GaAs quantum wells

GaAs quantum wells

Accepted Manuscript Electric field effects on nonlinear optical rectification in symmetric coupled AlxGa1−xAs/GaAs quantum wells Guanghui Liu, Kangxi...

6MB Sizes 1 Downloads 86 Views

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

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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

RI

d

SC

c

PT

University, Guangzhou 510006, P.R. China

AC C

EP T

ED

MA

NU

*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

PT

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

RI

optoelectronic devices.

AC C

EP T

ED

MA

NU

SC

Keywords:Optical rectification; Electric field; Symmetric coupled quantum wells.

ACCEPTED MANUSCRIPT

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

PT

over the frequency spectrum of laser light, generation of ultrashort pulses, all-optical signal processing and ultrafast switching [7]. In bulk materials, optical nonlinearities

RI

are inherently weak, thereby restricting their actual applications, while giant optical

SC

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

NU

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

MA

promising applications in photo-electronic devices such as high speed electro-optical modulators [19], far infrared photo detectors [20], semiconductor optical amplifiers

nanofabrication

ED

[21], four wave mixing and mode locking [22]. The great development of techniques

makes

it

possible

to

prepare

semi- conductor

optics.

EP T

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

AC C

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

PT

experimental observations of giant second-order nonlinear optical effects in asymmetric coupled quantum wells [25, 31, 38, 40]. To acquire larger second-order

RI

nonlinear optical effects in asymmetric coupled quantum wells, adjusting the barrier

SC

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

NU

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

MA

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

ED

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

EP T

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

AC C

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

PT

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

SC

H 

RI

structure is given by

with

z  (b / 2  L),  b / 2  z  b / 2, z  (b / 2  L),

NU

V V ( z)   0 0

o t h e r w, ise

(1)

(2)

MA

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

ED

wave functions n,k (r) and the energy levels En ,k are, respectively, given by

and

EP T

 n,k (r)  n ( z)uc (r// )eik r

// //

En , k 

 2 k //2  En , 2m *

(3)

(4)

AC C

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 )  Enn ( 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 mz cos[ m ] L0 L0

(6)

ACCEPTED MANUSCRIPT  0

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 )   cmm

(8)

m 1

In calculating the wave function φn (z), we ensure that En is independent on the chosen

PT

L0 , and that φn (z) is localized in the well region.

Using the energy levels and the wave functions together with the compact density

RI

matrix approach and the iterative procedure [27, 37, 42], nonlinear optical

 0( 2) 

SC

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)

NU

where 12  1 z  2 , 12   2 z  2  1 z 1 , 12 E 2 E1  E21 and   E .

MA

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

ED

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 = ωwhich is:

EP T

)  0( ,2max 

2q 3  s 2 .  0  2 Γ1 Γ 2 12 12

(10)

3.Results and discussions

AC C

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.

PT

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

RI

electric field, the resonant peaks of  0( 2) exhibit a blue shift. This feature can be

SC

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

NU

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,

MA

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

ED

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

EP T

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

AC C

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

PT

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

RI

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

SC

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

NU

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

MA

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

ED

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

EP T

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 .

AC C

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

PT

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

RI

red shift of  0( 2) are observed in Fig.9(a) and Fig.8, respectively, as L increases from 2

SC

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

NU

 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)

MA

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

ED

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

EP T

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

AC C

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

PT

competition between 12 an 122 . Our results will stimulate further theoretical and experimental investigations on second-order nonlinear optical effects in symmetric

RI

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,

SC

electro-optical modulators, and all optical switches.

Acknowledge ments: This work is supported by the National Natural Science

NU

Foundation of China (under Grant Nos. 61775043, 61475039), Guangdong Provincial Department of Science and Technology (under Grant No. 2017B010128001), the

AC C

EP T

ED

MA

Science and Technology Bureau of Zhongshan (under Grant No. 2015A2001).

