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Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field
Superlattices and Microstructures xxx (2017) 1e10
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field Keyin Li, Kangxian Guo*, Litao Liang Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 April 2017 Received in revised form 7 June 2017 Accepted 7 June 2017 Available online xxx
The shape effect of triangular quantum dots (TQDs) on the second-harmonic generations (SHG) coefficient and the optical rectification (OR) coefficient with the applied electric field has been theoretically investigated, by using the compact-density-matrix approach and iterative method. Within the effective-mass approximation, the energy levels and the wave functions of the system are obtained. The results show that both the strength of external electric field and the shape of triangular quantum dots have a significant influence on the second-harmonic generations and the optical rectification. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Triangular quantum dot Second-harmonic generations Optical rectification
1. Introduction In the last two decades, with the advances in semiconductor nanoelectronics, considerable attention has been paid to the nonlinear optical properties of quantum dots (QDs) due to the fact that quantum confinement can trigger lots of novel nonlinear optical properties [9e14]. Benefiting from the significant progresses in modern nanofabrication techniques, it became possible to fabricate new low-dimensional semiconductor structures in nanoscopic scale [9e14]. These controlling structures' optical properties are conducive to the development of semiconductor optoelectronics devices. The SHG and OR effect of QDs play the important role in nonlinear optical properties, especially in presence of noise. QDs can be made into different geometrical shapes such as cylindrical shape, spherical shape, and cubical shape [1e3], and there are many experimental and theoretical works focusing on the optical properties of QDs. For instance, Zhang and Guo studied the nonlinear optical rectification in cubical quantum dots with applied electric field and concluded that the OR coefficient is strongly dependent on the length of cubical QDs and the magnitude of electric field [4]. Liu et al. surveyed the polaron effects on the optical rectification and the second harmonic generation in cylindrical quantum dots with magnetic field [5]. Shao et al. surveyed second-harmonic generations in cubical quantum dots with applied electric field [6]. Li et al. investigated the influence of the shape of quantum dots on their third-harmonic generations for both elliptical QDs and triangular QDs and revealed that the shape of elliptical QDs and triangular QDs had different influence on third-harmonic generations [7]. These researches show that the geometrical size and shape of QDs, especially under an applied electric field, make a great difference to the optical properties. Apparently, considering its significance to the optical properties of QDs [15e18], the studies of realistic profile of the confinement potential deserves more attention.
* Corresponding author. E-mail address: [email protected] (K. Guo). http://dx.doi.org/10.1016/j.spmi.2017.06.028 0749-6036/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
2
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
In this paper, we intend to study on the second-harmonic generations coefficient and the optical rectification coefficient in triangular quantum dots with the applied electric field. We especially focus on the shape effect of TQDs under the applied electric field on the SHG and the OR. The outline of this paper is as follows. In Section 2, a model for TQDs is given and the analytical forms of the second-harmonic generations coefficient and the optical rectification coefficient for TQDs is obtained. In Section 3, the numerical results and discussions are presented. In Section4, a brief conclusion is acquired. 2. Theory 2.1. Theoretical model In this work, we consider that the electrons in the system are much more confined in the z-direction than in the x and y directions under an electric field, and they are assumed to be moving on the x-y plane. The confining potential is taken as the triangular bound potential form [7,8],
1 2 V ¼ m u20 r 2 1 þ b cos 3q ; 2 7
(1)
where m is the electron effective mass, u0 represents the confinement frequency associated with the confinement potential in the (x,y) plane, r and q are the polar radius and polar angle in the polar coordinate system, respectively, while b denotes a parameter to transform the shape of the triangular lateral confining potential. As shown in Fig. 1, we plot three-dimensional view and contour lines of the confining potentials in quantum dots modeled by potential V. From the graph, we find that when b ¼ 0, the lateral confining potential is a classical circle confining potential. More importantly, in the case of bs0, the lateral confining potential can become a triangular shaped one, and it can be seen that as b increases, a regular triangular QD turns into a triangle darts shaped QD, which reflects that the asymmetry of the triangular QDs enhances. Similar potential has also been used previously for the investigation of the electronic structures in circular, elliptic, and triangular shaped QDs containing single or a few electrons [15], and the study of the effects of Shannon entropy and electric field on polaron in RbCl triangular quantum dot [26]. Under the effective mass approximation, the Hamiltonian of an electron in the TQDs system under the action of an electric field F which is along the x direction, can be described as:
Fig. 1. The three-dimensional view and contour lines of the confining potentials in quantum dots modeled by potential V.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
H¼
3
p2 1 2 2 2 m 1 þ exF; þ u r b cos 3 q 0 7 2m 2
(2)
the effect of the terms 17m u20 r 2 b cos 3q exF in Eq. (2) remain to a small correction compared to the effect of the other terms. Therefore we take them in perturbation Hamiltonian, and the Hamiltonian in this system can be divided into two parts:
H ¼ H0 þ H 0 ;
(3)
2
p 1 2 2 0 1 2 2 where H0 ¼ 2m þ 2m u0 r , H ¼ 7m u0 r b cos 3 q exF. Through the above process, we can regard H0 as unperturbed Hamiltonian and H0 as the perturbation term. Clearly, H0 is the two-dimensional harmonic oscillator Hamiltonian, and its eigenenergy and eigenfunction is as follows [19],
ð0Þ
En0 m ¼ ðn þ 1ÞZu0 ¼ ð2n0 þ jmj þ 1ÞZu0 ;
(4)
and
eimq
12
2n0 !
