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Linear and nonlinear intersubband optical absorption in symmetric double semi-parabolic quantum wells
Physics Letters A 374 (2010) 2675–2680
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Linear and nonlinear intersubband optical absorption in symmetric double semi-parabolic quantum wells A. Keshavarz ∗ , M.J. Karimi Department of Physics, College of Science, Shiraz University of Technology, P.O. Box 313-71555, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 31 January 2010 Received in revised form 10 April 2010 Accepted 18 April 2010 Available online 28 April 2010 Communicated by R. Wu Keywords: Double quantum wells Optical properties Nanostructures
a b s t r a c t The linear and the nonlinear intersubband optical absorption in the symmetric double semi-parabolic quantum wells are investigated for typical GaAs/Alx Ga1−x As. Energy eigenvalues and eigenfunctions of an electron confined in finite potential double quantum wells are calculated by numerical methods from Schrödinger equation. Optical properties are obtained using the compact density matrix approach. In this work, the effects of the barrier width, the well width and the incident optical intensity on the optical properties of the symmetric double semi-parabolic quantum wells are investigated. Our results show that not only optical incident intensity but also structure parameters such as the barrier and the well width really affect the optical characteristics of these structures. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical properties of quantum wells (QWs), quantum well wires (QWWs) and quantum dots (QDs) have been studied by many researchers in recent years [1–11]. Since the susceptibility of these structures due to the strong quantum confinement effects are much stronger than that of the bulk materials [12,13], research in this field is noticeable in the literature. Optical properties of these structures also exhibit interesting applications in photo-electronic devices, such as semiconductor lasers, optical switching, infrared photodetectors, and so on [14–17]. Also applying the external electric field gives rise to polarization of the charge carrier distributions and cause an energy shift which modifies the optical properties of the electrons in the QWs [18–21]. Studies have shown that this modification sensitively depends on the structure of the quantum system and the applied electric field. In addition, investigation of the optical properties of double quantum wells (DQWs), as we know, have very interesting applications in quantum well laser [22]. In recent work, several theoretical studies of the linear and nonlinear intersubband optical absorption and the refractive index changes, based on the single quantum wells and multi-quantum wells have been presented [23–28]. Literature survey shows that the research in this field is often based on solving Schrödinger equation to find the wave functions and the energy spectrum analytically. By applying the density matrix method, optical coefficients are obtained and the effects of the changing quantum well
*
Corresponding author. Tel.: +98 711 7261392; fax: +98 711 7354523. E-mail address: [email protected] (A. Keshavarz).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.049
Fig. 1. Schematic diagram for a symmetric double semiparabolic quantum well. The center of the z axis coincides with the center of the barrier.
parameters are investigated. In this way one must restrict himself to well-known infinite potential wells such as parabolic and semi-parabolic QWs [26–34]. It should be noted that the infinite potential wells are not physically feasible. The reason is that the maximum barrier hight obtained in GaAs/Alx Ga1−x As system is 350 meV for Al0.4 , and above this concentration, the effective barrier decreases [3]. However progress in computational physics with high accuracy permits us to investigate the finite potential wells, which are important theoretically and experimentally. In this Letter, we have studied the optical properties of symmetric double semi-parabolic quantum confinement potential, which is shown schematically in Fig. 1. To do this, the numerical methods are employed to solve the Schrödinger equation
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A. Keshavarz, M.J. Karimi / Physics Letters A 374 (2010) 2675–2680
Fig. 2. The ground state and L B = 8.0 nm (d).