ACCEPTED MANUSCRIPT References

AC C

EP T

ED

MA

NU

SC

RI

PT

[1] P.A. Fran ken, A.E. Hill, C.W. Peters, G. Weinreich, Generation of optical harmonics, Phys. Rev. Lett. 7 (1961) 118-119. [2] H. Aouani, M. Rah mani, M. Navarro-Cia, S.A. Maier, Th ird-harmon ic-upconversion enhancement fro m a single semiconductor nanoparticle coupled to a plasmonic antenna, Nat. Nanotechnol. 9 (2014) 290-294. [3] O. Barelli, E. Grinvald, N. Meir, L. Neeman, O. Dan, Enhanced Third-Harmon ic Generation fro m a Metal/Semiconductor Core/Shell Hybrid Nanostructure, Acs Nano 9 (2015) 8064-8069. [4] H. Qian, Y. Xiao, Z. Liu, Giant Kerr response of ultrathin gold films fro m quantum size effect, Nat. Commun. 7 (2016) 13153. [5] M.A. Foster, A.C. Turner, J.E. Sharping, B.S. Sch midt, M. Lipson, A.L. Gaeta, Broad -band optical parametric gain on a silicon photonic chip, Nature 441 (2006) 960. [6] L.F. Mollenauer, R.H. Stolen, J.P. Go rdon, Experimental observation of pico second pulse narrowing and solitons in optical fibers, Phys. Rev. Lett. 45 (1980) 1095-1098. [7] R.W. Boyd, Nonlinear Optics, Academic Press, New York, 2008. [8] C.R. McDonald, K.S. A min, S. Aalmalki, T. Brabec, Enhancing high harmonic output in solids through quantum confinement, Phys. Rev. Lett. 119 (2017) 183902. [9] C. Sirtori, F. Capasso, D.L. Sivco, A.Y. Cho, Giant, trip ly resonant, third -order nonlinear susceptibility in coupled quantum wells, Phys. Rev. Lett. 68 (1992) 1010-1013. [10] L. Bou zaiene, H. Alamri, L. Sfaxi, H. Maaref, Simu ltaneous effects of hydrostatic pressure, temperature and electric field on optical absorption in InAs/GaAs lens shape quantum dot, J. A lloys. Compd. 655 (2016) 172-177. [11] L. Lu, W. Xie, H. Hassanabadi, Linear and nonlinear optical absorption coefficients and refractive index changes in a two-electron quantum dot, J. Appl. Phys. 109 (2011) 063108. [12] S. Baskoutas, E. Paspalakis, A.F. Terzis, Effects of excitons in nonlinear optical rectification in semiparabolic quantum dots, Phys. Rev. B 74 (2006) 153306. [13] R. Chen, D.L. Lin, B. Mendoza, Enhancement of the third -order nonlinear optical susceptibility in Si quantum wires, Phy. Rev. B 48 (1993) 11879-11882. [14] M.G. Barseghyan, C.A. Duque, E.C. Niculescu, A. Radu, Intense laser field effects on the linear and nonlinear optical p roperties in a semiconductor quantum wire with triangle cross section, Superlatt. Microstruct. 66 (2014) 10-22. [15] E.B. A l, F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sökmen, Effects of applied electric and magnetic fields on the nonlinear optical properties of asymmetric double inverse parabolic quantum well, Opt. Mater. 47 (2015) 1-6. [16] U. Yesilgul, E.B. Al, J.C. Martínez-Oro zco, R.L. Restrepo, M.E. Mora -Ramos, C.A. Duque, F. Ungan, E. Kasapoglu, Linear and nonlinear optical properties in an asymmetric double quantum well under intense laser field: Effects of applied electric and magnetic fields, Opt. Mater. 58 (2016) 107-112. [17] C.A. Duque, E. Kasapoglu, S. Şakiroglu, H. Sari, I. Sökmen, Intense laser effects on nonlinear optical absorption and optical rect ification in single quantum wells under applied electric and magnetic field, Appl. Surf. Sci. 257 (2011) 2313-2319. [18] P. Andreakou, S. Cronenberger, D. Scalbert, A. Nalitov, N.A. Gipp ius, A.V. Kavokin, M. Nawrocki, J.R. Leonard, L.V. Butov, K.L. Camp man, A.C. Gossard, M. Vladimirova, Nonlinear optical spectroscopy of indirect excitons in coupled quantum wells, Phys. Rev. B 91 (2015) 125437. [19] J.E. Roth, O. Fidaner, R.K. Schaevitz, Y.H. Kuo, T.I. Kamins, J.S. Harris, D.A. M iller, Optical modulator on silicon employing germanium quantum wells, Opt. Express 15 (2007) 5851. [20] S.R. Andrews, B.A. M iller, Experimental and theoretical studies of the performance of quantum well infrared photodetectors, J. Appl. Phys. 70 (1991) 993-1003. [21] A. V. Uskov, R.J.M. E. P. O’Reilly, R. P. Webb, D. Cotter, M. Laemmlin, N. N. Ledentsov, D.Bimberg, On u ltrafast optical switching based on quantum dot semicondctor optical amp lifier in nonlinear interferometer, IEEE Photon. Technol. Lett. 16 (2004) 1041. [22] D.S. Chemlia, D.A.B. M iller, P.W. Smith, Nonlinear optical properties of GaAs/GaAlAs mu ltiple quantum well material -phenomena and applications, Opt. Eng. 24 (1985) 556-564. [23] M.B. Panish, Molecular beam epitaxy, Prog. Solid. State. Ch. 208 (1980) 916-922. [24] P.D. Dapkus, Metalorganic chemical vapor deposition, Vacuum 39 (1982) 1002. [25] E. Rosencher, P. Bois, J. Nagle, E. Costard, Observation of nonlinear optical rectificat ion at 10.6 μm in co mpositionally asymmetrical AlGaAs mult iquantum wells, Appl. Phys. Lett. 55 (1989) 1597-1599. [26] B. Chen, K.X. Guo, R.Z. Wang, Z.H. Zhang, Z.L. Liu, Linear and nonlinear intersubband optical absorption in double triangular quantum wells, Solid. State. Commun. 149 (2009) 310-314.

