Linear and Nonlinear Optical Properties of Spherical Quantum Dots: Effects of Hydrogenic Impurity and Conduction Band Non-Parabolicity

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1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 3, March 15, 2012 Linear and Nonlinear Optical Properties of Spherical Quantum Dots: Effects of Hydrogenic Impurity and Conduction Band Non-Parabolicity G. Rezaei, 1, B. Vaseghi, 1 and N.A. Doostimotlagh 2 1 Department of Physics, College of Sciences, Yasouj University, Yasouj, , Iran 2 Young Researchers Club, Yasooj Branch, Islamic Azad University, Yasooj, Iran (Received July 11, 2011; revised manuscript received December 26, 2011) Abstract Simultaneous effects of an on-center hydrogenic impurity and band edge non-parabolicity on intersubband optical absorption coefficients and refractive index changes of a typical GaAs/Al xga 1 xas spherical quantum dot are theoretically investigated, using the Luttinger Kohn effective mass equation. So, electronic structure and optical properties of the system are studied by means of the matrix diagonalization technique and compact density matrix approach, respectively. Finally, effects of an impurity, band edge non-parabolicity, incident light intensity and the dot size on the linear, the third-order nonlinear and the total optical absorption coefficients and refractive index changes are investigated. Our results indicate that, the magnitudes of these optical quantities increase and their peaks shift to higher energies as the influences of the impurity and the band edge non-parabolicity are considered. Moreover, incident light intensity and the dot size have considerable effects on the optical absorption coefficients and refractive index changes. PACS numbers: La, Hb Key words: hydrogenic impurity, non-parabolic conduction band, quantum dots, absorption coefficient, refractive index 1 Introduction Novel, optical and electrical properties of semiconductor nanostructures have made them superior devices for applications in high speed electro-optical devices, farinfrared laser amplifiers and photo-detectors. [1 3] Thus a great deal of works have been done on the linear and nonlinear optical properties of semiconductor quantum wells, wires and dots iecent years, from practical and theoretical points of view. [4 12] In addition, confined hydrogenic impurities in semiconductor devices substantially modify the physical properties of these structures. After Bastard s pioneering work on the donor impurity in a semiconductor quantum well, [13] study of impurities in semiconductor quantum dots (QDs) have also been of great interest to researchers. Thus, many works have been devoted to understand the effects of hydrogenic impurity on the electronic structure and optical properties of QDs in the framework of effective mass and parabolic one band approximations. [14 26] Binding energy of the hydrogen-like impurity in a spherical QD has investigated by Porras Montenegro et al. [14] Barati et al. made their investigation on the impurity states in lens-shaped and semi-lens-shaped quantum dots. [15] The binding energy of an impurity located at the center of multilayered spherical quantum dot, under the influence of a magnetic field has reported by Boz et al. [16] Barati et al. have used the standard perturbation method to investigate the binding energy of hydrogenic impurity confined in an ellipsoidal finite potential QD. [17] Li et al. have calculated the binding energy of a hydrogenic donor impurity in a rectangular parallelepiped-shaped quantum dot in the framework of effective-mass envelope-function theory using the plane wave basis. [18] He et al., have used the non-degenerate and the degenerate perturbation methods to investigate the effects of an electric field on a hydrogenic impurity confined in a spherical parabolic quantum dot. [19] Nonlinear optical properties of a hydrogenic donor in lens-shaped quantum dots are studied by Vahdani et al. [20] Baskoutas, et al. used the potential morphing method to investigate the effects of impurities and external electric field on the electronic structure and nonlinear optical rectification in a quantum dot. [21] Xie has investigated the nonlinear optical properties of a hydrogenic donor in a disc-like parabolic QD using the matrix diagonalization method. [22] Nonlinear optical properties of a hydrogenic donor in an ellipsoidal finite potential QD are studied by Rezaei et al. [23] Beyond the above mentioned works, it is well known that the energy dispersioelation is parabolic only near the band edge and in the more realistic model, one has to consider the conduction band non-parabolicity due to the confinement effects. Therefore, several articles have been devoted to investigate the influence of conduction band non-parabolicity on the electronic and optical properties of semiconductor nanostructures. [27 31] To our knowledge, conduction band non-parabolicity effect on the linear and Corresponding author, grezaei@mail.yu.ac.ir c 2011 Chinese Physical Society and IOP Publishing Ltd

