Accepted Manuscript Electric field effects on nonlinear optical rectification in symmetric coupled AlxGa1−xAs/GaAs quantum wells
Guanghui Liu, Kangxian Guo, Zhongmin Zhang, Hassan Hassanbadi, Liangliang Lu PII: DOI: Reference:
S0040-6090(18)30491-7 doi:10.1016/j.tsf.2018.07.026 TSF 36783
To appear in:
Thin Solid Films
Received date: Revised date: Accepted date:
19 December 2016 5 July 2018 19 July 2018
Please cite this article as: Guanghui Liu, Kangxian Guo, Zhongmin Zhang, Hassan Hassanbadi, Liangliang Lu , Electric field effects on nonlinear optical rectification in symmetric coupled AlxGa1−xAs/GaAs quantum wells. Tsf (2018), doi:10.1016/ j.tsf.2018.07.026
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ACCEPTED MANUSCRIPT Electric field effects on nonlinear optical rectification in symmetric coupled AlxGa1−xAs/GaAs quantum wells Guanghui Liua, Kangxian Guo b*, Zhongmin Zhang b , Hassan Hassanbadic, Liangliang Lu*d a
State Key Laboratory of Optoelectronic Materials and Technologies, School of
Physics, Sun Yat-sen University, Guangzhou 510275, P.R. China b
Department of Physics, School of Physics and Electronic Engineering, Guangzhou
Department of Physics, Shahrood University of Technology, Shahrood, Iran School of Physics, Nanjing University, Nanjing 210093, PR China
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d
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University, Guangzhou 510006, P.R. China
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*Corresponding author: e-mail:
[email protected] (K.X. Guo). e-mail:
[email protected] (L. Lu).
ACCEPTED MANUSCRIPT Abstract Nonlinear optical rectification in symmetric coupled Alx Ga1−x As/GaAs quantum wells with external electric field is investigated using numerical method and compact density matrix approach. Our results reveal that for the resonant peaks of optical rectification, a blue shift is exhibited for increasing electric field, while a red shift followed by a blue shift is exhibited for increasing barrier or well widths. The resonant peak values of optical rectification can reach a maximum value by an
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appropriate choice for the electric field, the barrier or well widths. Our studies pave the way for the design, optimization and applications of quantum-sized nonlinear
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optoelectronic devices.
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Keywords:Optical rectification; Electric field; Symmetric coupled quantum wells.
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1. Introduction Nonlinear optics have been an extensive research field since the invention of the laser in the 1960s [1]. The nonlinear interaction of light with matter itself leads to many intriguing physical phenomena, such as harmonic generation [2, 3], optical Kerr effects [4], optical parametric oscillation [5] and optical soliton [6]. The nonlinear effects have an important role in modern photonic functionalities, including control
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over the frequency spectrum of laser light, generation of ultrashort pulses, all-optical signal processing and ultrafast switching [7]. In bulk materials, optical nonlinearities
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are inherently weak, thereby restricting their actual applications, while giant optical
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nonlinearities can be achieved via confining electrons in nanostructures with quantum sizes due to quantum confinement effects [4, 8, 9]. Giant optical nonlinearities in nonlinear optical
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nanostructures are of crucial importance for developing
nano-devices. In this context, semi-conductor quantum systems provide a good platform for obtaining obvious nonlinear optical effects [10-18], which shows
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promising applications in photo-electronic devices such as high speed electro-optical modulators [19], far infrared photo detectors [20], semiconductor optical amplifiers
nanofabrication
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[21], four wave mixing and mode locking [22]. The great development of techniques
makes
it
possible
to
prepare
semi- conductor
optics.
