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Linear and nonlinear intersubband optical absorption in double triangular quantum wells Bin Chen, Kang-Xian Guo ∗ , Rui-Zhen Wang, Zhi-Hai Zhang, Zuo-Lian Liu Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China

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Article history: Received 3 August 2008 Received in revised form 27 October 2008 Accepted 25 November 2008 by P. Sheng Available online 3 December 2008 PACS: 78.67.De 78.20.Ci 42.65.An 73.21.Fg Keywords: A. Quantum wells D. Optical properties D. Electronic states D. Applied electric field effects

a b s t r a c t The linear and the third-order nonlinear optical absorptions in the asymmetric double triangular quantum wells (DTQWs) are investigated theoretically. The dependence of the optical absorption on the right-well width of the DTQWs is studied, and the influence of the applied electric field on the optical absorption is also taken into account. The analytical expressions of the linear and the nonlinear optical absorption coefficients are obtained by using the compact density-matrix approach and the iterative method. The numerical calculations are presented for the typical GaAs/Alx Ga1−x As asymmetric DTQWs. The results show that the linear as well as the nonlinear optical absorption coefficients are not a monotonous function of the right-well width, but have complex relationships with it. Moreover, the calculated results also reveal that applying an electric field to the DTQWs with a thinner right-well can enhance the linear optical absorption but has no prominent influence on the nonlinear optical absorption. In addition, the total optical absorption is strongly dependent on the incident optical intensity. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction In the past few years, the nonlinear optical properties in the low-dimensional semiconductor quantum systems, such as quantum wells [1–22], quantum wires [23–26] and quantum dots [27–35], have attracted much attention both in practical applications and in theoretical research. For one reason, the nonlinear effects in these low-dimensional quantum systems can be enhanced more dramatically over those in bulk materials due to the existence of a strong quantum-confinement effect. For the other, these nonlinear properties have the potential for device application in far-infrared laser amplifiers, photodetectors, electro–optical modulators and all optical switches. In addition, the fast development of graving technologies such as molecularbeam epitaxy and metal-organic chemical vapor deposition has also accelerated research in this area. Recently, there has been a considerable interest in the linear and the nonlinear optical absorptions based on intersubband transitions in semiconductor quantum heterostructures [18–23, 27–29]. In 1987, Ahn and Chuang [18,19] calculated the linear and the nonlinear intersubband optical absorption coefficients in

∗

Corresponding author. Fax: +86 20 39366871. E-mail address: [email protected] (K.-X. Guo).

0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.11.032

a semiconductor quantum well, and the influences of the applied electric field as well as the incident optical intensity on the optical absorption were discussed in detail. In 1998, Goldys and Shi [21] studied the linear and the nonlinear intersubband optical absorptions in a AlGaAs/AlAs/InGaAs strained double barrier quantum well, and the subband nonparabolicity and elastic strain effect were also taken into account. In 2006, electron–phonon interaction effects on the linear and the nonlinear optical absorptions in cylindrical quantum wires were investigated by Yu et al. [23]. In 2008, Karabulut and Baskoutas [27] discussed the influences of impurities, applied electric field, and incident optical intensity on the linear and the nonlinear optical absorptions in spherical quantum dots. In some recent works [36–39], a few authors have focused their attention on researches into the optical absorption in coupled quantum wells. However, a systematic study on the linear and the nonlinear optical absorptions in this quantum system is still lacking. Therefore, the research in this field is still important both in theoretical research and in practical applications. It is well known that the quantum confinement of the electron in the triangular quantum well is much stronger than that in the square quantum well with the same width. Therefore, some novel optical properties can be expected in such a quantum system. Motivated by this idea, we will investigate the linear, third-order nonlinear, and total optical absorptions in the GaAs/Alx Ga1−x As