2 486 Communications in Theoretical Physics Vol. 57 nonlinear optical ACs and RI changes of a hydrogenic impurity confined in an SQD has not been investigated so far. To this aim, in this article we consider an on-center hydrogenic impurity confined in an SQD and using the matrix diagonalization method to find the energy eigenvalues and eigenfunctions. In addition, we investigate the effect of edge band non-parabolicity on the linear and the third-order nonlinear intersubband optical ACs and RI changes of the system. The organization of this paper afterward is as follows: In Sec. 2 we describe the theoretical framework. Our numerical results and a brief summary are presented in Secs. 3 and 4, respectively. 2 Theory Within the framework of effective-mass approximation the Hamiltonian of a system consisting of an electron bound to a donor inside a spherical finite confining potential QD in the presence of conduction band nonparabolicity effects, can be written as [27] Ĥ = Ĥ0 e2 ǫ r, (1) where e is the electron charge, ǫ is the dielectric constant and ( Ĥ 0 = 2 2 ) 2 2m 2 β 4 2m + V (r), (2) is the Hamiltonian of the system, without considering the hydrogenic impurity effects. Here, m is the effective mass of the electron, the non-parabolicity factor, β, is [27] ) β = (1 m y + 2y 2 1 m y + 2y 2, (3) E g where y = /E g, is the spin-orbit splitting energy, m 0 and E g are the free-electron mass and band gap energy of the semiconductor, respectively. The confinement potential, V (r), is { 0, r < r0, V (r) = (4) V 0, r > r 0. In what follows, the band edge non-parabolicity is considered only inside the dot and the outside effects are ignored. To obtain energy eigenvalues and eigenfunctions of Eq. (1), the total Hamiltonian of the system is diagonalized in the space spanned model as follows. [29] To obtain energy eigenvalues and eigenvectors of the system, the total Hamiltonian of the system is diagonalized in the space spanned model as follows [29] Ψ = { jlj (k j r), r < r 0, C j Y ljm j (θ, φ) (5) j h lj (iq j r), r > r 0, where Y ljm j (θ, φ) is the angular part of the wave function, j lj (k j r) and h lj (iq j r) are the spherical Bessel and Hankle functions of the first order with ( m ) 1/2 k j = w [1 1 4βEj β 2 ] 1/2, (6) 2m q j = b (U 0 E j ) 2. (7) m w (m b ) is the electron effective mass inside (outside) the dot and E j, is the j-th eigenvalue of Ĥ 0. Energy eigenvalues and corresponding eigenvectors of the system are obtained by diagonalization of the matrix representation of the total Hamiltonian, Ĥ, in the basis of Ĥ0. After the energies and their corresponding wave functions are obtained, the linear, the third-order nonlinear and the total optical absorption coefficients (ACs) and refractive index (RI) changes for the intersubband transitions can be calculated by the density matrix approach and the perturbation expansion method. Therefore, within a two-level system approach, one can find the following expression for the total optical ACs [11] α(i, ω) = α (1) (ω) + α (3) (I, ω), (8) where µ α (1) M 12 2 σ v Γ 21 (ω) = ω ǫ R (E 21 ω) 2 + ( Γ 21 ) 2, (9) α (3) (ω) µ ( I ) M 12 2 σ v Γ [ 21 = ω ǫ R 2ǫ 0 c [(E 21 ω) 2 + ( Γ 21 ) 2 ] 2 4 M 12 2 M 22 M 11 2 [3E21 2 4E 21 ω + 2 (ω 2 Γ 2 21 )] ] E ( Γ 21) 2, (10) are the linear and the third-order nonlinear optical ACs, respectively. In the above equations, σ ν is the carrier density, M ij = χ nim i eˆx χ njm j (i, j = 1, 2) are the dipole moment matrix elements and E 21 = E 2 E 1 is the energy difference between two lowest electronic states. c is the speed of light in free space, µ is the permeability, and I = 2ǫ 0 c E 2 is the intensity of electromagnetic field. Using the same procedure we find the following expression for the total optical RI changes n(ω) = n(1) (ω) + n(3) (ω), (11) where n (1) (ω) = σ v M 12 2 [ E 21 ω ] 2n 2 r ǫ 0 (E 21 ω) 2 + ( Γ 21 ) 2, (12) n (3) (ω) = σ v M 12 2 µci 4n 3 r ǫ 0 [(E 21 ω) 2 + ( Γ 21 ) 2 ] 2 [4(E 21 ω) M 12 2 (M 22 M 11 ) 2 (E 21 ) 2 + ( Γ 21 ) { 2 (E 21 ω)[e 21 (E 21 ω) }] ( Γ 21 ) 2 ] ( Γ 21 ) 2 (2E 21 ω), (13) are the linear and the third-order nonlinear RI changes, where is the refractive index.