ϕ1 (full curves) and the first excited state ϕ2 (dashed curves) of DSQW with the confining potential for L B = 0 (a), L B = 2.0 nm (b), L B = 4.0 nm (c)
for the symmetric double semi-parabolic quantum well (DSQW). Then the results are used to calculate the linear, the third order nonlinear optical absorbtion coefficient, and the refractive index changes. In order to check the validity of our numerical method, we compared our numerical results with the analytical solution of a few infinite and finite QWs such as square, parabolic, semiparabolic and double square QWs. The outcome confirms the validity of our numerical method. The Letter is organized as follows: In Section 2, we review and summarize the formalism of the optical properties of the system. Section 3 is allocated to the numerical calculation for the energy eigenvalues and eigenfunctions of electron in the DSQW. Also, we have investigated the behavior of refractive index changes and optical absorption coefficient for the DSQW. Finally, the conclusions are given in Section 4. 2. Theory Consider the time independent Schrödinger equation in onedimensional case as
−
h¯ 2
d2
2m∗ dz2
ϕ (z) + V (z)ϕ (z) = E ϕ (z),
(1)
in which z represents the growth direction, ϕ ( z) is the wave function, m∗ is the effective mass, which is assumed to be constant, and V ( z) is the confining potential introduced as
⎧ 1 ∗ 2 ⎪ m ω0 ( z + ⎪ ⎨2 V ( z) = V 0 , ⎪ ⎪ ⎩1 ∗ 2 m ω0 ( z − 2
LB 2 ) , 2
− L2B − L W < z < − L2B , | z|
LB 2 ) , 2
LB 2
LB , 2
| z| LB 2
LB 2
+ LW ,
(2)
+ LW ,
where ω0 is the frequency of the semi-parabolic confining potential in the QWs, L B is the width of the barrier, and V0 is the profile of the conduction-band potential in the introduced QW. The well width L W is determined by V0 . Our problem is now to use a suitable numerical method to find the solution of both the energy eigenvalues E, and the eigenfunctions ϕ ( z), which will be done in the next section. To calculate the changes of the refractive index and absorption coefficients corresponding to the optical transitions between two subbands, we have used the density matrix approach method [3,35]. Suppose our system is excited by an electromagnetic field such as
E (t ) = E 0 cos ωt = Ee i ωt + Ee −i ωt ,
(3)
where ω is the frequency of the external incident field with a polarization vector normal to the QW. On the other hand, the elec-
A. Keshavarz, M.J. Karimi / Physics Letters A 374 (2010) 2675–2680
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tronic polarization P (t ) and susceptibility χ (t ) are defined by the dipole operator M, and the density matrix ρ respectively
P (t ) = ε0 χ (ω) Ee −i ωt + ε0 χ (−ω) Ee i ωt 1 = trace(ρ M ), V
(4)
where V is the volume of the system, ε0 is the permittivity of free space, and trace means the summation over the diagonal elements of the matrix. Using the same density matrix formalism, the analytic expressions of the linear and the third order nonlinear susceptibilities for a two-level quantum system are given as follows [24,25]:
ε0 χ (1) (ω) =
σ V | M 21 |2 , E 21 − h¯ ω − ih¯ Γ12
(5)
and
ε0 χ (3) (ω) σ V | M 21 |2 | E |2 4| M 21 |2 =− E 21 − h¯ ω − ih¯ Γ12 ( E 21 − h¯ ω)2 + (h¯ Γ12 )2 −
( M 22 − M 11 ) ( E 21 − ih¯ Γ12 )( E 21 − h¯ ω − ih¯ Γ12 ) 2
,
Fig. 3. The confinement energies of the two lowest states of a symmetric double quantum well as a function of the central barrier width.
(6)
where σ V is the carrier density, E i j = E i − E j is the energy interval of the two level system. M i j is the dipole matrix element, which is defined by M i j = |ϕi |ez|ϕ j | (i , j = 1, 2), and Γ shows the damping due to electron–phonon interaction. The change in the refractive index and the absorbtion coefficient is given by
n(ω) nr
= Re
α (ω) = ω
χ (ω) 2nr2
,
μ Im ε0 χ (ω) , εR
(7)
where nr is the refractive index. So the total refractive index change and the total absorption coefficient α (ω, I ) can be written as
n(ω) nr
=
n(1) (ω) nr
+
n(3) (ω) nr
,
α (ω, I ) = α (1) (ω) + α (3) (ω, I ),
(8)
where I is the incident optical intensity and is defined as
I =2
2 2nr ε R E (ω)2 . E (ω) = μ μc
(9)
Here c is the speed of light in free space, μ is the permeability of the system, and ε R is the real part of the permittivity. 3. Results and discussion We have solved numerically the Schrödinger equation for the symmetric DSQW potential. Then, we have investigated the linear, the third-order nonlinear, and the total refractive index and absorption coefficient changes in a GaAs/Alx Ga1−x As semiconductor quantum wells with finite confinement potential. The parameters used in our calculations are as follows [24]: m∗GaAs = 0.067m0 , m∗AlGaAs = 0.0919m0 , where m0 is the mass of a free electron,
σ v = 3.0 × 1022 m−3 , nr = 3.2, Γ12 = 1/ T 12 where T 12 = 0.14 ps, μ = 4π × 10−7 Hm−1 , V 0 = 228 meV (corresponding to Al con-
centration x = 0.3). In Fig. 2, the first two wave functions of DSQW are plotted as a function of the growth direction for different values of the barrier width with ω0 = 1.0 × 1014 s−1 . If one plots the probability