r 2 l2 2
jn0 m ¼ pffiffiffiffiffiffi l 0 ðlrÞjmj e 2p ðn þ jmj!Þ
jmj
L n0
2
r2 l
;
(5) 2
where n ¼ 0; 1; 2; 3/, n0 ¼ 0; 1; 2; 3/, m ¼ 0; ±1; ±2/, l ¼ m u20 =Z. n means the principal quantum number, n0 is the radial jmj
quantum number, and m represents the magnetic quantum number, respectively. Ln0 is about the Laguerre polynomial. According to the perturbation theory, the wave function and eigenenergy in this system can be obtained by
X < ji H 0 jj > j > : fi ¼ ji þ ð0Þ ð0Þ j isj Ei Ej
(6)
ð1Þ
By the reason of the first order of the perturbation energy Ei ð0Þ
Ei ¼ Ei
þ
2 X < ji H 0 jj > isj
ð0Þ
ð0Þ
Ej
Ei
¼ 0,
; i; j ¼ 1; 2; 3/:
(7)
We give the energies of ground and first excited state energies as follows:
E1 ¼ Zu0
25pm2 u30 b2
E2 ¼ 2Zu0
4
12544Zl
e2 F 2 2l2 Zu0
7275pm2 u30 b2 401408Zl4
;
3e2 F 2 4l2 Zu0
(8)
:
(9)
2.2. The second-harmonic generations coefficient and optical rectification coefficient Using the compact density-matrix approach and the iterative method, we can get the SHG coefficient and OR coefficient. Assuming that the system is excited by an electromagnetic field of frequency u, such as ~ expðiutÞ þ E ~ expðiutÞ. The time-dependent Schro €dinger equation that the evolution of the density EðtÞ ¼ E0 cosðutÞ ¼ E matrix operator r obeys is as follows [20,21]:
vrij 1
¼ H00 erEðtÞ; r ij Gij r rð0Þ ; ij iZ vt
(10)
where H00 represents the Hamiltonian of the system excluding the effect of EðtÞ, rð0Þ expresses the unperturbed density matrix
operator, e signifies the electronic charge, and Gij is the phenomenological relaxation rate. In this paper, we make Gij ¼ G0 ¼ T10 for isj. Eq. (8) can be solved by means of the iterative method [22,23], Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
4
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
rðtÞ ¼
X
rðnÞ ðtÞ;
(11)
o 1h i 1 nh 0 ðnþ1Þ i ðnþ1Þ H0 ; r er; rðnÞ EðtÞ: Gij rij ij ij iZ iZ
(12)
n
with [14,24] ðnþ1Þ
vrij
vt
¼
The electric polarization of the system can be expressed as [22]:
ð1Þ ~ iut ð2Þ ~ 2 2iut ð2Þ ~ 2 PðtÞ ¼ ε0 cu Ee þ ε 0 c 2u E e ; þ c, c, þ ε0 c0 E ð1Þ
ð2Þ
(13)
ð2Þ
where cu , c2u and c0 are the linear susceptibility, second-harmonic generation and optical rectification susceptibility, respectively. ε0 is the permittivity of vacuum. The nth order electric polarization is given by Refs. [9,13]
P ðnÞ ðtÞ ¼
1
U
Tr rðnÞ er ;
(14)
where U represents the volume of the system and Tr means the trace symbol of the matrix rðnÞ er. In our work, the SHG coefficient and the OR coefficient are given as [24,25,28,29]:
c2ð2Þ u ¼
e3 sU
M12 M23 M31 ; ε0 Z2 ðu u21 þ iG21 Þð2u u31 þ iG31 Þ
(15)
and
c0ð2Þ
u221 1 þ GG01 þ u2 þ G20 GG01 1 4e3 sU 2 ih i: ¼ M12 d12 h ε0 Z2 ðu21 uÞ2 þ G20 ðu21 þ uÞ2 þ G20
(16)
In the above equation, sU denotes the electron density, u is the incident photon frequency. Mij ¼ < fi r cos q fj > is the offdiagonal matrix element and d12 ¼ < f2 jr cos qjf2 > < f1 jr cos qjf1 > . E21 ¼ Zu21 ¼ E2 E1 and E31 ¼ Zu31 ¼ E3 E1 denote the energy difference. 3. Results and discussions In this section, the SHG and OR are numerically calculated for the typical GaAs/AlAs semiconductor. The parameters used in our calculations are as follows [25]: m ¼ 0:067m0 (where m0 is the mass of free electron), sU ¼ 5:0 1022 m3 , Zu0 ¼ 3:0 meV, ε0 ¼ 8:85 1012 Fm1 , G0 ¼ 1=0:14 ps, and G1 ¼ 1=0:1 ps. 3.1. The influences of TQDs with an electric field F on the second-harmonic generations coefficient ð2Þ
In Fig. 2, we plot the SHG coefficient c2u as a function of the incident photon frequency u, with b ¼ 1 and five different values of applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/cm. From the figure, we can clearly find two features. One is that the ð2Þ
resonant peak of c2u is not a monotonic function of the electric field F, but decreases firstly and then increases with the ð2Þ increasing electric field F. In order to show the relationship between the resonant peak of c2u and the electric field F, the transition moments (M12 , M23 , and M31 ) and the their product M12 M23 M31 are drawn in Fig. 3a and Fig. 3b. Fig. 3b shows that with the increment of F, M12 M23 M31 , at the very start, decreases to minimum value when F is about 1.922 KV/cm which means ð2Þ
