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Nonlinear intersubband optical absorption in semiparabolic quantum wells Guang-hui Wang ∗ Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510006, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 22 May 2013 Accepted 16 October 2013 Keywords: Nonlinear optical absorption Semiparabolic quantum well Density matrix approach

a b s t r a c t The linear and third-order nonlinear optical absorptions in semiparabolic quantum wells are studied in detail. Analytic formulas for the linear and third-order nonlinear optical absorption coefﬁcients are obtained using the compact density matrix approach. Based on this model, numerical results are presented for typical GaAs/AlGaAs semiparabolic quantum wells. The results show that the factors of the incident optical intensity and the semiparabolic conﬁnement frequency have great inﬂuences on the total optical absorption coefﬁcients. © 2014 Published by Elsevier GmbH.

PACS: 42.65. −k 71.35.Cc 78.67.De

1. Introduction In the past few years, the nonlinear optical properties of semiconductor quantum well and quantum dot nanostructures, in particular the second- and third-order optical nonlinearities have attracted much attention in the theoretical and applied physics sides [1–7], because of their novel physical properties and promise for potential applications. Furthermore, quantum conﬁnement of carriers in these low-dimensional semiconductor nanostrucures lead to the formation of discrete energy levels and the drastic change of optical absorption spectra [8]. One of the most remarkable properties of these low-dimensional electronic systems is that the optical transitions between the size-quantized subbands are feasible. Recently, the linear intersubband optical absorption within the conduction band of a GaAs quantum well has been studied experimentally without an electric ﬁeld [9], and with an electric ﬁeld [10]. The fact that a very large dipole strength and a narrow band width were observed suggests that the intersubband optical transitions in a quantum well may have very large optical nonlinearities. Nonlinear intersubband optical absorption in a semiconductor quantum well was calculated

∗ Tel.: +86 2039310083; fax: +86 2039310083. E-mail address: [email protected] 0030-4026/$ – see front matter © 2014 Published by Elsevier GmbH. http://dx.doi.org/10.1016/j.ijleo.2013.10.116

by Ahn et al. [11]. Intersubband optical absorption in coupled quantum wells under an applied electric ﬁeld was studied by Yuh and Wang [12]. In 1993,Cui et al. [13] studied absorption saturation of intersubband optical transitions in GaAs/Alx Ga1-x As multiple quantum wells in experiment, and they tested the saturation optical intensity for Is = 0.67 MW/cm2 . In 2003, refractive index changes induced by the incident optical intensity in semiparabolic quantum wells were investigated by the present author [14]. In 2010,Zhang et al. [15] studied nonlinear optical absorption coefﬁcients and refractive index changes in a two-dimensional system. Optical absorption and refractive index changes in a twodimensional quantum ring with an applied magnetic ﬁeld were investigated by Xie [16]. From fundamental and practical points of view, These linear and nonlinear size-quantized transitions have the potential for device applications in far-infrared laser ampliﬁers, photodetectors, and high-speed electrooptical modulators [17,18]. In this paper, the linear and third-order nonlinear intersubband optical absorptions in semiparabolic quantum wells are studied in detail. In Section 2, analytical formulas for the linear, third-order nonlinear and total optical absorption coefﬁcients are derived by using the compact-density-matrix method and an iterative procedure. Numerical results and discussion are presented in Section 3. We ﬁnd that the incident optical intensity and the semiparabolic conﬁnement frequency have great inﬂuence on the total optical absorption coefﬁcients.

