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Electric field effect on the linear and nonlinear intersubband refractive index changes in asymmetrical semiparabolic and symmetrical parabolic quantum wells Li Zhang∗ Department of Mechanism and Electron, Panyu Polytechnic, Panyu 511483, PR China Received 16 May 2004; received in revised form 16 December 2004; accepted 20 December 2004 Available online 8 February 2005

Abstract By using the displacement harmonic variant method and the compact density matrix approach, the linear and nonlinear intersubband refractive index changes (RICs) in a semiparabolic quantum well (QW) with applied electric field have been investigated in detail. The simple analytical formulae for the linear and nonlinear RICs in the system were also deduced. The symmetrical parabolic QWs with applied electric fields were taken into account for comparison. Numerical calculations on typical GaAs QWs were performed. The dependence of the linear and nonlinear RICs on the incident optical intensity, the frequencies of the confined potential of the QWs and the strength of the applied electric field were discussed. Results reveal that the RICs in the semiparabolic quantum well system sensitively depend on these factors. The calculation also shows that the semiparabolic QW is a more ideal nonlinear optical system relative to the symmetric parabolic QW systems. © 2005 Elsevier Ltd. All rights reserved. PACS: 68.65.Fg; 78.20.Ci; 78.66.Fd; 42.65.-k Keywords: Refractive index changes; Semiparabolic quantum wells; Electric field; Density matrix approach

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E-mail address: [email protected]. 0749-6036/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2004.12.010

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1. Introduction Quantum confinement of the carriers in the semiconductor quantum well (QW) leads to the formation of discrete energy levels within the well, which results in significant optical nonlinearity in the semiconductor QW system compared with that in the bulk material [1,2]. These nonlinear optical properties, such as electro-optic effect [3–5], harmonic generation [1,2,5–11], optical absorption effect [12–17] and refractive index changes [3,15–18] (RICs), have the potential for device applications in far-infrared laser amplifiers [19], photo-detectors [20], and high-speed electro-optical modulators [21]. Therefore, from both the viewpoints of foundation and the application, the nonlinear optical properties of semiconductor QWs have attracted much attention in recent years [1–19]. After understanding sufficiently the essence of the optical nonlinearity, in order to obtain a significant nonlinear optical effect, much attention is focusing on finding the most suitable nonlinear optical material and the most optimized quantum confined systems [22]; at the same time, the applied field effects have also been noticed [4,7,8, 11,14–16]. For example, with regard to the investigation of the RICs in various quantum well systems, Kan et al. [15] investigated experimentally and theoretically the RICs in an electric-field-biased AlGaAs QW system; results reveal that the applied electric field can modulate about 4% of the maximum variation of the refractive index near the lowest excitonic transition gap; Chang et al. [3] studied theoretically the RIC in semiconductor superlattic systems, and the parabolic band model and effective mass approach had been adopted; Chuang and co-worker [16] calculated the influences of applied electric field on the RICs induced by the interband and intersubband optical transitions in rectangular QWs and parabolic QWs, but only the linear RIC had been considered, while the nonlinear RIC had been neglected in their study. Taking the third-order nonlinear RIC into account, Kuhn’s group [17] discussed systematically the RICs and optical absorption in Alx Ga1−x As/GaAs/Alx Ga1−x As rectangular QWs; their study results reveal that the RICs sensitively depend on the incident optical intensity, the carrier concentration and the width of the QW system. In all of the works mentioned above, very large RICs have been obtained, and results of theoretical calculation agree well with correlative experiments [3]. In some recent works [4,8,11], a semiparabolic QW system has been brought forward, and the exact electronic states for the semiparabolic QW with applied electric field have been solved via the methods of displacement harmonic oscillation and numerical calculation. It is obvious that the semiparabolic QW is an asymmetrical QW system, and the applied electric field can also adjust its symmetry. So, the nonlinear optical properties in a semiparabolic QW can be anticipated as having a more significant enhancement than that in symmetrical parabolic QWs [4,7,8,14–16]. On the other hand, the RIC in a semiparabolic QW system has not been investigated. Bearing this idea in mind, we will investigate the RIC properties in the semiparabolic QW system in this paper. In order to make the readers understand more explicitly the relative physical properties of the semiparabolic QWs with an applied electric field from many different profiles, it is necessary to briefly explain the differences and relationships between Ref. [11] and the present paper. (3) (i) The third harmonic generation (THG) coefficient, χ3ω , in the semiparabolic QWs with applied electric fields was investigated in Ref. [11]. The semiparabolic QW systems