densities |ϕ ( z)|2 , it is seen that the overlap between |ϕ1 ( z)|2 and |ϕ2 (z)|2 is increases by increasing the barrier width. This leads to the same values of energy for these states at larger values of the barrier width as shown in Fig. 3. Also, this overlap increases the dipole matrix element. The energy of the ground state and that of the first excited state are presented in Fig. 3 as a function of the barrier width. This figure shows that the difference between the energies of these states decreases as the barrier width increases. This result is in agreement with the results of the double square quantum well [36]. It should be noted that our results for L B = 0 are equal to the results of a finite parabolic QW. In Fig. 4, the linear, third-order nonlinear and total refractive index changes ( n/nr ) are plotted as a function of the photon energy for different values of barrier width with an incident optical intensity of I = 0.4 MW/cm2 . As can be seen from this figure, the linear change generated by the χ (1) term is the opposite in sign of the nonlinear change generated by the χ (3) term. Therefore, the total refractive index change will be reduced by the nonlinear contribution. This figure also shows that the magnitude of the resonant peaks of the linear and nonlinear refractive index changes increase with increasing barrier width. But, the increase in the nonlinear term ( n(3) /nr ) is larger than that of linear term ( n(1) /nr ). Moreover, for larger values of the barrier width, the nonlinear refractive index change is greater than that of the linear term. The linear α (1) (ω), third-order nonlinear α (3) (ω) and total α (ω) optical absorption coefficients are plotted as a function of incident photon energy h¯ ω for different values of the barrier width in Fig. 5. The linear absorption coefficient is positive and decreases with increasing barrier width. Whereas, the nonlinear absorption coefficient is negative and has a complicated behavior with the variation of the barrier width. The resonant peak of the linear and the nonlinear absorption occurs at different values of incident photon energy h¯ ω . The position of the resonant peak for the total absorption coefficient is determined by the larger term (linear or nonlinear). When the difference between the magnitude of the linear and the nonlinear absorbtion coefficient is small, the resonant peak splits up into two separate peaks known as bleaching effect [1]. The bleaching effect can be seen in Fig. 5(b). Fig. 6, illustrates the linear, third-order nonlinear and the total optical absorption coefficients as a function of the photon energy
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A. Keshavarz, M.J. Karimi / Physics Letters A 374 (2010) 2675–2680
Fig. 4. Variations of the linear (dashed curves), third-order (dotted curves) and total (full curves) refractive index changes versus the photon energy for L B = 0 (a), L B = 2.0 nm (b), L B = 4.0 nm (c) and L B = 8.0 nm (d).
for three different values of semi-parabolic confinement frequencies ω0 = 0.5 × 1014 s−1 , ω0 = 1.0 × 1014 s−1 , ω0 = 1.5 × 1014 s−1 respectively, with the incident optical intensity I = 0.4 MW/cm2 and the barrier width L B = 2.0 nm. From this figure, we can see that the magnitude of resonant peak of the linear and the total optical absorption coefficients increase while that of the nonlinear term decreases by increasing the semi-parabolic confinement frequency. In addition, the resonant peaks of the linear, the nonlinear and the total absorption coefficients suffer an obvious blue-shift with the increase of the semi-parabolic confinement frequency. The reason is that the energy interval increases as the semiparabolic confinement frequency increases. The physical meaning of the above statements are as follows: At fixed values of I and Γ12 , the magnitudes of the resonant peaks of the linear and the nonlinear absorption coefficients depend on the values of the energy interval E 21 and the dipole matrix element M 21 . For the lower values of the barrier width or larger values of semi-parabolic confinement frequency, E 21 has a higher value while M 21 has a smaller value. For the linear absorption coefficient the variation of E 21 has an important role with respect to the variation of M 21 . The situation for nonlinear absorption coefficient is more complicated. In this case, for the lower values of the
barrier width, the effect of M 21 is dominant, elsewhere the dominant term is E 21 . Fig. 7 shows the total optical absorption coefficient as a function of the incident photon energy for ω0 = 1.0 × 1014 s−1 and L B = 2 nm, for four different incident optical intensities. It can be clearly seen that the total optical absorption coefficient will be reduced significantly with increasing the incident optical intensity. This is due to the contribution of the magnitude of the nonlinear optical absorption coefficient which is highly related to the incident optical intensity. In Fig. 8, the maximum values of the total refractive index change ( n/nr )max and the resonant peak values of the total optical absorption coefficient of the symmetric DSQW have been compared with those of a finite parabolic QW. This figure shows that the corresponding maximum values of the total refractive index change of DSQW are larger than those of a finite parabolic QW. In addition, the value of ( n/nr )max decreases (increases) with increasing of ω0 (L B ). We can also see that the resonant peak values of the total optical absorption coefficient of DSQWare lower than those of a finite parabolic QW. It is seen that the value of the resonant absorption peak increases (decreases) with increasing ω0 (L B ).
