state binding energy of two electrons in a GaAs spherical QD have been studied by Sivakami and Mahendran [19]. Eerdunchaolu et al. [20] investigated the influence of the temperature and polaron effects on the effective potential of the weak-coupling exciton in semiconductor quantum dots, based on the Lee-Low-Pines-Huybrechts variational method. Ramos and co-workers have studied the combined effects of pressure and temperature on the exciton binding energy in a ¨ cylindrical QD with Poschl–Teller (PT) confining potential with the use of the effective mass approximation and a variational calculation procedure. To our knowledge the combined effects of external electric field, hydrostatic pressure and temperature on the binding energy
G. Rezaei et al. / Physica E 44 (2012) 1562–1566
of an off-center hydrogenic impurity in a spherical Gaussian QD have not been studied so far. So, the present paper is devoted to investigate the simultaneous effects of electric field, hydrostatic pressure and temperature on the off-center donor impurity binding energy in a spherical Gaussian QD, using the matrix diagonalization method. To this end, the Hamiltonian, the binding energy of an off-center hydrogenic impurity confined in a typical GaAs spherical Gaussian QD, under the influence of external electric field, hydrostatic pressure, and temperature are described in Section 2. Numerical results and a brief summery are presented in Sections 3 and 4 respectively.
2. Theory Within the framework of effective-mass approximation the Hamiltonian for a hydrogenic off-center impurity in a GaAs QD, under the influence of external electric field is given by ^ ¼ H
_2 2mn ðP,TÞ
r2
e2 þeFr cos y þ Vðr,PÞ, EðP,TÞ9rD9
ð2Þ
,
ð3Þ
ð5Þ
where V 0 40 denotes the depth of the potential well and R(P) is the pressure-dependent range of the confinement potential (the dot radius) which is obtained from the fractional change in the volume of the sample [37] RðPÞ ¼ Rð0Þ½13PðS11 þ 2S12 Þ1=3 :
ð6Þ 1
Here Rð0Þ is the dot radius at zero pressure, S11 ¼ 1:16 103 kbar 1 and S12 ¼ 3:7 104 kbar are the elastic constants of the GaAs [34]. We are interested to find the ground state energy eigenvalue of the Hamiltonian of Eq. (1), as this is needed for the calculation of the binding energy. To obtain energy eigenvalue associated with the hydrogenic off-center impurity, the Hamiltonian of Eq. (1) is diagonalized in the space spanned model by expanding the total wave function as follows [5]: X Fm ¼ ci ci ðrÞ, ð7Þ i
ð8Þ
where r o ¼ minðr,DÞ and r 4 ¼ maxðr,DÞ. Therefore, the matrix ^ in the above bases, are given elements of the total Hamiltonian, H, by the following expressions: X ^ Fm S ¼ /Fm 9H9 f½2ni þ li þ3=2_odij þ V ij g, ð9Þ with V ij ¼
2 1 2 Rni li ðrÞ V 0 er =R ðPÞ mn ðP,TÞo2 r 2 Rnj lj ðrÞr 2 dr 2 0 " Z Z 1 1 e2 X r lo dO Rni li ðrÞY nli ,m ðy, jÞ P ðcos yÞ lþ1 l E ðP,TÞ 0 l ¼ 0r4 eFr cos y Rnj lj ðrÞY lj ,mj ðy, jÞr 2 dr: Z
1
ð10Þ
Diagonalization of the matrix representation of the total ^ gives energy eigenvalues and corresponding Hamiltonian, H, eigenvectors of the system. Using the above energy eigenvalues one can find the binding energy of a hydrogenic impurity as ð11Þ 0
where b ¼ 1:26 101 eV GPa1 and c ¼ 3:77 103 eV GPa2 . In Eq. (1), Vðr,PÞ is the pressure dependent Gaussian confinement potential 2 =R2 ðPÞ
1 X 1 r lo ¼ P ðcos yÞ, lþ1 l 9rD9 r l¼0 4
Eb ¼ E0 E,