B. Chen et al. / Solid State Communications 149 (2009) 310–314

311

respectively. Here, kk and rk are the wave vector and coordinate in the xy plane and uc (r) is the periodic part of the Bloch function in the conduction band at k = 0. ϕn (z ) and En are the solutions of the one-dimensional schrödinger equation H0 ϕn (z ) = E ϕn (z ),

(5)

where H0 is the z component of the whole Hamiltonian H, and it is given by h¯ 2

d2

+ V (z ) − qFz . 2m∗ dz 2 By solving Eq. (5), the bound states can be given as follows, H0 = −

Fig. 1. Schematic diagram for an asymmetric double triangular quantum well.

c Ai(η1 ) 1 c Ai(η2 ) + d2 Bi(η2 ) ϕ(z ) = 2 c3 Ai(η3 ) + d3 Bi(η3 ) c4 Ai(η4 )

(6)

(7)

with double triangular quantum wells (DTQWs) in the present paper. An schematic diagram for the DTQW is shown in Fig. 1. We keep the width of the left-well (denoted by the symbol WL ) unchanged, and restrict our attention to the dependence of the optical absorption on the width of the right-well (denoted by the symbol WR ). Also, the influence of the applied electric field (F ) on the optical absorption has been taken into account. This paper is organized as follows: In Section 2, the Hamiltonian, relevant eigenstates and eigenenergies are discussed in the GaAs/Alx Ga1−x As DTQWs, and the analytical expression of the linear and the third-order nonlinear optical absorption coefficients are obtained with the compact density matrix approach and the iterative method. In Section 3, numerical calculations for the typical GaAs/Alx Ga1−x As DTQWs are performed, and the dependence of optical absorption coefficients on WR and F is analyzed in detail. Finally, brief conclusions are given in Section 4. 2. Theory In this section, we will discuss the eigenstates and the eigenenergies in the DTQWs, and will present the formalism for the derivation of linear and third-order nonlinear optical absorption coefficient. For simplicity, we suppose an idealized DTQW heterostructure model, where we neglect band nonparabolicity and variable effective mass. By the effective mass approximation, the electron Hamiltonian in this DTQW is well described by H =−

η1,2 =

4m∗ V0 /WL h¯ 2

1/3

(−qFWL /2V0 ∓ 1)2 −qFWL WL E × z ∓1 − ±1 , 2V0

η3,4 =

2

V0

1/3

4m V0 /WR h¯ (−qFWR /2V0 ∓ 1)2 −qFWR WR E × z ∓1 − ∓1 . 2

∗

2V0

2

V0

(8)

Here Ai and Bi are the regular and the irregular Airy functions, E is the corresponding eigenenergy, and c1 , c2 , c3 , c4 , d2 and d3 are the normalized coefficients of the wave function. All of these normalized coefficients and the eigenenergy E can be numerically solved by the standard boundary condition of the electronic bound state. Next, a brief derivation of the linear and third-order nonlinear optical absorption coefficient in DTQWs will be presented by the compact density matrix method and the iterative procedure. Assuming a monochromatic incident electromagnetic field E (t ) = E˜ exp(−iωt ) + E˜ exp(iωt ) is applied to the system with a polarization vector normal to the quantum wells, the evolution of the one-electron density matrix ρ is given by the time-dependent Schrödinger equation

∂ρij 1 = [H0 − qzE (t ), ρ ]ij − Γij (ρ − ρ (0) )ij , ∂t ih¯

(9)

∂2 ∂2 ∂2 + + + V (z ) − qFz , ∂ x2 ∂ y2 ∂ z2

(1)

2z + WL Θ (−z ) + V0 2z − WR Θ (z ). W W

where H0 is the Hamiltonian for this system without the incident field E (t ), ρ (0) is the unperturbed density matrix and Γij is the relaxation rate. For simplicity, we will assume that Γij = Γ0 = 1/T0 . Eq. (9) is solved using the usual iterative method [4,5]:

(2)

ρ(t ) =

h¯ 2

2m∗

with V (z ) = V0

L

R

X

ρ (n) (t )