A. Keshavarz, M.J. Karimi / Physics Letters A 374 (2010) 2675–2680
2679
Fig. 5. Variations of the linear (dashed curves), third-order (dotted curves) and total (full curves) absorption coefficients versus the photon energy for L B = 0 (a), L B = 2.0 nm (b), L B = 4.0 nm (c) and L B = 8.0 nm (d).
Fig. 6. The linear (dashed curves), third-order (dotted curves) and total (full curves) absorption coefficients versus the photon energy for a fixed barrier width, L B = 2.0 nm, and three different values of the semi-parabolic confinement frequencies.
Fig. 7. The total optical absorption coefficient versus the photon energy for four different values of the incident optical intensities I with ω0 = 1.0 × 1014 s−1 and L B = 2 nm.
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Fig. 8. The maximum values of the total refractive index change (a) and the resonant peak values of the total optical absorption coefficient (b) versus of the barrier width with incident optical intensity I = 0.2 MW/cm2 .
4. Conclusions In conclusion, we have presented the optical properties of the symmetric DSQW, using compact density matrix approach. Numerical calculations are performed to solve the Schrödinger equation in order to find energy eigenvalues and eigenfunctions of symmetric DSQW. Results are employed to investigate the linear and the nonlinear optical absorption coefficients and refractive index changes, and the following results are deduced. It is found that the energy interval decreases and the dipole matrix element increases with increasing the barrier width. This causes the resonant peaks of the refractive index changes and optical absorption coefficients experience a blue-shift. Results show that the total absorption coefficient increases by decreasing the barrier width or increasing the well width. Also, the higher values of the refractive index change occur for the larger values of the barrier width. Finally our calculations show that the results of the total refractive index change and the total optical absorption coefficient of the symmetric DSQW have the substantial difference with those of a finite parabolic QW. The difference between the results of these two system increases by increasing the barrier width. Acknowledgements Financial support from the Shiraz University of Technology research council is gratefully acknowledged. We would like to thank Dr. S. Poostforush for useful discussions and comments. References [1] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron 23 (1987) 2196. [2] M.M. Fejer, S.J.B. Yoo, R.L. Byer, A. Harwit, J.S. Harris, Phys. Rev. Lett. 62 (1989) 1041.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
ω0 for different values
E. Rosencher, P. Bois, Phys. Rev. B 44 (1991) 11315. K.J. Kuhn, G.U. Lyengar, S. Yee, J. Appl. Phys. 70 (1991) 5010. G.H. Wang, K.X. Guo, Physica B 315 (2002) 234. L. Zhang, H.J. Xie, Physica E 22 (2004) 791. D.M. Sedrakian, A.Z. Khachatrian, G.M. Andresyan, V.D. Badalyan, Opt. Quantum Electron. 36 (2004) 893. Y.B. Yu, K.X. Guo, S.N. Zhu, Physica E 27 (2005) 62. G.H. Wang, K.X. Guo, Physica E 28 (2005) 14. Y.B. Yu, S.N. Zhu, K.X. Guo, Phys. Lett. A 335 (2005) 175. Y.B. Yu, S.N. Zhu, K.X. Guo, Solid State Commun. 139 (2006) 76. L. Zhang, Y.M. Chi, J.J. Shi, Phys. Lett. A 366 (2007) 256. B. Chen, K.X. Guo, R.Z. Wang, Y.B. Zheng, B. Li, Eur. Phys. J. B 66 (2008) 227. I˙ . Karabulut, S. Baskoutas, J. Appl. Phys. 103 (2008) 073512. C.H. Liu, B.R. Xu, Phys. Lett. A 372 (2008) 888. B. Chen, K.X. Guo, Z.L. Liu, R.Z. Wang, Y.B. Zheng, B. Li, J. Phys.: Condens. Matter 20 (2008) 255214. X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. T. Brunhes, P. Boucaud, S. Sauvage, Phys. Rev. B 61 (2000) 5562. X.F. Zhao, C.H. Liu, Eur. Phys. J. B 53 (2006) 209. S. Sauvage, P. Boucaud, Phys. Rev. B 59 (1999) 9830. G. Wang, Phys. Rev. B 72 (2005) 155329, (5pp). J.T. Verdeyen, Laser Electronic, Prentice-Hall, Inc., New Jersey, 1989, pp. 361– 412. I˙ . Karabulut, Ü. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283. S. Ünlü, I˙ . Karabulut, H. Safak, Physica E 33 (2006) 319. B. Chen, K.X. Guo, R.Z. Wang, Z.H. Zhang, Z.L. Liu, Solid State Commun. 149 (2009) 310. L. Zhang, Opt. Quantum Electron. 36 (2004) 665. L. Zhang, Superlatt. Microstruct. 37 (2005) 261. C.J. Zhang, K.X. Guo, Physica E 39 (2007) 103. I˙ . Karabulut, H. Safak, M. Tomak, Solid State Commun. 135 (2005) 735. S. Baskoutas, E. Paspalakis, A.F. Terzis, Phys. Rev. B 74 (2006) 153306. L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315. C.J. Zhang, K.X. Guo, Physica B 383 (2006) 183. C.J. Zhang, K.X. Guo, Physica E 33 (2006) 363. C.J. Zhang, K.X. Guo, Physica B 387 (2007) 276. R.W. Boyd, Nonlinear Optics, second ed., Academic Press, New York, 2003. P. Harrison, Wells Quantum, Wires and Dots, second ed., Wiley, New York, 2006.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Linear and nonlinear intersubband optical absorption in symmetric double semi-parabolic quantum wells A. Keshavarz ∗ , M.J. Karimi Department of Physics, College of Science, Shiraz University of Technology, P.O. Box 313-71555, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 31 January 2010 Received in revised form 10 April 2010 Accepted 18 April 2010 Available online 28 April 2010 Communicated by R. Wu Keywords: Double quantum wells Optical properties Nanostructures
a b s t r a c t The linear and the nonlinear intersubband optical absorption in the symmetric double semi-parabolic quantum wells are investigated for typical GaAs/Alx Ga1−x As. Energy eigenvalues and eigenfunctions of an electron confined in finite potential double quantum wells are calculated by numerical methods from Schrödinger equation. Optical properties are obtained using the compact density matrix approach. In this work, the effects of the barrier width, the well width and the incident optical intensity on the optical properties of the symmetric double semi-parabolic quantum wells are investigated. Our results show that not only optical incident intensity but also structure parameters such as the barrier and the well width really affect the optical characteristics of these structures. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical properties of quantum wells (QWs), quantum well wires (QWWs) and quantum dots (QDs) have been studied by many researchers in recent years [1–11]. Since the susceptibility of these structures due to the strong quantum confinement effects are much stronger than that of the bulk materials [12,13], research in this field is noticeable in the literature. Optical properties of these structures also exhibit interesting applications in photo-electronic devices, such as semiconductor lasers, optical switching, infrared photodetectors, and so on [14–17]. Also applying the external electric field gives rise to polarization of the charge carrier distributions and cause an energy shift which modifies the optical properties of the electrons in the QWs [18–21]. Studies have shown that this modification sensitively depends on the structure of the quantum system and the applied electric field. In addition, investigation of the optical properties of double quantum wells (DQWs), as we know, have very interesting applications in quantum well laser [22]. In recent work, several theoretical studies of the linear and nonlinear intersubband optical absorption and the refractive index changes, based on the single quantum wells and multi-quantum wells have been presented [23–28]. Literature survey shows that the research in this field is often based on solving Schrödinger equation to find the wave functions and the energy spectrum analytically. By applying the density matrix method, optical coefficients are obtained and the effects of the changing quantum well
*
Corresponding author. Tel.: +98 711 7261392; fax: +98 711 7354523. E-mail address: [email protected] (A. Keshavarz).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.049
Fig. 1. Schematic diagram for a symmetric double semiparabolic quantum well. The center of the z axis coincides with the center of the barrier.