that the c2u effect is very week now, and then enhances, which shows good agreement with the Fig. 2. The second is that the ð2Þ
resonant peak of c2u exhibits a little red shift. The reason for this trait is due to the fact the energy intervals decline with the increasing F. The variation degree of energy intervals with F can be seen from Fig. 4. Fig. 4 reveals that the energy intervals decrease with F slowly, which indicates that the red shift is not really obvious. ð2Þ
The c2u as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three different values of b, b ¼ 1, b ¼ 2,
b ¼ 3, are portrayed in Fig. 5. In this diagram, the resonant peak of cð2Þ 2u shifts to the lower incident photon frequency region. This is because of the mixing of many eigenstates with various angular momenta when b increases, which results in energy intervals decrease with the increasing asymmetry of TQDs. As seen from Fig. 6, the energy intervals decrease monotonously ð2Þ
with increasing b. Moreover, we can also find from Fig. 5 that the resonant peak of c2u declines with the augment of b (i.e., Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
5
ð2Þ
Fig. 2. The SHG coefficient c2u as a function of the incident photon frequency u, with b ¼ 1 and five different values of the applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/cm.
a
b
Fig. 3. a. The matrix elements as function of electric field F with b ¼ 1. b. The matrix elements' product M12 M23 M31 as function of electric field F with b ¼ 1.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
Fig. 4. The energy intervals as function of the applied electric field F with b ¼ 1.
with the increasing asymmetry of TQDs). These two characteristics correspond to the conclusions made by Li et al. [7]. The reason is that with increasing b, a regular triangular disc QD evolves into a triangle darts QD and the confining potential energy gradually decreases. For different TQDs shape in Fig. 1, the confining potential energy on the three corner position is lower than in the middle region, which means that the electrons are more constrained in the middle region than the corner location. Under the circumstances, we plot Fig. 7a and Fig. 7b. Fig. 7a and b show that the transition moments (M12 , M23 , and M31 ) decrease, and their product M12 M23 M31 decreases as well, respectively, which is consistent with Fig. 5. Hence, if a larger SHG coefficient is desired, a more regular TQD should be adopted. 3.2. The influences of TQDs with an electric field F on the optical rectification coefficient ð2Þ
Fig. 8 shows the OR coefficient c0 as a function of the incident photon frequency u, with b ¼ 1 and five different values of ð2Þ
applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/cm. It can be clearly seen from this graph that the c0 has the similar characteristic with the
cð2Þ 2u
in the same conditions. That is, the resonant peak of
which coincides with the Fig. 4. But the peak of
cð2Þ 0
cð2Þ 0
displays a slight red shift when F increases,
increases with the enhancement of F, and it is worth mentioning that ð2Þ
when the electric field F ¼ 0 KV/cm, the OR coefficients c0 will disappear. This trait is decided by the dipolar matrix elements
ð2Þ
Fig. 5. The c2u as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three different values of b, b ¼ 1, b ¼ 2, b ¼ 3.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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Fig. 6. The energy intervals as function of b with F ¼ 9 KV/cm.
a
b
Fig. 7. a. The matrix elements as function of b with F ¼ 9 KV/cm. b. M12 M23 M31 as function of b with F ¼ 9 KV/cm.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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8
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
ð2Þ
Fig. 8. The OR coefficient c0 as a function of the incident photon frequency u, with b ¼ 1 and five different values of applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/ cm.