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Fig. 1. The maximum RICs n/nr as functions of the strength of the applied electric field F when the confined potential frequency ω0 and the optical beam intensity I kept as 3 × 1014 s−1 and 3 × 1010 W m−2 , respectively. The solid lines and the dashed lines correspond to the RICs in asymmetrical semiparabolic QWs and the symmetrical parabolic QWs, respectively. (3)

with four energy-levels were taken in account. Due to the χ3ω being a complex quantity, (3) only the modulus of χ3ω was discussed. In the present paper, the linear and nonlinear RICs are studied in the models. In essence, the linear and nonlinear RICs are related to the real parts of the linear susceptibility, χ (1) (ω), and the third-order susceptibility, χ (3) (ω), respectively (refer to Eqs. (4), (5), (8) and (9)). Furthermore, it is well known that the physical processes for the linear susceptibility, χ (1)(ω), and the third-order susceptibility, χ (3) (ω) (also termed as the optical Kerr coefficient), are obviously different from the (3) process for THG, χ3ω (Ref. [23]). Thus, the behavior of χ (3) (ω) can not be obtained directly from the investigation for THG in Ref. [11]. On the other hand, a semiparabolic QW with two energy levels is considered in the present work, which is also different from the case in Ref. [11]. (ii) From the discussed results of the RICs and the THG in the two papers, there exist obviously differences. For example, with the increasing of the magnitude of the applied electric fields, the absolute values of RICs increase monotonically, but the absolute values of THG decrease monotonically [11]. These conclusions maybe have significant meaning for guiding relative experimental researches. (iii) Not only the asymmetric semiparabolic QWs, but also the symmetric parabolic QWs with electric fields are investigated in the present paper. Detailed comparison for the RICs in the two types of systems are performed, and the results show that the semiparabolic QW is a more ideal nonlinear optical system due to its self-asymmetries and adjustment

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of applied electric field relative to the symmetrical QW system. The linear and nonlinear RICs in the parabolic QWs are insensitive to the applied electric field since all the subbands are shifted by an equal amount and the wavefunctions are all displaced with an equal distance [16]. Furthermore, under the same effective widths and a relatively low confined frequency (such as ω0 < 2 × 1014 s−1 in Fig. 2), the RICs in the semiparabolic QWs are larger than those in the parabolic QWs. (iv) Though the density matrix approach and the process for solving the electronic states of semiparabolic QWs with electric fields are the same in these two papers, there are substantial differences in the two papers due to the reasons given above, and they reflect the physical properties of the semiparablic QWs from two different profiles. The paper is organized as follows: In Section 2, the exact electronic states in the semiparabolic QW with applied electric field [4,8] are first briefly described; then, under the compact density matrix approach, simple analytical formulae for the linear and thirdorder nonlinear RICs in the systems are deduced. In Section 3, numerical calculations on typical GaAs material are performed. The calculations are focused on the dependence of the RIC on the incident optics beam intensity, the carrier concentration, the applied electric field strength, and the confined potential frequency. Results reveal that the RICs of the semiparabolic QW sensitively depend on these factors, which illustrates that, by selecting reasonable parameters, one can obtain a large RIC in the semiparabolic QW systems. 2. Theory Under the effective mass approximation, by adopting the methods of envelope wavefunction and displacement harmonic oscillation [4,8,11], it is easy to give the electron eigen-wavefunction in the investigated semiparabolic systems, and it is given by ψn,k// (r) = exp(ik// · r// )Uc (r)φn (z)