here, m0 is the free electron mass and EG g ðP,TÞ, the variation of the energy band gap for a GaAs quantum dot at G-point with hydrostatic pressure and temperature, is given by " # T2 4 ðP,TÞ ¼ 1:5195:405 10 ð4Þ EG þbP þcP 2 , g T þ204
Vðr,PÞ ¼ V 0 er
where ci ðrÞ ¼ Rni li ðrÞY li mi ðy, jÞ is the ith spherical harmonic oscillator eigenvector with the corresponding eigenvalue, Eið0Þ ¼ ð2ni þ li þ3=2Þ_o0 (here ni, li and mi, respectively denote the principal, orbital, and magnetic quantum numbers). Without any loss of generality, we make the assumption that the impurity ion locates on the z-axis. Then the magnetic quantum number, m, is a good one and thus the summation in Eq. (7) includes only the terms with a fixed magnetic quantum numbers m (i.e., mi ¼m). In order to calculate the matrix elements of the total Hamiltonian, we use the relation
i,j
ð1Þ
where rðDÞ is the position vector of the electron (impurity) originating from the center of the dot. F is the electric field strength making an angle y with z-axis. P is the hydrostatic pressure in GPa and T is the temperature in Kelvin. EðP,TÞ and mn ðP,TÞ are, respectively, the pressure and temperature dependent static dielectric constant and electron effective mass. For GaAs EðP,TÞ and mn ðP,T) are given by Refs. [34–36] ( 12:74 expð1:73 103 PÞexp½9:4 105 ðT75:6Þ EðP,TÞ ¼ 13:18 expð1:73 103 PÞexp½20:4 105 ðT300Þ
For T o 200 K and T Z200 K. " !#1 2 1 n þ m ðP,TÞ ¼ 1 þ7:51 G m0 , Eg ðP,TÞ EG g ðP,TÞ þ 0:341
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where E (E ) is the lowest eigenvalue of the system with (without) hydrogenic impurity.
3. Numerical results and discussion In this section we have calculated the influence of an external electric field, hydrostatic pressure, temperature and the dot parameters on the binding energy of an off-center donor impurity confined in a typical GaAs spherical Gaussian QD. The physical parameters used for numerical computation are: The effective mass of an electron and the dielectric constant at zero pressure and temperature are, respectively, taken to be mn ð0Þ ¼ 0:067me , where me is the free electron mass, and E ¼ 12:74. Figs. 1–3 show the lowest binding energy of an off-center hydrogenic impurity as a function of the dot radius for different values of external electric field, hydrostatic pressure and temperature, respectively. These figures show that, the binding energy decreases as the dot radius, R, is increased and finally, approaches to the binding energy of a free hydrogen atom at large values of R. This is a consequence of the decreasing confinement effects. From Fig. 1, it is obvious that the binding energies strongly depends on the external electric field strength, i.e., the smaller the electric field strength is, the larger the binding energy will be. This confirms the fact that the increase of the electric field causes the electron to be less confined to the impurity and accordingly reduces the binding energy. The dependencies of the donor binding energy upon the dot radius are shown in Fig. 2 for several values of the applied hydrostatic pressure. When the hydrostatic pressure increases, the dot radius and dielectric constant decrease, the effective mass of electron increases, leading to more confinement of the electron in the dot. This leads to enhance the impurity binding energy.
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G. Rezaei et al. / Physica E 44 (2012) 1562–1566
Fig. 1. Binding energy as a function of the dot radius and different values of external electric field strength.