3 No. 3 Communications in Theoretical Physics Numerical Results and Discussion Intersubband optical ACs and RI changes of a spherical GaAs QD surrounded by Al x Ga 1 x As, where x is the Al concentration, are calculated numerically. The parameters used in this work are as follows: x = 0.3, σ v = m 3, = 3.2, T 2 = 0.2 ps, Γ 2 = 1/T 2. [11] The barrier height is V 0 = 1.02x ev and the value of the static dielectric constant ǫ is assumed to be the same in GaAs and Al x Ga 1 x As (ǫ = 13.18). The effective masses are m 1 = 0.067m 0 in GaAs and m 2 = ( x)m 0 in Al x Ga 1 x As, where m 0 is the free electron mass. In addition, for GaAs we have, = 0.34 ev, E g = ev and β = (ev) 1. [27] To investigate the influence of the hydrogenic impurity on the optical properties of the system, we have presented the variations of the linear, the third-order nonlinear and the total optical ACs and RI changes of an SQD with non-parabolic conduction band, as a function of the photon energy with I = 0.2 MW/cm 2 and the dot radius r 0 = 0.5a B = 5 nm in Figs. 1 and 2. An important feature of these figures is that, the resonant peak values of the optical ACs and RI changes in the presence of the hydrogenic impurity are greater and occur at higher incident photon energy compared to the case in which hydrogenic impurity is not considered. This is due to the fact that the transition energy (E 21 ) and the dipole moment matrix elements of these two cases are different. These results are in a good agreement with those reported in Ref. [32]. and RI changes increase, as the influence of the conduction band non-parabolicity is considered. In addition, according to the results of, [29] the non-parabolic conduction band leads to the change of the energy difference between the subbands. Thus, a blue shift of absorption peak appears from to mev. These results confirm the fact that, the conduction band non-parabolicity effect decreases the energy eigenvalues, enhances the effective overlap between the wave functions and consequently increases the transition dipole moment matrix elements of the system. Therefore, the magnitudes of optical ACs and RI changes increase and their peaks exhibit a blue shift as the influence of conduction band non-parabolicity is considered. Fig. 2 The linear, the third-order nonlinear and the total RI changes of a non-parabolic band quantum dot, with and without hydrogenic impurity. Fig. 1 The linear, the third-order nonlinear and the total ACs of a non-parabolic band quantum dot, with and without hydrogenic impurity. The linear, the third-order nonlinear and the total optical ACs and RI changes of a hydrogenic impurity in an SQD with and without considering the band edge nonparabolicity effect, as a function of the photon energy with I = 0.2 MW/cm 2 and r 0 = 0.5a B = 5 nm are plotted in Figs. 3 and 4 respectively. These two figures show that the magnitudes of the linear, absolute value of the third order nonlinear and the total optical ACs Fig. 3 Conduction band non-parabolicity effect on the linear, the third-order nonlinear and the total ACs of a hydrogenic impurity quantum dot. In Figs. 5 and 6 the linear, the third order nonlinear and the total optical ACs and RI changes of a hydrogenic impurity in an SQD, with non-parabolic conduction band, are plotted as a function of the photon energy for different values of r 0, with I = 0.2 MW/cm 2. As it is seen, optical ACs and RI changes are strongly size dependent and exhibit a blue shift with decreasing r 0. This is due

4 488 Communications in Theoretical Physics Vol. 57 to the fact that the transition probability will increase meanwhile, the energy difference between the subbands will decrease with increasing r 0. Thus, the third-order nonlinear optical AC increases considerably and the magnitude of the total optical AC decreases. Also, the linear, the third-order nonlinear and the total RI changes are increased with increasing r 0. Fig. 6 The linear, the third order nonlinear and the total RI changes of a hydrogenic impurity quantum dot as a function of the photon energy and different values of the dot radius, in the presence of non-parabolicity effect. Fig. 4 Variations of the linear, the third order nonlinear and the total RI changes of a hydrogenic impurity quantum dot, with and without band non-parabolicity effect. Fig. 7 Variations of the total AC of a hydrogenic impurity quantum dot as a function of the photon energy and different values of incident light intensity, in the presence of non-parabolicity effect. Fig. 5 Absorption coeffiecients of a hydrogenic impurity quantum dot as a function of the photon energy and different values of the dot radius, in the presence of nonparabolicity effect. According to Eqs. (10) and (13) the nonlinear terms of optical AC and RI changes depend on the incident light intensity and the contribution of these terms enhances with increasing I. To illustrate this effect, the total optical AC and RI changes of an impurity in an SQD, with nonparabolic conduction band, are plotted in Figs. 7 and 8 as a function of the photon energy for different values of incident optical intensity with r 0 = 0.5a B = 5 nm. It is obvious that, an increase in the intensity decreases the total optical AC and RI changes. Fig. 8 The total RI changes of a hydrogenic impurity quantum dot as a function of the photon energy and different values of incident light intensity, in the presence of non-parabolicity effect.