(10)

n

Here z represents the growth direction of this quantum well, h¯ is Planck’s constant, m∗ is the effective mass of the conduction-band, q is the electronic charge, F is the applied electric field, V0 is the profile of the conduction-band potential in this quantum well, and Θ (z ) is the Heaviside step function, respectively. By solving the Schrödinger equation H ψn,k (r) = en,k ψn,k (r), the eigenfunctions ψn,k (r) and the eigenenergies en,k are given by

ψn,k (r) = ϕn (z )uc (r)eikk •rk ,

(3)

and en,k = En +

h¯ 2 2m∗

|kk |2 ,

(4)

with

∂ρij(n+1) ∂t

=

o 1 n (n+1) H0 , ρ (n+1) ij − ih¯ Γij ρij ih¯ −

1 qz , ρ (n) ij E (t ).

ih¯

(11)

The electronic polarization of the quantum wells can be expanded as Eq. (10). Neglecting the terms which only induce a little contribution to our calculations, we can get a concise expression of the electronic polarization: P (t ) ≈ ε0 χ (1) (ω)E (iωt ) + ε0 χ (3) (ω)E (iωt ),

(12)

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B. Chen et al. / Solid State Communications 149 (2009) 310–314

Fig. 2a. The linear, the nonlinear, and the total optical absorption coefficients as a function of incident photon energy h¯ ω for four different right-well width, WR = 3, 5, 7 and 9 nm, with WL = 10 nm, F = 0 kV/cm and I = 0.2 MW/cm2 .

where χ (1) (ω) and χ (3) (ω) denote the linear and third-order nonlinear susceptibilities, respectively. ε0 is the vacuum permittivity. With the same compact density matrix approach and the iterative procedure as [10,23], the analytical expressions of the linear and the third-order nonlinear susceptibilities for a two-level quantum system are given as follows [10,11,23]: For the linear term,

ε0 χ

(1)

ρs |M21 |2 (ω) = . E21 − h¯ ω − ih¯ Γ0

(13)

For the third order term

ρs |M21 |2 |E˜ |2 4|M21 |2 E21 − h¯ ω − ih¯ Γ0 (E21 − h¯ ω)2 + (h¯ Γ0 )2 |M22 − M11 |2 . (14) − (E21 − ih¯ Γ0 )(E21 − h¯ ω − ih¯ Γ0 ) Here, ρs is the density of electrons in the QWs, Mij = hi|qz |ji is dipole matrix element, E21 = E2 − E1 is the energy interval of two different electronic states, and h¯ ω is the incident photon energy. The susceptibility χ (ω) is related to the absorption coefficient α(ω, I ) by r µ α(ω) = ω Im (ε0 χ (ω)) , (15) εR ε0 χ (3) (ω) = −

where µ is the permeability of the material and εR = n2r ε0 (nr is the refractive index) is the real part of the permittivity. Using Eqs. (13)– (15), the analytic forms of the linear and the third-order nonlinear optical absorption coefficients are obtained. For the linear optical absorption coefficient, (1)

α (ω) = ω

r

µ |M21 |2 ρs h¯ Γ0 , εR (E21 − h¯ ω)2 + (h¯ Γ0 )2

(16)

and for the third-order nonlinear one,

µ I ε R ε 0 nr c |M21 |4 ρs h¯ Γ0 |M22 − M11 |2 × 1 − [(E21 − h¯ ω)2 + (h¯ Γ0 )2 ]2 4|M21 |2 [(E21 − h¯ ω)2 − (h¯ Γ0 )2 + 2E21 (E21 − h¯ ω)] × (17) 2 E21 + (h¯ Γ0 )2

α (3) (ω, I ) = −2ω

r

where c is the light velocity in free space and I is the incident optical intensity which is given as I = 2ε0 nr c |E˜ |2 .