G.-h. Wang / Optik 125 (2014) 2374–2377

2. Theory The effective-mass Hamiltonian for an electron in a semiparabolic quantum well can be written as

2 H=− 2m∗

∂2 ∂2 ∂2 + 2 + 2 ∂x2 ∂y ∂z

+ V (z)

(1)

where z represents the growth direction, is Planck’s constant, and m* is the conduction-band effective mass, which will be taken to be constant in the rest of the paper. V(z) is the conﬁning potential, namely

⎧ ⎨ 1 m∗ ω2 z2 , z > 0 0 2 V (z) = ⎩ ∞, z < 0

(2)

n,k (r)

= A−1/2 ϕn (z)Uc (r)eik.r ,

(3)

and 2 |k |2 , 2m∗

εn,k = En +

(4)

where A is the area of the well, k|| and r|| are the wave vector and coordinates in the xy plane, and Uc (r) is the periodic part of the Bloch function in the conduction band at k = 0. ϕn and En are the envelope wave function and the transverse energy of the nth subband, solutions of one-dimensional Schrödinger equation H0 ϕn (z) = En ϕn (z), where H0 is the z part of the Hamiltonian H in Eq. (1), i.e., H0 = −(2 /2 m* )(d2 /dz2 ) + V(z), which eigenfunction and eigenenergy are given by

En = 2n +

3

ω0

2

(n = 0, 1, 2, . . ., )

ϕn (z) = Nn exp − 12 ˛2 z

2

H2n+1 (˛z),

n ||q|z|

m

= A−1

e

(z > 0),

(6)

˛−1

−i(k||n −k||m ).r||

dr|| ϕn ||q|z|ϕm

(8)

˜ iωt + Ee ˜ −iωt . = Ee

Let the sign denote the one-electron density matrix for this regime. Then the evolution of the density matrix obeys the following time-dependent Schrödinger equation ∂t

−1

= (i)

[H0 − qzE(t), ]ij − ij ( − (0) )ij,

(9)

where (0) is the unperturbed density matrix, and ij is the relaxation rate. For simplicity, we will assume in the following only two different ij values: 1 = 1/T1 for i = j is the diagonal relaxation rate, and 2 = 1/T2 for i = / j is the off-diagonal relaxation rate. T1 is a population relaxation time, which can be enhanced by storing the excited electrons on a metastable level. T2 is certainly governed by intrinsic mechanisms such as electron–electron interaction or optical-phonon emission for an excitation energy. Eq. (9) is solved by using the usual iterative method [1], then (t) =

(n) (t),

(10)

n

with (n+1) ∂ij

∂t

=

1 i −

(n+1)

[H0 , (n+1) ]ij − iij ij

1 [qz, (n) ]ij E(t). i

(11)

The electronic polarization can be expanded as ˜ iωt + ε0 (2) E˜ 2 e2iωt + ε0 (2) E˜ 2 + ε0 (3) E˜ 3 eiωt P(t) = ε0 (1) Ee 2ω 0 (3)

+ ε0 3ω E˜ 3 e3iωt + · · · + c.c.

(12)

(2) The dc optical rectiﬁcation term ε0 x0 E˜ 2 will be neglected due

(7)

= ıkn ,km ϕn ||q|z|ϕm , ||

E(t) = E0 cos(ωt)

(5)

− 1 √ 2n where ˛ = = 2 (2n + 1)! 2 is the normalization constant, and H2n+1 (˛z) are the Hermite polynomials. Therefore the dipolar transitional matrix element can be written as [1] m∗ ω0 /, Nn

where ı is the Kronecker delta function, and q is the electronic charge. Next we will derive the expression of the linear and third-order nonlinear optical absorption coefﬁcients in the model. Let us consider an electromagnetic ﬁeld with frequency ω, which is incident with a polarization vector normal to the quantum well. The system is excited by an electromagnetic ﬁeld

∂ij

where ω0 is the semiparabolic conﬁnement frequency (see Fig. 1). The eigenfunctions n,k (r) and eigenenergies εn,k are solutions of the Schrödinger equation H n,k (r) = εn,k n,k (r) and are given by

2375

||

(2)

to the small size of 0 . Contributions due to higher order harmonic terms in ω will also be neglected. These assumptions yield the approximate form for P(t) as ˜ iωt + ε0 (3) E˜ 3 eiωt + c.c.. P(t) ≈ ε0 (1) Ee

(13)

For simplicity, we shall conﬁne our attention to two-level systems only for electronic transitions. Hereafter, the ground state will be denoted by g, the ﬁrst excited state by e, respectively. The analytical forms of the linear (1) and the nonlinear (3) susceptibilities are given as follows by the same procedure as Ref. [6]. First, for the linear term ε0 (1) (ω) =

N|Meg |2 . ωeg − ω − ige

(14)

For the third-order term

ε0 (3) (ω) =

−N|Meg |2 |E|2 ωeg − ω − ige

4|Meg |2 (ωeg − ω)2 + (ge )

Fig. 1. Schematic diagrams for energy levels and wave functions of a semiparabolic quantum well.

