Optical absorptions in asymmetrical semi-parabolic quantum wells

Superlattices and Microstructures 62 (2013) 225–232

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Optical absorptions in asymmetrical semi-parabolic quantum wells X.Q. Yu a, Y.B. Yu b,⇑ a b

Physics Department, Southeast University, Nanjing 211189, China School of Materials, Ningbo University of Technology, Ningbo 315211, China

a r t i c l e

i n f o

Article history: Received 3 April 2013 Received in revised form 21 July 2013 Accepted 29 July 2013 Available online 7 August 2013 Keywords: Optical absorption Parabolic potential Nonlinear Quantum wells

a b s t r a c t Optical absorptions in semi-parabolic quantum wells are investigated. By using density-matrix approach and iterative procedure, the analytical formula of optical absorption coefficients in this semi-parabolic quantum wells are deduced. The numerical results on typical GaAs/AlGaAs materials are calculated. The relations between the optical absorptions and the quantum confinement from the semi-parabolic quantum wells, the incident optical intensity, and the relaxation rate are investigated. The results indicate that the absorption peaks bule-shift and the values of the peaks increase with the enhancement of the parabolic confining frequency. The values of the absorption peaks increase quickly with the increase of the relaxation time. In addition, the total absorption coefficient decrease with the increase of the incident optical intensity. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In the past few years, the nonlinear optical properties of semiconductor low-dimensional quantum system, such as quantum well, wire, and dot, are of considerable interest because of their relevance for studying practical applications [1,2]. Different from the bulk structures, in the low-dimensional quantum system, the electron energy levels are discrete in the confined direction [3]. The previous studies have shown that the low-dimensional quantum system can lead to a enhancement of optical nonlinearities. The influences of intense laser fields on the nonlinear optical properties of donor impurities in a quantum dot was investigated [4–9]. The results showed that

⇑ Corresponding author. Tel./fax: +86 574 87617798. E-mail address: [email protected] (Y.B. Yu). 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.07.021

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the confinement potential and the incident intense laser radiation can strongly influence the nonlinear optical properties. The optical absorptions and refractive index changes in a spherical quantum dot with parabolic confinement subjected to an external electric field were studied [10]. They showed that the peak positions of the total absorption coefficient and total refractive index changes red-shifts with the increase of the position of the impurity or the quantum dot radius. Liu et al. investigated nonlinear optical absorption and refractive index changes with three-dimensional ring-shaped pseudoharmonic potential and they found that the optical absorption coefficients and the refractive index changes are strongly affected by the chemical potential and parameter of the ring-shaped potential [11]. The optical absorptions and the refractive index changes in a quantum box were also investigated and the results showed that both the incident optical intensity and the structure parameters have a great effect on the optical absorptions and refractive index changes [12]. Yakar et al. investigated the optical absorptions in a spherical parabolic quantum dot and they showed that the optical absorption coefficients are strongly influenced by the impurity, incident optical intensity, relaxation time, and parabolic potential [13]. The optical absorption coefficients and binding energy in a spherical quantum dot were investigated and the results showed that the parabolic potential and the incident optical intensity have a great effect the optical absorption coefficients [14]. The linear and third-order nonlinear optical absorption coefficients in the inverse parabolic quantum wells with an external electric field were studied and the results showed that the incident optical intensity considerably affects the total absorption coefficient [15]. The linear and nonlinear optical absorption coefficients and refractive index changes in a two-electron quantum dot were studied and the results showed that the peak positions of optical absorptions blue-shifts with the enhancement of the confinements [16]. They also investigated the optical properties of donor impurities in quantum dots under the influence of laser field with Gaussian potential [17]. The linear and nonlinear intersubband optical absorption in a symmetric double semi-parabolic quantum wells were investigated and the results showed that both the optical incident intensity and the structure parameters really influence the optical properties [18]. The optical absorption coefficients in a disk-shaped parabolic quantum dot with a static magnetic field were strongly affected not only by a static magnetic field but also the confinement frequency [19]. Karabulut et al. investigated the nonlinear optical rectification in asymmetrical semi-parabolic quantum wells without and with an applied electric field [20]. However, the optical absorptions in semi-parabolic quantum wells have not been studied. In this paper, we will investigate the optical absorptions in semi-parabolic quantum wells. The analytical expressions of the optical absorption coefficients were obtained by using density-matrix method and iterative method. Then, the numerical calculations are presented in GaAs/AlGaAs quantum wells. Conclusions are given in Section 4. 2. Theory The electron is confined in the quantum wells with semi-parabolic confined potential. The effective-mass Hamiltonian for this system can be written as

