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The second-harmonic generation susceptibility in semiparabolic quantum wells with applied electric ﬁeld Jian-Hui Yuan a,n, Yan Zhang a, Hua Mo a, Ni Chen a, Zhihai Zhang b,n a b

The Department of Physics, Guangxi Medical University, Nanning, Guangxi 530021, China School of Physics and Electronics, Yancheng Teachers University, Yancheng 224051, China

art ic l e i nf o

a b s t r a c t

Article history: Received 19 June 2015 Received in revised form 8 August 2015 Accepted 11 August 2015

The second-harmonic generation susceptibility in semiparabolic quantum wells with applied electric ﬁeld is investigated theoretically. For the same topic studied by Zhang and Xie [Phys. Rev. B 68 (2003) 235315] [1], some new and reliable results are obtained by us. It is easily observed that the second harmonic generation susceptibility decreases and the blue shift of the resonance is induced with increasing of the frequencies of the conﬁned potential. Moreover, a transition from a two-photon resonance to two single-photon resonances will appear adjusted by the frequencies of the conﬁned potential. Similar results can also be obtained by controlling the applied electric ﬁeld. Surprisingly, the second harmonic generation susceptibility is weakened in the presence of the electric ﬁeld, which is in contrast to the conventional case. Finally, the resonant peak and its corresponding resonant energy are also taken into account. & 2015 Elsevier B.V. All rights reserved.

Keywords: Second-harmonic generation Quantum well Electric ﬁeld

1. Introduction The study of the low-dimensional semiconductor quantum systems with quantum conﬁnement, such as quantum wells (QWs), quantum wires, quantum rings and quantum dots [1–12], has attracted much attention in experimental and theoretical subject because of the enhanced nonlinear effect. To fully understand and predict experimental phenomenon, the nonlinear optical properties in these semiconductor structures, such as optical absorption and refractive index changes, second-harmonic generation (SHG), third-harmonic generation and optical rectiﬁcation, have been so intensively studied theoretically [1–17]. These enhance nonlinear effects that have great potential for developing semiconductor quantum micro-device, such as high-speed electrooptical modulators, far-infrared photo detectors, and semiconductor optical ampliﬁers [18–21]. In the past few years, the study of the optical properties in semiconductor QWs has been so intensively studied [1–3,13–17]. This is because the nonlinear effects can be enhanced more dramatically in QWs than in bulk materials. For the bulk susceptibility, it is not very large because of the symmetry of the crystal structure. For nanomaterials also with symmetric structure, evenn

Corresponding authors. E-mail addresses: [email protected] (J.-H. Yuan), [email protected] (Z. Zhang). http://dx.doi.org/10.1016/j.optcom.2015.08.030 0030-4018/& 2015 Elsevier B.V. All rights reserved.

order nonlinear optical effects are usually vanishing in theory. Thus the contributions to the second order nonlinear optical susceptibilities are zero for a symmetrical QW, but as the symmetry is broken, nonvanishing contributions to second order nonlinear optical susceptibilities are expected to appear [2]. Consequently, in order to obtain the enhanced second order nonlinear optical susceptibilities in QWs, externally applied electric ﬁelds are used to remove the symmetry [1,2,15,16,22] or the QW structures are produced with a built-in asymmetry using advanced material growing technology [5,23,24]. Recently, Guo and Du [25] reported their results for linear and nonlinear optical absorption coefﬁcients and refractive index changes in asymmetrical Gaussian potential QWs with applied electric ﬁeld. After this moment, the other optical properties in the asymmetrical Gaussian potential QWs are investigated, such as nonlinear optical rectiﬁcation [26], SHG[27], and nonlinear optical absorption via two-photon process [28]. But these are serious errors in these above reports for the optical properties in asymmetrical Gaussian potential QWs because of the replacement of the asymmetrical Gaussian potential with the semiparabolic potential under the condition of a certain limit [29]. Also, for the model of semiparabolic QWs with the electric ﬁeld, some bad wavefunctions and the energy levels are chosen by those authors [25–28]. Electric ﬁeld effect on the second-order nonlinear optical properties of parabolic and semiparabolic quantum wells has been reported by Zhang and Xie [1], however their results for the SHG susceptibility in electric-ﬁeld-biased parabolic QWs have