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nanostructures with quantum sizes [23, 24], advancing the development of nonlinear
In the past decades, nonlinear optical effects in coupled quantum wells have been intensively studied [9, 16, 18, 25-40]. This is closely associated with the fact that the
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coupling between the two quantum wells can be controlled effectively by tuning the structure parameters so that coupled quantum wells exhibit novel, enhanced, or technologically applicable nonlinear response [9, 25, 27, 40, 41]. Furthermore, to optimize the performances of coupled quantum wells, researchers investigated the influences of various physical factors on optical nonlinearities of coupled quantum wells such as the application of electric and magnetic fields [15, 16, 26, 28], strong laser field [16, 26], excitonic effects [18], temperature and pressure [29, 30]. Among nonlinear optical effects, second-order nonlinear optical effects have received more attention because the magnitudes of second-order nonlinear susceptibility are stronger than those of higher-order ones [7], showing more significance for the practical applications. Second-order nonlinear optical effects can only be observed for
ACCEPTED MANUSCRIPT semi-conductor nanostructures with inversion-symmetry breaking [7]. There are usually two main means to break the inversion-symmetry including tailoring the confinement potential of symmetric nanostructure to a signature of asymmetry by applying external electric field [42, 43] and preparing asymmetric nanostructures using sophisticated material growing technologies such as molecular beam epitaxy and metal-organic chemical vapor deposition [23, 24]. Theoretical studies have been extensively carried out [16, 27-29, 31, 34], since the
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experimental observations of giant second-order nonlinear optical effects in asymmetric coupled quantum wells [25, 31, 38, 40]. To acquire larger second-order
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nonlinear optical effects in asymmetric coupled quantum wells, adjusting the barrier
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and well widths is usually performed [27, 34, 41]. For example, Wang et al revealed that the peak values of nonlinear optical rectification (NOR) coefficients show an
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increase followed by a decrease with the increase of the barrier width, and therefore more obvious NOR can be obtained by an appropriate choice of the barrier width [27]. In addition, appropriate electric or magnetic fields can also induce more remarkable
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second-order nonlinear optical effects in asymmetric coupled quantum wells [28, 29, 36, 38, 39]. For example, Karabulut et al reported that two maximum values can be
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found for the peak values of NOR by manipulating electric or magnetic fields [36]. Although second-order nonlinear optical effects in asymmetric coupled quantum wells are fully analyzed and discussed, second-order nonlinear optical effects in symmetric
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coupled quantum wells are few reported due to the inversion-symmetry. In previous work, third-order nonlinear optical effects in symmetric coupled quantum wells have been reported [30, 32, 33, 39], and novel and desirable results are exhibited through
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controlling the structure parameters. Here, in order to produce second-order nonlinear optical effects in symmetric coupled quantum wells, external electric field is applied to the system due to breaking the inversion-symmetry of the confinement potential. In this paper, numerical method is employed to calculate the electronic quantum states. Then, electric field effects on nonlinear optical rectification in symmetric coupled quantum wells are investigated. Our results indicate that the resonant peak values of NOR coefficients can be increased to a maximum value by adopting an appropriate choice for the electric field, the barrier width or the well width. For the resonant peaks of optical rectification, a blue shift is observed for increasing electric field, while a red shift followed by a blue shift is observed for increasing barrier or well widths. Our investigations pave the way for exploring second order nonlinear
ACCEPTED MANUSCRIPT optical effects in symmetric coupled quantum systems. 2.Theory In this section, we will calculate energy levels and wave functions of an electron confined in symmetric coupled Alx Ga1−xAs/GaAs quantum wells (CQW) under an external electric field. The schematic diagram for electronic confined potential profile is shown in Fig. 1. The growth direction of the quantum wells is along the z-direction. The external electric field F is along the growth direction. The origin for z is taken to
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be the centre of the structure. V0 ,b and L are the potential height, the barrier width and the well width, respectively. The Hamiltonian for the electron confined in the 2 2 2 2 ( 2 2 2 ) V ( z ) qFz 2m * x y z
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H
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structure is given by
with
z (b / 2 L), b / 2 z b / 2, z (b / 2 L),
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V V ( z) 0 0
o t h e r w, ise
(1)
(2)
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where , m* and q are the Planck constant, the conduction-band effective mass and the electron charge. By solving the Schrödinger equation H n,k (r) En,k n,k (r) , the
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wave functions n,k (r) and the energy levels En ,k are, respectively, given by
and
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n,k (r) n ( z)uc (r// )eik r
// //
En , k
2 k //2 En , 2m *
(3)
(4)
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Where r//=(x,y), k//= (k x,k y) and u(r//) is the periodic part of the Bloch function in the conduction band at k // = 0. The wave functions φn (z) and the energy levels En satisfy the following one-dimensional Schrödinger equation: [
2 2 V ( z ) qFz ]n ( z ) Enn ( z ) . 2m * z 2
(5)
To solve Eq.(5), eigenfunctions of the infinite potential well are taken as the base functions. L0 is the width of the infinite potential well. These base functions are formed as[37, 44] m
with
2 mz cos[ m ] L0 L0
(6)
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if m is odd.