(18)

Fig. 2b. The resonant peak of the linear, the nonlinear and the total optical absorption coefficients as a function of the right-well width WR for WL = 10 nm, F = 0 kV/cm and I = 0.2 MW/cm2 .

Therefore, the total absorption coefficient can be given as

α(ω, I ) = α (1) (ω) + α (3) (ω, I ).

(19)

3. Results and discussions In the present section, numerical calculations are carried out for a typical GaAs/Alx Ga1−x As quantum well. The parameters adopted in our calculations are as follows [22]: m∗ = 0.067m0 (m0 is the free-electron mass), V0 = 228 meV, ρs = 3 × 1022 m−3 , T0 = 0.14ps, and nr = 3.2. In Fig. 2a, the linear α (1) (ω), the third-order nonlinear α (3) (ω, I ) and the total α(ω, I ) optical absorption coefficients are plotted as a function of incident photon energy h¯ ω for four different values of WR as WR = 3, 5, 7, 9 nm with WL = 10 nm, F = 0 kV/cm and I = 0.2 MW/cm2 . It should be noted that the barrier width, denoted by WB as shown in Fig. 1, also varies with the increase of WR , and the corresponding values of WB is 6.5, 7.5, 8.5 and 9.5 nm, respectively. From this figure, it can be clearly seen that for each WR , the α (1) (ω), α (3) (ω, I ) and α(ω, I ) as a function of h¯ ω has an prominent peak, respectively, at the same position, which occurs due to the one-photon resonance enhancement, i.e. h¯ ω ≈ E21 . Moreover, the large linear absorption coefficient α (1) (ω) is positive whereas the nonlinear one α (3) (ω, I ) is negative. As a result, the total absorption coefficient α(ω, I ) is reduced due to the contributions of α (3) (ω, I ). In addition, the resonant peaks of the linear, the nonlinear, and the total absorption coefficients suffer an obvious red-shift with the increase of WR . This feature can be understood as that with the increase of WR , the quantumconfinement of the electron decreases quickly and simultaneously WB increases, which weakens the coupling between the two wells. These two factors together result in that the energy levels become very close each other, i.e. the energy intervals are reduced, and as a result, the resonant peak of absorption coefficient appears at the low-energy direction, i.e. suffers a red-shift. More importantly, the absorption coefficient is not a monotonous function of the right well width WR , but has a complex relationship with it as shown in Fig. 2a. To show the dependence of the absorption coefficients on the WR more clearly, we have plotted Fig. 2b, which presents the resonant peak of linear, third-order nonlinear and total absorption coefficients as a function of WR with WL = 10 nm, F = 0 kV/cm and I = 0.2 MW/cm2 . From these two figures we can see that while the right-well is thinner (WR < 7 nm as shown in Figs. 2a and 2b), the linear absorption resonant peak decreases quickly with the increase of WR , but the third-order nonlinear one is not sensitive to WR . While WR approaches WL , the resonant peaks both of the linear absorption coefficient and of the third-order nonlinear one