2 d . In Fig. 9, we draw the dipolar matrix elements M 2 d M12 12 12 12 versus the applied electric field F with b ¼ 1. From Fig. 9, it can 2 d be observed that the dipolar matrix elements M12 12 is zero in the condition of F ¼ 0 KV/cm, and subsequently increases with the augment of the electric field F. Therefore, an applied electric field can excite the OR effect for the TQDs. ð2Þ
In Fig. 10, we present the OR coefficient c0 as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three ð2Þ
different values of b, b ¼ 1, b ¼ 2, b ¼ 3. From Fig. 10, one can see that the resonant peak of c0 will move to lower incident photon frequency region with increasing asymmetry parameter b of TQDs. The physical origin of this characteristic is, under ð2Þ
the same condition, same to the explanation for that of the SHG coefficient c2u . What's more, we find the OR peak intensity increases when the asymmetry parameter b of TQDs enhances, which is obviously different from that of the SHG coefficient 2 cð2Þ 2u . The origin is from the dipolar matrix elements M12 d12 . According to the specialty of the confining potential explained in
the SHG effect and the description in Fig. 3a, one can see that although the probability of the electrons in the ground state and the first excited state (M11 and M22 ) decreases when the confining potential energy gradually reduces, the state of the electrons keep more in the ground state than the first excited state, so d12 increases. And in Fig. 11, we also can find that the 2 d dipolar matrix elements M12 12 increases with the augment of b. The influence of the asymmetry parameter b of TQDs to the
2 d Fig. 9. The dipolar matrix elements M12 12 versus the applied electric field F with b ¼ 1.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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9
ð2Þ
Fig. 10. The OR coefficient c0 as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three different values of b, b ¼ 1, b ¼ 2, b ¼ 3.
2 d Fig. 11. The dipolar matrix elements M12 12 as function of b with F ¼ 9 KV/cm.
ð2Þ
resonant peak of c0 is similar to the conclusion summarized by Hassan Hassanabadi et al. [27], which researched the nonlinear optical rectification and the second-harmonic generation in semi-parabolic and semi-inverse squared quantum wells. Hence, through all of the above findings, we can obtain the larger SHG coefficient and OR coefficient when the asymmetry parameter b of TQDs and the electric field F is adjusted properly. But it is also worth pointing out that if the external electric field is too large, the structure of material may be broken down.
4. Conclusion In this paper, we study the SHG and OR coefficients in TQDs with the applied electric field. Our results show that the peak positions of the SHG and the OR will present a red shift whether the applied electric field or the asymmetry of TQDs increases. Moreover, the SHG coefficient is not a monotonic function of the applied electric field F for TQDs, but the OR coefficient increases with the augment of F. More importantly, the resonant peak of the SHG in the TQDs under an electric field declines with the increasing asymmetry of TQDs, but those for the OR in TQDs is opposite. Therefore, the magnitude of external electric field and the shape of TQDs have a great influence on the SHG and OR coefficients, and the study of the shape effect of QDs with the external factors on the nonlinear optical properties plays an important role in semiconductor physics because they can greatly modulate and optimize the performance of low-dimensional quantum devices. Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
Acknowledgments The work is supported by the National Natural Science Foundation of China (under Grant Nos. 61178003, 61475039), Guangdong Provincial Department of Science and Technology (under Grant No.2013 B090500012), the Science and Technology Bureau of Zhongshan (under Grant No. 2015A2001). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