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been proved to be wrong [2]. Factually, there are some serious errors reported in Ref. [1] for the SHG susceptibility in a semiparabolic QW with an applied electric ﬁeld. So it is very necessary for us to investigate the nonlinear optical properties in semiparabolic quantum wells with applied electric ﬁeld. In this paper, the electric-ﬁeld-induced SHG susceptibility in semiparabolic QWs is investigated theoretically. Also, the inﬂuence of the applied electric ﬁeld and the frequencies of the conﬁned potential on the SHG susceptibility has been taken into account. Compared with Ref. [1], the new results show that (1) the SHG susceptibility in semiparabolic QWs depend dramatically on the frequencies of the conﬁned potential and the external electric ﬁeld; (2) the SHG susceptibility decreases and the blue shift of the resonance is induced with increasing of the frequencies of the conﬁned potential; (3) the SHG susceptibility is weakened in the presence of the electric ﬁeld, which is in contrast to the conventional case. This paper is organized as follows: Hamiltonian, the relevant wave functions and energy levels are brieﬂy described in Section 2. Also the analytical expressions of the SHG susceptibility in semiparabolic QWs are presented in this section. Numerical calculations and detailed discussions for typical AlxGa1 xAl/GaAs materials are given in Section 3. Finally, a brief summary is presented in Section 4.

and

Φtn (z ) = Nn exp ( − α 2 (z + β )2) Htn (α (z + β )),

(7)

with

me⁎ ω 0 , =

α=

β=

qF me⁎ ω02

(8)

where Htn is the Hermite functions and tn is real, Nn is the normalization constant. tn is determined by Φtn (z = 0) ≡ 0, that is to say, the relation always should be satisﬁed as Htn (αβ ) ≡ 0. Obviously, tn = 2n + 1 as the electric ﬁeld is absent, where n = 0, 1, 2…. 2.2. The second-harmonic generation susceptibility The formulas of the SHG susceptibility in the two models will be derived by using the compact-density-matrix method and the iterative procedure. The system is excited by electromagnetic ﬁeld ˜ iωt + Ee ˜ −iωt . Let us denote ρ as the one-electron density E (t ) = Ee matrix for this regime. Then the evolution of density matrix is given by the time-dependent Schrödinger equation:

∂ρij ∂t

=

1 [H0 − qzE (t ), ρ]ij − Γij (ρ − ρ (0) )ij . =

(9)

where H0 is the Hamiltonian for this system without the electromagnetic ﬁeld E(t ); ρ(0) is the unperturbed density matrix; and Γij is the relaxation rate. Eq. (9) is calculated by the following iterative method [1,29,30]:

2. Theory 2.1. Electronic structure Within the framework of effective-mass approximation, the Hamiltonian of an electron conﬁned in semiparabolic QWs in the presence of electric ﬁeld along the z-axis can be written as

= 2 ⎛ ∂2 ∂2 ∂2 ⎞ H= − + + 2 ⎟ + V (z ) + qFz. ⎜ 2me⁎ ⎝ ∂x2 ∂y2 ∂z ⎠

(1)

with

ρ (t ) =

∑ ρ(n) (t ), n

(10)

with

∂ρij(n + 1) ∂t

=

1 1 {[H0, ρ (n + 1) ]ij − i=Γij ρij(n + 1) } − [qz, ρ (n) ]ij E (t ). i= i=

(11)

The electric polarization of the quantum system due to E(t ) can be expressed as

1 ⁎ 2 2 m ω z , z ≥ 0, V (z ) = { 2 e 0 z < 0, ∞,

(2)

˜ iωt + ε0 χ (1) E˜ 2e2iωt + c . c . + ε0 χ (2) E˜ 2, P (t ) ≈ ε0 χω(1) Ee 2ω 0

(12)

where z represents the growth direction of theQWs. me⁎ is the effective mass in materials. ℏ is the Planck constant, ω0 is the frequency of the semiparabolic conﬁned potential in QWs, F is the strength of the electric ﬁeld and q is the absolute value of the electric charge. Under the envelope wave-function approximation, the eigenfunctions Ψtn, k (r ) and eigenenergies εtn, k are the solutions of the Schrödinger equation for H and are given by [1,25–28]

where χω(1), χ2(ω2) and χ0(2) are the linear, SHG and optical rectiﬁcation susceptibility respectively. With the same compact density matrix approach and the iterative procedure as [30], the analytical expression of the SHG susceptibility is given as [1,2,22,27]

