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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

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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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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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