V.A. Harutyunyan, E.M. Kazaryan, Phys. E 36 (2007) 114. R.C. Ashoori, H.L. Stormer, J.S. Weiner, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. B 189 (1993) 117. Q. Ye, R. Tsu, E.H. Nicollian, Phys. Rev. B 44 (1991) 1806. Zhihai Zhang, Kangxian Guo, Binchen, Ruizhen Wang, Minwu Kang, Phys. B 404 (2009) 2332e2335. Guanghui Liu, Kangxian Guo, Qingjie Wu, Jing-He Wu, Superlatt. Microstruct. 53 (2013) 173e183. Shuai Shao, Kang-Xian Guo, Zhi-Hai Zhang, Ning Li, Chao Peng, Phys. B 406 (2011) 393e396. Keyin Li, Kangxian Guo, Litao Liang, Superlatt. Microstruct. 102 (2017) 300e306. T. Ezaki, Y. Sugimoto, N. Mori, C. Hamaguchi, Semicond. Sci. Technol. 13 (1998) A1. K.X. Guo, S.W. Gu, Phys. Rev. B Condens. Matter 47 (1993) 16322e16325. Z.W. Zuo, H.J. Xie, J. Phys. Condens. Matter 22 (2010) 025403e025409. R. Safari, S. Shojaei, Optik 125 (2014) 6926e6929. N. Li, K.X. Guo, S. Shao, Opt. Commun. 285 (2012) 2734e2738. G.H. Wang, Optik 125 (2014) 2374e2377. L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315e235320. T. Ezaki, N. Mori, C. Hamaguchi, Phys. Rev. B 56 (1997) 6428. € rg Magnúdo tir, Vidar Gudmundsson, Phys. Rev. B 60 (1999) 16591. Ingibjo A.J. Peter, K. Navaneethakrishnan, Phys. E 40 (2008) 2747. S. Rajashabala, K. Navaneethakrishnan, Braz. J. Phys. 37 (2007) 1134. B. Li, K.X. Guo, Ch.J. Zhang, Y.B. Zheng, Phys. Lett. A 367 (2007) 493. € €nül, O. Ozer, €nül, F. Üzgün, Mod. Phys. Lett. A 17 (37) (2002) 2453. B. Go B. Go G.H. Liu, K.X. Guo, C. Wang, Phys. B 407 (2012) 2334e2339. W.F. Xie, Phys. Lett. A 372 (2008) 5498e5500. B. Chen, K.X. Guo, R.Z. Wang, Z.H. Zhang, Z.L. Liu, Solid State Commun. 149 (2009) 310e314. G.H. Wang, Phys. Rev. B 72 (2005) 155329e155333. Shuai Shao, Kang-Xian Guo, Zhi-Hai Zhang, Ning Li, Chao Peng, Phys. B 406 (2011) 393e396. M. Tiotsop, A. J Fotue, S. C Kenfack, N. Issofa, H. Fotsin, L. C Fai, Chin. Phys. B 25 (4) (2016) 048401. Hassan Hassanabadi, Guanghui Liu, Liangliang Lu, Solid State Commun. 152 (2012) 1761e1766. Liangliang Lu, Wenfang Xie, Hassan Hassanabadi, Qinghu Zhong, Superlatt. Microstruct. 50 (2011) 501e510. Liangliang Lu, Wenfang Xie, Hassan Hassanabadi, Phys. B 406 (2011) 4129e4134.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field Keyin Li, Kangxian Guo*, Litao Liang Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 April 2017 Received in revised form 7 June 2017 Accepted 7 June 2017 Available online xxx
The shape effect of triangular quantum dots (TQDs) on the second-harmonic generations (SHG) coefficient and the optical rectification (OR) coefficient with the applied electric field has been theoretically investigated, by using the compact-density-matrix approach and iterative method. Within the effective-mass approximation, the energy levels and the wave functions of the system are obtained. The results show that both the strength of external electric field and the shape of triangular quantum dots have a significant influence on the second-harmonic generations and the optical rectification. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Triangular quantum dot Second-harmonic generations Optical rectification
1. Introduction In the last two decades, with the advances in semiconductor nanoelectronics, considerable attention has been paid to the nonlinear optical properties of quantum dots (QDs) due to the fact that quantum confinement can trigger lots of novel nonlinear optical properties [9e14]. Benefiting from the significant progresses in modern nanofabrication techniques, it became possible to fabricate new low-dimensional semiconductor structures in nanoscopic scale [9e14]. These controlling structures' optical properties are conducive to the development of semiconductor optoelectronics devices. The SHG and OR effect of QDs play the important role in nonlinear optical properties, especially in presence of noise. QDs can be made into different geometrical shapes such as cylindrical shape, spherical shape, and cubical shape [1e3], and there are many experimental and theoretical works focusing on the optical properties of QDs. For instance, Zhang and Guo studied the nonlinear optical rectification in cubical quantum dots with applied electric field and concluded that the OR coefficient is strongly dependent on the length of cubical QDs and the magnitude of electric field [4]. Liu et al. surveyed the polaron effects on the optical rectification and the second harmonic generation in cylindrical quantum dots with magnetic field [5]. Shao et al. surveyed second-harmonic generations in cubical quantum dots with applied electric field [6]. Li et al. investigated the influence of the shape of quantum dots on their third-harmonic generations for both elliptical QDs and triangular QDs and revealed that the shape of elliptical QDs and triangular QDs had different influence on third-harmonic generations [7]. These researches show that the geometrical size and shape of QDs, especially under an applied electric field, make a great difference to the optical properties. Apparently, considering its significance to the optical properties of QDs [15e18], the studies of realistic profile of the confinement potential deserves more attention.