parameters are investigated. In this way one must restrict himself to well-known infinite potential wells such as parabolic and semi-parabolic QWs [26–34]. It should be noted that the infinite potential wells are not physically feasible. The reason is that the maximum barrier hight obtained in GaAs/Alx Ga1−x As system is 350 meV for Al0.4 , and above this concentration, the effective barrier decreases [3]. However progress in computational physics with high accuracy permits us to investigate the finite potential wells, which are important theoretically and experimentally. In this Letter, we have studied the optical properties of symmetric double semi-parabolic quantum confinement potential, which is shown schematically in Fig. 1. To do this, the numerical methods are employed to solve the Schrödinger equation
2676
A. Keshavarz, M.J. Karimi / Physics Letters A 374 (2010) 2675–2680
Fig. 2. The ground state and L B = 8.0 nm (d).
ϕ1 (full curves) and the first excited state ϕ2 (dashed curves) of DSQW with the confining potential for L B = 0 (a), L B = 2.0 nm (b), L B = 4.0 nm (c)
for the symmetric double semi-parabolic quantum well (DSQW). Then the results are used to calculate the linear, the third order nonlinear optical absorbtion coefficient, and the refractive index changes. In order to check the validity of our numerical method, we compared our numerical results with the analytical solution of a few infinite and finite QWs such as square, parabolic, semiparabolic and double square QWs. The outcome confirms the validity of our numerical method. The Letter is organized as follows: In Section 2, we review and summarize the formalism of the optical properties of the system. Section 3 is allocated to the numerical calculation for the energy eigenvalues and eigenfunctions of electron in the DSQW. Also, we have investigated the behavior of refractive index changes and optical absorption coefficient for the DSQW. Finally, the conclusions are given in Section 4. 2. Theory Consider the time independent Schrödinger equation in onedimensional case as
−
h¯ 2
d2
2m∗ dz2
ϕ (z) + V (z)ϕ (z) = E ϕ (z),
(1)
in which z represents the growth direction, ϕ ( z) is the wave function, m∗ is the effective mass, which is assumed to be constant, and V ( z) is the confining potential introduced as
⎧ 1 ∗ 2 ⎪ m ω0 ( z + ⎪ ⎨2 V ( z) = V 0 , ⎪ ⎪ ⎩1 ∗ 2 m ω0 ( z − 2
LB 2 ) , 2
− L2B − L W < z < − L2B , | z|
LB 2 ) , 2
LB 2
LB , 2
| z| LB 2
LB 2
+ LW ,
(2)
+ LW ,
where ω0 is the frequency of the semi-parabolic confining potential in the QWs, L B is the width of the barrier, and V0 is the profile of the conduction-band potential in the introduced QW. The well width L W is determined by V0 . Our problem is now to use a suitable numerical method to find the solution of both the energy eigenvalues E, and the eigenfunctions ϕ ( z), which will be done in the next section. To calculate the changes of the refractive index and absorption coefficients corresponding to the optical transitions between two subbands, we have used the density matrix approach method [3,35]. Suppose our system is excited by an electromagnetic field such as
E (t ) = E 0 cos ωt = Ee i ωt + Ee −i ωt ,
(3)
where ω is the frequency of the external incident field with a polarization vector normal to the QW. On the other hand, the elec-
A. Keshavarz, M.J. Karimi / Physics Letters A 374 (2010) 2675–2680
2677
tronic polarization P (t ) and susceptibility χ (t ) are defined by the dipole operator M, and the density matrix ρ respectively
P (t ) = ε0 χ (ω) Ee −i ωt + ε0 χ (−ω) Ee i ωt 1 = trace(ρ M ), V
(4)
where V is the volume of the system, ε0 is the permittivity of free space, and trace means the summation over the diagonal elements of the matrix. Using the same density matrix formalism, the analytic expressions of the linear and the third order nonlinear susceptibilities for a two-level quantum system are given as follows [24,25]:
ε0 χ (1) (ω) =
σ V | M 21 |2 , E 21 − h¯ ω − ih¯ Γ12
(5)
and
ε0 χ (3) (ω) σ V | M 21 |2 | E |2 4| M 21 |2 =− E 21 − h¯ ω − ih¯ Γ12 ( E 21 − h¯ ω)2 + (h¯ Γ12 )2 −
( M 22 − M 11 ) ( E 21 − ih¯ Γ12 )( E 21 − h¯ ω − ih¯ Γ12 ) 2
,
Fig. 3. The confinement energies of the two lowest states of a symmetric double quantum well as a function of the central barrier width.