B. Chen et al. / Solid State Communications 149 (2009) 310–314

increase gradually with the increase of WR . However, the resonant peak of total absorption coefficient decreases monotonously with the increase of WR . These complicated features can be understood as follows. From Eqs. (16) and (17), we can see that the resonant peak of absorption coefficient depends not only on the dipole matrix element but also on the energy interval of the quantum system. It is well known that increasing WR can induce a larger overlap between the electronic wave functions, which results in a larger dipole matrix element but reduce the energy interval quickly. The calculated results can be attributed to the complicated competitions between these two factors. While the right-well is thinner, the change of the energy interval is the main factor influencing the varieties of the linear optical absorption. And as a result, the resonant peak of linear optical absorption coefficient decreases quickly with the increase of WR . However, due to the fact that the third-order nonlinear optical absorption coefficient is proportional to |M21 |4 while the linear one to |M21 |2 as shown in Eqs. (16) and (17), which results in that the variety of nonlinear optical absorption coefficient generated by the decrease of energy interval can be greatly compensated by that generated by the increase of dipole matrix element, the nonlinear resonant peak is not sensitive to WR . While the width of the right-well approaches the one of the left-well, the increase of dipole matrix element predominates, which results in a large increase of the resonant peak both of the linear and of the nonlinear optical absorption coefficients. Comparing Figs. 2a and 2b with Fig. 1(a), Fig. 1(b) of [39], we can observe a very different nonlinear property for the absorption peak in DTQW and in DSQW (double square quantum well). The results presented in [39] show that the values of the nonlinear absorption peak will decrease quickly with the increase of the system size, such as increasing Lb (barrier width) or Lw (well width), and that for a wider DSQW (Lb > 1 nm and Lw > 2 nm) the nonlinear optical absorption can be ignored. However, in our results as shown in Figs. 2a and 2b the nonlinear absorption effect is even stronger in a wider quantum system. Therefore, it should be considered in this case. More importantly, the peak value of the total absorption in our results is more than 10 times higher than the one obtained in [39]. In Figs. 3a and 3b, the influences of the applied electric field on optical absorption coefficients is researched in detail. Fig. 3a shows the linear α (1) (ω), the nonlinear α (3) (ω, I ), and the total α(ω, I ) optical absorption coefficient as a function of incident photon energy h¯ ω for three different applied electric field value, F = 0, 40, 80 kV/cm with WL = 10 nm, (WR , WB ) = (4, 7) nm and I = 0.2 MW/cm2 . From this figure, it can be clearly seen that for each F , a resonant peak for the linear, the nonlinear and the total optical absorption coefficient occurs, respectively, at the same position (h¯ ω = E21 ). More importantly, we can see that the resonant peak both of the linear and of the total optical absorption coefficient increase simultaneously with the increase of the applied electric field F , while the one of the nonlinear optical absorption coefficient is not sensitive to F . Fig. 3b, which shows the resonant peak of the linear, the nonlinear and the total optical absorption coefficients as a function of F for WL = 10 nm, (WR , WB ) = (4, 7) nm and I = 0.2 MW/cm2 , presents these important features more clearer. The physical origin for these features is that: applying an electric field to a DTQW can weaken the coupling between the two wells. As a result, the energy levels are separated greatly, i.e. the energy interval increases, and the overlap integral is reduced largely, i.e. the dipole matrix element |M21 | decreases. The complicated competition between the energy interval and the dipole matrix element determines these features for the same reason given in the explanations for Figs. 2a and 2b. In addition, Fig. 3a also shows us that with the increase of F , a significant blue-shift of the absorption resonant peak can be induced, for increasing F can lead

313

Fig. 3a. The linear, the nonlinear, and the total optical absorption coefficients as a function of incident photon energy h¯ ω for three different values of applied electric field F = 0, 40, and 80 kV/cm, with WL = 10 nm, (WR , WB ) = (4, 7) nm and I = 0.2 MW/cm2 .

Fig. 3b. The resonant peak of the linear, the nonlinear and the total optical absorption coefficients as a function of the applied electric field F for WL = 10 nm, (WR , WB ) = (4, 7) nm and I = 0.2 MW/cm2 .

to a larger energy interval, which results in that the resonant peak of the absorption coefficient occurs at the high-energy direction, i.e. suffers a blue-shift. In Fig. 4, the total optical absorption coefficient α(ω, I ) is plotted as a function of incident photon energy h¯ ω for five different values of incident optical intensity, I = 0.0, 0.5, 1.09, 1.5, and 2.1 MW/cm2 with WL = 10 nm, (WR , WB ) = (4, 7) nm and F = 0 kV/cm. From this figure, it can be clearly seen that the peak of the total absorption coefficient decreases prominently with the increase of I. The strong absorption saturation occurs at I = 1.09 MW/cm2 . More importantly, the figure also shows that the resonant peak of absorption coefficient can be bleached at sufficiently high intensities. For example, the bleaching effect can be clearly observed when I = 1.5 MW/cm2 , and at I = 2.1 MW/cm2 , the resonant peak is significantly split up into two peaks due to the strong bleaching effect. From this figure, we can conclude that the total absorption coefficient, especially for the resonant peak, is strongly dependent on the incident optical intensity due to the contributions of the third-order nonlinear term. Therefore, the nonlinear effects should be considered for the research of the optical absorption in low-dimensional quantum systems, in particular for the case with a strong incident optical intensity. Finally, it should be noted that the linear infinite confinement in the external walls of the DTQWs used in this paper is not physically feasible because the maximum barrier height obtained

