Superlattices and Microstructures

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1 Superlattices and Microstructures 51 (2012) Contents lists available at SciVerse ScienceDirect Superlattices and Microstructures journal homepage: Intersubband optical refractive index changes in an asymmetric quantum dot underlying an external static magnetic field Xue Chao Li, An Min Wang, Zhao Liang Wang, Yang Yang Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui , China article info abstract Article history: Received 2 December 2011 Received in revised form 7 February 2012 Accepted 8 March 2012 Available online 19 March 2012 Keywords: Refractive index change Quantum dot Magnetic field We theoretically investigate the refractive index (RI) changes in an asymmetric quantum dot (QD) underlying an external static magnetic field. We obtain the confined wave functions and energies of an electron in QD by the effective-mass approximation. Using the compact-density-matrix approach and iterative method, we obtain the analytical expressions of linear, nonlinear and total RI changes. The results of numerical calculations for the typical GaAs/AlGaAs QD show that the RI changes are sensitive to the parameters of the asymmetric potential and incident optical intensity. Moreover, the resonance peaks of the RI changes shift with the value of magnetic field B or the radius of the QD changing. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The study of confined quantum systems has become an interesting field of investigation with the development of the fabrication technique of nanostructures in the last decades. These confined structures, such as quantum wells, quantum wires and quantum dots (QDs) [1 7], have discrete energy levels (subbands) and particular optical properties, so the nonlinear effects in these structures are much stronger than the bulk materials [8 20]. These particular properties can lead to interesting applications in optoelectronic devices, for instance, the QDs that confine electrons in all three spatial dimensions have a variety of important technological applications, such as infrared photodetectors, cellular automata, electro-optical modulators, all optical switches and infrared lasers [21 24]. Vahdani and Rezaei [25] investigated the refractive index (RI) changes in a parabolic cylinder QD Corresponding author. Tel./fax: addresses: xuechao@mail.ustc.edu.cn (X.C. Li), anmwang@ustc.edu.cn (A.M. Wang) /$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

2 X.C. Li et al. / Superlattices and Microstructures 51 (2012) and found that RI changes are strongly affected by the size of the dot, the optical intensity and the electromagnetic field polarization. Karimi and Rezaei [16] studied the effects of external electric and magnetic fields on the linear and nonlinear intersubband optical properties of finite semiparabolic QDs and found that external electric and magnetic fields have a great influence on these optical quantities. Xie and Liang [19] investigated the optical properties of a donor impurity in a two-dimensional quantum pseudodot and found that the optical properties of a donor impurity in a two-dimensional pseudoharmonic QD are strongly affected by the zero point of the pseudoharmonic potential, the chemical potential of the electron gas and the Coulomb interaction. In the numerical simulation [23,25], the size of QD has influence on absorption and refraction index change in the cylindrical QD and parabolic cylinder quantum dot. Moreover, the third harmonic generation depends obviously on the radii of the cylindrical QDs [26]. However, the RI changes in an asymmetric QD have not been researched before. Considering that the application of magnetic field can modify the transition and optical properties of electron in QD, we study the RI changes in asymmetric QD underlying an external magnetic field. In this work, the refractive index changes are strongly affected by the size of the dot, asymmetric potential, incident optical intensity and magnetic field. This will provide more means to control the nonlinear optical properties. Therefore, this system may have some reference value for optoelectronic devices. In this paper, we investigate the RI changes for the typical GaAs/AlAGaAs QDs which are confined by radial potential, asymmetrical potential and external magnetic field. This paper is organized as follows. In Section 2, the eigenfunctions and eigenenergies of electron states are obtained by using the effective-mass approximation. The analytical expressions for the linear, third-nonlinear and total RI changes are derived by means of the compact-density-matrix approach and the iterative method. In Section 3, the numerical results and discussions are presented in GaAs/AlGaAs asymmetric QDs. A brief summary is given in Section Model and analysis The system we study is that an electron moves in the QD which is confined by the radial potential of the form 1 2 m x 2 0q 2, an asymmetrical potential VðzÞ and a perpendicular magnetic field. Using the effective-mass approximation, we can write the Hamiltonian of the system as [18,19,27,28] bh ¼ 1 P e 2 2m c A 1 þ 2 m x 2 0q 2 þ VðzÞ ð1þ where m is the effective mass, P is momentum of the electron, A is the vector potential of the magnetic field BðB ¼ r A, where A q ¼ A z ¼ 0; A u ¼ Bq=2Þ; x 0 ¼ h=m R 2 is the frequency of electron [29]. The form of asymmetrical potential VðzÞ is as follows [18,30,31] L VðzÞ ¼U 0 z z 2 ; ð2þ L where U 0 and L are the parameters of asymmetrical potential. In cylindrical coordinates, the corresponding eigenfunctions of this system are W ¼ f ðq; uþvðzþ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðq; uþ ¼ 1 ðjmjþnþ! 1 a 1þjmj 2p2 jmj n! m! expðimuþqjmj exp F q2 n; jmjþ1; q2 ; ð4þ 4a 2 2a 2 vðzþ ¼C nz z m exp sffiffiffiffiffiffiffiffiffiffiffiffi! sffiffiffiffiffiffiffiffiffiffiffiffi m U 0 z 2 F n 2h 2 L 2 z ; m þ 1 2 ; m U 0 z!; 2 ð5þ 2h 2 L 2 qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h where a ¼ m X is the effective length scale, X ¼ x 2 c þ 4x2 0, x c ¼ eb=m c is the cyclotron frequency, Fða; b; xþ is the confluent hypergeometric function, n is the radial quantum number, m is the magnetic ð3þ

