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S0749-6036(15)30024-0 http://dx.doi.org/10.1016/j.spmi.2015.06.003 YSPMI 3804

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Superlattices and Microstructures

Received Date: Revised Date: Accepted Date:

13 April 2015 4 June 2015 4 June 2015

Please cite this article as: Z-H. Zhang, L. Zou, C. Liu, J-H. Yuan, Electric field effect on the nonlinear optical properties in asymmetrical Gaussian potential quantum wells, Superlattices and Microstructures (2015), doi: http:// dx.doi.org/10.1016/j.spmi.2015.06.003

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Electric field eﬀect on the nonlinear optical properties in asymmetrical Gaussian potential quantum wells Zhi-Hai Zhang1∗, LiLi Zou1 , Chenglin Liu1 , Jian-Hui Yuan2∗∗ 1

School of Physics and Electronics, Yancheng Teachers University,Yancheng, 224051,PR China 2

The department of Physics, Guangxi medical university, Nanning, Guangxi, 530021, China

Abstract In the present work, the optical absorption coeﬃcients (OACs) and the changes in the refractive index (RI) in asymmetrical Gaussian potential quantum wells (QWs) with the applied electric field are studied in detail. We find both energy and wavefunction for low-lying state are wrong in the previous work[1, 2, 3]. Simultaneously, we obtain new and reliable results via the diﬀerential method. Finally, the applied electric field, well width, and well depth have great influence on the optical absorption coeﬃcients and refractive index changes in this system. Keywords: Optical properties; finite diﬀerence method; Quantum wells 1. Introduction In recent years, the rapid development of the crystal growth technique such as the molecular beam epitaxy, liquid phase epitaxy and chemical vapor deposition has become possible to produce a variety of dimensionality semiconductor nanostructures[4, 5, 6]. These structures are interesting because the charge carrier motion can be restricted in one, two and three dimensions, ∗

Corresponding author Corresponding author Email addresses: [email protected]; Tel: 183-6114-8893 (Zhi-Hai Zhang1 ), [email protected]; Tel:150-7882-3937 (Jian-Hui Yuan2 ) ∗∗

Preprint submitted to elsevier

June 4, 2015

and the energy levels of electrons are quantized in the material growth confining direction, which gives the possibility for the eﬀective control of the physical characteristics of these structures. Much attention has been paid to the nonlinear optical properties of low-dimensional semiconductor structures [7, 8, 9, 10, 11] because of the potential application to electronic and optoelectronic devices. Ever increasing interest in these quantum structures is ascribed not only to their potential applications but also the fundamental physics involved. For instance, V. Ustoglu Unal et al in 2013, have studied the applied electric field on the linear and nonlinear changes in the RI of the Modified-P¨oschlTeller QW where they treat the applied electric field as a perturbation[12]. O. Aytekin et al have studied the nonlinear optical properties of a P¨oschlTeller QW under electric and magnetic fields.Also they have give the optimizing nonlinear optical coeﬃcients using adjustable confinement and the electric and magnetic field eﬀects[13]. Using the finite diﬀerence method[14], GH. Safarpour et al have studied the linear and third order nonlinear OACs and RI changes in a GaAs/GaAlAs nanowire superlattice which involved anisotropic quantum dots, under the eﬀect of a high frequency laser radiation. Recently, Guo and Du have reported their results for linear and nonlinear OACs and RI changes in asymmetrical Gaussian potential QW with applied electric field[1]. After this moment, the other optical properties in this system are investigated, such as nonlinear optical rectification and second-harmonic generation[2, 3]. But we find both energy and wavefunction for the low-lying state in this model are chosen wrongly to applied in these works above [15]. Therefore, it is very necessary to investigate the nonlinear optical properties for electron confined in the asymmetrical Gaussian potential quantum wells in the presence of electric field. In the present work, we present a numerical study of the linear and nonlinear OACs and RI changes in an asymmetrical Gaussian potential QWs. In the section 2, the eigenfunctions and eigenenergies of electron states are obtained using finite diﬀerence method, and the analytical expression for the OACs and RI changes are derived by means of the compact-density-matrix approach and an iterative method. In the section 3, the numerical results and discussions are presented for asymmetrical Gaussian potential QWs with applied electric field. A brief summary is given in section 4.