* Corresponding author. E-mail address: [email protected] (K. Guo). http://dx.doi.org/10.1016/j.spmi.2017.06.028 0749-6036/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
2
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
In this paper, we intend to study on the second-harmonic generations coefficient and the optical rectification coefficient in triangular quantum dots with the applied electric field. We especially focus on the shape effect of TQDs under the applied electric field on the SHG and the OR. The outline of this paper is as follows. In Section 2, a model for TQDs is given and the analytical forms of the second-harmonic generations coefficient and the optical rectification coefficient for TQDs is obtained. In Section 3, the numerical results and discussions are presented. In Section4, a brief conclusion is acquired. 2. Theory 2.1. Theoretical model In this work, we consider that the electrons in the system are much more confined in the z-direction than in the x and y directions under an electric field, and they are assumed to be moving on the x-y plane. The confining potential is taken as the triangular bound potential form [7,8],
1 2 V ¼ m u20 r 2 1 þ b cos 3q ; 2 7
(1)
where m is the electron effective mass, u0 represents the confinement frequency associated with the confinement potential in the (x,y) plane, r and q are the polar radius and polar angle in the polar coordinate system, respectively, while b denotes a parameter to transform the shape of the triangular lateral confining potential. As shown in Fig. 1, we plot three-dimensional view and contour lines of the confining potentials in quantum dots modeled by potential V. From the graph, we find that when b ¼ 0, the lateral confining potential is a classical circle confining potential. More importantly, in the case of bs0, the lateral confining potential can become a triangular shaped one, and it can be seen that as b increases, a regular triangular QD turns into a triangle darts shaped QD, which reflects that the asymmetry of the triangular QDs enhances. Similar potential has also been used previously for the investigation of the electronic structures in circular, elliptic, and triangular shaped QDs containing single or a few electrons [15], and the study of the effects of Shannon entropy and electric field on polaron in RbCl triangular quantum dot [26]. Under the effective mass approximation, the Hamiltonian of an electron in the TQDs system under the action of an electric field F which is along the x direction, can be described as:
Fig. 1. The three-dimensional view and contour lines of the confining potentials in quantum dots modeled by potential V.
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K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
H¼
3
p2 1 2 2 2 m 1 þ exF; þ u r b cos 3 q 0 7 2m 2
(2)
the effect of the terms 17m u20 r 2 b cos 3q exF in Eq. (2) remain to a small correction compared to the effect of the other terms. Therefore we take them in perturbation Hamiltonian, and the Hamiltonian in this system can be divided into two parts:
H ¼ H0 þ H 0 ;
(3)
2
p 1 2 2 0 1 2 2 where H0 ¼ 2m þ 2m u0 r , H ¼ 7m u0 r b cos 3 q exF. Through the above process, we can regard H0 as unperturbed Hamiltonian and H0 as the perturbation term. Clearly, H0 is the two-dimensional harmonic oscillator Hamiltonian, and its eigenenergy and eigenfunction is as follows [19],
ð0Þ
En0 m ¼ ðn þ 1ÞZu0 ¼ ð2n0 þ jmj þ 1ÞZu0 ;
(4)
and
eimq
12
2n0 !
r 2 l2 2
jn0 m ¼ pffiffiffiffiffiffi l 0 ðlrÞjmj e 2p ðn þ jmj!Þ
jmj
L n0
2
r2 l
;
(5) 2
where n ¼ 0; 1; 2; 3/, n0 ¼ 0; 1; 2; 3/, m ¼ 0; ±1; ±2/, l ¼ m u20 =Z. n means the principal quantum number, n0 is the radial jmj
quantum number, and m represents the magnetic quantum number, respectively. Ln0 is about the Laguerre polynomial. According to the perturbation theory, the wave function and eigenenergy in this system can be obtained by
X < ji H 0 jj > j > : fi ¼ ji þ ð0Þ ð0Þ j isj Ei Ej
(6)
ð1Þ
By the reason of the first order of the perturbation energy Ei ð0Þ
Ei ¼ Ei
þ
2 X < ji H 0 jj > isj
ð0Þ
ð0Þ
Ej
Ei
¼ 0,
; i; j ¼ 1; 2; 3/:
(7)
We give the energies of ground and first excited state energies as follows:
E1 ¼ Zu0
25pm2 u30 b2
E2 ¼ 2Zu0
4
12544Zl
e2 F 2 2l2 Zu0
7275pm2 u30 b2 401408Zl4
;
3e2 F 2 4l2 Zu0
(8)
:
(9)
2.2. The second-harmonic generations coefficient and optical rectification coefficient Using the compact density-matrix approach and the iterative method, we can get the SHG coefficient and OR coefficient. Assuming that the system is excited by an electromagnetic field of frequency u, such as ~ expðiutÞ þ E ~ expðiutÞ. The time-dependent Schro €dinger equation that the evolution of the density EðtÞ ¼ E0 cosðutÞ ¼ E matrix operator r obeys is as follows [20,21]:
vrij 1
¼ H00 erEðtÞ; r ij Gij r rð0Þ ; ij iZ vt
(10)
where H00 represents the Hamiltonian of the system excluding the effect of EðtÞ, rð0Þ expresses the unperturbed density matrix
operator, e signifies the electronic charge, and Gij is the phenomenological relaxation rate. In this paper, we make Gij ¼ G0 ¼ T10 for isj. Eq. (8) can be solved by means of the iterative method [22,23], Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
rðtÞ ¼
X
rðnÞ ðtÞ;
(11)
o 1h i 1 nh 0 ðnþ1Þ i ðnþ1Þ H0 ; r er; rðnÞ EðtÞ: Gij rij ij ij iZ iZ
(12)
n
with [14,24] ðnþ1Þ
vrij
vt
¼
The electric polarization of the system can be expressed as [22]:
ð1Þ ~ iut ð2Þ ~ 2 2iut ð2Þ ~ 2 PðtÞ ¼ ε0 cu Ee þ ε 0 c 2u E e ; þ c, c, þ ε0 c0 E ð1Þ
ð2Þ
(13)
ð2Þ