" # 2  h @2 @2 @2 1 H¼ þ þ þ m x20 z2 ; ðz > 0Þ 2 2m @x2 @y2 @z2

ð1Þ

where m⁄ is the effective mass of the electron. x0 is the frequency of the semi-parabolic confining potential. z axis is the growth direction of the semi-parabolic quantum wells. The eigen-functions and eigen-energies can be obtained by solutions of the Schrödinger equation for H as

wn;k ðrÞ ¼ /n ðzÞU c ðrÞexpðikk  rk Þ;

ð2Þ

2

en;k

h  ¼ En þ jkk j2 ; 2m

ð3Þ

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227

where kk and rk are the wave vector and coordinate in the xy plane and Uc(r) is the periodic part of the Bloch function. /n(z) and En are the solutions of one-dimensional Schrödinger equation Hz/n(z) = En/n(z). The /(z) and En can be solved as

  1 /n ðzÞ ¼ N n Exp  a2 z2 H2nþ1 ðazÞ; 2

ð4Þ

and

  3 hx0 ðn ¼ 0; 1; 2; . . .Þ En ¼ 2n þ 2 h

ð5Þ i12

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi where a ¼ m x0 = h; N n ¼ 1a  p22n ð2n þ 1Þ! . H2n+1(az) is the Hermite functions.One can find that En is different from the energy levels En ¼ ð2n þ 12Þ hx0 ðn ¼ 0; 1; 2; . . .Þ obtained from parabolic quantum confinement potential. Next one can present a formalism for the derivation of the optical absorption coefficients in semiparabolic quantum wells by using compact density-matrix approach and iterative method. The system   is excited by an electromagnetic field EðtÞ ¼ E eixt þ E eixt . Let us denote qas the one-electron density matrix for this regime. The evolution of the density matrix qobeys the following time-dependent equation:

@ qij 1 ¼ ½H0  ezEðtÞ; qij  Cij ðq  qð0Þ Þij ; ih @t

ð6Þ

where H0 is the Hamiltonian for this system without the electromagnetic field E(t);Cij is the relaxation rate; q(0) is the unperturbed density matrix. Above equation can be solved by using the usual iterative method P qðtÞ ¼ n qðnÞ ðtÞ, with ðnþ1Þ

@ qij

@t

¼

o 1  1 n ðnþ1Þ  ½ez; qðnÞ ij EðtÞ: H0 ; qðnþ1Þ ij  ihCij qij ih ih

ð7Þ

We only consider the first three orders of the expansion of electronic polarization, i.e., 



ixt 3 ixt PðtÞ ¼ e0 vð1Þ þ e0 vð3Þ þ c:c; x Ee x E e

ð8Þ

ð3Þ

where v(1) and vx are the linear and third-order nonlinear susceptibility coefficients, respectively. The nth order electronic polarization is given by P(n)(t) = V1Tr(q(n)ez), where V is the volume of interaction. One can obtain the expression of the linear and third-order nonlinear susceptibility coefficients in the quantum wells as

vð1Þ ðxÞ ¼

rs jMij j2 ; e0 hðx  xij Þ þ ihCij

vð3Þ ðxÞ ¼ 

4jM ij j4 rs Cij 3

ð9Þ 

E2

e0 h Cii ðxij  x  iCij Þ ðxij  xÞ2 þ C2ij

;