Ψtn, k (r ) = Φtn (z ) Uc (r∥) exp (ik∥·r∥)

where s is the surface density of electrons in the QWs, Γ0 is the phenomenological relaxation rate, Eij is the energy interval of two different electronic states, and Mij is the off-diagonal matrix element which is given by Mij = 〈i|z|j〉 where (i, j = 1, 2, 3). The SHG susceptibility has a resonant peak for the condition as =ω = E21 = E31/2 given by

(3)

and

εtn, k = Etn +

= 2k∥2 2me⁎

(4)

Here, k∥ and r∥ are respectively 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. Φtn (z ) and Etn can be obtained by solving the following Schrödinger equation:

⎡ ⎤ = 2 ∂2 + V (z ) + qFz⎥ Φtn (z ) = Etn Φtn (z ). Hz Φtn (z ) = ⎢ − ⎣ 2me⁎ ∂z2 ⎦

(5)

The electronic energy levels and corresponding wave functions are given as follows [1]:

Etn = (2tn + 1 − α 2β2)

=ω 0 , 2

n = 1, 2, 3…,

(6)

χ2(ω2) =

q3σM12 M23 M31 1 × ε0 (E31 − 2=ω + i=Γ0 )(E21 − =ω + i=Γ0 )

χ2(ω2), max =

q3σM12 M23 M31 ε0 (=Γ0 )2

(13)

(14)

3. Results and discussions In this section, the electric-ﬁeld-induced SHG susceptibility in semiparabolic quantum wells is investigated theoretically. Numerical calculations are carried out on typical AlxGa1 xAl/GaAs materials. The parameters adopted in the present work are as

J.-H. Yuan et al. / Optics Communications 356 (2015) 405–410

407

Fig. 1. The SHG susceptibility χ2(2ω) (a) versus the incident photon frequency ω with four different strengths of the electric ﬁeld F = 0, 2.0 × 107, 5.0 × 107 and 10.0 × 107 V/m , the low-lying energy levels of E1, E2 and E3 (b), the product of M21M32 M31 of matrix elements Mij (c) and the energy space ΔE of E21 and E31/2 (d) versus the electric ﬁeld strength for the conﬁned potential frequency ω0 = 3.6 × 1014 Hz .

follows: me⁎ = 0.067m0 (m0 is the electron mass), σ = 5.0 × 1024 m3, and Γ0 = 1/0.14 ps. In Fig. 1, the SHG susceptibility χ2(ω2) (a) is plotted as a function of the incident photon frequency ω with four different strengths of the electric ﬁeld F = 0, 2.0 × 107 , 5.0 × 107 and 10.0 × 107 V/m . It is easily seen that (1) the SHG susceptibility will decrease and the blue shift of resonance of photon is induced as the strength of electric ﬁeld increases as reported by Ref. [16], which is contrast to the conventional case as reported by Refs. [12,22,27]; (2) a transition from a two-photon resonance to two single-photon resonances will appear adjusted by the electric ﬁeld; (3) the single peak of the SHG susceptibility corresponding to the two-photon resonance occurs for the weak electric ﬁeld about the photon energy =ω ≈ E21 ≈ E31/2, and for the peaks of the two single-photon resonances, the main and the weak peaks are always related to the resonant energy Eres ≈ E31/2 and Eres ≈ E21; (4) the line width of photon spectrum becomes more wider as the increase of the electric ﬁeld. In order to interpret these properties above, the lowlying energy levels of E1, E2 and E3 (b), the product of M21M32 M31 of matrix elements Mij (c) and the energy space ΔE of E21 and E31/2 (d) are depicted as a function of the electric ﬁeld strength for the