3 582 X.C. Li et al. / Superlattices and Microstructures 51 (2012) quantum number, C nz is the normalization constant, m ¼ 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8m U 0 L 2 þ 1 þ 1 and n h 2 z is the quantum number. The electron eigenenergies E are given by E ¼ hx n þ jmjþ1 þ mhx c þ 2h rffiffiffiffiffiffiffiffi2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 2U 0 4n 2 2 L m z þ 1 2 8m U 0 L 2 8m þ 1 U 0 L 2 A5; ð6þ h 2 h 2 In this work, the Zeeman splitting is too small to be negligible, which is described in previous works [19,26,32]. By the compact-density-matrix method and the iterative procedure [33], the total RI change can written as where and DnðxÞ n r ¼ Dnð1Þ ðxþ þ Dnð3Þ ðxþ ; ð7þ n r n r Dn ð1þ ðxþ n r ¼ r mjm 21 j 2 2n 2 r e 0 " # E 12 hx ; ð8þ ðe 12 hx Þ 2 þ ðhc 12 Þ 2 Dn ð3þ ðxþ lcjm 21 j 2 r m I ¼ h i n 2 r 4n 3 r e 0 ðe 12 hx Þ 2 þ ðhc 12 Þ 2 ( 4ðE 12 hx ÞjM 21 j 2 ð M 22 M 11 Þ 2 n h i o ) ðe ð Þ 2 þðhc 12 Þ 2 12 hxþ E 12 ðe 12 hxþ ðhc 12 Þ 2 ðhc 12 Þ 2 ½2E 12 hxš E 12 ð9þ are the linear and third-order nonlinear RI change, respectively. In the above equations, n r is the refractive index, r m is the carrier density, M ij ¼jhW i jezjw j ij (i; j ¼ 1; 2) is the dipole matrix elements, e 0 is the permittivity of free space, E ij ¼ E j E i is the energy interval of two different electronic states, C 12 is the relaxation rate, l is the permeability of the system, c is the speed of light and I is the incident optical intensity. 3. Results and discussions In the following, we will discuss the RI changes in a GaAs/AlGaAs QD underlying an external static magnetic field. In calculation we use the following parameters [34,35]: n r = 3.2, r m ¼ m 3, l ¼ 4p 10 7 Hm 1, C 12 ¼ 1=T 12 and T 12 = 0.2 ps is the intersubband relaxation time. In Fig. 1, we plot the linear, third-order nonlinear and total RI changes as a function of the incident photon energy hx with R ¼ 7:0 nm;b ¼ 5:0 T;L ¼ 2:0 nm; U 0 ¼ 0:1 ev and I ¼ 3:0 MW=cm 2. We can clearly see that the large linear change is the opposite in sign of the nonlinear change. Therefore, the total RI change will be reduced. The larger the incident optical intensity is, the larger contribution of nonlinear term to the total RI is. This can be found from Eq. (9). As shown in Fig. 2, the total RI change is plotted as a function of the incident photon energy hx with R ¼ 7:0nm;L ¼ 2:0 nm; U 0 ¼ 0:1eV;I ¼ 3:0 MW=cm 2 for three different values of the static magnetic field B; B ¼ 0; 5 and 10 T, respectively. The resonant peaks occur at hx ¼ 0:0428 ev; 0:0435 ev and 0:0458 ev, respectively. We can find that the bigger magnetic field B is, the higher energy regions of the resonance peak is. The reason is that the effective confinement is enhanced and energy intervals of electron are bigger when the magnetic field B increases. In Fig. 3, we plot the total RI change as a function of the incident photon energy hx with R ¼ 7:0nm;B ¼ 5:0 T;L ¼ 2:0 nm; U 0 ¼ 0:1 ev for three different values of incident optical intensity I; I ¼ 1:0; 3:0 and 5.0 MW=cm 2, respectively. We observe from the figure that the total RI change will be visibly reduced as the incident optical intensity increases. This is because that the higher optical