ð10Þ

where rs is the density of electrons in the quantum wells, 0 is the vacuum permittivity, xij = (Ei  Ej)/  . Mij = —h/j—z—/ii— (i, j = 0, 1, 2, 3) is the off-diagonal matrix element. h The optical absorption coefficients are related to the optical susceptibility by [21]

aðxÞ ¼ x

rffiffiffiffiffi l Imðe0 vðxÞÞ;

eR

ð11Þ

where l is magnetic permeability, eR ¼ n2r e0 is the real dielectric constant, nr is the refractive index of the medium. Therefore, the linear and third-order nonlinear optical absorption coefficients can be obtained as

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rffiffiffiffiffi

að1Þ ðxÞ ¼ x

l rs jMij j2 Cij ; eR h ðx  xij Þ2 þ hC2ij

ð12Þ

and

rffiffiffiffiffi

að3Þ ðx; IÞ ¼ x

4jMij j4 rs C2ij l I ; h eR 2e0 nr c h3 C ðx  xÞ2 þ C2 i2 ii ij ij

ð13Þ

where c is the velocity of light in vacuum, I = 2e0nrc—E—2 is the intensity of light. The total absorption coefficient can be obtained by

aðx; IÞ ¼ að1Þ ðxÞ þ að3Þ ðx; IÞ ¼x

rffiffiffiffiffi

rffiffiffiffiffi

2jM ij j4 rs C2ij l rs jMij j2 Cij l I x : 2 2 3 eR hðx  xij Þ þ hCij eR e0 nr c h Cii ½ðxij  xÞ2 þ C2 2

ð14Þ

ij

3. Results and discussions We will carry out the numerical calculations of the linear and third-order nonlinear optical absorption coefficients for a typical GaAs/AlGaAs semi-parabolic quantum wells. Where rs = 5.0  1024 m3,eR = 13.1,T1 = 1 ps, T2 = 0.2ps,l = 4p  107 H m1, nr = 3.2, m⁄ = 0.067 m0 (m0 is the mass of a free electron) [2]. The linear optical absorption coefficient a(1) versus the photon energy hm is plotted for three different parabolic confining frequencies of x0: x0 = 1  1014 S1,x0 = 1.2  1014 S1, and x0 = 1.5  1014 S1, respectively. From Fig. 1 one can see that there are three peaks of the linear optical absorption for the three confining frequencies. One can see that the absorption peaks buleshift with the increasing of the semi-parabolic potential confining frequency x0. However, the values of the absorption peaks do not change with the increase of confining frequency x0. The enhancement of the parabolic confined potential can only influence the energy levels of electron in the quantum wells. It does not influence the linear optical absorption coefficient. Fig. 2 plot the third-order nonlinear optical absorption coefficient a(3) versus the photon energy hm for three different parabolic confining frequencies of x0: x0 = 1  1014 S1, x0 = 1.2  1014 S1, and x0 = 1.5  1014 S1, respectively. From Fig. 2 one can see that three are also three absorption peaks for different confining frequencies. The third-order nonlinear optical absorption peaks buleshift with the increase of parabolic confining frequency x0. However, it is different from the case of linear optical

Fig. 1. The linear optical absorption coefficient a(1) versus the photon energy hm for three different parabolic confining frequencies of x0: x0 = 1  1014 S1, x0 = 1.2  1014 S1, and x0 = 1.5  1014 S1, respectively.