5 No. 3 Communications in Theoretical Physics Conclusion In conclusion, we have investigated the linear and the third-order nonlinear optical ACs and RI changes of an SQD, using the compact density matrix approach. Matrix diagonalization method is used to find the simultaneous effects of hydrogenic impurity and band edge nonparabolicity on energy eigenvalues and eigenfunctions of the dot. Energy eigenvalues and eigenfunctions are used to investigate the influences of impurity and band edge non-parabolicity on the optical ACs and RI changes of the system. It is found that the energy intervals and the dipole moment matrix elements in the presence of hydrogenic impurity are larger than those of without impurity. Therefore, the magnitudes of optical ACs and RI changes enhance and their resonant peaks move to higher energy regions with impurity. Furthermore, the transition energy between the two subbands and the dipole moment matrix elements are increased as the influence of conduction band non-parabolicity is considered. This leads to increase the values of these optical properties and their peaks have an obvious blue-shift. Finally our results show that the incident light intensity and the dot size have a great influence on the optical properties of the system. Acknowledgements This work is partially supported by the Yasouj University. References [1] T.H. Hood, J. Lightwave Technol. 6 (1988) 743. [2] R.F. Kazarinov and R.A. Suris, Sov. Phys. Semicond. 5 (1971) 707. [3] D.A.B. Miller, Int. J. High Speed Electron. Syst. 1 (1990) 19. [4] E. Rosencher and Ph. Bois, Phys. Rev. B 44 (1991) [5] Kelin J. Kuhn, Gita U. lyengar, and Sinclair Yee, J. Appl. Phys. 70 (1991) [6] N. Sfina, S. Abdi-Ben Nasrallah, S. Mnasri, and M. Said, J. Phys. D: Appl. Phys. 42 (2009) (5pp). [7] E. Ozturk, H. Sari and I. Sokmen, J. Phys. D: Appl. Phys. 38 (2005) 935. [8] M.G. Barseghyan, A. Kh. Manaselyan and A.A. Kirakosyan, J. Phys.: Condens. Matter 18 (2006) S2161. [9] C. González-Santander and F. Domínguez-Adame, Phys. Lett. A 374 (2010) [10] R. Chen, H.Y. Liu, and H.D. Sun, Solid State Communications 150 (2010) 707. [11] M.R.K. Vahdani and G. Rezaei, Phys. Lett. A 374 (2010) 637. [12] B. Li, K.X. Guo, Z.L. Liu, and Y.B. Zheng, Phys. Lett. A 372 (2008) [13] G. Bastard, Phys. Rev. B 24 (1981) [14] N. Porras-Montenegro and S.T. Pérez-Merchancano, Phys. Rev. B 46 (1992) [15] M. Barati, M.R.K. Vahdani, and G. Rezaei, J. Phys.: Condens. Matter 19 (2007) (14pp). [16] F.K. Boz, S. Aktas, A. Bilekkay, and S.E. Okan, Applied Surface Science 256 (2010) [17] M. Barati, G. Rezaei, and M.R.K. Vahdani, Phys. Stat. Sol. b 244 (2007) [18] S.S. Li and J.B. Xia, J. Appl. Phys. 101 (2007) [19] L.L. He and W.F. Xie, Superlattices and Microstructures 47 (2010) 266. [20] M.R.K. Vahdani and G. Rezaei, Phys. Lett. A 373 (2009) [21] S. Baskoutas, E. Paspalakis, and A.F. Terzis, J. Phys.: Condens. Matter 19 (2007) (9pp). [22] W.F. Xie, Phys. Lett. A 372 (2008) [23] G. Rezaei, M.R.K. Vahdani, and B. Vaseghi, Curr. Appl. Phys. 11 (2011) 176. [24] İbrahim Karabulut and Sotirios Baskoutas, J. Appl. Phys. 103 (2008) [25] G.H. Wang and K.X. Guo, J. Phys.: Condens. Matter 13 (2001) [26] Mehmet Şahin, J. Appl. Phys. 106 (2009) [27] M.K. Bose, K. Midya, and C. Bose, J. Appl. Phys. 101 (2007) [28] A. Sivakami and M. Mahendran, Physica B 405 (2010) [29] G. Rezaei, N.A. Doostimotlagh, and B. Vaseghi, Physica E 43 (2011) [30] D.J. Newson and A. Kurobe, Semicond. Sci. Technol. 3 (1988) 786. [31] G. Rezaei, M.R.K. Vahdani, and B. Vaseghi, Physica B 406 (2011) [32] Bekir Çakir, Yusuf Yakar, Ayhan Özmena, M. Özgür Sezer, and Mehmet Şahin, Superlattices and Microstructures 47 (2010) 556.

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