conﬁned potential frequency ω0 = 3.6 × 1014 Hz . From Fig. 1(b), these low-lying energy levels of E1, E2 and E3 are increasing with the increase of the electric ﬁeld, which is different from the case of the electron conﬁned in the parabolic QWs with electric ﬁeld [1,2]. Furthermore, the energy space ΔE of E21 and E31/2 always increase with the increase of the electric ﬁeld (see in Fig. 1(d)), and it is easily observed that the energy space of E21 is always larger than that of E31/2. It is found that the change of the energy space of E21 is more obvious than that of E31/2. Thus we can easily understand the cause of the blue shift of resonance of photon and a broad line width of photon spectrum induced by the strength of electric ﬁeld. When the energy space of E21 and E31/2 differs too much, the condition of the two-photon resonance is not satisﬁed under the condition of approximation. However, the single photon resonance condition is easy to meet, so a transition from a two-photon resonance to two single-photon resonances will appear as the electric ﬁeld increases. From Eq. (13), it is easily understood that the SHG susceptibility χ2(ω2) is proportional to the product of M21M32 M31 of matrix elements Mij. We ﬁnd that the product of M21M32 M31 always decreases as the electric ﬁeld increases (see in Fig. 1(c)), which is the cause of the decrease of SHG susceptibility

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J.-H. Yuan et al. / Optics Communications 356 (2015) 405–410

transition from a two-photon resonance to two single-photon resonances will appear adjusted by the frequencies of the conﬁned potential (see in Fig. 2(a) and (b)). It is easily seen that the singlephoton resonance of photon is occurred under the smaller frequencies of the conﬁned potential, i.e. ω0 = 1.0 × 1014 Hz . And the larger the electric ﬁeld is, the more obvious the case is. It is because the difference of energy space between E21 and E31/2 for a small frequency of the conﬁned potential is more obvious than that for a large frequency of the conﬁned potential especially for the large electric ﬁeld, which can be easily seen in Fig. 3. To better understand the peak of the SHG susceptibility χ2(ω2) and its corresponding resonant energy Eres, the peak of the SHG susceptibility χ2(ω2) and the resonant energy of photon Eres are plotted in Fig. 4 as a function of the electric ﬁeld strength for the conﬁned potential frequency ω0 = 3.6 × 1014 Hz . The theoretical peaks for the SHG susceptibility χ2(ω2) are calculated according to Eq. (14). This will not generally be the case because the condition of two-photon resonance is only satisﬁed in the absence of the electric ﬁeld. So it is wrong to obtain the peaks for χ2(ω2) using Eq. (14) as reported by Refs. [1,27]. It is easily observed that the theoretical peaks for χ2(ω2) are always larger than the actual peak for χ2(ω2) . From Fig. 4(a), a transition from a two-photon resonance to two single-photon resonances will appear when the electric ﬁeld strength is more than 8.0 × 107 V/m . It is easily seen in Fig. 4(b) that the main (or weak) peak can be measured approximately as the value of the SHG susceptibility χ2(ω2) for the photon energy =ω = E31/2 (or =ω = E21).

Fig. 2. The SHG susceptibility χ2(2ω) versus the incident photon frequency ω with four different conﬁned potential frequencies ω0 = 1.0 × 1014 , 2.0 × 1014 , 4.0 × 1014 and 5.0 × 1014 Hz with the electric ﬁeld strengths F ¼0 and 2.0 × 107 V/m (a), 5.0 × 107 and 10.0 × 107 V/m (b) where in the inset of (a) and (b) we plot the product of M21M32 M31 of matrix elements Mij as a function of the conﬁned potential frequency for different electric ﬁeld strengths.

with increasing of the strength of electric ﬁeld. In Fig. 2(a) and (b), the SHG susceptibility χ2(ω2) is plotted as a function of the incident photon frequency ω with four different conﬁned potential frequencies ω0 = 1.0 × 1014 , 2.0 × 1014 , 4.0 × 1014 and 5.0 × 1014 Hz with the electric ﬁeld strengths F¼ 0 and 2.0 × 107 V/m (a), 5.0 × 107 and 10.0 × 107 V/m (b) where in the inset of Fig. 2(a) and (b) we plot the product of M21M32 M31 of matrix elements Mij as a function of the conﬁned potential frequency for different electric ﬁeld strengths. For a ﬁxed electric ﬁeld strength, it is easily observed that the SHG susceptibility χ2(ω2) will decrease with the increase of the frequency of the conﬁned potential because the product of M21M32 M31 of matrix elements Mij decreases monotonously with the increase of the frequency of the conﬁned potential. That is to say, if one wants to obtain a large SHG susceptibility, a smaller frequency of the conﬁned potential should be adopted. Moreover, the blue shift of resonance of photon is induced by the frequency of the conﬁned potential. In order to interpret it, the energy spaces ΔE of E21 and E31/2 are depicted in Fig. 3 as a function of the conﬁned potential frequency ω0 for four different electric ﬁeld strengths F = 0, 2.0 × 107 , 5.0 × 107 and 10.0 × 107 V/m. From Fig. 3, both the energy spaces ΔE of E21 and E31/2 are increasing with the increase of the frequency of the conﬁned potential. Furthermore, a