2

2. Theory Let us consider an electron confined in the asymmetrical Gaussian potential QWs in the presence of the z-direction external electric field. Within the eﬀective-mass approximation, the Hamiltonian of this system can be written as ~2 ∂ 2 ∂ ∂2 H = − ∗ ( 2 + 2 + 2 ) + V (z) + qF z, (1) 2m ∂x ∂y ∂z where m∗ is the eﬀective mass of electron. z represents the growth direction of the QWs. F is the strength of the external electric field and q is the absolute value of the electric charge. And the confinement potential V (z) is given by { −V0 exp(−z 2 /2L2 ) z≥0 V (z) = (2) ∞ z < 0, here V0 and L are the height of the Gaussian potential QWs and the range of the confinement potential, respectively. Using the finite diﬀerence method[16], the electron energy levels En and their corresponding wave function ϕn (z) in asymmetrical Gaussian potential quantum wells can be obtained by solving the following Schr¨odinger equation: ~2 ∂ 2 + V (z) + qF z]ϕn (z) = En ϕn (z), (3) 2m∗ ∂z 2 After the energy levels and the corresponding wave functions are obtained, the linear and nonlinear OACs and RI changes for intersubband transitions can be calculated by the compact-density-matrix method and the iterative procedure. The analytical expressions of the linear and the thirdorder nonlinear susceptibilities for a two-level quantum system are given as follows[17, 18, 19]. [−

ε0 χ(1) (ω) =

N |M21 |2 . E21 − ~ω − i~Γ12

N |M21 |2 E21 − ~ω − i~Γ12 4|M21 |2 ×[ (E21 − ~ω)2 + (~ω)2 (M22 − M11 )2 − ]. (E21 − ~ω − i~Γ12 )

(4)

ε0 χ(3) (ω) = −

3

(5)

The susceptibility χ(ω) is related to the change in the refractive index as follow: ∆n(ω) χ(w) = Re 2 , nr 2nr

(6)

where nr is the refractive index. By using Eqs.(4-6), the linear and the third-order nonlinear refractive index changes are obtained by ∆n(1) (ω) 1 E21 − ~ω = 2 |M21 |2 N [ ] nr 2nr ε0 (E21 − ~ω)2 + (~Γ12 )2

(7)

and ∆n(3) (ω) µc NI = − 3 |M21 |2 [ (E21 − ~ω)2 + (~Γ12 )2 ]2 nr 4nr ε0 E21 − ~ω (M22 − M11 )2 ×[4(E21 − ~ω)|M21 |2 − {(E21 − ~ω) (E21 )2 + (~Γ12 )2 ×[E21 (E21 − ~ω) − (~Γ12 )2 ] − (~Γ12 )2 (2E21 − ~ω)}], (8) where N is the carrier density in this system, µ is the permeability of the system, Eij = Ei − Ej is the energy interval of two diﬀerent electronic states, Mij is the matrix elements which is defined by Mij = |⟨φi |ex|φj ⟩|(i, j = 1, 2). I is the incident optical intensity and defined as √ εR 2nr I=2 |E(ω)|2 = |E(ω)|2 , (9) µ µc where c is the speed of light in free space. Therefore, the total refractive index change can be written as ∆n(ω) ∆n(1) (ω) ∆n(3) (ω) = + . nr nr nr

(10)

In addition, the susceptibility χ(ω) is related to the absorption coeﬃcient α(ω) by √ µ |M21 |2 N ~Γ12 (1) α (ω) = ω . (11) εR (E21 − ~ω)2 + (~Γ12 )2

4

√

µ I |M21 |2 N ~Γ12 α (ω, I) = −ω ( ) [4|M21 |2 2 2 2 εR 2ε0 nr c [(E21 − ~ω) + (~Γ12 ) ] 2 |M22 − M11 |2 [3E21 − 4E21 ~ω + ~2 (ω 2 − Γ212 )] − ]. (12) 2 E21 + (~Γ12 )2 (3)

So, the total absorption coeﬃcient α(ω, I) is given by α(ω, I) = α(1) (ω) + α(3) (ω).