2

−

(ωeg

(Mee − Mgg )2 − ige )(ωeg − ω − ige )

(15)

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G.-h. Wang / Optik 125 (2014) 2374–2377

where N is the carrier density in this system; ωeg = (E e − Eg )/ is the electronic transition frequency; Mnm = ϕn ||q|z|ϕm , (n, m = g, e) is the relaxation rate from the ﬁrst excited state to the ground state. The susceptibility (ω) is related to the absorption coefﬁcients as

˛(ω) = ω

Im[ε0 (ω)]. εR

(16)

where is the permeability of the system, εR is the real part of the permittivity and (ω) is the Fourier component of (t) with e−iωt dependence. From Eqs. (14)–(16), the linear and the third-order nonlinear optical absorption coefﬁcients ˛(1) (ω)and ˛(3) (ω)can be obtained as follows, respectively[6]

(1)

˛

(ω) = ω

|Meg |2 Nge

. εR (ωeg − ω)2 + (ge )2

˛(3) (ω, I) = −ω

εR

2 2

2

2nr ε0 c[(ω − ωeg ) + (ge ) ] 2

|Mee − Mgg |2 [(ω − ωeg ) − (ge ) + 2ωeg (ωeg − ω)] 2

(ωeg ) + (ge )

(18)

2

where nr is the medium refractive index; εR is the real part of the dielectric constant, deﬁned as εR = n2r ε0 , here ε0 is vacuum dielectric constant, and c is the speed of light in free space; I is the incident optical intensity, which is deﬁned as

I=2

2nr ER |E () |2 = |E () |2 .

c

(19)

Since the third-order nonlinear optical absorption coefﬁcient ˛(3) (ω, I)is negative and is proportional to the incident optical intensity I, the total absorption coefﬁcient ˛(ω, I) decreases as I increases. ˛(ω, I) is reduced by one-half when I reaches a value Is called the saturation intensity. So we have the relation ˛(ω, Is ) = ˛(1) (ω)/2, or equivalently ˛(3) (ω, Is ) = −˛(1) (ω)/2, so we obtain the saturation intensity Is as follows: 2

ε0 nr c[(ω) − ωeg )2 + (ge ) ]

Is = 4|Meg |2 −

|Mee −Mgg |2 [(ω−ωeg )2 −(ge )2 +2ωeg (ωeg −ω)] (ωeg )2 +(ge )2

(20)

So the total optical absorption coefﬁcient ˛(ω, I) can be written as ˛(ω, I) = ˛(1) (ω) + ˛(3) (ω, I)

Fig. 2. The total optical absorption coefﬁcient ˛(ω, I) versus the photon energy ω for ﬁve different values of the incident optical intensity I: (a) I = 0, (b) I = 0.3 MW/cm2 , (c) I = 0.63 MW/cm2 , (d) I = 0.8 MW/cm2 , (e) I = 1.0 MW/cm2 , with ω0 = 1.0 × 1014 s−1 .

