Polarizabilities of shallow donors in spherical quantum dots with parabolic confinement

Physics Letters A 355 (2006) 59–62 www.elsevier.com/locate/pla

Polarizabilities of shallow donors in spherical quantum dots with parabolic confinement A. John Peter Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China Received 30 August 2005; accepted 25 January 2006 Available online 28 February 2006 Communicated by J. Flouquet

Abstract The binding energy and impurity polarizability of hydrogenic donors in a spherical quantum dot of Cd1−xin Mnxin Te/Cd1−xout Mnxout Te structure are presented assuming parabolic confinement. Within the effective mass approximation, the effects of electric field, as a function of dot size, are discussed on the results obtained using a variational method. The results show that the impurity binding energy (i) increases with the reduction in dot sizes, (ii) decreases in an electric field—this decrease is more prominent for wider dots whereas for narrow dots the electric field seems to be unimportant, (iii) increases to a maximum value around the effective Bohr radius, i.e., 60 Å, and then decreases as the size of the dot increases, and (iv) the polaronic effects are suppressed when an electric field is coupled. Spin polaronic shifts are estimated using a mean field theory. The results show that the spin polaronic shift decreases as the electric field and dot radius increase. These results are compared with the existing literatures. © 2006 Elsevier B.V. All rights reserved. PACS: 71.38.+i; 73.20.Dx; 73.40.Lq; 76.40.+b Keywords: Donor binding; Polaron; Quantum dot

The fabrication of semiconductor heterostructures, especially after the invention of advanced techniques such as MBE (Molecular Beam Epitaxial) growth and MOCVD (Metal Organic Chemical Vapour Deposition) combined with electron lithography, with quantum confinement in all three directions is expected to exhibit some exotic electronic behaviour. The impurities in semiconductors affect the electrical, optical and transport properties. The work of Bastard [1] on the binding energy of a hydrogenic impurity in a quantum well with an infinite potential barrier has been extended to QWW and quantum dots (QDs) [2,3]. Chayanika Bose obtained the binding energies of shallow hydrogenic impurities in spherical QDs with parabolic confinement using variational and perturbation approaches, assuming infinite barriers [4,5]. The energy levels of shallow donor impurities in QD systems in the presence of magnetic fields have drawn considerable attention [6,7]. Both

E-mail address: [email protected] (A. John Peter). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.107

from the theoretical and experimental point of view, the effects of an applied electric field on the physical properties of lowdimensional systems have become a considerable interest in the development of innovative semiconductor devices. In particular, there occurs an energy shift in quantum states and a polarization of the carrier distribution, if the application of electric field is in the direction of growth of the heterostructure. These effects are used to control and modulate the intensity output of optoelectronic devices. López-Gondar et al. [8] and Santiago et al. [9] have reported on the electric field effects on shallow impurity states in GaAs/GaAlAs QWs. Using the Hasse variational method, Ilaiwi [10] has made a theoretical calculation that disclosed between the effect of screening and the polarizabilities of shallow donors and acceptors in infinite barrier of QWs. Binding energies of shallow donors in an electric field have been studied on quantum boxes and the results show that as the size of the quantum box is reduced both the energy of the ground state and the binding energy increase [11]. Theoretical calculations of the effects of the electric field on the energy lev-

60

A. John Peter / Physics Letters A 355 (2006) 59–62

els in a GaAs/GaInAs surface quantum well wire (QWW) and on the geometry-dependent polarizability of a shallow donor in a rectangular cross section QWW have been reported as well [12,13]. As a general feature the polarizability increases with the size of the structure and on increasing the electric field. However, the literature is sparse with the results of diluted magnetic quantum dots on the effects of electric field. In the present work, we investigate the effects of a static electric field on the donor binding energies for a spherical QD with parabolic confinement. A variational calculation of binding energies of the donor impurities in Cd1−x Mnx Te quantum dots in the presence of electric field applied along the growth direction is considered. We work within the effective mass approximation and adopt a variational envelope wave function for the donor electron. A systematic study with variation of electric fields as well as dot radii has been attempted for the finite barrier confinement. We also investigate theoretically the donor bound spin polaron in a quantum dot. The mean field theory with modified Brillouin function that is already used for bulk and quantum well case has been extended to the case of a QD and we estimate the spin polaronic shifts to the impurity ionization energies. To our knowledge, this is the first calculation on a hydrogenic impurity in a parabolic diluted magnetic semiconductor quantum dot in the presence of electric fields. We consider a parabolic QD (depth VD , and radius R) of the magnetically non-uniform “spin-doping” super-lattice system such as Cd1−xout Mnxout Te/Cd1−xin Mnxin Te/Cd1−xout Mnxout Te. Such a QD may be fabricated by the method of evolution of self-assembled quantum dots (QDs) in the Stransky–Krastanow mode as in the case of Cd1−x Mnx Se QDs or by electron beam lithography and wet chemical etching which is used to fabricate quantum wires [14]. In the effective mass approximation, the Hamiltonian of an electron in a parabolic QD in the presence of a electric field along the z direction, may be written as HD = −

h¯ 2 ∇ 2 + |e|F z + VD , 2m∗

(1)