(13)

3. Results and discussions In this section, we will discuss the optical absorption coeﬃcient and refractive index changes in asymmetrical Gaussian potential QWs under the external electric fields. The parameter adopted in our calculation are as follow [20, 21, 22]: m∗ = 0.067m0 (m0 is the free-electron mass), N = 5 × 1016 cm−3 , nr = 3.2, T12 = 0.14 ps, Γ12 = 1/T12 , I = 0.6 M W/cm2 . In Fig.1, the variations of the confinement potential profile, ground and first excited state energies and the envelope wave functions for related energy levels for some the well depths, well widths and applied electric field parameters are schematically presented. Fig.1(a),(b) and (c) show that the change in the wavefunctions is clearly visible and energy levels are also shifted. As can be seen from the figure, the ground and first excited energy levels become closer to each other(see Fig.1(b)). Therefore, in some regions, the eﬀect of the variations of L and V0 on the ground state and first excited state are comparable, and thus they eliminate the eﬀects of each other. This feature can be explained by Fig.2, which presents the energy diﬀerence E21 between ground states E1 and first excited states E2 as a function of the well width L and well depth V0 with L = 2 nm and V0 = 100 meV, respectively. It can be seen from the figure that the variation of well width L is dominant and E21 decreases with increasing L when L takes a small value. Meanwhile, it can be clearly seen that the energy diﬀerence E21 increases with the enhancement of the applied electric field, which will lead to a blue shift of resonant peak. In Fig.3, we show the linear α(1) (ω), the third-order nonlinear α(3) (ω), and the total OACs α(ω) as a function of the photon frequency ω, with L = 2 nm, F = 100 kV/cm and three diﬀerent values of V0 . From this figure, it can be clearly seen that the α(1) (ω), α(3) (ω) and α(ω) as a function of ω has an 5

prominent peak, at the same position, which occurs due to the one-photon resonance enhancement. As the well depth increases, the magnitude of the linear and the third-order nonlinear absorption coeﬃcients decreases slightly. Moreover, the large linear change generated by the χ(1) term is opposite in sign of the nonlinear change generated by the χ(3) term. The total absorption coeﬃcient is increased by the linear term, but it is significantly reduced by the third-order nonlinear term. Therefore, the total absorption coeﬃcient decreases as the well depth increases. In addition, the resonant peaks suﬀer an obvious blue-shift with the increase of well depth. As seen in Fig.2(a), the energy diﬀerence E21 between the two diﬀerent electronic states increases when the well depth increases. Therefore, the resonant peaks shift toward higher energies with the increase of the well depth. Meanwhile, the value of the total OACs α(ω) decreases with increasing well depth V0 (see the inset in Fig.3). In Fig.4, the linear α(1) (ω), the third-order nonlinear α(3) (ω), and the total OACs α(ω) are plotted as a function of the photon frequency ω for three diﬀerent well width L with V0 = 100 meV, F = 100 kV/cm. The parameter L characterizes the change of well width. When the well depth V0 is fixed, we noted that the bigger the value of the parameter L is, the wider the well will be. It can be observed that the well width have a great influence on the optical absorption coeﬃcients. First, the magnitude of the total OACs α(ω) decreases rapidly as the well width increases. And the third-order nonlinear α(3) (ω) does not change monotonously with the increasing well width. The other property is the resonant peak positions have an obvious red-shift with the well width increases. These features can be understood easily as follows: when the well width increases, the quantum confinement becomes weaker, so the energy diﬀerence E21 between the first excited state and the ground state of the system decreases, and also the corresponding peak position shifts toward lower energies (see Fig.2(b)). Meanwhile, the value of the total OACs α(ω) decreases with increasing well width L (see the inset in Fig.4). In Fig.5, we present the linear α(1) (ω), the third-order nonlinear α(3) (ω), and the total OACs α(ω) as a function of the photon frequency ω with L = 2 nm, V0 = 100 kV/cm and for three diﬀerent applied electric field values as F = 0, 50, 100 kV/cm. From Fig.5, we can see that the peak values of the linear α(1) (ω) and the third-order nonlinear α(3) (ω) OACs decrease by an increase in the applied electric field F . One should also note that the total OACs α(ω) will be strong bleached at linear center for some applied electric field values, and the collapse at the center of total OACs α(ω) will gradually 6