In Fig. 3 we show our results for the impurity binding energy as a function of the dot radius for different values of temperature. It is well known that, the dielectric constant increases and the electron effective mass decreases with increasing the temperature. Also, an increase in the temperature weakens the electron localization near the impurity. Consequently, the binding energy decreases for all values of temperature. To make the above mentioned results more clear, we have plotted the variation of the binding energy versus the hydrostatic pressure and temperature in Fig. 4. It is seen that the binding energy increases (decreases) with increasing the pressure (temperature). The results of this figure are in a good agreement with those of Figs. 2 and 3. Fig. 5 displays the dependence of the lowest binding energy upon the impurity location for different values of the dot radius (a), the potential depth (b), the applied electric field (c), the hydrostatic pressure and temperature (d). It is obvious that the binding energy decreases as the impurity position is shifted away from the center. This is attributed to the fact that, due to the quantum confinement effects the electron tends to localize in the center of the dot. Moreover, from Fig. 5(b), it is clearly seen that the binding energy increases when the potential depth is increased. Increasing the potential depth, V0, causes the stronger confinement of carriers in the QD. This leads to enhancement of the binding energy. This figure clearly shows the quantum confinement effect, i.e., the stronger the confinement is, the
Fig. 2. Variations of the binding energy versus the dot radius and different values of the hydrostatic pressure.
Fig. 3. Dependence of the biding energy upon the dot size for several values of temperature.
Fig. 4. Binding energy as a function of hydrostatic pressure (a) and temperature (b) with F ¼ 50 keV=cm, D ¼ 5 nm, R ¼ 10 nm and V 0 ¼ 280 meV.
G. Rezaei et al. / Physica E 44 (2012) 1562–1566
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Fig. 5. Variation of the binding energy with respect to the impurity position and different values of the dot radius (a), the potential depth (b), the external electric field strength (c), the hydrostatic pressure and temperature (d).
bigger the binding energy will be. These results are in a good agreement with those of Ref. [5].
4. Conclusion In conclusion, we have studied the effects of the dot radius, potential depth, impurity position, external electric field, hydrostatic pressure, and temperature on the hydrogenic impurity ground state binding energy of a spherical Gaussian QD, by using the matrix diagonalization method. Our results show that the binding energy strongly depends on the values of the dot parameters, R and V0, the impurity location, D, and the external perturbations, F, P and T. We found that the binding energy decreases with increasing (decreasing) R (V0). Furthermore, an increase (decrease) in P (F and T) enhances the donor binding energy.
Acknowledgments This work is supported by the Yasouj University. References [1] G. Bastard, Physical Review B 24 (1981) 4714. [2] M.K. Bose, K. Midya, C. Bose, Journal of Applied Physics 101 (2007) 054315. [3] R.P.A. Lim, M. Amado, F. Domı´nguez-Adame, Nanotechnology 19 (2008) 135402.