4 X.C. Li et al. / Superlattices and Microstructures 51 (2012) Fig. 1. The linear, third-order nonlinear and total RI changes as a function of the incident photon energy hx with R ¼ 7:0 nm; B ¼ 5:0 T; L ¼ 2:0 nm; U 0 ¼ 0:1 ev and I ¼ 3:0 MW=cm 2. Fig. 2. The total RI change as a function of incident photon energy hx for three different values of B, B ¼ 0; 5 and 10 T, respectively. intensity increases the nonlinear RI change, as be seen from Eq. (9), but the linear term does not change with incident optical intensity, as Eq. (8). Therefore, it is necessary to employ a relatively weaker incident optical intensity if one want to obtain a larger total RI change. In Fig. 4, we show the total RI change as a function of incident photon energy hx with R ¼ 7:0 nm;b ¼ 5:0 T; U 0 ¼ 0:1eV;I ¼ 3:0 MW=cm 2 for three different values of L; L ¼ 1:0; 1:5 and 2:0 nm, respectively. From Fig. 4, we can find that there is no visible shift of the resonance peak position with changing L but the maximum value of the total RI change increases significantly as L increases. Therefore, to obtain a larger total RI change, one should employ a reasonably larger L. As shown in Fig. 5, the total RI change is plotted as a function of the incident photon energy hx with R ¼ 7:0 nm;b ¼ 5:0 T;L ¼ 2:0 nm;i ¼ 3:0 MW=cm 2 for three different values of U 0 ; U 0 ¼ 60, 80 and 100 mev, respectively. It can be easily found that the resonance peak position of total RI change does not move when we change U 0 and the maximum peak value of the total RI change reduces with increasing U 0.

5 584 X.C. Li et al. / Superlattices and Microstructures 51 (2012) Fig. 3. The total RI change as a function of incident photon energy hx for three different values of I; I ¼ 1:0; 3:0 and 5:0 MW=cm 2, respectively. Fig. 4. The total RI change as a function of incident photon energy hx for three different values of L; L ¼ 1:0; 1:5 and 2.0 nm, respectively. In order to investigate the influence of R on the total RI change, we plot the total RI change as a function of incident photon energy hx with B ¼ 5:0 T;L ¼ 2:0 nm; U 0 ¼ 0:1eV;I ¼ 3:0 MW=cm 2 for three different values of R; R ¼ 6; 7 and 8 nm, respectively. From Fig. 6, we can clearly find that three resonant peaks occur at hx ¼ 0:0601 ev, ev and ev, respectively. As R increases, the resonant peak positions shift to lower energies (redshift) and the peak value of total RI change does not alter. This is because of the fact that the quantum confinement effects will result in the separation of energy levels, the stronger the confinement effects are, the broader the separation will be. 4. Summary We make a study of the RI changes for a GaAs/AlGaAs asymmetric QD underlying an external static magnetic field. The results show that the RI changes are sensitive to the incident optical intensity and the characteristic parameters of asymmetric potential. Moreover, the redshift occurs when the size of QD increases. Interestingly, the blueshift of peak position can be found when the static magnetic field

