Shallow Donor Impurity Ground State in a GaAs/AlAs Spherical Quantum Dot within an Electric Field

Commun. Theor. Phys. (Beijing, China) 52 (2009) pp. 710 714 c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 4, October 15, 2009 Shallow Donor Impurity Ground State in a GaAs/AlAs Spherical Quantum Dot within an Electric Field YUAN Jian-Hui, XIE Wen-Fang, and HE Li-Li Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, China (Received October 20, 2008) Abstract Using the configuration-integration methods (CI) [Phys. Rev. B 45 (1992) 19], we report the results of the Hydrogenic-impurity ground state in a GaAs/AlAs spherical quantum dot under an electric field. We discuss the variations of the binding energies of the Hydrogenic-impurity ground state as a function of the position of impurity D, the radius R of the quantum dot, and also as a function of electric field F. We find that the ground energy and binding energy of impurity placed anywhere depend strongly on the position of impurity. Also, electric field can largely change the Hydrogenic-impurity ground state only limiting to the big radius of quantum dot. And the differences in energy level and binding energy are observed from the center donor and off-center donor. PACS numbers: 73.20.Dx, 73.20.Hb, 71.15.Nc, 71.55.Eq Key words: donor, quantum dots, binding energy, electric field 1 Introduction In the past decades, the advances in crystal-growth techniques, such as etching or molecular beam epitaxy etc., has now become possible to produce quantum dots (QDs). The new, unusual properties of the lowdimensional nanometer-sized semiconductor (the three dimensional nanoscale confinements of the charged carriers) give rise to a full quantum nature. [2] Many new phenomena have been discovered in this low-dimensional semiconductor system. Thus, more and more attentions are attracted to study the nature of QDs, especially, because of their potential applications in the optical devices. QDs may be fabricated in different shapes, for example, disklike (cylindrical) shape, spherical shape, and so on. The spherical shape QDs are fabricated from semiconductor nanocrystals embedded in either an insulating or a semiconducting matrix. From the point of view of quantum confinement, impurities in semiconductors can affect the electrical, optical, and transport properties. So, an understanding of the nature of impurity states in semiconductor structures is one of the crucial problems. The study of the impurity states in semiconductor nanostructures was initiated only in the early 1980s through the pioneering works of Bastard. [3] In spite of growing interest in the topic of impurity doping in nanocrystallites, most theoretical work have been carried out on shallow donors in spherical QDs employing perturbation methods, [4 8] or variational approaches. [9 16,18 24] For example, using the perturbation methods, Bose el al. [4 7] obtained the binding energy of a shallow hydrogenic impurity in a spherical quantum dot. Based on variational approaches, Zhu [9 11] studied the energies of an off-center hydrogenic donor confined by a spherical QD with a finite rectangular potential well. Using the plane wave method, Li [17] calculated the electronic states of a hydrogenic donor impurity in low-dimensional semiconductor nanostructures in the framework of effectivemass envelope-function theory. Recently, using perturbation methods, we [8] has reported the binding energy of an off-center hydrogenic donor confined by a spherical parabolic potential. However, the above-mentioned authors do not discuss the electric field effect on the impurity states. As we known, the electric field can destroy the symmetry of the system. Thus, the energy levels of shallow impurities in QDs can draw a considerable attention. Also, both from the theoretical and technological point of view, external fields have become an interesting probe for studying the physical properties of low-dimensional systems. It is because that an electric field results in an energy Stark shift of the quantum states. Thus, the energy spectrum of the carriers will make respectable changes. However, until now, few other authors [19 24] have studied how electric field affects the energy levels and binding energies of a donor impurity. Murillo [19] calculated the binding energy of an on-center donor impurity in a spherical GaAs-(Ga,Al)As QD with parabolic confinement as a function of the radius of the QD and as a function of the intensity of an applied electric field by variational approaches. The quantum size, impurity position, and electric field effects on the energy of a shallow donor placed anywhere in a GaAs spherical quantum crystallite embedded in Ga 1 x Al x As matrix was discussed theoretically by Assaid et al. [20] Recently, based on the same variational approaches, Peter et al. [24] have studied the binding en- Supported by the National Natural Science Foundation of China under Grant No. 10775035 E-mail: jianhui831110@163.com