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Fig. 2. The third-order nonlinear optical absorption coefficient a(3) versus the photon energy hm for three different parabolic confining frequencies of x0: x0 = 1  1014 S1, x0 = 1.2  1014 S1, and x0 = 1.5  1014 S1, respectively.

absorption in Fig. 1, the absolute values of third-order nonlinear optical absorption peaks decrease with the increase of parabolic confining frequency. It indicate that the parabolic confining potential can influence the values of third-order nonlinear optical absorption coefficient. Fig. 3 plot the total optical absorption coefficient a versus the photon energy hm also for three different parabolic confining frequencies of x0: x0 = 1  1014 S1,x0 = 1.2  1014 S1, and x0 = 1.5  1014 S1, respectively. From Fig. 3 one can see that the total absorption peak buleshift with the increase of confining frequency. The values of the peaks increase with the enhancement of the parabolic confining frequency. In Fig. 4, we plot the linear optical absorption coefficient a(1) versus the photon energy hm with T1 = 5T2 for three different relaxation rates T2: T2 = 0.1 ps, T2 = 0.2 ps, and T2 = 0.3 ps, respectively. One can found from Fig. 4 that there are three peaks at the same photon energy for there relaxation rates. The values of the peaks increase quickly with the increase of the relaxation time. However, the values of the linear absorption coefficient decrease with the increase of the relaxation time for the photon energy far away from the absorption peak. Fig. 5 plot the third-order nonlinear optical absorption coefficient a(3) versus the photon energy hm with T1 = 5T2 for three different relaxation rates T2: T2 = 0.1 ps, T2 = 0.2 ps, and T2 = 0.3 ps, respectively. The absolute values of third-order nonlinear optical absorption peaks increase with the increase of relaxation time. In Fig. 6 we plot the total optical absorption coefficient a versus the photon energy hm with T1 = 5T2 for three different relaxation rates T2: T2 = 0.1 ps, T2 = 0.2 ps, and T2 = 0.3 ps,

Fig. 3. The total optical absorption coefficient a versus the photon energy hm also for three different parabolic confining frequencies of x0: x0 = 1  1014 S1, x0 = 1.2  1014 S1, and x0 = 1.5  1014 S1, respectively.

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Fig. 4. (Color online) The linear optical absorption coefficient a(1) versus the photon energy hm with T1 = 5T2 for three different relaxation rates T2: T2 = 0.1 ps, T2 = 0.2 ps, and T2 = 0.3 ps, respectively.

Fig. 5. (Color online) The third-order nonlinear optical absorption coefficient a(3) versus the photon energy hm with T1 = 5T2 for three different relaxation rates T2: T2 = 0.1 ps, T2 = 0.2 ps, and T2 = 0.3 ps, respectively.

Fig. 6. (Color online) The total optical absorption coefficient a versus the photon energy hm with T1 = 5T2 for three different relaxation rates T2: T2 = 0.1 ps, T2 = 0.2 ps, and T2 = 0.3 ps, respectively.

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Fig. 7. (Color online) The third-order nonlinear optical absorption coefficient versus the photon energy hm for three incident optical intensities I: I = 1.0  109 W/m2, I = 1.5  109 W/m2, and I = 2.0  109 W/m2, respectively.

Fig. 8. (Color online) The total optical absorption coefficient versus the photon energy also for three incident optical intensities.

respectively. It is similar to the case in Fig. 4, the absorption peaks increase with the increase of the relaxation time. However, one found that there are two peaks for the relaxation time T2 = 0.2 ps which is different from the case in Fig. 4. In the above calculations we set the incident optical intensity I = 1.0  109 W/m2. From Eqs. (15) and (16) one can see that the third-order nonlinear optical absorption coefficient and the total absorption coefficient are related to the incident optical intensity I. Fig. 7 plot the third-order nonlinear optical absorption coefficient versus the photon energy hm for three incident optical intensities I: I = 1.0  109 W/m2, I = 1.5  109 W/m2, and I = 2.0  109 W/m2, respectively. From Fig. 7 one can see that the absolute values of the third-order nonlinear optical absorption peaks increase with the increasing of the incident optical intensities. In Fig. 8 we plot the total optical absorption coefficient versus the photon energy also for three incident optical intensities. One can see that the total absorption coefficient decrease with the increase of the incident optical intensity. Moreover, the two absorption peaks will be absence when the incident optical intensity increases.