ACCEPTED MANUSCRIPT

AC C

EP T

ED

MA

NU

SC

RI

PT

[27] R.Z. Wang, K.X. Guo, Z.L. Liu , B. Chen, Y.B. Zheng, Nonlinear optical rect ification in asymmetric coupled quantum wells, Phys. Lett. A 373 (2009) 795-798. [28] Í. Karabulut, C.A. Duque, Nonlinear optical rectificat ion and optical absorption in GaAs/Ga1−xAlxAs double quantum wells under applied electric and magnetic fields, Physica E 43 (2011) 1405-1410. [29] M.J. Karimi, A. Keshavarz, Second harmonic generation in asymmetric double semi -parabolic quantum wells: Effects of electric and magnetic fields, hydrostatic pressure and temperature, Physica E 44 (2012) 1900-1904. [30] U. Yesilgul, F. Ungan, E.B. Al, E. Kasapoglu, H. Sari, I. Sökmen, Effects of magnetic field, hydrostatic pressure and temperature on the nonlinear optical properties in symmetric double semi-V-shaped quantum well, Opt. Quant. Electron. 48 (2016) 560. [31] S. Janz, F. Chatenoud, R. Normandin, Quasi-phase-matched second-harmonic generation fro m asymmetric coupled quantum wells, Opt. Lett. 19 (1994) 622-624. [32] U. Yesilgul, Linear and nonlinear intersubband optical absorption coefficients and refract ive index changes in symmetric double semi-V-shaped quantum wells, J. Lumin. 132 (2012) 765-773. [33] A. Keshavarz, M .J. Karimi, Linear and nonlinear intersubband optical absorption in symmetric double semi-parabolic quantum wells, Phys. Lett. A 374 (2010) 2675-2680. [34] J. Khurgin, Second order susceptibility o f asymmetric coupled quantum well structures, Appl. Phys. Lett. 51 (1987) 2100-2102. [35] M.J. Karimi, A. Keshavarz, A. Poostforush, Linear and nonlinear intersubband optical absorption and refractive index changes of asymmetric double semi-parabolic quantum wells, Superlatt. Microstruct. 49 (2011) 441-452. [36] İ. Karabulut, M.E. Mora -Ramos, C.A. Duque, Nonlinear optical rectification and optical absorption in GaAs/Ga 1–xAlxAs asymmetric double quantum wells: Co mbined effects of applied electric and magnetic fields and hydrostatic pressure, J. Lumin. 131 (2011) 1502-1509. [37] U. Yesilgul, F. Ungan, E. Kasapoglu, H. Sari, I. Sökmen, Linear and nonlinear optical propert ies in asymmetric double semi-V-shaped quantum well, Physica B 475 (2015) 110-116. [38] F. Capasso, C. Sirtori, A.Y. Cho, Coupled quantum well semiconductors with giant electric field tunable nonlinear optical properties in the infrared, IEEE J. Quantum. Elect. 30 (1994) 1313-1326. [39] M .J. Karimi, A. Keshavarz, Electric field effects on the linear and nonlinear intersubband optical properties of double semi-parabolic quantum wells, Superlatt. Microstruct. 50 (2011) 572-581. [40] E. Rosencher, P. Bo is, B. Vinter, J. Nagle, D. Kap lan, Giant nonlinear optical rectification at 8– 12 μm in asymmetric coupled quantum wells, Appl. Phys. Lett. 56 (1990) 1822-1824. [41] B. Chen, K.X. Guo, Z.L. Liu, R.Z. Wang, Y.B. Zheng, B. Li, Second-order nonlinear optical susceptibilities in asymmetric coupled quantum wells, J. Phys:. Condensed Matter 20 (2008) 255214. [42] K.X. Guo, S.W. Gu , Nonlinear optical rectification in parabolic quantum wells with an ap plied electric field, Phy. Rev. B 47 (1993) 16322. [43] L. Zhang, H.J. Xie, Electric field effect on the second -order nonlinear optical propert ies of parabolic and semiparabolic quantum wells, Phys. Rev. B 68 (2003) 235315. [44] E. Ozturk, I. Sokmen, Intersubband transitions in an asymmetric double quantum well, Superlatt. Microstruct. 41 (2007) 36-43.

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)

PT

correspond to L=6 nm, b=3 nm, and both (c) and (d) correspond to L=4 nm, b=4 nm.

RI

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,

SC

and (b) corresponds to L=4 nm, b=4 nm.

NU

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)

MA

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 .

ED

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)

EP T

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

AC C

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

PT

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

AC C

EP T

ED

MA

NU

SC

RI

value of 122 12 .

ACCEPTED MANUSCRIPT Highlights

EP T

ED

MA

NU

SC

RI

PT

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.

AC C

  

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9