(6)
where σ V is the carrier density, E i j = E i − E j is the energy interval of the two level system. M i j is the dipole matrix element, which is defined by M i j = |ϕi |ez|ϕ j | (i , j = 1, 2), and Γ shows the damping due to electron–phonon interaction. The change in the refractive index and the absorbtion coefficient is given by
n(ω) nr
= Re
α (ω) = ω
χ (ω) 2nr2
,
μ Im ε0 χ (ω) , εR
(7)
where nr is the refractive index. So the total refractive index change and the total absorption coefficient α (ω, I ) can be written as
n(ω) nr
=
n(1) (ω) nr
+
n(3) (ω) nr
,
α (ω, I ) = α (1) (ω) + α (3) (ω, I ),
(8)
where I is the incident optical intensity and is defined as
I =2
2 2nr ε R E (ω)2 . E (ω) = μ μc
(9)
Here c is the speed of light in free space, μ is the permeability of the system, and ε R is the real part of the permittivity. 3. Results and discussion We have solved numerically the Schrödinger equation for the symmetric DSQW potential. Then, we have investigated the linear, the third-order nonlinear, and the total refractive index and absorption coefficient changes in a GaAs/Alx Ga1−x As semiconductor quantum wells with finite confinement potential. The parameters used in our calculations are as follows [24]: m∗GaAs = 0.067m0 , m∗AlGaAs = 0.0919m0 , where m0 is the mass of a free electron,
σ v = 3.0 × 1022 m−3 , nr = 3.2, Γ12 = 1/ T 12 where T 12 = 0.14 ps, μ = 4π × 10−7 Hm−1 , V 0 = 228 meV (corresponding to Al con-
centration x = 0.3). In Fig. 2, the first two wave functions of DSQW are plotted as a function of the growth direction for different values of the barrier width with ω0 = 1.0 × 1014 s−1 . If one plots the probability
densities |ϕ ( z)|2 , it is seen that the overlap between |ϕ1 ( z)|2 and |ϕ2 (z)|2 is increases by increasing the barrier width. This leads to the same values of energy for these states at larger values of the barrier width as shown in Fig. 3. Also, this overlap increases the dipole matrix element. The energy of the ground state and that of the first excited state are presented in Fig. 3 as a function of the barrier width. This figure shows that the difference between the energies of these states decreases as the barrier width increases. This result is in agreement with the results of the double square quantum well [36]. It should be noted that our results for L B = 0 are equal to the results of a finite parabolic QW. In Fig. 4, the linear, third-order nonlinear and total refractive index changes ( n/nr ) are plotted as a function of the photon energy for different values of barrier width with an incident optical intensity of I = 0.4 MW/cm2 . As can be seen from this figure, the linear change generated by the χ (1) term is the opposite in sign of the nonlinear change generated by the χ (3) term. Therefore, the total refractive index change will be reduced by the nonlinear contribution. This figure also shows that the magnitude of the resonant peaks of the linear and nonlinear refractive index changes increase with increasing barrier width. But, the increase in the nonlinear term ( n(3) /nr ) is larger than that of linear term ( n(1) /nr ). Moreover, for larger values of the barrier width, the nonlinear refractive index change is greater than that of the linear term. The linear α (1) (ω), third-order nonlinear α (3) (ω) and total α (ω) optical absorption coefficients are plotted as a function of incident photon energy h¯ ω for different values of the barrier width in Fig. 5. The linear absorption coefficient is positive and decreases with increasing barrier width. Whereas, the nonlinear absorption coefficient is negative and has a complicated behavior with the variation of the barrier width. The resonant peak of the linear and the nonlinear absorption occurs at different values of incident photon energy h¯ ω . The position of the resonant peak for the total absorption coefficient is determined by the larger term (linear or nonlinear). When the difference between the magnitude of the linear and the nonlinear absorbtion coefficient is small, the resonant peak splits up into two separate peaks known as bleaching effect [1]. The bleaching effect can be seen in Fig. 5(b). Fig. 6, illustrates the linear, third-order nonlinear and the total optical absorption coefficients as a function of the photon energy
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Fig. 4. Variations of the linear (dashed curves), third-order (dotted curves) and total (full curves) refractive index changes versus the photon energy for L B = 0 (a), L B = 2.0 nm (b), L B = 4.0 nm (c) and L B = 8.0 nm (d).