2
if m is e v e n .
m
(7)
The wave function φn (z) is expanded in a set of basis functions as follows:
n ( z ) cmm
(8)
m 1
In calculating the wave function φn (z), we ensure that En is independent on the chosen
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L0 , and that φn (z) is localized in the well region.
Using the energy levels and the wave functions together with the compact density
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matrix approach and the iterative procedure [27, 37, 42], nonlinear optical
0( 2)
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rectification coefficient 0( 2) is given by
4q 3 s 2 122 (1 Γ 2 / Γ1 ) ( 2 Γ 22 )( Γ 2 / Γ1 1) , 12 12 0 2 [(12 ) 2 Γ 22 ][(12 ) 2 Γ 22 ]
(9)
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where 12 1 z 2 , 12 2 z 2 1 z 1 , 12 E 2 E1 E21 and E .
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Here, φ1 is the ground state wave function, φ2 is the first excited state wave function, E21 is the energy level interval between the ground state energy level E1 and the first-excited state energy level E2 , ρs is the electron density in the system, 0 is the
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vacuum permittivity. 1 is the diagonal relaxation rate and 2 is the off-diagonal relaxation rate. 0( 2) has a resonant peak value for ω12 = ωwhich is:
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) 0( ,2max
2q 3 s 2 . 0 2 Γ1 Γ 2 12 12
(10)
3.Results and discussions
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In this section, we discuss the effects of electric field on nonlinear optical rectification in symmetric coupled Alx Ga1−xAs/GaAs quantum wells. The materials used for the model is AlxGa1−xAs/GaAs heterostructures with the Al concentration x = 0.3. The parameters adopted in this work are as follows [26,27,34]: m*= 0.067 m0 (m0 is the free electron mass), V0 =228 meV, 1 =1/1ps, 2 =1/0.2 ps, L0 =60 nm. Figure 2 demonstrates the ground state wave function φ1 and the first excited state wave function φ2 of symmetric CQW for different values of external electric field F. In the absence of the electric field, φ1 and φ2 have even and odd parities, respectively. Therefore, φ1 and φ2 in the left and right wells have the same localization. However, applying the electric field to the symmetric CQW leads to the fact that with the increase of the electric field, φ1 is more localized in the right well and φ2 is more
ACCEPTED MANUSCRIPT localized in the left well. The reason for the feature is that increasing the electric field strengthens the left well and weakens the right well. Therefore, the lower energy quantum state (φ1) is more localized in the right (weaker) well while the higher energy quantum state (φ2 ) is more localized in the left (stronger) well [35, 39]. The application of the electric field breaks the parities of φ1 and φ2 so that nonlinear optical rectification in symmetric CQW can be obviously acquired.
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In Fig.3, we plot the NOR coefficients 0( 2) of symmetric CQW for different values of the electric field F. It is clearly seen from Fig.3(a,b) that with the increase of the
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electric field, the resonant peaks of 0( 2) exhibit a blue shift. This feature can be
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explained as follows. From the discussion on Fig. 3, we know that the increase of the electric field strengthens the left well and weakens the right well so that the
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first-excited state energy level E2 is heightened and the ground state energy level E1 is lowered (see Fig.5). Therefore, increasing the electric field induces an enlargement of the energy level interval E21 (see Fig.4), leading to the blue shift of 0( 2) . In addition,
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Fig.3(a,b) shows that the resonant peak values of 0( 2) feature an increase followed by a decrease as the electric field increases. According to Eq.(10), the resonant peak values
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of 0( 2) are determined by 122 12 . In Fig.4, we plot the variation of the geometric factor 122 12 of symmetric CQW as a function of the electric field. Figure 4 demonstrates that
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122 12 increases firstly and then decreases with the increase of the electric field,
thereby inducing a maximum value of 122 12 (see the dashed black lines in Fig.4), the feature for which is dependent on the competition between 12 and 122 . Figure 5
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shows that 12 and 122 increase and decrease, respectively, as the electric field increases. It is apparent that 12 is the main factor influencing 122 12 before 122 12 reaches its maximum value, and then 122 plays an important role in influencing 122 12 . This is linked to the fact that the increase of the electric field makes the ground state (φ1) be more localized over the right (weaker) well and the first-excited state (φ2) be more localized over the left (stronger) well. Therefore, increasing the electric field enlarges δ12 and reduces the overlap 12 between the ground and first-excited state wave functions. We can conclude that an appropriate choice of the external electric field can induce a maximum value for the resonant peak values of 0( 2) .