[4] Z. Barticevic, M. Pacheco, C.A. Duque, L.E. Oliveira, Journal of Physics: Condensed Matter 21 (2009) 405801. [5] W. Xie, Physica B 403 (2008) 2828. [6] I.F.I. Mikhail, I.M.M. Ismail, Physica Status Solidi (b) 244 (2007) 3647. [7] M. Barati, G. Rezaei, M.R.K. Vahdani, Physica Status Solidi (b) 244 (2007) 2605. [8] D.S. Kr¨ahmery, W.P. Schleichy, V.P. Yakovlevz, Journal of Physics A: Mathematical and General 31 (1998) 4493. [9] R.S. Daries Bella, K. Navaneethakrishnan, Solid State Communication 130 (2004) 773. [10] M. Revathi, A. John Peter, Solid State Communication 150 (2010) 816. [11] S.-S. Li, J.-B. Xia, Journal of Applied Physics 100 (2006) 083714. [12] J. Pe rez-Conde, A.K. Bhattacharjee, Solid State Communication 135 (2005) 496. [13] M. Barati, M.R.K. Vahdani, G. Rezaei, Journal of Physics: Condensed Matter 19 (2007) 136208. [14] C.M. Duque, M.G. Barseghyan, C.A. Duque, Physica B 404 (2009) 5177. [15] G. Rezaei, N.A. Doostimotlagh, Physica E 44 (2012) 833. [16] A. John Peter, Solid State Communication 147 (2008) 296. [17] Sr. Gerardin Jayam, K. Navaneethakrishnan, Solid State Communication 126 (2003) 681. [18] S.T. Perez-Merchancano, H. Paredes-Gutierrez, J. Silva-Valencia, Journal of Physics: Condensed Matter 19 (2007) 026225. [19] A. Sivakami, M. Mahendran, Superlattices and Microstructures 47 (2010) 530. [20] E. Wuyunqimuge, W. Xin, Superlattices and Microstructures 48 (2010) 373. [21] M.E. Mora-Ramos, M.G. Barseghyan, C.A. Duque, Physica E 43 (2010) 338. [22] C. Xia, Y. Liu, S. Wei, Applied Surface Science 254 (2008) 3479. [23] L. Jiang, H. Wang, H. Wu, Q. Gong, S. Feng, Journal of Applied Physics 105 (2009) 053710. [24] D. Nasri, N. Sekkal, Physica E 42 (2010) 2257. [25] G. Rezaei, B. Vaseghi, J. Ebrahimi, Superlattices and Microstructures 49 (2011) 591. [26] C.A. Duque, E. Kasapoglu, S. Sakiroglu, H. Sari, I. Sokmen, Applied Surface Science 256 (2010) 7406. [27] A. Sivakami, M. Mahendran, Physica B 405 (2010) 1403. [28] S. Sahoo, Y.C. Lin, Y.K. Ho, Physica E 40 (2008) 3107. [29] K. Chang, J.-B. Xia, Journal of Applied Physics 84 (1998) 1454. [30] S.-S. Li, J.-B. Xia, Journal of Applied Physics 101 (2007) 093716.
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[31] A. John Peter, K. Gnanasekar, K. Navaneethakrishnan, European Physical Journal B 53 (2006) 283. [32] A.L. Vartanian, L.A. Vardanyan, E.M. Kazaryan, Physica E 40 (2008) 1513. [33] L. He, W. Xie, Superlattices and Microstructures 47 (2010) 266. ¨ [34] E. Kasapoglu, F. Ungan, H. Sari, I. Sokmen, Physica E 42 (2010) 1623.
¨ [35] M.G. Barseghyan, A. Hakimyfard, S.Y. Lopez, C.A. Duque, A.A. Kirakosyan, Physica E 42 (2010) 1618. ¨ [36] U. Yesilgul, S. S- akirogˇlu, E. Kasapogˇlu, H. Sari, I. Sokmen, Superlattices and Microstructures 48 (2010) 106. [37] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 1996.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
External electric field, hydrostatic pressure and temperature effects on the binding energy of an off-center hydrogenic impurity confined in a spherical Gaussian quantum dot G. Rezaei n, S.F. Taghizadeh, A.A. Enshaeian Department of Physics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 March 2012 Accepted 26 March 2012 Available online 1 April 2012
Based on the effective-mass approximation within a matrix diagonalization scheme, simultaneous effects of external electric field, hydrostatic pressure and temperature on the binding energy of an offcenter hydrogenic donor confined by a spherical Gaussian potential have been calculated. The binding energy dependencies on the dot radius, the potential depth, the impurity position, the electric field strength, the hydrostatic pressure and the temperature are reported. We found that, not only the internal parameters (such as: the dot radius, the potential depth and the impurity position) but also the external perturbations (such as: electric field strength, hydrostatic pressure and temperature) have a great influence on the impurity binding energy. & 2012 Elsevier B.V. All rights reserved.