No. 4 Shallow Donor Impurity Ground State in a GaAs/AlAs Spherical Quantum Dot within an Electric Field 711 ergies of shallow hydrogenic impurity in a GaAs/GaAlAs quantum dot with spherical confinement, parabolic confinement, and rectangular confinement as a function of dot radius in the influence of electric field. However, the donor position effect on the level and binding energy of donor impurity in presence of external electric field was not taken into account by above-mentioned authors but the author of Refs. [20,21]. And mostly all of study about this topic was carried out by the variational approaches. In the present paper, we will employ parabolic confining potential to study the electronic properties of a donor placed anywhere in a spherical QD in the presence of an external electric field by CI methods. We will attempt to study the influence of position of impurity, the external electric field on ground state of shallow donor impurity. Our numerical calculation is carried out for one of the typical semiconducting materials, GaAs/AlAs. In Sec. 2, the Hamiltonian and the calculation method are presented. Main results are presented in Sec. 3. A simple summary is presented in Sec. 4 2 Model and Theory Within the framework of effective-mass approximation, the Hamiltonian of an off-center hydrogenic donor confined by a spherical QD with a parabolic potential can be written as follow: H = p2 e e 2 2m + V (r) + qf r, (1) e ε r D where the hydrogenic impurity locates at D, r(p) is the position vector (the momentum vector) of the electron originating from the center of the dot, q is the electronic charge, F is the external electric field strength, m e is the effective mass of an electron and V (r) is the confining potential in the form of V (r) = 1 2 m e ω2 0 r2, (2) in where, ω 0 is the strength of the confinement potential frequency. Considering the strong confinement potential, the Hamiltonian of single hydrogenic impurity in a spherical QD can be expressed as the sum of the original harmonic Hamiltonian oscillator term H (0) and a Coulomb interaction term H (1) that acts as a perturbation term over H (0), i.e., where H = H (0) + H (1) + H (2), (3) H (0) = p2 e 2m + 1 e 2 m eω0r 2 2, (4) e2 H (1) = ε r D, (5) H (2) = qf r. (6) We introduce the Spherical coordinates. In order to obtain the ground eigenfunction and eigenenergy associated with the off-center hydrogenic donor within an external electric field in a spherical QD, let us consider a linear function of the form f Ψ m = c i ψ i. (7) i=1 In there, ψ i (r) is a 3D harmonic oscillator state with the frequency ω 0 and an energy (2n i +l i +3/2) hω 0. The principal, orbital, and magnetic quantum numbers of ψ i (r) are n i, l i, and m i, respectively. We make the assumption that the impurity ion locates on the Z-axis. Then the magnetic quantum number is a good one. The summation in Eq. (7) includes only the terms with a fixed magnetic quantum numbers m, (i.e., m 1 = m 2 = = m = 0). According to Schrödinger Equation H Ψ m = E Ψ m. (8) We can easily obtain the equation f [H i,j Eδ ij ]c j = 0, i = 1, 2,...,f. (9) j=1 In order to calculate the matrix elements of Hamiltonian, we adopt the relation 1 r D = l=0 r< l r> l+1 P l (cos Θ), (10) where r < = min(r, D), r > = max(r, D), and Θ is the angle between r and D. The matrix elements of Hamiltonian are then given by the following expressions, with u I i,j H i,j = [N + 3/2] hω 0 δ i,j + u I i,j + u II i,j, (11) = qf i r cosθ j, (12) u II i,j = e2 ε l i+l j l= l i l j ( li l j l 0 0 0 [ D 1 D l+1 0 ( 1) m (2l i + 1) (2l j + 1) ) ( ) li l j l m m 0 R nil i R njl j r l+2 dr ] + D l R nil i R njl j r 1 l dr, (13) D where ( l 1 l 2 l 3 m 1 m 2 m 3 is the Winger 3-j symbol. In order to gain the level E of impurity ground state, we only need to diagonalize the matrix elements of H by using computer, and abstract the minimum value. )