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B. Chen et al. / Solid State Communications 149 (2009) 310–314

to the DTQWs with a thinner right-well can result in a larger total absorption coefficient, and can induce a prominent blueshift of the resonant peak. In addition, the total optical absorption coefficients α(ω, I ) are strongly dependent on the incident optical intensity I as shown in our calculated results. With the increase of I, the total optical absorption coefficient α(ω, I ) decreases quickly, and the resonant peak is bleached greatly at stronger incident optical intensity. These results are important both in experimental research and in practical applications, and may have significant influences on the improvements of optical devices, such as ultrafast optical switches, optical bistability devices and so on. Acknowledgments

Fig. 4. The total optical absorption coefficient α(ω, I ) as a function of incident photon energy h¯ ω for five different values of incident optical intensity I for WL = 10 nm, (WR , WB ) = (4, 7) nm and F = 0 kV/cm.

in GaAs/Alx Ga1−x As system is 350 meV for x = 40%. Above this Al concentration, the Γ and X valleys cross and the effective barrier decreases [4]. Nevertheless, the infinite confinement model in our calculation is still a good approximation of the physical case, for the contribution of the evanescent part of the electron wave functions can be ignored reasonably due to the fact that the electron is strongly confined in the triangular well. When such a contribution is considered, the linear and nonlinear absorption peak will increase slightly but the total absorption peak will decrease due to the fact that the nonlinear absorption peak increases more quickly than the linear one. For the same reason, the bleaching effect can be observed at a relatively low incident optical intensity when compared with the one obtained in our present calculation. Moreover, it should also be noted that the strength of the applied electric field F used in the calculation can not be too large (All the value of F in our calculations is reasonable.), otherwise the electron can not be confined effectively in the triangular well for a real case. 4. Conclusion In this paper, the linear α (1) (ω), nonlinear α (3) (ω, I ) and total α(ω, I ) optical absorption coefficients have been studied in detail for a typical asymmetric GaAs/Alx Ga1−x As double triangular quantum well. The calculations mainly focus on the dependence of optical absorption coefficients on the widths of the right-well and the applied electric field. The calculated results show that, for a DTQW with a relatively thin right-well, the nonlinear optical absorption is not sensitive to the WR , so the total optical absorption coefficient mainly depends on the linear optical absorption coefficient. But for a DTQW with a relatively thick right-well, the nonlinear optical absorption coefficient can be changed greatly by increasing WR . As a result, the total optical absorption coefficient can be reduced significantly. Moreover, increasing WR can induce an obvious red-shift of the resonant peak. The results also reveal that the applied electric field has a significant influence on the optical absorption coefficients. Applying a strong electric field