1 2 = Nn exp(ik// · r// )Uc (r) exp − [α(z + β) ] Htn [α(z + β)], (1) 2 √ ∗ m ω0 /, β = q F/m ∗ ω02 , k// and r// are the electronic wavevector where α = and coordinate in the x y plane, Uc (r) is the periodic part of the Bloch function in the conduction band at k = 0, and Htn (x) is a Hermite functions [24]. The eigen-energy corresponding to the wavefunction ψn,k// (r) is written as 1 1 2 |k// |2 + ω0 (2tn + 1) − m ∗ ω02 β 2 . (2) ∗ 2m 2 2 In the same way, the wavefunctions and corresponding energy levels in the symmetrical parabolic QW [4,8] with electric fields can also be obtained. They take similar forms as in Eqs. (1) and (2), respectively, and the only distinction is that the subscript tn of Htn (x) should take a serial of integers, namely, tn = 0, 1, 2, . . .. Next the formulae of the linear and nonlinear RICs in the models will be briefly exp(−iωt) + E ∗ exp(iωt) described, assuming a monochromatic incident field E(t) = E is applied to the system. In the case of neglecting the optical rectification term and the higher than third-order terms [17,18], the electronic polarization of the QW will be a series n,k// =

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Fig. 2. The maximum RICs n/nr versus the confined potential frequency ω0 when F = 2 × 107 V/m for both of the semiparabolic QWs (solid lines) and parabolic QWs (dashed lines).

expansion as follows, + ε0 χ (3) (ω) E(iωt). P(t) ≈ ε0 χ (1) (ω) E(iωt)

(3)

Although the optical rectification term could be neglected rigidly only for the rigidly symmetric quantum systems, for many of the asymmetrical quantum confined systems, such as the asymmetric QWs which consist of symmetric QWs with applied electric field [14,17], the asymmetric double barrier QW systems [12], the formulae of the linear and nonlinear intersubband optical absorptions and RICs deduced by using Eq. (3) have approximately correctly described the physical properties in these systems due to the small sizes of the second-order susceptibilities. For example, the calculation of the linear and nonlinear optical absorptions and RICs in a symmetric QW with electric field [14,17] agreed well with the relative experimental results [25]; the investigation on the linear and nonlinear optical absorptions in the asymmetric step Al0.3 Ga0.7 As/AlAs/GaAs and Al0.3 Ga0.7 As/AlAs/In0.15 Ga0.85 As QWs [12] are also in excellent agreement with the experimental results [26]. The symmetry of the semiparabolic QW systems seems between that of symmetrical parabolic QWs and asymmetric step QWs, thus the Eq. (3) should be a reasonably approximation for the semiparabolic QW. Furthermore, for the investigation of the RICs in the semiparabolic QW without electric fields, this approximation has been adopted [18]. So it is appropriate to employ Eq. (3) to investigate the RICs in semiparabolic and parabolic QWs with applied electric fields in the present work. By using the same compact-density matrix approach and iterative procedure as Refs. [4–8,12–18], for a two-level quantum system, the analytical forms for the linear and

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nonlinear susceptibilities are given as follows [16–18]. For the linear term, ρs q 2 µ21 . E 21 − ω − iΓ0 For the third order term 2 ρs q 4 µ221 | E| 4µ221 (3) ε0 χ (ω) = − E 21 − ω − iΓ0 (E 21 − ω)2 + (Γ0 )2 2 σ21 . − (E 21 − iΓ0 )(E 21 − ω − iΓ0 ) ε0 χ (1) (ω) =

(4)

(5)

Here, ρs is the density of electrons in the semiparabolic QWs, E i j = i,ki − j,k j is the //

//

energy interval of two different electronic states, µi j is the matrix element which is defined by µi j = |φi (z)|z|φ j (z)|,