712 YUAN Jian-Hui, XIE Wen-Fang, and HE Li-Li Vol. 52 According to Refs. [8,10,25], the binding energy of an off-center hydrogenic donor ground state confined by a spherical QD can be defined by not more than 1% even the radius of QD is bigger than 25 nm. E B = E 0 E, (14) where the energy level E 0 is equivalent to the case that there is no Coulomb potential in the Hamiltonian of Eq. (1), which can be obtained exactly. For center donor impurity, we only take the position of donor D = 0 to discuss. 3 Results and Discussions Our numerical calculation is carried out for one of the typical semiconducting materials, GaAs/AlAs, as an example with the material parameters shown in the following: ǫ = 13.1, m e = 0.067m 0, where m 0 is the mass of the free electron. Then, the length unit is nm, and the effective Rydberg (the energy unit) is R y = e2 /2a B ǫ = 5.31 mev. In the case of QD with parabolic confinement, the energy of electrons is hω = h 2 β/m. The parameter β, thus scales as 1/R 2 (i.e., R β 1/2 ), where R is the radius of the spherical QD. In the following discussion, we assume that the radius R of the QD is equivalent to the parameter β. Also according to Ref. [21], we introduce a parameter ξ = D/R to describe the position of donor impurity. Fig. 2 (a) The energy of an off-center impurity ground state E as a function of R with three different F values and ξ = 0.5. (b) The energy of a center impurity ground state E as a function of R with three different F values. Fig. 1 The ground energy E as a function of an external electric field strength F for an electric in the parabolic QDs. Figure 1 shows the ground energy E as a function of an external electric field strength F for an electron in the parabolic QDs. We find that the ground energy E decreases with the increase of the radius of QD. It is very obvious that the effect of the electric field F on the ground energy E, especially to the big radius of QD. Compared electric field strength F = 500.0 KV/m. With F = 0.0 KV/m, we find the ground energy E descends about m eq 2 F 2 R 4 /(2 h 2 ) that is similar to the result of the accurate values. We note that the error of our result is To our study upon the energy of the donor ground state, we present in Fig. 2(a) the ground energy E of an off-center donor as a function of R with three different F values and ξ = 0.5, where the values of F are set from F = 0.0 to F = 500.0 KV/m. The ground energies of a center donor with the same parameters are showed in Fig. 2(b). It is observed that the ground energy increases with the decrease of radius of QD. Also, we find that the electric field effect on the ground state of donor impurity, factually, can be neglected only limiting to the smaller radius of QD. For center donor ground energy, in the presence of an external electric field, our results in Fig. 2(b) is similar to the Ref. [19]. In comparison with Ref. [19], a difference is that we select some weaker electric field to discuss. Our aim is that a large negative energy does not want to appear. Compared Fig. 2(a) with Fig. 2(b), as the impurity is shifted off the center, we notice that the ground energy ascends that is very different to the center donor energy. And the bigger the electric field is, the more obvious the effect is. This is a consequence of the quantum confinement [8,25] and the effect of the electric field on

No. 4 Shallow Donor Impurity Ground State in a GaAs/AlAs Spherical Quantum Dot within an Electric Field 713 the electronic probability density. [19] In Fig. 3, we show the binding energy of ground state E B as a function of R with three different F values, where the values of F are set from F = 0.0 to F = 500.0 KV/m. We discuss the ground binding energy of both an off-center donor impurity showed in Fig. 3(a) and a center donor impurity showed in Fig. 3(b). It is observed that the ground bind energy of donor impurity decreases as the electric field increases for fixed the radius of QD in both an off-center donor and a center donor. It is interpreted that the presence of electric field will destroy the distribution of ground wavefunction of the electron. Thus, the Coulomb force between the donor ion and electron will weaken. The result is also similar to previous reports. [19] With the same above discussion, the electron field effect on binding energy of both an off-center donor and a center donor can be neglected in a smaller radius of QD because the quantum size confinement is very large. It is noticeable that for larger F the slope of the curve (dot) firstly charge quickly in the region of smaller radius of QD, then almost unchange in the region of bigger radius of QD. It implies that the Coulomb force between the donor ion and electron will become weaker in the region of bigger radius of QD. Compared Fig. 3(a) with Fig. 3(b), the binding energy will descend as the impurity is moved off the center. Thus, the effect of the position of the donor should be considered in experimental