ACCEPTED MANUSCRIPT In Fig.6, we show the NOR coefficients 0( 2) of symmetric CQW for different barrier width b with L=4 nm and F=20 kV/cm. As shown in Fig.6, the resonant peaks of 0( 2) exhibit a red shift followed by a blue shift as the barrier width increases. This feature can be physically explained as follows. The coupling between the two wells gets weak with the increase of barrier width, which will reduce the energy level interval E21 . However, the increase of the barrier width makes the ground state be
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more localized in the right (weaker) well and the first-excited state be more localized in the left (stronger) well, which will increase E21 . The competition between the
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reduced coupling and the increased localization determines the relationship between b and E21 . A maximum value of E21 can be found at b=5.4 nm in Fig.7(a) (see the
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dashed black line). For b<5.4 nm, the reduced coupling is prominent, and thus, a decrease of E21 is observed in Fig.7(a) as b increases from 1 nm to 5.4 nm, leading to
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the red shift of 0( 2) in Fig.6. However, for b>5.4 nm, the increased localization of the ground and first-excited states becomes prominent, and thus, an increase of E21 is
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observed in Fig.7(a) as b increases from 5.4 nm, leading to the blue shift of 0( 2) in Fig.6. From Fig.6, we also see that the resonant peak values of 0( 2) climb up firstly
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and then decline, which is clearly shown in Fig.7. In Fig.7, we plot the geometrical factor 12 , 122 and 122 12 of symmetric CQW as a function of the barrier width with L=4
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nm and F=20 kV/cm. Fig.7(b) clearly shows an increase of 122 12 followed by a decrease as L increases, leading to a maximum value of 122 12 at b=4.3 nm (see the red black line), the feature for which is dependent on the competition between 12 and 122 .
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Fig.7(c) depicts that 12 increases with the increase of b, while 122 climbs up firstly and then declines. It is apparent that 12 is the main factor determining 122 12 for b<4.3 nm, while 122 becomes the main factor determining 122 12 for b>4.3 nm. We can conclude that an appropriate choice of the barrier width under external electric field can induce a maximum value for the resonant peak values of 0( 2) . Figure 8 displays the NOR coefficients 0( 2) of symmetric CQW for different well width L with b=4 nm and F=20 kV/cm. From the figure, It is clearly seen that the resonant peaks of 0( 2) experience a red shift followed by a blue shift with the increase of the well width. The nonmonotonic well width dependence of the resonant peaks of
ACCEPTED MANUSCRIPT 0( 2) can be attributed to the competition between the reduced quantum confinement effect and the increased localization of the ground and first-excited states. The increase of the well width reduces the quantum confinement effect, which will decrease E21 . However, the increase of the well width makes the ground state (φ1) be more localized in the right (weaker) well and the first-excited state (φ2 ) be more localized in the left (stronger) well, which will increase E21 . A maximum value of E21
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can be found at L=5.4 nm in Fig.9(a) (see the dashed black line). For L<5.4 nm, the reduced quantum confinement effect is prominent, and hence, a decrease of E21 and a
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red shift of 0( 2) are observed in Fig.9(a) and Fig.8, respectively, as L increases from 2
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nm to 5.4 nm, However, for L>5.4 nm, the increased localization of the ground and first-excited states becomes prominent, and hence, a decrease of E21 and a red shift of
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0( 2) are observed in Fig.9(a) and Fig.8, respectively, as L increases from 5.4 nm. In addition, Figure 8 demonstrates an increase of the resonant peak values of 0( 2)
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followed by a decrease with the increase of L, the feature for which is also clearly exhibited in Fig.9(b). Figure 9(b) shows that 122 12 increases for L<4.4 nm and decreases for L>4.4 nm as L increases, leading to a maximum value of 122 12 at L=4.4
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nm (see the dashed red line), which is explained as the competition between 122 and 12 . Fig.9(c) depicts that 12 increases and 122 decreases as L increases. 12 plays an
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important role in 122 12 for L<4.4 nm, while 122 has a significant impact on 122 12 for L>4.4 nm. Therefore, we can conclude that a maximum value for the resonant peak
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values of 0( 2) can be found by an appropriate choice of the well width under external electric field.