1. Introduction Hydrogenic impurity states in semiconductor nano-structures are a subject of interest due to possible technological applications in electronic and optoelectronic devices associated with these systems. Impurity states in these structures are very essential as they govern the thermal, optical and electrical properties. After Bastard’s pioneering work on the donor impurity in a semiconductor quantum well [1], the study of impurities in semiconductor quantum dots (QDs) has also been of great interest to researchers. Thus, a great number of theoretical works has been devoted to the understanding of confined impurity states in low-dimensional semiconductor QDs with different shapes and various confinement potentials [2–13]. Furthermore, external electric field, hydrostatic pressure and temperature effects on the electronic structure of the semiconductor QDs have also attracted much attention due to practical and fundamental points of view. There are several reports on the electric field, hydrostatic pressure and temperature dependence of binding energy of hydrogenic impurities in semiconductor QDs [14–33]. For example: Duque et al. [14] have used the effective mass and parabolic band approximations and a variational scheme to calculate the combined effects of intense laser radiation, hydrostatic pressure, and applied electric field on shallowdonor impurity confined in cylindrical-shaped single and double
n
Corresponding author. Tel./fax: þ 98 741 222 3048. E-mail addresses: [email protected], [email protected] (G. Rezaei).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.03.028
GaAs2Ga1x Alx As QD. Using the direct diagonalization method and by means of the Luttinger–Kohn effective mass equation, Rezaei and Doostimotlagh [15], have investigated the effects of electric field, hydrostatic pressure and conduction band nonparabolicity on the binding energies of the lower-lying states and the diamagnetic susceptibility of an on-center hydrogenic impurity confined in a typical GaAs2Alx Ga1x As spherical QD. Simultaneous effects of pressure and magnetic field on the donor states in a parabolic quantum dot was studied by John Peter [16]. Gerardin Jayam and Navaneethakrishnan have calculated the effects of electric field and hydrostatic pressure on the donor binding energies in a spherical quantum dot with parabolic confinement potential [17]. Perez-Merchancano et al. [18] have studied the influences of hydrostatic-pressure on the donor binding energy in GaAs–(Ga, Al)As quantum dots . Based on the variational method and the effective-mass approximation, combined effects of hydrostatic pressure and temperature on the ground state binding energy of two electrons in a GaAs spherical QD have been studied by Sivakami and Mahendran [19]. Eerdunchaolu et al. [20] investigated the influence of the temperature and polaron effects on the effective potential of the weak-coupling exciton in semiconductor quantum dots, based on the Lee-Low-Pines-Huybrechts variational method. Ramos and co-workers have studied the combined effects of pressure and temperature on the exciton binding energy in a ¨ cylindrical QD with Poschl–Teller (PT) confining potential with the use of the effective mass approximation and a variational calculation procedure. To our knowledge the combined effects of external electric field, hydrostatic pressure and temperature on the binding energy
G. Rezaei et al. / Physica E 44 (2012) 1562–1566
of an off-center hydrogenic impurity in a spherical Gaussian QD have not been studied so far. So, the present paper is devoted to investigate the simultaneous effects of electric field, hydrostatic pressure and temperature on the off-center donor impurity binding energy in a spherical Gaussian QD, using the matrix diagonalization method. To this end, the Hamiltonian, the binding energy of an off-center hydrogenic impurity confined in a typical GaAs spherical Gaussian QD, under the influence of external electric field, hydrostatic pressure, and temperature are described in Section 2. Numerical results and a brief summery are presented in Sections 3 and 4 respectively.