(6)

and σi j = |µii −µ j j |(i, j = 1, 2). The susceptibility χ is related to the change in refractive index n(ω)/nr by n(ω) χ(ω) , (7) = Re nr 2nr2 where nr is the refractive index, and χ(ω) is the Fourier component of χ(t) with exp(−iωt) dependence. Using Eqs. (4), (5) and (7), the analytic forms of the linear and nonlinear changes in the refractive index are obtained. For the linear term, ρs q 2 µ221 (E 21 − ω) n (1)(ω) = , nr 2nr2 ε0 (E 21 − ω)2 + (Γ0 )2 and, for the third-order term, ρs q 4 µ221 n (3)(ω) µcI =− 3 nr 4nr ε0 [(E 21 − ω)2 + (Γ0 )2 ]2 −

2 σ21

(8) 4µ221 (E 21 − ω)

E 21 (E 21 − ω)2 − (Γ0 )2 (3E 21 − 2ω) 2 + (Γ )2 E 21 0

,

(9)

where ε0 is the vacuum dielectric constant, c is the light velocity in free space, µ is the permeability of the systems, and I is the optical beam intensity which is given as 2 = 2nr /µc| E| 2. I = 2 ε R /µ| E| (10) So, the total RIC n(ω, I )/n r is given by n(ω, I )/nr = n (1) (ω)/nr + n (3)(ω, I )/nr .

(11)

In particular, for the symmetric parabolic QW systems with electric fields, the analytical RIC formulae can be further simplified via the following relations: 1 µ21 = |φ2 (z)|z|φ1(z)| = √ , 2α

(12)

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E 21 = ω0 ; σ21 = 0.

267

(13)

Thus, the nonlinear and nonlinear RICs for the parabolic QWs with applied electric fields are given as (ω0 − ω) ρs q 2 n (1) (ω) = 2 2 , nr 4nr α ε0 (ω0 − ω)2 + (Γ0 )2

(14)

n (3) (ω) ρs q 4 (ω0 − ω) µcI =− 3 4 . nr 4nr α ε0 [(ω0 − ω)2 + (Γ0 )2 ]2

(15)

and

From Eqs. (14) and (15), it is obviously observed that the linear and nonlinear RICs are independent of the electric field since all the subbands are shifted by an equal amount and the wave functions are all displaced by an equal distance, which is analogous to the absorption spectrum in the symmetrical parabolic QWs [16] because of the same reasons. 3. Numerical results and discussion In the present section, numerical calculations are performed on typical GaAs semiparabolic QW and parabolic QW systems. As for the discussions in Ref. [8], the same effective widths for semiparabolic QWs and parabolic QWs are assumed, which means ω A = ω S /2 (where ω S and ω A denote the confined potential frequencies of the symmetrical parabolic QW and the asymmetrical semiparabolic QW, respectively). In the following discussion, we denote ω A = ω S /2 = ω0 for convenience. The material parameters chosen are as follows [14–17]: m ∗ = 0.067m 0 (m 0 is the bare electron mass), ρs = 5 × 1024 m−3 and T = 0.14 ps. Fig. 1 depicts the maximum RIC n/nr as a function of the strength of the applied electric field F when the confined potential frequencies ω0 and the optical beam intensity I are kept as 3 × 1014 s−1 and 3 × 1010 W m−2 , respectively. In fact, the solid line and the dashed line correspond to the RICs in asymmetrical semiparabolic QWs and the symmetrical parabolic QWs, respectively (the same as in Fig. 2). From the figure, it can be seen that, as the applied electric field F varies from −1 × 107 to 9 × 107 V/m, both the absolute values of the linear and nonlinear RICs, |n (1) /nr | and |n (3)/nr | for the semiparabolic QWs, increase monotonically, while those for the symmetric parabolic QWs are kept constant. These features can be understood easily as follows: for the semiparabolic QWs, when the strength of the applied electric field increases, the asymmetry of the quantum well becomes stronger, so the matrix elements µ21 increase, which makes both |n (1)/nr | and |n (3) /nr | increase with the increase of the electric field; but for the parabolic QW, the matrix elements µ21 are independent of the electric field in terms of Eq. (12), thus |n (1)/nr | and |n (3) /nr | are kept at certain constants. Detailed calculation finds that the matrix element |µ21 | for semiparabolic QWs is over that for parabolic QWs when F > 3.8 × 107 V/m, which results in the crossings of corresponding solid lines and dashed lines at about F = 3.8 × 107 V/m. This characteristic specifies that, in order to obtain the same RICs in the parabolic and semiparabolic QWs, the magnitude of applied electric field to the semiparabolic QWs should be over than 3.8 × 107 V/m. On the