for three different values of semi-parabolic confinement frequencies ω0 = 0.5 × 1014 s−1 , ω0 = 1.0 × 1014 s−1 , ω0 = 1.5 × 1014 s−1 respectively, with the incident optical intensity I = 0.4 MW/cm2 and the barrier width L B = 2.0 nm. From this figure, we can see that the magnitude of resonant peak of the linear and the total optical absorption coefficients increase while that of the nonlinear term decreases by increasing the semi-parabolic confinement frequency. In addition, the resonant peaks of the linear, the nonlinear and the total absorption coefficients suffer an obvious blue-shift with the increase of the semi-parabolic confinement frequency. The reason is that the energy interval increases as the semiparabolic confinement frequency increases. The physical meaning of the above statements are as follows: At fixed values of I and Γ12 , the magnitudes of the resonant peaks of the linear and the nonlinear absorption coefficients depend on the values of the energy interval E 21 and the dipole matrix element M 21 . For the lower values of the barrier width or larger values of semi-parabolic confinement frequency, E 21 has a higher value while M 21 has a smaller value. For the linear absorption coefficient the variation of E 21 has an important role with respect to the variation of M 21 . The situation for nonlinear absorption coefficient is more complicated. In this case, for the lower values of the
barrier width, the effect of M 21 is dominant, elsewhere the dominant term is E 21 . Fig. 7 shows the total optical absorption coefficient as a function of the incident photon energy for ω0 = 1.0 × 1014 s−1 and L B = 2 nm, for four different incident optical intensities. It can be clearly seen that the total optical absorption coefficient will be reduced significantly with increasing the incident optical intensity. This is due to the contribution of the magnitude of the nonlinear optical absorption coefficient which is highly related to the incident optical intensity. In Fig. 8, the maximum values of the total refractive index change ( n/nr )max and the resonant peak values of the total optical absorption coefficient of the symmetric DSQW have been compared with those of a finite parabolic QW. This figure shows that the corresponding maximum values of the total refractive index change of DSQW are larger than those of a finite parabolic QW. In addition, the value of ( n/nr )max decreases (increases) with increasing of ω0 (L B ). We can also see that the resonant peak values of the total optical absorption coefficient of DSQWare lower than those of a finite parabolic QW. It is seen that the value of the resonant absorption peak increases (decreases) with increasing ω0 (L B ).
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Fig. 5. Variations of the linear (dashed curves), third-order (dotted curves) and total (full curves) absorption coefficients versus the photon energy for L B = 0 (a), L B = 2.0 nm (b), L B = 4.0 nm (c) and L B = 8.0 nm (d).
Fig. 6. The linear (dashed curves), third-order (dotted curves) and total (full curves) absorption coefficients versus the photon energy for a fixed barrier width, L B = 2.0 nm, and three different values of the semi-parabolic confinement frequencies.
Fig. 7. The total optical absorption coefficient versus the photon energy for four different values of the incident optical intensities I with ω0 = 1.0 × 1014 s−1 and L B = 2 nm.
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Fig. 8. The maximum values of the total refractive index change (a) and the resonant peak values of the total optical absorption coefficient (b) versus of the barrier width with incident optical intensity I = 0.2 MW/cm2 .
4. Conclusions In conclusion, we have presented the optical properties of the symmetric DSQW, using compact density matrix approach. Numerical calculations are performed to solve the Schrödinger equation in order to find energy eigenvalues and eigenfunctions of symmetric DSQW. Results are employed to investigate the linear and the nonlinear optical absorption coefficients and refractive index changes, and the following results are deduced. It is found that the energy interval decreases and the dipole matrix element increases with increasing the barrier width. This causes the resonant peaks of the refractive index changes and optical absorption coefficients experience a blue-shift. Results show that the total absorption coefficient increases by decreasing the barrier width or increasing the well width. Also, the higher values of the refractive index change occur for the larger values of the barrier width. Finally our calculations show that the results of the total refractive index change and the total optical absorption coefficient of the symmetric DSQW have the substantial difference with those of a finite parabolic QW. The difference between the results of these two system increases by increasing the barrier width. Acknowledgements Financial support from the Shiraz University of Technology research council is gratefully acknowledged. We would like to thank Dr. S. Poostforush for useful discussions and comments. References [1] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron 23 (1987) 2196. [2] M.M. Fejer, S.J.B. Yoo, R.L. Byer, A. Harwit, J.S. Harris, Phys. Rev. Lett. 62 (1989) 1041.
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