disappear with increasing applied electric field. To show the relation between the resonant peaks value of the total OACs α(ω) and the applied electric field F more clearly, we have plotted the inset in Fig.5, it is easily observed that the bleaching begins approximately at the applied electric field F < 80 kV/cm in our study. While the eﬀect of the applied electric field is to shift the peak positions to a higher energy. The reason is that the applied electric field leads to the bigger energy diﬀerence E21 (see Fig.2). In Fig.6, we show the linear ∆n(1) /nr , the third-order nonlinear ∆n(3) /nr , and the total RI changes ∆n/nr with L = 2 nm and F = 100 kV/cm as a function photon frequency ω for three diﬀerent values of well depth V0 = 50, 100, 150. Because the sign of the linear and nonlinear RI changes is opposite, the total RI changes will be reduced. It clearly seems that the well depth V0 increases the peak value of the RI changes decreases. To look at this behavior accurately, we plot the inset in Fig.6, in which the total RI changes ∆n/nr decreases monotonously with increasing well depth V0 . Additionally the resonant peaks shift to higher energies (blue shift). It is due to the increment in energy diﬀerence between ground and first states (see Fig.2). In Fig. 7, we show the linear ∆n(1) /nr , the third-order nonlinear ∆n(3) /nr , and the total RI changes ∆n/nr with V0 = 100 meV and F = 100 kV/cm for three diﬀerent values of well width L = 1 nm, L = 1.63 nm and L = 2 nm. Where the inset in Fig.7 we plot the total RI changes ∆n/nr as a function of well width. We can clearly see that the resonant peak of the linear ∆n(1) /nr , the third-order nonlinear ∆n(3) /nr , and the total RI changes ∆n/nr show red shift with increase in well width. In addition, one readily observes that the resonant peak of the RI changes do not change monotonously. The maximum value of total RI changes ∆n/nr is approximate at L = 1.63 nm in our study. In Fig.8, we present the linear ∆n(1) /nr , the third-order nonlinear ∆n(3) /nr , and the total RI changes ∆n/nr with L = 2 nm and V0 = 100 meV for three diﬀerent values of applied electric field F = 0 kV/cm, F = 50 kV/cm and F = 100 kV/cm. It is seen from the figure that the resonant peaks will move to the right of the curve when the applied electric field F increases. The larger the applied electric field is, the smaller the value of the peak is. The resonant peaks will move to higher energy when F increases, which predicts a strong confinement-induced blue shift. All these phenomena agree well with above results discussed in Fig.2.

7

4. Conclusions In this paper, we investigated the intersubband OACs and the RI change in asymmetrical Gaussian potential QWs. We obtained the energy eigenvalues and the energy eigenfunctions by the finite diﬀerent method. The optical properties have been calculated using compact density matrix approach. Numerical calculations show that the OACs and RI changes depend dramatically on the shape of QWs and the applied electric field. It was found that the total OACs decreases with the increase of well width L and well depth V0 , but increases with increase of the applied electric field. Whereas the total RI changes decreases with the increase of the well depth V0 and the applied electric field F , and we observed that the maximum of the total RI changes when the well depth and the applied electric field take a fixed value. Besides, when the applied electric field is increased, the absorption peaks show blue shift. Similar eﬀects are observed in case of the RI changes dispersion areas. Our theoretical results may make a contribution to experimental studies and provide a kind of approximative modeling for the practical application such as optoelectronics devices and optical communication, etc. Acknowledgments: Project supported by the National Science Fundation of China (under Grant No. 11447193, 11447101 and 61475039), the Youth Science Foundation of Guangxi Medical University in China (under Grant No. GXMUYSF201313), the Guangxi Department of Education Research Projects in China (under Grant No. KY2015LX046) and the University National Science Foundation of Jiangsu Province of China (under Grant No. 14KJB140015). References [1] A.X. Guo, and J.F. Du, Superlatt. and Microstruct. 64 (2013) 158-166. [2] J.H. Wu, K.X. Guo, and G.H. Liu, Physica B 446 (2014) 59-62. [3] W.J. Zhai, Physica B 454 (2014) 50-55. [4] M. Sundaram, S.A. Chalmers, P.E. Hopkins, and A.C. Gossard, Science 254 (1991) 1326. [5] M.R. Panish, Science 208 (1980) 916.