studies. Fig. 3 (a) The binding energy of ground state E B as a function of R with three different F values and ξ = 0.5. (b) The binding energy of a center impurity ground state E B as a function of R with three different F values. Fig. 4 (a) The binding energy of ground state E B as a function of an external electric field strength F with three different ξ values and R = 10nm. (b) The binding energy of ground state E B as a function of an external electric field strength F with three different ξ values and R = 5 nm. Fig. 5 The binding energy of ground state E B as a function of ξ with three different F values and R = 5 nm. To understand the effect of electric field on the ground binding energy in the presence of an external electric field, we depict the ground binding energy E B as a function of an external electric field strength F with three different ξ values and R = 10 nm that is showed in Fig. 4(a). In Fig. 4(b), we show the same case but R = 5 nm. It is

714 YUAN Jian-Hui, XIE Wen-Fang, and HE Li-Li Vol. 52 observed that as the impurity is shifted off the center for fixed electric field, the ground binding energy of impurity decreases. Also, the ground binding energy decreases with the increase of electric field for fixed the position of impurity. It is important that the position of impurity effect on binding energy is obvious in both bigger and smaller radius of QD for fixed electric field F. However, electric field effect on binding energy is observed obviously only limiting to bigger radius of QD. In Fig. 5, we plot the ground binding energy E B as a function of ξ with three different F values and a smaller radius R = 5 nm. As we expected, for smaller radius of QD, the position of impurity effect on binding energy is more obvious than the electric field effect mainly because of quantum size confinement. We notice that as the impurity is shifted off the center, the ground binding energy decreases quickly with smaller radius QD for fixed electric field. However, it is inconspicuous that the binding energy ascends with the decrease of electric field for fixed the position of the donor impurity. 4 Summary In conclusion, we have applied the parabolic confining potential to a description of the off-center donor impurity in spherical QDs within an external electric field. Based on CI methods, the ground energy and binding energy of donor impurity located anywhere, have been studied by us. We discuss the variations of the energy level and binding energies of Hydrogenic-impurity ground state as a function of the position of impurity D, the radius R of the quantum dot, and also as a function of electric field F. We find that it is very obvious that the ground energy and binding energy of impurity placed anywhere depends strongly on the position of impurity. Also, electric field can largely change the Hydrogenic-impurity ground state only limiting the bigger radius of QD. And the differences in energy level and binding energy are observed from the center donor and off-center donor. References [1] W.M. Que, Phys. Rev. B 45 (1992) 19. [2] T. Chakraborty, Quantum Dots, Elsevier Science B.V., Amsterdam (1999). [3] G. Bastard, Phys. Rev. B 24 (1981) 4714. [4] C. Bose, Physica E 4 (1999) 180. [5] C. Bose and C.K. Sarkar, Soli. Stat. Electron 42 (1998) 1661. [6] C. Bose and C.K. Sarkar, Physica B 253 (1998) 238. [7] C. Bose, J. Appl. Phys. 83 (1998) 3089. [8] J.H. Yuan and C. Liu, Physica E 41 (2008) 41. [9] J.L. Zhu, J.J. Xiong, and B.L. Gu, Phys. Rev. B 41 (1990) 6001. [10] J.L. Zhu and X. Chen, Phys. Rev. B 50 (1994) 4497. [11] J.L. Zhu, J.H. Zhao, and J.J. Xiong, J. Phys. Condens. Matter. 6 (1994) 5097. [12] A. John Peter, Physica E 28 (2005) 225. [13] S. Kang, J. Li, and T.Y. Shi, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 3491. [14] A.K. Manaselyan and A.A. Kirakosyan, Physica E 28 (2005) 462. [15] Y.P. Varshni, Superlatt. Microstruct. 29 (2001) 233. [16] C. Dane, H. Akbas, S. Minez, and A. Guleroglu, Physica E 41 (2008) 278. [17] S.S. Li and J.B. Xia, Phys. Lett. A 366 (2007) 120 [18] S. Baskoutasa and A.F. Terzis, Physica E 40 (2008) 1367. [19] G. Murillo and N.P. Montenegro, Phys. Stat. Sol. (b) 220 (2000) 187. [20] E. Assaid, E. Feddi, M. Khaidar, F. Dujardin, and B. Stëbé, Phys. Scr. 63 (2001) 329. [21] M. Ulas, E. Cicek, and S. Senturk Dalgic, Phys. Stat. Sol. (b) 241 (2004) 2968. [22] E.C. Niculescu, Moder. Phys. Lett. B 15 (2001) 545. [23] S.G. Jayam and K. Navaneethakrishnan, Soli. Stat. Commun. 126 (2003) 681. [24] A.J. Peter and V. Lakshminarayana, Chin. Phys. Lett. 25 (2005) 3021 [25] W.F. Xie, Physica B 403 (2008) 2828.