I|Meg |2 N ge 2

4|Meg |2 −

(17)

= ˛(1) (ω) 1 −

I 2Is

(21)

occur at around I = 0.63 MW/cm2 , which is identical with the saturation optical intensity 0.67 MW/cm2 tested experimently by Cui et al. [13] in GaAs/Alx Ga1−x As multiple quantum wells. When the incident optical intensity I exceeds the value of 0.63 MW/cm2 , the exciton absorption peak will be signiﬁcantly split up into two peaks. Fig. 3 shows the peak absorption coefﬁcient ˛(ω, I) as a function of the incident optical intensity I with Is = 0.63 MW/cm2 , and ω0 = 1.0 × 1014 s−1 . From Fig. 3, we can see that the peak absorption coefﬁcient ˛(ω, I) will decrease linearly with the incident optical intensity I increasing. In low intensity limit (I → 0), the peak absorption coefﬁcient ˛(ω, I) = 420.31 cm−1 , but when the incident optical intensity I increases to I = 1.0 MW/cm2 , the peak absorption coefﬁcient ˛(ω, I) will decrease to 86.73 cm−1 , which is only one ﬁfth of the linear absorption coefﬁcient in the semiparabolic quantum well with conﬁnement frequency ω0 = 1.0 × 1014 s−1 . Fig. 4 shows the peak absorption coefﬁcient ˛(ω, I) versus semiparabolic conﬁnement frequency ω0 for three different values of the incident optical intensity I: (a) I = 0; (b) I = 0.3 MW/cm2 ; (c) I = 0.6 MW/cm2 . From Fig. 4, we can see that the peak absorption coefﬁcient when I = 0; (namely, the linear absorption coefﬁcient), will keep invariable, but when I = / 0; the peak absorption coefﬁcient will increase with semiparabolic conﬁnement frequency ω0 increasing, and with the increase of semiparabolic conﬁnement frequency ω0 , the variation of the peak absorption coefﬁcient will gradually become not very obvious. In summary, by using the compact density matrix approach, we have studied the total optical absorption coefﬁcient for typical GaAs/AlGaAs semiparabolic quantum wells. The results show

3. Results and discussion In the following, we will discuss total optical absorption coefﬁcients in the GaAs/AlGaAs semiconductor quantum well. The parameters used in our numerical work are adopted as [4–6]: T2 = 0.2 ps, nr = 3.2, N = 1.0 × 1016 cm−3 , m* = 0.067 m0 (m0 is the mass of a free electron). Fig. 2 shows the total optical absorption coefﬁcient ˛(ω, I) as a function of the incident photon energy ω with ω0 = 1.0 × 1014 s−1 , while the incident optical intensity I for ﬁve various values: (a) I = 0; (b) I = 0.3 MW/cm2 ; (c) I = 0.63 MW/cm2 ; (d) I = 0.8 MW/cm2 ; (e) I = 1.0 MW/cm2 . It can be seen from Fig. 2 that the total optical absorption coefﬁcient will decrease signiﬁcantly with the incident optical intensity I increasing. Since the incident optical ﬁeld only contributes to the ˛(3) term, at sufﬁciently high incident optical intensities the absorption at linear canter can be bleached. This is illustrated by the series of total optical absorption curves given in Fig. 2. We can see that the strong absorption saturation begins to

Fig. 3. The peak absorption coefﬁcient ˛(ω, I) versus the incident optical intensity I with Is = 0.63 MW/cm2 , and ω0 = 1,0 × 1014 s−1 .

G.-h. Wang / Optik 125 (2014) 2374–2377

2377

References

Fig. 4. The peak absorption coefﬁcient ˛(ω, I) versus semiparabolic conﬁnement frequency ω0 for three different values of the incident optical intensity I: (a) I = 0, (b) I = 0.3 MW/cm2 , (c) I = 0.6 MW/cm2 .

that the factors of the incident optical intensity and the semiparabolic conﬁnement frequency have great inﬂuences on the total optical absorption coefﬁcients. As we know, with recent advances in nanofabrication technology it has become possible to fabricate such a semiparabolic quantum well. Therefore, theoretical study on the nonlinear optical absorption may make a great contribution to experimental studies, may have profound consequences as regards practical application of electrooptical devices, for example, nonlinear optical absorption effects can be applied in Q switch, self mode-locking of lasers with saturable absorbers, laser stable frequence and absorption spectroscopy. Acknowledgments This work was supported by the National Natural Science Youth Foundation of China (Grant no. 60906042) and the National Natural Science Foundation of China (Grant nos. 10974058 and 61178003).

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