This work is supported by the National Natural Science Foundation of China (under Grant No. 60878002) and the Science and Technology Committee of Guangdong Province (under Grant Nos. 2007B010600061, 2008B010200043, 2008B010600050 and 8251009101000002). References [1] M.M. Fejer, S.J.B. Yoo, R.L. Byer, A. Harwit, J.S. Harris, Phys. Rev. Lett. 62 (1989) 1041. [2] E. Rosencher, P. Bois, J. Nagle, S. Delaître, Electron. Lett. 25 (1989) 1063. [3] K.X. Guo, S.W. Gu, Phys. Rev. B 47 (1993) 16322. [4] E. Rosencher, P. Bois, Phys. Rev. B 44 (1991) 11315. [5] L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315. [6] B. Chen, K.X. Guo, Z.L. Liu, R.Z. Wang, Y.B. Zheng, B. Li, J. Phys.: Condens. Matter 20 (2008) 255214. [7] C.J. Zhang, K.X. Guo, Physica B 383 (2006) 183. [8] L. Zhang, H.J. Xie, Physica E 22 (2004) 791. [9] Z.E. Lu, K.X. Guo, Commun. Theor. Phys. 45 (2006) 171. [10] G.H. Wang, Q. Guo, K.X. Guo, Chinese J. Phys. 41 (2003) 296. [11] L. Zhang, Superlatt. Microstruct. 37 (2005) 261. [12] D. Indjin, A. Mirčetić, Z. Ikonić, V. Milanović, G. Todorović, Physica E 4 (1999) 119. [13] C.J. Zhang, K.X. Guo, Physica E 33 (2006) 363. [14] C.J. Zhang, K.X. Guo, Physica B 387 (2007) 276. [15] D.M. Sedrakian, A.Z. Khachatrian, G.M. Andresyan, V.D. Badalyan, Opt. Quantum Electron. 36 (2004) 893. [16] K.X. Guo, C.Y. Chen, J. Phys.: Condens. Matter 7 (1995) 6583. [17] K.X. Guo, C.Y. Chen, Solid State Commun. 99 (1996) 363. [18] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE 23 (1987) 2196. [19] D. Ahn, S.L. Chuang, J. Appl. Phys. 62 (1987) 3052. [20] K.J. Kuhn, G.U. Lyengar, S. Yee, J. Appl. Phys. 70 (1991) 5010. [21] E.M. Goldys, J.J. Shi, Phys. Status Solidi (B) 210 (1998) 237. [22] İ. Karabulut, Ü. Atav, H. S.afak, M. Tomak, Eur. Phys. J. B 55 (2007) 283. [23] Y.B. Yu, S.N. Zhu, K.X. Guo, Solid State Commun. 139 (2006) 76. [24] Y.B. Yu, K.X. Guo, Physica E 18 (2003) 492. [25] Y.B. Yu, K.X. Guo, S.N. Zhu, Physica E 27 (2005) 62. [26] K.X. Guo, C.Y. Chen, Microelctron. Eng. 52 (2000) 127. [27] İ. Karabulut, S. Baskoutas, J. Appl. Phys. 103 (2008) 073512. [28] G.H. Wang, K.X. Guo, Physica E 28 (2005) 14. [29] C.H. Liu, B.R. Xu, Phys. Lett. A 372 (2008) 888. [30] Y.B. Yu, S.N. Zhu, K.X. Guo, Phys. Lett. A 335 (2005) 175. [31] W.F. Yang, X.H. Song, S.Q. Gong, Y. Cheng, Z.Z. Xu, Phys. Rev. Lett. 99 (2007) 133602. [32] B. Li, K.X. Guo, C.J. Zhang, Y.B. Zheng, Phys. Lett. A 367 (2007) 493. [33] B. Li, K.X. Guo, Z.L. Liu, Y.B. Zheng, Phys. Lett. A 372 (2008) 1337. [34] G.H. Wang, K.X. Guo, Physica B 315 (2002) 234. [35] G.H. Wang, K.X. Guo, J. Phys.: Condens. Matter 13 (2001) 8197. [36] M. Bedoya, A.S. Camacho, Phys. Rev. B 72 (2005) 155318. [37] P.F. Yuh, K.L. Wang, Phys. Rev. B 38 (1988) 8377. [38] E. Ozturk, I. Sokmen, Superlatt. Microstruct. 41 (2007) 36. [39] L. Zhang, Commun. Theor. Phys. 49 (2008) 786.

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