where cu , c2u and c0 are the linear susceptibility, second-harmonic generation and optical rectification susceptibility, respectively. ε0 is the permittivity of vacuum. The nth order electric polarization is given by Refs. [9,13]
P ðnÞ ðtÞ ¼
1
U
Tr rðnÞ er ;
(14)
where U represents the volume of the system and Tr means the trace symbol of the matrix rðnÞ er. In our work, the SHG coefficient and the OR coefficient are given as [24,25,28,29]:
c2ð2Þ u ¼
e3 sU
M12 M23 M31 ; ε0 Z2 ðu u21 þ iG21 Þð2u u31 þ iG31 Þ
(15)
and
c0ð2Þ
u221 1 þ GG01 þ u2 þ G20 GG01 1 4e3 sU 2 ih i: ¼ M12 d12 h ε0 Z2 ðu21 uÞ2 þ G20 ðu21 þ uÞ2 þ G20
(16)
In the above equation, sU denotes the electron density, u is the incident photon frequency. Mij ¼ < fi r cos q fj > is the offdiagonal matrix element and d12 ¼ < f2 jr cos qjf2 > < f1 jr cos qjf1 > . E21 ¼ Zu21 ¼ E2 E1 and E31 ¼ Zu31 ¼ E3 E1 denote the energy difference. 3. Results and discussions In this section, the SHG and OR are numerically calculated for the typical GaAs/AlAs semiconductor. The parameters used in our calculations are as follows [25]: m ¼ 0:067m0 (where m0 is the mass of free electron), sU ¼ 5:0 1022 m3 , Zu0 ¼ 3:0 meV, ε0 ¼ 8:85 1012 Fm1 , G0 ¼ 1=0:14 ps, and G1 ¼ 1=0:1 ps. 3.1. The influences of TQDs with an electric field F on the second-harmonic generations coefficient ð2Þ
In Fig. 2, we plot the SHG coefficient c2u as a function of the incident photon frequency u, with b ¼ 1 and five different values of applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/cm. From the figure, we can clearly find two features. One is that the ð2Þ
resonant peak of c2u is not a monotonic function of the electric field F, but decreases firstly and then increases with the ð2Þ increasing electric field F. In order to show the relationship between the resonant peak of c2u and the electric field F, the transition moments (M12 , M23 , and M31 ) and the their product M12 M23 M31 are drawn in Fig. 3a and Fig. 3b. Fig. 3b shows that with the increment of F, M12 M23 M31 , at the very start, decreases to minimum value when F is about 1.922 KV/cm which means ð2Þ
that the c2u effect is very week now, and then enhances, which shows good agreement with the Fig. 2. The second is that the ð2Þ
resonant peak of c2u exhibits a little red shift. The reason for this trait is due to the fact the energy intervals decline with the increasing F. The variation degree of energy intervals with F can be seen from Fig. 4. Fig. 4 reveals that the energy intervals decrease with F slowly, which indicates that the red shift is not really obvious. ð2Þ
The c2u as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three different values of b, b ¼ 1, b ¼ 2,
b ¼ 3, are portrayed in Fig. 5. In this diagram, the resonant peak of cð2Þ 2u shifts to the lower incident photon frequency region. This is because of the mixing of many eigenstates with various angular momenta when b increases, which results in energy intervals decrease with the increasing asymmetry of TQDs. As seen from Fig. 6, the energy intervals decrease monotonously ð2Þ
with increasing b. Moreover, we can also find from Fig. 5 that the resonant peak of c2u declines with the augment of b (i.e., Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
5
ð2Þ
Fig. 2. The SHG coefficient c2u as a function of the incident photon frequency u, with b ¼ 1 and five different values of the applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/cm.
a
b
Fig. 3. a. The matrix elements as function of electric field F with b ¼ 1. b. The matrix elements' product M12 M23 M31 as function of electric field F with b ¼ 1.
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K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
Fig. 4. The energy intervals as function of the applied electric field F with b ¼ 1.
with the increasing asymmetry of TQDs). These two characteristics correspond to the conclusions made by Li et al. [7]. The reason is that with increasing b, a regular triangular disc QD evolves into a triangle darts QD and the confining potential energy gradually decreases. For different TQDs shape in Fig. 1, the confining potential energy on the three corner position is lower than in the middle region, which means that the electrons are more constrained in the middle region than the corner location. Under the circumstances, we plot Fig. 7a and Fig. 7b. Fig. 7a and b show that the transition moments (M12 , M23 , and M31 ) decrease, and their product M12 M23 M31 decreases as well, respectively, which is consistent with Fig. 5. Hence, if a larger SHG coefficient is desired, a more regular TQD should be adopted. 3.2. The influences of TQDs with an electric field F on the optical rectification coefficient ð2Þ
Fig. 8 shows the OR coefficient c0 as a function of the incident photon frequency u, with b ¼ 1 and five different values of ð2Þ
applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/cm. It can be clearly seen from this graph that the c0 has the similar characteristic with the
cð2Þ 2u
in the same conditions. That is, the resonant peak of
which coincides with the Fig. 4. But the peak of
cð2Þ 0
cð2Þ 0
displays a slight red shift when F increases,
increases with the enhancement of F, and it is worth mentioning that ð2Þ
when the electric field F ¼ 0 KV/cm, the OR coefficients c0 will disappear. This trait is decided by the dipolar matrix elements
ð2Þ
Fig. 5. The c2u as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three different values of b, b ¼ 1, b ¼ 2, b ¼ 3.