6 X.C. Li et al. / Superlattices and Microstructures 51 (2012) Fig. 5. The total RI change as a function of incident photon energy hx for three different values of U 0; U 0 ¼ 60; 80 and 100 mev, respectively. Fig. 6. The total RI change as a function of incident photon energy hx for three different values of R; R ¼ 6; 7 and 8 nm, respectively. B increases. Our results may make a contribution to experimental studies and provide a new model for the practical application such as all optical switches, infrared photodetectors and electro-optical modulators, etc. Acknowledgments This work has been supported by the National Natural Science Foundation of China under Grant No References [1] M.B. Panish, Science 208 (1980) 916. [2] J. Khurgin, Phys. Rev. B 38 (1988) 4056.

7 586 X.C. Li et al. / Superlattices and Microstructures 51 (2012) [3] S. Sauvage, P. Boucaud, Phys. Rev. B 59 (1999) [4] M. Sundaram, S.A. Chalmers, P.F. Hopkins, A.C. Gossard, Science 254 (1991) [5] C. Sirtori, F. Capasso, D.L. Sivco, A.Y. Cho, Phys. Rev. Lett. 68 (1992) [6] T. Brunhes, P. Boucaud, S. Sauvage, Phys. Rev. B 61 (2000) [7] W. Yang, X. Song, S. Gong, Y. Cheng, Z. Xu, Phys. Rev. Lett. 99 (2007) [8] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) [9] C.J. Zhang, Z.X. Wang, Y. Liu, K.X. Guo, Physica E 43 (2010) 372. [10] Í Karabulut, Ü Atav, H. Sßafak, M. Tomak, Physica B 393 (2007) 133. [11] R. Atanasov, F. Bassani, V.M. Agranovich, Phys. Rev. B 50 (1994) [12] L. Tsang, D. Ahn, S.L. Chuang, Appl. Phys. Lett. 52 (1988) 697. [13] W. Yang, S. Gong, R. Li, Z. Xu, Phys. Lett. A 362 (2007) 37. [14] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) [15] C.J. Zhang, X. Song, W. Yang, Z. Xu, J. Phys. B 42 (2009) [16] M.J. Karimi, G. Rezaei, Physica B 406 (2011) [17] W. Yang, X. Song, R. Li, Z. Xu, Phys. Rev. A 78 (2008) [18] C.J. Zhang, Z.X. Wang, Y. Liu, C.D. Peng, K.X. Guo, Phys. Lett. A 375 (2011) 484. [19] W. Xie, S. Liang, Physica B 406 (2011) [20] B. Li, K.X. Guo, Z.L. Liu, Y.B. Zheng, Phys. Lett. A 372 (2008) [21] M. Troccoli, A. Belyanin, F. Capasso, E. Kubukcu, D.J. Sinco, A.Y. Cho, Nature 433 (2005) 845. [22] X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. [23] C. Liu, B. Xu, Phys. Lett. A 372 (2008) 888. [24] C.S. Lent, P.D. Tougaw, W. Porod, G.H. Bernstein, Nanotechnology 4 (1993) 49. [25] M.R.K. Vahdani, G. Rezaei, Phys. Lett. A 374 (2010) 637. [26] S. Shao, K. Guo, Z. Zhang, N. Li, C. Peng, Solid State Commun. 151 (2011) 289. [27] C.G. Darwin, Proc. Cambridge Philos. Soc. 27 (1930) 86. [28] M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan, Physica E 31 (2006) 85. [29] W. Que, Phys. Rev. B 45 (1992) [30] I.I. Goldman, V.D. Krivchenkov, Problems in Quantum Mechanics, Pergamon Press, New York, [31] Y. Weissman, J. Jortner, Phys. Lett. A 70 (1979) 177. [32] Í Karabulut, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 131 (2011) [33] E. Rosencher, P. Bois, Phys. Rev. B 44 (1991) [34] K.X. Guo, C.Y. Chen, J. Phys.: Condens. Matter 7 (1995) [35] S. Adachi, J. Appl. Phys. 58 (1985) 3.