2

where VD = V0 (B)r for |r|  R while VD = V0 (B) for |r| > R, R2 and V0 (B) is the barrier height of the parabolic dot, which is taken to be 70% of the difference in the band gap between Cd1−xout Mnxout Te and Cd1−xin Mnxin Te. The barrier height decreases as the magnetic field increases as a result of variation of band offsets. The ground state energy of an electron in a parabolic quantum dot with magnetic and electric fields is estimated by variational method. We have assumed the trial functions as ψin (r) = Ain e−ξ r (1 + νF r cos θ ), (0)

2

(0)

e−δr

ψout (r) = −Aout

r

(1 + νF r cos θ ),

r < R, r  R.

(2)

Here Ain and Aout are normalization constants. By matching the wave functions and their derivatives at the boundaries of the QD, and along with the normalization, we fix the values of Ain , Aout , and ξ = R1 ( R1 + δ). We take δ and ν as the variational parameters. By introducing the effective Rydberg

R ∗ = m∗ e4 /(2h¯ 2 ε 2 ) as the unit of energy and the effective Bohr radius a ∗ = h¯ 2 ε/(m∗ e2 ) as the length unit, the Hamiltonian given in Eq. (1) becomes |e|a ∗ F r cos θ VD + ∗, (3) ∗ R R where VD is the parabolic confinement. The ground state energy of the conduction electron in a parabolic QD in the external electric field, ED , is obtained by minimizing the expectation value of HD with respect to the variational parameters δ and ν for various electric fields using Eq. (2). The Hamiltonian for a donor situated at the center of the parabolic dot in the presence of external electric field applied along the growth direction is HD = −∇ 2 +

2 HID = HD − . (4) r The ionization energy of the donor in the presence of electric field, Eion (F ) = ED − ψ|HID |ψmin , is obtained by variational method using the following trial wave functions with α as the variational parameter: (0) αr e , ψin (r) = ψin (0) ψout (r) = ψout eαr ,

r < R, r  R.

(5)

The exchange interaction arising between the spin of a conduction electron and the Mn2+ spins is described by the Hamiltonian Hm as  Hm = − (6) J (r, R j )s · S j . Here S j is the spin of the Mn2+ ion at position R j and s is the spin of the conduction electron centered at r. The exchange interaction J (r, R j ) is dependent on the overlap between the orbital of the conduction electron and of the 3d electrons. Kasuya and Yanase [15], who explained the transport properties of magnetic semiconductors, originally developed the theory of spin polaron (SP). This mean field theory, which invokes the exchange interaction between the carrier and impurity given in Eq. (6), yields the spin polaronic shift, Esp , with the modified Brillouin function [16]    1 Sβ|ψ|2 dτ, Esp = βN0 xS0 (x)|ψ|2 Bs (7) 2 2k[T + T0 (x)] where β is the exchange coupling parameter, S is the Mn2+ spin, and xN0 is the Mn ion concentration. S0 (x), the effective spin, and T0 (x), the effective temperature are the semiphenomenological parameters, which describe the paramagnetic response of the Mn2+ ions in the bulk Cd1−x Mnx Te [16]. In Eq. (7), ψ is the envelope function as given in Eq. (5) with the appropriate values of the variational parameters, k is the Boltzmann constant and Bs (η) is the modified Brillouin function. The parameters used in our calculations are N0 = 2.94 × 1022 cm−3 , βN0 ≈ 220 meV, and the semi-phenomenological parameters S0 (Xin = 0.02) = 1.97, S0 (Xout = 0.1) = 1.08, T0 (Xin = 0.02) = 0.94, and T0 (Xout = 0.1) = 3.84 [17]. Using the envelop function given in Eq. (5) with the appropriate

A. John Peter / Physics Letters A 355 (2006) 59–62

Fig. 1. Variation of the lowest binding energy with dot radius for various electric fields for V0 = 111.09 meV.

61

Fig. 3. Variation of ionization energy with dot radius for different electric fields.