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K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
Fig. 6. The energy intervals as function of b with F ¼ 9 KV/cm.
a
b
Fig. 7. a. The matrix elements as function of b with F ¼ 9 KV/cm. b. M12 M23 M31 as function of b with F ¼ 9 KV/cm.
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8
K. Li et al. / Superlattices and Microstructures xxx (2017) 1e10
ð2Þ
Fig. 8. The OR coefficient c0 as a function of the incident photon frequency u, with b ¼ 1 and five different values of applied electric field F ¼ 0; 3; 6; 9; 12; 15 KV/ cm.
2 d . In Fig. 9, we draw the dipolar matrix elements M 2 d M12 12 12 12 versus the applied electric field F with b ¼ 1. From Fig. 9, it can 2 d be observed that the dipolar matrix elements M12 12 is zero in the condition of F ¼ 0 KV/cm, and subsequently increases with the augment of the electric field F. Therefore, an applied electric field can excite the OR effect for the TQDs. ð2Þ
In Fig. 10, we present the OR coefficient c0 as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three ð2Þ
different values of b, b ¼ 1, b ¼ 2, b ¼ 3. From Fig. 10, one can see that the resonant peak of c0 will move to lower incident photon frequency region with increasing asymmetry parameter b of TQDs. The physical origin of this characteristic is, under ð2Þ
the same condition, same to the explanation for that of the SHG coefficient c2u . What's more, we find the OR peak intensity increases when the asymmetry parameter b of TQDs enhances, which is obviously different from that of the SHG coefficient 2 cð2Þ 2u . The origin is from the dipolar matrix elements M12 d12 . According to the specialty of the confining potential explained in
the SHG effect and the description in Fig. 3a, one can see that although the probability of the electrons in the ground state and the first excited state (M11 and M22 ) decreases when the confining potential energy gradually reduces, the state of the electrons keep more in the ground state than the first excited state, so d12 increases. And in Fig. 11, we also can find that the 2 d dipolar matrix elements M12 12 increases with the augment of b. The influence of the asymmetry parameter b of TQDs to the
2 d Fig. 9. The dipolar matrix elements M12 12 versus the applied electric field F with b ¼ 1.
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9
ð2Þ
Fig. 10. The OR coefficient c0 as a function of the incident photon frequency u, with F ¼ 9 KV/cm and three different values of b, b ¼ 1, b ¼ 2, b ¼ 3.
2 d Fig. 11. The dipolar matrix elements M12 12 as function of b with F ¼ 9 KV/cm.
ð2Þ
resonant peak of c0 is similar to the conclusion summarized by Hassan Hassanabadi et al. [27], which researched the nonlinear optical rectification and the second-harmonic generation in semi-parabolic and semi-inverse squared quantum wells. Hence, through all of the above findings, we can obtain the larger SHG coefficient and OR coefficient when the asymmetry parameter b of TQDs and the electric field F is adjusted properly. But it is also worth pointing out that if the external electric field is too large, the structure of material may be broken down.
4. Conclusion In this paper, we study the SHG and OR coefficients in TQDs with the applied electric field. Our results show that the peak positions of the SHG and the OR will present a red shift whether the applied electric field or the asymmetry of TQDs increases. Moreover, the SHG coefficient is not a monotonic function of the applied electric field F for TQDs, but the OR coefficient increases with the augment of F. More importantly, the resonant peak of the SHG in the TQDs under an electric field declines with the increasing asymmetry of TQDs, but those for the OR in TQDs is opposite. Therefore, the magnitude of external electric field and the shape of TQDs have a great influence on the SHG and OR coefficients, and the study of the shape effect of QDs with the external factors on the nonlinear optical properties plays an important role in semiconductor physics because they can greatly modulate and optimize the performance of low-dimensional quantum devices. Please cite this article in press as: K. Li et al., Shape effect on the second order nonlinear optical properties in triangular quantum dots with applied electric field, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.06.028
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Acknowledgments The work is supported by the National Natural Science Foundation of China (under Grant Nos. 61178003, 61475039), Guangdong Provincial Department of Science and Technology (under Grant No.2013 B090500012), the Science and Technology Bureau of Zhongshan (under Grant No. 2015A2001). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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