2. Theory Within the framework of effective-mass approximation the Hamiltonian for a hydrogenic off-center impurity in a GaAs QD, under the influence of external electric field is given by ^ ¼ H
_2 2mn ðP,TÞ
r2
e2 þeFr cos y þ Vðr,PÞ, EðP,TÞ9rD9
ð2Þ
,
ð3Þ
ð5Þ
where V 0 40 denotes the depth of the potential well and R(P) is the pressure-dependent range of the confinement potential (the dot radius) which is obtained from the fractional change in the volume of the sample [37] RðPÞ ¼ Rð0Þ½13PðS11 þ 2S12 Þ1=3 :
ð6Þ 1
Here Rð0Þ is the dot radius at zero pressure, S11 ¼ 1:16 103 kbar 1 and S12 ¼ 3:7 104 kbar are the elastic constants of the GaAs [34]. We are interested to find the ground state energy eigenvalue of the Hamiltonian of Eq. (1), as this is needed for the calculation of the binding energy. To obtain energy eigenvalue associated with the hydrogenic off-center impurity, the Hamiltonian of Eq. (1) is diagonalized in the space spanned model by expanding the total wave function as follows [5]: X Fm ¼ ci ci ðrÞ, ð7Þ i
ð8Þ
where r o ¼ minðr,DÞ and r 4 ¼ maxðr,DÞ. Therefore, the matrix ^ in the above bases, are given elements of the total Hamiltonian, H, by the following expressions: X ^ Fm S ¼ /Fm 9H9 f½2ni þ li þ3=2_odij þ V ij g, ð9Þ with V ij ¼
2 1 2 Rni li ðrÞ V 0 er =R ðPÞ mn ðP,TÞo2 r 2 Rnj lj ðrÞr 2 dr 2 0 " Z Z 1 1 e2 X r lo dO Rni li ðrÞY nli ,m ðy, jÞ P ðcos yÞ lþ1 l E ðP,TÞ 0 l ¼ 0r4 eFr cos y Rnj lj ðrÞY lj ,mj ðy, jÞr 2 dr: Z
1
ð10Þ
Diagonalization of the matrix representation of the total ^ gives energy eigenvalues and corresponding Hamiltonian, H, eigenvectors of the system. Using the above energy eigenvalues one can find the binding energy of a hydrogenic impurity as ð11Þ 0
where b ¼ 1:26 101 eV GPa1 and c ¼ 3:77 103 eV GPa2 . In Eq. (1), Vðr,PÞ is the pressure dependent Gaussian confinement potential 2 =R2 ðPÞ
1 X 1 r lo ¼ P ðcos yÞ, lþ1 l 9rD9 r l¼0 4
Eb ¼ E0 E,
here, m0 is the free electron mass and EG g ðP,TÞ, the variation of the energy band gap for a GaAs quantum dot at G-point with hydrostatic pressure and temperature, is given by " # T2 4 ðP,TÞ ¼ 1:5195:405 10 ð4Þ EG þbP þcP 2 , g T þ204
Vðr,PÞ ¼ V 0 er
where ci ðrÞ ¼ Rni li ðrÞY li mi ðy, jÞ is the ith spherical harmonic oscillator eigenvector with the corresponding eigenvalue, Eið0Þ ¼ ð2ni þ li þ3=2Þ_o0 (here ni, li and mi, respectively denote the principal, orbital, and magnetic quantum numbers). Without any loss of generality, we make the assumption that the impurity ion locates on the z-axis. Then the magnetic quantum number, m, is a good one and thus the summation in Eq. (7) includes only the terms with a fixed magnetic quantum numbers m (i.e., mi ¼m). In order to calculate the matrix elements of the total Hamiltonian, we use the relation
i,j
ð1Þ
where rðDÞ is the position vector of the electron (impurity) originating from the center of the dot. F is the electric field strength making an angle y with z-axis. P is the hydrostatic pressure in GPa and T is the temperature in Kelvin. EðP,TÞ and mn ðP,TÞ are, respectively, the pressure and temperature dependent static dielectric constant and electron effective mass. For GaAs EðP,TÞ and mn ðP,T) are given by Refs. [34–36] ( 12:74 expð1:73 103 PÞexp½9:4 105 ðT75:6Þ EðP,TÞ ¼ 13:18 expð1:73 103 PÞexp½20:4 105 ðT300Þ
For T o 200 K and T Z200 K. " !#1 2 1 n þ m ðP,TÞ ¼ 1 þ7:51 G m0 , Eg ðP,TÞ EG g ðP,TÞ þ 0:341
1563
where E (E ) is the lowest eigenvalue of the system with (without) hydrogenic impurity.