The peak of the χ2(ω2) will be split into two peaks as the electric ﬁeld strength is more than 8.0 × 107 V/m and the resonant energies Eres of the main peak and weak peak always increase as the electric ﬁeld strength increases (see in Fig. 4(c)). Moreover, the resonant energies Eres related to the main peak are always smaller than that related to the weak peak. In Fig. 5, the peak of the SHG susceptibility χ2(ω2) (a) and the resonant energy of photon Eres (b) are plotted as a function of the conﬁned potential frequency ω0 for two different electric ﬁeld strengths F = 0, 5 × 107 V/m . It is easily found that the peak of χ2(ω2) related to a small electric ﬁeld strength has a large peak value of χ2(ω2) for ﬁxed frequency of the conﬁned potential. For the electric ﬁeld strength F¼0 V/m, there is only one peak because the condition of two-photon resonance is always satisﬁed no matter how much the frequency of the conﬁned potential ω0 is. It is easily seen that the peak of χ2(ω2) always decreases as the frequency of the conﬁned potential increases. For the electric ﬁeld strength F = 5 × 107 V/m , the single peak related to two-photon resonance will split into two peaks related to two single-photon resonances as the frequency of the conﬁned potential is less than ω0 = 2.0 × 1014 Hz . In Fig. 5(b), the resonant energy of photon Eres for the electric ﬁeld strength F = 5 × 107 V/m is always larger than that for the electric ﬁeld strength F ¼0 V/m, which shows the blue shift of the resonant energy of photon Eres. It is easily observed that the resonant energy of photon Eres corresponding to the weak peak is larger than that corresponding to the main peak in the presence of two single-photon resonances.

4. Conclusion In this paper, we have derived the analytical expressions of the SHG susceptibility in the semiparabolic QWs with applied electric ﬁeld by using the compact density matrix approach. Numerical results on typical AlxGa1 xAl/GaAs materials show that the SHG susceptibility in this semiparabolic quantum well depend dramatically on the frequencies of the conﬁned potential and the

J.-H. Yuan et al. / Optics Communications 356 (2015) 405–410

409

Fig. 3. The energy space ΔE of E21 and E31/2 versus the conﬁned potential frequency ω0 for four different electric ﬁeld strengths F = 0, 2.0 × 107, 5.0 × 107 and 10.0 × 107 V/m .

Fig. 4. The peak of the SHG susceptibility ω0 = 3.6 × 1014 Hz .

χ2(2ω) and the resonant energy of photon Eres versus the electric ﬁeld strength for the conﬁned potential frequency

external electric ﬁeld. We ﬁnd that SHG susceptibility decreases and the blue shift of the resonance is induced with increasing of the frequencies of the conﬁned potential. Moreover, a transition from a two-photon resonance to two single-photon resonances will appear adjusted by the frequencies of the conﬁned potential. Similar results can also be obtained by controlling the applied

electric ﬁeld. Surprisingly, the SHG susceptibility is weakened in the presence of the electric ﬁeld, which is in contrast to the conventional case. As we know, with the recent advances in nanofabrication technology, it is possible to produce such semiparabolic QWs. Our theoretical results may make a contribution to experimental studies and provide a kind of approximative

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J.-H. Yuan et al. / Optics Communications 356 (2015) 405–410

Fig. 5. The peak of the SHG susceptibility χ2(2ω) (a) and the resonant energy of photon Eres (b) versus the conﬁned potential frequency ω0 for two different electric ﬁeld strengths F = 0, 5 × 107 V/m .

modeling for the practical application such as optoelectronics devices and optical communication.

Acknowledgments This research was supported by Project supported by the National Science Foundation of China under Grant nos. 11447193 and 11447101, the Youth Science Foundation of Guangxi Medical University in China under Grant no. GXMUYSF201313, the University National Science Foundation of Jiangsu Province of China under Grant no. 14KJB140015 and Guangxi Department of Education Research Projects in China under Grant no. KY2015LX046.

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