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other hand, when F = 9 × 107 V/m, the total RIC in the semiparabolic QW is larger by nearly 21.9% than that in the parabolic QW, which illustrates that the asymmetrical semiparabolic QW system is a more ideal quantum confined system for RICs relative to the symmetric parabolic one. It is also noticed that the linear RIC n (1) /nr is positive, while the nonlinear RIC n (3) /nr is negative. Thus, calculation of the RIC using only the linear χ (1) term and neglecting the nonlinear χ (3) term may be excessively optimistic for the systems. For the semiparabolic QWs, though the sign of n (3)/nr is opposite to that of n (1) /nr , and the absolute value of n (3) /nr increases with the increasing of F, the total RIC still has a significant enhancement for the electric field effect because the increment of n (1) /nr is greater that of n (3)/nr . For example, relative to absent electric field, the total RIC enhances by about 39.6% induced by a 9 × 107 V/m electric field. Fig. 2 plots the maximum RICs n/nr versus the confined potential frequency ω0 when the applied electric field F is kept at 2 × 107 V/m for both of the semiparabolic QWs (solid lines) and parabolic QWs (dashed lines). It is observed that, as ω0 decreases, the absolute values of the linear RIC, third-order RIC and the total RIC increase monotonically. Comparing the solid lines with corresponding dashed lines, it is found that the lower the frequencies of the confined potential ω0 is, the more obvious the differences of the corresponding RICs in semiparabolic QWs and in parabolic QWs become. This feature illustrates that, in order to obtain a large RIC in the semiparabolic QW system, a relatively low confined potential frequency ω0 should be chosen. For instance, when the confined potential frequency ω0 = 1.4×1014 s−1 , the linear and nonlinear RICs in the semiparabolic QW are enhanced by 12.8% and 28.9% relative to the parabolic QW, respectively. Fig. 3 shows the results by illustrating the change in the two components of the RIC (n (1) /nr and n (3) /nr ) for an ω0 = 3 × 1014 s−1 semiparabolic QW at the incident optical intensity of 3 × 1010 W m−2 (Fig. 3(A)) and 18 × 1010 W m−2 (Fig. 3(B)). In fact, the solid line, the dashed line and the dotted line denote the linear RIC, the third order nonlinear RIC and the total RIC, respectively. The largest RIC is not at the line center but has a shift from the center [17], which is obviously different from the harmonics generation properties in QW systems [5–10]. (The maximum of the harmonic generations, such as the second-harmonic generations [5–8] or third-harmonic generations [9,10], usually occur at the resonant frequency points.) For example, though the resonant frequency of the photon is 291.6 meV in Fig. 3(A), the positive peaks for the linear, the nonlinear and the total RICs are at 286.9 meV, 294.3 meV and 285.1 meV, respectively. Comparing Fig. 3(A) with Fig. 3(B), it is found that, as the incident optical intensity I changes from 3 × 1010 to 18 ×1010 W m−2 , the linear RIC stays nearly unchanged, but the third-order RIC increases apparently. This feature can be observed and understood directly from Eqs. (8) and (9). When I is small (such as I 3 × 1010 W m−2 in Fig. 3(A)), the total RIC is positive and it is contributed mainly by the linear term n (1) /nr ; furthermore, as I increases, the total RIC will reduce and its peak position will shift away from the center. This results specify that, if one wants to get a large RIC with positive sign, a relatively low incident optical intensity should be adopted. When I is relatively large (such as when I is about 18 × 1010 W m−2 in Fig. 3(B)), the total RIC is contributed mainly by the nonlinear term n (3) /nr , so it has the negative sign. This characteristic means that, for the calculation of the RIC in semiparabolic QW systems with strong incident optical intensity, the nonlinear term cannot be neglected.