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[6] C. Sirtori, F. Capasso, D.L. Sivco, and A.Y. Cho, Phys. Rev. Lett. 68 (1992) 1010. [7] M.J. Karimi, and H. Vafaei, Superlatt. and Microstruct. 78 (2015) 1-11. [8] R. Khordad, Physica B 406 (2011) 3911-3916. [9] R. Khordad, and H. Bahramiyan, Superlatt. and Microstruct. 76 (2014) 163-173. [10] I. Karabulut, M.E. Mora-Ramos, and C.A. Duque, J. Lumin. 131 (2011) 1502. [11] I. Tsang, S.-L. Chuang, and S.M. Lee, Phys. Rev. B 41 (1990) 5942. [12] V. Ustoglu Unal, E. Aksahin, and O. Aytekin, Physica E 47 (2013) 103-108. [13] O. Aytekin, S. Turgut, and M. Tomark, Physica E 44 (2012) 1612-1616. [14] Gh. Safarpour, M.A. Izadi, M. Novzari, and S. Yazdanpanahi, Superlatt. and Microstruct. 75 (2014) 936-947. [15] J.H. Yuan, and Z.H. Zhang, Superlatt. and Microstruct.(2015), doi: http://dx.doi.org/10.1016/j.spmi.2015.04.001. [16] Datta.S. Quantum Transport Atom to TransistorCambridge University Press (2005) pp38-40 ¨ u , I. ˙ Karabulut, and H. Safak, Physica. E 33 (2006) 319. [17] S. Unl¨ [18] C.H. Liu, and B.R. Xu. Phys. Lett. A 33 (2006) 324. [19] Y.B. Yu, S.N. Zhu, and K.X. Guo, Solid State Commun. 139 (2006) 76. [20] K.X. Guo, and C.Y. Chen, Physica B 269 (1999) 139. [21] Z.H. Zhang, K.X. Guo, B. Chen, R.Z. Wang, M.W. Kang, and S. Shao, Superlatt. Microstruct. 47 (2010) 325. [22] C.J. Zhang, and K.X. Guo, Physica E 39 (2007) 103.

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Figure Captions Figure 1 The variation of the confinement potential profile, ground and first excited state energies and the envelope wave functions as a function of the position z for diﬀerent values of (a)the well depth with F = 100 kV/cm and V0 = 100 meV, (b)well width with L = 2 nm and F = 100 kV/cm and (c)applied electric field with L = 2 nm and V0 = 100 meV. Figure 2 The energy diﬀerence E21 between ground states E1 and first excited states E2 are plotted as a function of well depth V0 (a) and the well width L(b) with L = 2 nm and V0 = 100 meV, respectively. Figure 3 The linear α(1) (ω), the third-order nonlinear α(3) (ω) and the total OACs α(ω) as a function of the incident photon frequency for three diﬀerent values V0 = 50, 100, 150 meV with L = 2 nm, and F = 100 KV/cm where the inset we plot the total OACs α(ω) as a function of the well depth V0 . Figure 4 The linear α(1) (ω), the third-order nonlinear α(3) (ω) and the total OACs α(ω) as a function of the incident photon frequency for three diﬀerent values L = 2, 1.5, 1 nm with V0 = 100 meV, and F = 100 KV/cm where the inset we plot the total OACs α(ω) as a function of the well width L. Figure 5 The linear α(1) (ω), the third-order nonlinear α(3) (ω) and the total OACs α(ω) as a function of the incident photon frequency for three diﬀerent values F = 0, 50, 100 kV/cm with V0 = 100 meV, and F = 100 KV/cm where the inset we plot the total OACs α(ω) as a function of the applied electric field F . Figure 6 Variations of the linear ∆n(1) /nr , the third-order nonlinear (3) ∆n /nr , and the total RI changes ∆n/nr versus the photon frequency with L = 2 nm and F = 100 kV/cm for diﬀerent values of V0 where the inset we plot the total RI changes ∆n/nr as a function of the well depth V0 . Figure 7 Variations of the linear ∆n(1) /nr , the third-order nonlinear ∆n(3) /nr , and the total RI changes ∆n/nr versus the photon frequency with V0 = 100 meV and F = 100 kV/cm for diﬀerent values of L where the inset we plot the total RI changes ∆n/nr as a function of the well width L. Figure 8 Variations of the linear ∆n(1) /nr , the third-order nonlinear (3) ∆n /nr , and the total RI changes ∆n/nr versus the photon frequency with L = 2 nm and V0 = 100 meV for diﬀerent values of F where the inset we plot the total RI changes ∆n/nr as a function of the applied electric field F .