3. Numerical results and discussion In this section we have calculated the influence of an external electric field, hydrostatic pressure, temperature and the dot parameters on the binding energy of an off-center donor impurity confined in a typical GaAs spherical Gaussian QD. The physical parameters used for numerical computation are: The effective mass of an electron and the dielectric constant at zero pressure and temperature are, respectively, taken to be mn ð0Þ ¼ 0:067me , where me is the free electron mass, and E ¼ 12:74. Figs. 1–3 show the lowest binding energy of an off-center hydrogenic impurity as a function of the dot radius for different values of external electric field, hydrostatic pressure and temperature, respectively. These figures show that, the binding energy decreases as the dot radius, R, is increased and finally, approaches to the binding energy of a free hydrogen atom at large values of R. This is a consequence of the decreasing confinement effects. From Fig. 1, it is obvious that the binding energies strongly depends on the external electric field strength, i.e., the smaller the electric field strength is, the larger the binding energy will be. This confirms the fact that the increase of the electric field causes the electron to be less confined to the impurity and accordingly reduces the binding energy. The dependencies of the donor binding energy upon the dot radius are shown in Fig. 2 for several values of the applied hydrostatic pressure. When the hydrostatic pressure increases, the dot radius and dielectric constant decrease, the effective mass of electron increases, leading to more confinement of the electron in the dot. This leads to enhance the impurity binding energy.
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Fig. 1. Binding energy as a function of the dot radius and different values of external electric field strength.
In Fig. 3 we show our results for the impurity binding energy as a function of the dot radius for different values of temperature. It is well known that, the dielectric constant increases and the electron effective mass decreases with increasing the temperature. Also, an increase in the temperature weakens the electron localization near the impurity. Consequently, the binding energy decreases for all values of temperature. To make the above mentioned results more clear, we have plotted the variation of the binding energy versus the hydrostatic pressure and temperature in Fig. 4. It is seen that the binding energy increases (decreases) with increasing the pressure (temperature). The results of this figure are in a good agreement with those of Figs. 2 and 3. Fig. 5 displays the dependence of the lowest binding energy upon the impurity location for different values of the dot radius (a), the potential depth (b), the applied electric field (c), the hydrostatic pressure and temperature (d). It is obvious that the binding energy decreases as the impurity position is shifted away from the center. This is attributed to the fact that, due to the quantum confinement effects the electron tends to localize in the center of the dot. Moreover, from Fig. 5(b), it is clearly seen that the binding energy increases when the potential depth is increased. Increasing the potential depth, V0, causes the stronger confinement of carriers in the QD. This leads to enhancement of the binding energy. This figure clearly shows the quantum confinement effect, i.e., the stronger the confinement is, the
Fig. 2. Variations of the binding energy versus the dot radius and different values of the hydrostatic pressure.
Fig. 3. Dependence of the biding energy upon the dot size for several values of temperature.
Fig. 4. Binding energy as a function of hydrostatic pressure (a) and temperature (b) with F ¼ 50 keV=cm, D ¼ 5 nm, R ¼ 10 nm and V 0 ¼ 280 meV.
G. Rezaei et al. / Physica E 44 (2012) 1562–1566
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Fig. 5. Variation of the binding energy with respect to the impurity position and different values of the dot radius (a), the potential depth (b), the external electric field strength (c), the hydrostatic pressure and temperature (d).
bigger the binding energy will be. These results are in a good agreement with those of Ref. [5].
4. Conclusion In conclusion, we have studied the effects of the dot radius, potential depth, impurity position, external electric field, hydrostatic pressure, and temperature on the hydrogenic impurity ground state binding energy of a spherical Gaussian QD, by using the matrix diagonalization method. Our results show that the binding energy strongly depends on the values of the dot parameters, R and V0, the impurity location, D, and the external perturbations, F, P and T. We found that the binding energy decreases with increasing (decreasing) R (V0). Furthermore, an increase (decrease) in P (F and T) enhances the donor binding energy.
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