4.Conclusion
In this paper, the dependence of nonlinear optical rectification in symmetric coupled AlxGa1−xAs/GaAs quantum wells on the electric field, the barrier and well widths are revealed and elucidated. We have shown that the resonant peaks of optical rectification experience a blue shift for increasing electric field, which comes as a result of stronger electric field making the ground and the first-excited states be more localized in the weaker and stronger quantum wells, respectively. We also have shown that the resonant peaks of optical rectification exhibit a red shift followed by a blue shift for increasing barrier width or well width, which results from the competition
ACCEPTED MANUSCRIPT between the reduced coupling and the increased localization of the ground and first-excited states for increasing barrier width and the competition between the reduced coupling and the increased localization of the ground and first-excited states for increasing well width, respectively. Furthermore, an appropriate choice for the electric field, the barrier or well widths can induce a maximum value for the resonant peak values of optical rectification, the origin for which is highly dependent on the
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competition between 12 an 122 . Our results will stimulate further theoretical and experimental investigations on second-order nonlinear optical effects in symmetric
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coupled Alx Ga1−x As/GaAs quantum wells, and therefore are of more importance for the development of novel nonlinear optoelectronic devices such as photo-detectors,
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electro-optical modulators, and all optical switches.
Acknowledge ments: This work is supported by the National Natural Science
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Foundation of China (under Grant Nos. 61775043, 61475039), Guangdong Provincial Department of Science and Technology (under Grant No. 2017B010128001), the
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Science and Technology Bureau of Zhongshan (under Grant No. 2015A2001).
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ACCEPTED MANUSCRIPT Captions Fig.1. Schematic diagram for electronic confined potential profile.
Fig.2. The ground state wave function φ1 and the first excited state wave function φ2 of symmetric CQW for F=0, 10 kV/cm, 20 kV/cm, 30 kV/cm. both (a) and (b)
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correspond to L=6 nm, b=3 nm, and both (c) and (d) correspond to L=4 nm, b=4 nm.
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Fig.3. The NOR coefficients 0( 2) of symmetric CQW as a function of the photon energy for different values of the electric field F. (a) corresponds to L=6 nm, b=3 nm,
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and (b) corresponds to L=4 nm, b=4 nm.
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Fig.4. The geometrical factor 122 12 and the energy level interval E21 of symmetric CQW as a function of the photon energy for different values of the electric field F. (a)
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corresponds to L=6 nm, b=3 nm, and (b) corresponds to L=4 nm, b=4 nm. The dashed black lines in (a,b) indicates a maximum value of 122 12 .
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Fig.5. The ground state energy level E1 , the first excited state energy level E2 , and the geometrical factors 12 and 122 as a function of the electric field F.
(a)
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corresponds to L=6 nm, b=3 nm, and (b) corresponds to L=4 nm, b=4 nm. Fig.6. The NOR coefficients 0( 2) of symmetric CQW as a function of the photon
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energy for different barrier width b with F=20 kV/cm and L=4 nm. Fig.7. (a) The ground state energy level E1 , the first-excited state energy level E2 and and the energy level interval E21 of symmetric CQW as a function of the barrier width b (b) The geometrical factor 122 12 of symmetric CQW as a function of the barrier width b. (c) The geometrical factors 12 and 122 of symmetric CQW as a function of the barrier width b. In (a-c), F=20 kV/cm and L=4nm are adopted. The dashed black line in (a) indicates a maximum value of E21 , and the dashed red line in (b) indicates a maximum value of 122 12 . Fig.8. The NOR coefficients 0( 2) of symmetric CQW as a function of the photon
ACCEPTED MANUSCRIPT energy for different well width L with F=20 kV/cm and b=4 nm. Fig.9. (a) The ground state energy level E1 , the first-excited state energy level E2 and and the energy level interval E21 of symmetric CQW as a function of the well width L. (b) The geometrical factor 122 12 of symmetric CQW as a function of the well width L. (c) The geometrical factors 12 and 122 of symmetric CQW as a function of the well
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width L. In (a-c), F=20 kV/cm and b=4 nm are adopted. The dashed black line in (a) indicates a maximum value of E21 , and the dashed red line in (b) indicates a maximum
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value of 122 12 .
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Electric field-dependent optical rectification is revealed. Well width-dependent optical rectification is revealed under electric field. Barrier width-dependent optical rectification is revealed under electric field.
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