10

Fig.1 300

Energy (meV)

250

300 (a)

250

350

(b)

300

200

250

150

200

100

150

50

100

50

0

50

0

-50

200 150 100

-50

0

2

4

6

8

10

12

V =100 meV

-50

V =150 meV

-100

F=100 kV/cm

0

-150

L=2 nm

-100

0

-100

L=1.5 nm

(c)

14

-200 0

0

2

4

6

z (nm)

8

10 12 14 16 18 20

z (nm)

11

-150 0

F=150 kV/cm

2

4

6

8

10 12 14

z (nm)

16 18 20

Fig.2 200

160 F=0 kV/cm

meV)

140 120

(a)

F=50 kV/cm

180

F=0 kV/cm

(b)

F=50 kV/cm

160

F=100 kV/cm F=150 kV/cm

F=100 kV/cm F=150 kV/cm

140

F=200 kV/cm

F=200 kV/cm

120

100

100 80

80

60

60

40 20

40 40

80

120

160 V

200

240

280

0

1

2

3

4

5

6

L (nm)

0

12

7

8

9

10

Fig.3

1.4 V =50 meV

Absorption coefficient

V =100 meV 0

0.9

V =150 meV 0

0.8

10 /m)

(10

1.0

7

7

/m)

0

1.2

0.8

0.7

0.6

0.5

30

60

90

120

V

0.6

0

150

180

210

240

270

300

(meV)

0.4 0.2 0.0 -0.2 (3)

-0.4 -0.6 0.6

0.8

1.0

1.2

Photon Frequency

13

1.4

1.6

14

(10

/s)

1.8

Fig.4

1.8

L=1 nm L=1.5 nm

1.6 1.4 1.2

/m )

1.4 7

L=2 nm

(10

(10

7

/m)

1.6

1.2

0.8 0.6 0.4 0.2

1.0 Absorption coefficient

1.0

0.0 1.0

1.5

2.0

2.5

3.0

L (nm)

0.8 0.6 0.4 0.2 0.0 -0.2 (3)

-0.4 -0.6 0

1

2 Photon Frequency

14

3 14

(10

/s)

4

Fig.5

1.8 F=0 kV/cm

F=50 kV/cm

1.0

F=100 kV/cm 7

/m)

1.2

(10

(10

7

/m)

1.5

0.9

0.8 0.7 0.6 0.5

0.6 Absorption coefficient

0.9

0

50

100

150

200

250

300

F kV/cm

0.3 0.0 -0.3 -0.6 -0.9 -1.2

(3)

-1.5 0.0

0.2

0.4

0.6

0.8

1.0

Photon Frequency

15

1.2 14

(10

/s)

1.4

1.6

1.8

Fig.6

1.0

V =50 meV 0 V =100 meV 0

0.8

Refractive Index Change

0.4

n/n r

0.7

V =150 meV 0

r

n/n

0.6

0.8

0.6

0.5

0.4

0.3 50

100

150

200

250

300

V0 (meV)

0.2

0.0

-0.2

-0.4 (1)

n

-0.6

(3)

n

-0.8

/n

r

/n

r

n/n

r

-1.0 0.0

0.5

1.0 Photon Frequency

16

1.5

2.0 s

2.5

Fig.7

Refractive Index Change

0.8

L=1.63 nm

0.8

0.7

L=2 nm r

0.6

n /n

n/n

r

1.0

0.6 L=1 nm

0.4

0.5 0.4 0.3 0.2 0.1 0.0

1

2

3

4

5

L (nm)

0.2 0.0 -0.2 -0.4 (1)

n

-0.6

/n

r

(3)

n

-0.8

/n

r

n/n

r

-1.0 0.5

1.0

1.5

2.0

Photon Frequency

17

2.5

3.0 s

3.5

Fig.8

F=0 kV/cm

1.4

2 F=50 kV/cm

n/n Refractive Index Change

n/n r

r

1.2 1.0 0.8 0.6 0.4

1

F=100 kV/cm

0.2

0

50

100

150

200

250

300

F (kV/cm)

0

-1

(1)

n

(3)

n

-2

/n

r

/n

r

n/n

r

0.0

0.2

0.4

0.6

0.8

1.0

Photon Frequency

18

1.2 s

1.4

1.6

1.8

Highlights

1、 The OACs and RI in asymmetrical Gaussian QWs with the applied electric field

are studied in detail 2、 Some new and reliable results are obtained using the differential method. 3、The applied electric field, well width, and well depth have great influence on the OACs and RI

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