4. Conclusions The optical absorptions in semi-parabolic quantum wells have been investigated. The linear and third-order nonlinear optical absorption coefficient are obtained by using compact density-matrix approach and iterative method. The numerical calculations are presented for special GaAs/AlGaAs

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semi-parabolic quantum wells. From the above results, we have found that the optical absorption coefficients are strongly influenced by the confining frequency, relaxation time, and the incident optical intensity, respectively. However, their influences for the linear optical absorption, third-order nonlinear optical absorption, and the total optical absorption are different. We think our theoretical study may make a great contribution to experimental studies and may open up new opportunities for practical exploitation of the quantum-size effect in optical devices. Acknowledgments This work was supported by the National Natural Science Foundations of China (Nos. 11004030, 10804059, and 11274187) and the Sate Key Program for Basic Research of China (No. 2012CB326407). References [1] H. Hassanabadi, G. Liu, L. Lu, Solid State Commun. 152 (2012) 1761. [2] Y.B. Yu, S.N. Zhu, K.X. Guo, Phys. Lett. A 335 (2005) 175. [3] H. Hassanabadi, M. Hamzavi, S. Zarrinkamar, A.A. Rajabi, Few-Body System 48 (2010) 53; H. Hassanabadia, Eur. Phys. J.B 74 (2010) 415; H. Hassanabadi, H. Rahimov, L.l. Lu, C. Wang, J. Luminescence 132 (2012) 1095; H. Hassanabadi, M. Solaimani, H. Rahimov, Solid State Commun. 151 (2011) 1962. [4] L.L. Lu, W.F. Xie, H. Hassanabadi, J. Luminescence 131 (2011) 2538. [5] L.L. Lu, W.F. Xie, H. Hassanabadi, Q.H. Zhong, Superlattices Microstruct. 50 (2011) 501. [6] K.X. Guo, S.W. Gu, Phys. Rev. B 47 (1993) 16322. [7] E. Ozturk, Y. Ozdemir, Opt. Commun. 294 (2013) 361. [8] E. Rosencher, Ph. Bois, Phys. Rev. B 44 (1991) 11315. [9] Y.B. Yu, H.J. Wang, Superlattices Microstruct. 50 (2011) 252. [10] I. Karabulut, S. Baskoutas, J. Appl. Phys. 103 (2008) 073512. [11] G.H. Liu, K.X. Guo, H. Hassanabadi, L.L. Lu, B.H. Yazarloo, Physica B 415 (2013) 92. [12] S. Ünlü, I. Karabulut, H. S ß afak, Physica E: Low-Dimensional Syst. Nanostruct. 33 (2006) 319. [13] Y. Yakar, B. Çakir, A. Özmen, Opt. Commun. 283 (2010) 1795. [14] B. Çakir et al, Superlattices Microstruct. 47 (2010) 556. [15] S. Baskoutas, C. Garoufalis, A.F. Terzis, Eur. Phys. J. B 84 (2011) 241. [16] L.L. Lu, W.F. Xie, H. Hassanabadi, J. Appl. Phys. 109 (2011) 063108. [17] L.L. Lu, W.F. Xie, H. Hassanabadi, Physica B 406 (2011) 4129. [18] A. Keshavarz, M.J. Karimi, Phys. Lett. A 374 (2010) 2675. [19] G.H. Liu, K.X. Guo, C. Wang, Phys. B: Condens. Matter 407 (2012) 2334. [20] I. Karabulut, H. Sßafak, M. Tomak, Solid State Commun. 135 (2005) 735; I. Karabulut, H. Sßafak, Physica B 368 (2005) 82. [21] S.H. Pan, S.M. Feng, Phys. Rev. B. 44 (1991) 8165.