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Fig. 3. The linear (solid line), nonlinear (dashed line) and the total RIC (dotted line) as functions of the photon energy ω for ω0 = 3 × 1014 s−1 semiparabolic QW at I = 3 × 1010 W m−2 (A) and 18 × 1010 W m−2 (B).

The total RICs as functions of the photon energy ω for the semiparabolic QWs are plotted in Fig. 4. In fact, Fig. 4(A) corresponds to the ω0 = 3 × 1014 s−1 semiparabolic QW with three different electric fields: (a) F = 0, (b) F = 3 × 107 V/m, (c) F = 6 × 107 V/m. Fig. 4(B) corresponds to three semiparabolic QWs with different confined potential frequency: (d) ω0 = 1.7 × 1014 s−1 , (e) ω0 = 2 × 1014 s−1 , (f) ω0 = 2.5 × 1014 s−1 . The incident optical intensity I is kept a constant at 3 × 1010 W m−2 . A very important feature for Fig. 4(A) is that, for a certain frequency semiparabolic QW, the stronger the applied electric field is, the larger the total RIC is, and the more obviously the red-shift of the RIC curve moves. The reason for the enhancement of RIC due to the electric field has been explained in Fig. 1. The reason for the red-shift is due to the smaller transition energy E 21 for a stronger electric field [8]. A very important feature for Fig. 4(B)

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Fig. 4. The total RICs as functions of the photon energy ω for the semiparabolic QWs. (A) correspond to the ω0 = 3 × 1014 s−1 semiparabolic QW with three different electric fields: (a) F = 0, (b) F = 3 × 107 V/m, (c) F = 6 × 107 V/m. (B) correspond to three semiparabolic QWs with different confined potential frequency: (d) ω0 = 1.7 × 1014 s−1 , (e) ω0 = 2 × 1014 s−1 , (f) ω0 = 2.5 × 1014 s−1 . The incident optical intensity I kept at 3 × 1010 W m−2 .

is that, the stronger the semiparabolic confined frequency is, on the contrary, the smaller the total RIC becomes, and the more obviously the blue-shift of the RIC curve moves. It is well known that the energy interval becomes wider when the confinement of a QW system becomes stronger, which results in the blue-shift in Fig. 4(B). 4. Summary In conclusion, by using the displacement harmonic variant [4,8] and compact density matrix approach [13–18], the RICs in the semiparabolic QW with applied fields have

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been deduced and investigated in detail. Numerical calculations on the typical GaAs QWs are performed, and the calculation is mainly focused on the dependence of the RIC on the strength of the applied electric field, the confined potential frequency of the semiparabolic QW and the incident optical intensity. Detailed comparison for the RICs in the semiparabolic QWs and parabolic QWs are also carried out. Results reveal that, in order to obtain a large RIC in the semiparabolic QW system, a semiparabolic QW with low confined potential frequency should be chosen, a relatively strong electric field may be applied and a relatively weak incident optical intensity should be adopted. Furthermore, the absolute values of the linear and nonlinear RICs in semiparabolic QWs increase monotonically with the increasing of magnitude of the applied electric fields, but those in the parabolic QWs are insensitive to the applied electric field [16]. Under the same effective widths and a relatively low confined frequencies ω0 , the RICs in the semiparabolic QWs are larger than those in the parabolic QWs. These calculations also specify that, relative to the symmetrical QW system, the semiparabolic QW is an ideal nonlinear optical system due to its self-asymmetries and adjustment of applied electric field. Therefore, a theoretical study may make a great contribution to experimental studies. We hope that these theoretical results described in the present work could stimulate more experimental work and be proved by relative experimental work. Acknowledgments This work was supported by the Natural Science Foundation of Guangzhou Education Bureau under Grant No. 2060, P. R. China. The author would like to thank Prof. H.J. Xie for helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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