Fig. 4. Variation of ionization energy with the square of electric field strengths. Fig. 2. Variation of the lowest binding energy with electric field for different dot radii.

variational parameters, we obtain   1 Esp = βN0 Xin S0 (Xin )I1 + Xout S0 (Xout )I2 , (8) 2   where I1 = |ψin |2 Bs (η1 ) dτ , I2 = |ψout |2 Bs (η2 ) dτ , and

ηj 2S + 1 2S + 1 1 Bs (ηj ) = (9) coth ηj − coth , 2S 2S 2S 2S Sβ Sβ |ψin |2 and η2 = 2k[T +T |ψout |2 . with η1 = 2k[T +T 0 (xin )] 0 (xout )] In Fig. 1 we have displayed the variation of the binding energy of the electron corresponding to the lowest level in the dot for different electric field strengths as a function of dot radius. We observe the following: the lowest binding energy (i) decreases as dot radius increases and approaches the bulk value for large dot sizes and (ii) does not show much variation when an electric field is applied, unlike in the case of a quantum well [18]. This situation is clearly brought out in Fig. 2 where the binding energy decreases with the electric fields. The binding energy is higher for smaller dot radii due to the confinement. The donor binding energy as a function of dot radius with different electric fields is shown in Fig. 3. The ionization energy decreases in an electric field. The decrease is more for wider dots. For narrow dots, the electric field effect seems to

be unimportant. This is attributed to the fact that the confinement effects are appreciable for a smaller dot. As the electric field is increased the electron is pulled toward one side of the quantum dot as a result the binding energies decrease as a function of electric field. To bring about an appreciable change in the ionization energy one has to apply field strength in excess of 1 × 106 V m−1 . An interesting feature that is seen in Fig. 5 is that for dot radii a ∗ , the ionization energies in strong electric fields (106 V m−1 ) become negative. The reason being enhanced tunneling due to narrowing (asymmetry) of the well in strong electric fields working against impurities binding. In an attempt to estimate the donor polarizability we have plotted the donor ionization energy versus F 2 in Fig. 4. The polarizabilities are estimated for different dot radii. The slopes, namely, ∂Eion /∂F 2 were found and the polarizabilities are estimated for a dot radii, R = 1a ∗ and R = 5a ∗ . The value of polarizability is higher for smaller radii than for the larger radii. At present we do not have any experimental data of polarizabilities to compare our results. The variation of spin polaronic shift with dot size for different electric fields is brought out in Fig. 5. This figure shows that for a narrow dot the polaronic effects are important as compared to a dot of large radius. For narrow dots the spin polaronic shift decreases with electric field. In this work, we have considered the effects of electric and fields, applied along the growth direction of the system, in the

62

A. John Peter / Physics Letters A 355 (2006) 59–62

Acknowledgements The author fondly remembers the help and hospitality rendered by Prof. You Quan Li, Institute of Modern Physics, Zhejiang University, Hangzhou 310027, PR China. References

Fig. 5. Variation of spin polaronic shift with electric fields for different dot radii.

estimation of binding energies of the donor and impurity polarizability in Cd1−xin Mnxin Te/Cd1−xout Mnxout Te quantum dots. We do not have at present sufficient experimental data to compare our results. However, the problem of a harmonic oscillator under a confined geometry has drawn the attention of several physicists like Landau, Fock and Darwin [19]. Since at present quantum dots are drawing more attention in areas like spintronics and quantum computers we hope that the present work will stimulate more experimental activity on impurity states on quantum dots. Experimental efforts are encouraged to lend support to our calculations.

[1] G. Bastard, Phys. Rev. B 24 (1981) 4714. [2] J. Brown, N. Spector, J. Appl. Phys. 59 (1986) 1179. [3] N. Porras-Montenegro, S.T. Pérez-Merchancano, Phys. Rev. B 46 (1992) 9780. [4] C. Bose, J. Appl. Phys. 83 (1998) 3089. [5] C. Bose, C.K. Sarka, Physica B 253 (1998) 238. [6] Z. Xiao, J. Zhu, F. He, J. Appl. Phys. 79 (1996) 9181. [7] Z. Xiao, J. Appl. Phys. 86 (1999) 4509. [8] J. López-Gondar, J. d’Albuquerque e Castro, L.E. Oliveira, Phys. Rev. B 42 (1990) 7069. [9] R.B. Santiago, L.E. Oliveira, J. d’Albuquerque e Castro, Phys. Rev. B 46 (1992) 4041. [10] K.F. Ilaiwi, Superlattices Microstruct. 20 (1996) 173. [11] J.C. Lozano-Cetina, N. Porras-Montenegro, Phys. Status Solidi B 210 (1998) 717. [12] V. Narayani, B. Sukumar, Solid State Commun. 90 (1994) 579. [13] C.A. Duque, A. Montes, A.L. Morales, Physica B 302 (2001) 84. [14] N. Takahashi, K. Takabayashi, I. Souma, J. Shen, Y. Oka, J. Appl. Phys. 87 (2000) 6469. [15] T. Kasuya, A. Yanase, Rev. Mod. Phys. 40 (1968) 684. [16] J.A. Gaj, R. Planel, G. Fishman, Solid State Commun. 29 (1979) 435. [17] K. Gnanasekar, K. Navaneethakrishnan, Mod. Phys. Lett. B 18 (2004) 419. [18] G. Bastard, E.E. Mendez, L.L. Chang, L. Esaki, Phys. Rev. B 28 (1983) 3241. [19] T. Chakraborthy, Quantum Dots, North-Holland, Amsterdam, 1999.