- Email: [email protected]
Studies on the second-harmonic generations in cubical quantum dots with applied electric field
Physica B 406 (2011) 393–396
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Studies on the second-harmonic generations in cubical quantum dots with applied electric field Shuai Shao, Kang-Xian Guo , Zhi-Hai Zhang, Ning Li, Chao Peng Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China
a r t i c l e in f o
abstract
Article history: Received 9 August 2010 Received in revised form 28 October 2010 Accepted 29 October 2010
The second-harmonic generation (SHG) coefficient for cubical quantum dots (CQDs) with the applied electric field is theoretically investigated. Using the compact density-matrix approach and the iterative method, we get the analytical expression of the SHG coefficient. And the numerical calculations for the typical GaAs/AlAs CQDs are presented. The results show that the SHG coefficient can reach the magnitude of 10 5 m/V, about two orders higher than that in spherical quantum dot system. More importantly, the SHG coefficient is not a monotonic function of the length L of CQDs as well as the applied field F. If we select suitable values of F and L, we will get a higher value of the SHG coefficient. In addition, the relaxation rate also affects the SHG coefficient obviously. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quantum dots Second-harmonic generation
1. Introduction With the rapid development of semiconductor nanoelectronics in the past two decades, it became possible to produce quantum dots (QDs). These structures are very useful because of the fact that charge carrier motions are restricted in all three directions, which gives the possibility for physical characteristics effective control of those structures. The controlling quantum dots physical properties are attractive not only from the fundamental science point of view, but also because of its potential application in the development of semiconductor optoelectronics devices [1–4]. Modern methods for semiconductor nanostructures growth make it possible to obtain QDs of different geometrical shapes and sizes, such as spherical QDs, cubical QDs and cylindrical QDs. In the previous studies, considerable attentions had been paid to the spherical QDs, because they are easier to investigate taking into account their symmetry, which allows to obtain analytical solutions for the energy spectrum, coefficient of absorption, charge carriers mobility, etc. [5–8]. Recently, different authors discussed the electronic and optical properties in the CQDs [9–15]. For example, Feng [10] studied size-dependent emission properties and intersubband (ISB) transitions in cubic InN (c-InN) quantum dots (QDs). In 2008, Yang [12] calculated quantum states of a hydrogenic donor impurity in a cubic quantum dot by the infinite difference method. Dane et al. [13] discussed the effect of spatial electric field on the sub-band energy in a cubic GaAs/AlAs quantum dot. The result showed that the electron sub-band energy E decreased with the spatial electric field strength F increasing. What is more, the value of the electron sub-band energy E also
Corresponding author. Tel.: +86 13 342886687; fax: + 86 20 39366871.
E-mail address: [email protected] (K.-X. Guo). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.10.078
related to the orientation of the spatial electric field. Zhang et al. [14,15] discussed the nonlinear optical rectification and third harmonic generation in cubical quantum dot with applied electric field. Bryant et al. [16] focused on the correlation effects and quantum confinement of excitons in cubic quantum dot. The purpose of this paper is to discuss the second-harmonic generation in cubical quantum dot with applied electric field. In Section 2, the eigenfunctions and eigenenergies of electron states are obtained using the effective mass approximation and the analytical expression for SHG coefficient is derived by the compact density matrix approach and an iterative method. In Section 3, the numerical results and discussions are presented for GaAs/AlAs CQDs. And in Section 4, we lay out conclusion of this paper.
2. Model and theory Under the effective mass approximation, the Hamiltonian of an electron in the CQD system at the presence of an electric field ~ F can be described as [14] 2
^ ¼ ‘ r2 þ e~ H F ~ r þVðx,y,zÞ, 2m
ð1Þ
where Vðx,y,zÞ ¼
8 <0 :
1
L jxj,jyj, and jzj o , 2 elsewhere,
ð2Þ
and ~ F is the electric field vector, mn and e are the electron effective mass and charge, respectively. ~ r is the position vector of the electron, L is the length of the CQDs, and V(x,y,z) is the confining potential. In spherical coordinates, the external spatial electric field can be written ~ F ¼ Fðsinycosje^1 þ sinysinje^2 þ cosye^3 Þ. We take the effective
394
S. Shao et al. / Physica B 406 (2011) 393–396 2
Rydberg constant R ¼ m e4 =2‘ e2 as the unit of the energy and the 2 effective Born radius a ¼ ‘ e=m e2 as the unit of length. And e is the static dielectric constant of the material. Since we choose an infinite potential, the electron will be limited to movement within the cubic quantum dot. Therefore, the Hamiltonian can be rewritten as 2
H ¼ r þ Zðxsinycosj þysinysinj þ zcosyÞ,
ð3Þ
where Z ¼ ea F=R is the dimensional measure of the electric field. The ¨ numerical method of solving the Schrodinger equation is well given in Refs. [17,18]. Using the same method, we can give the corresponding solution
Ci ðei Þ ¼ C1 Aiðei Þ þ C2 Biðei Þ,
ð4Þ
2=3 , i ¼ ðAi xi Ei Þ=Ai
A1 ¼ Zsinycosj, A2 ¼ Zsinysinj, A3 ¼ where e Zcosy, Aiðei Þ and Biðei Þ are Airy functions. C1 and C2 are the normalized coefficients of the wave functions. In this paper, we set y ¼ 03 , j ¼ 03 . All of these normalized coefficients and eigenenergies can be numerically solved by the standard boundary condition of the electronic bound state. And then, we can get the formula of the SHG coefficient in CQDs through the compact-density-matrix method and an iterative procedure [20,21]. The system is excited by an electronmagnetic ~ iot þ Ee ~ iot . Let us denote r as the one-electron field ~ EðtÞ ¼ Ee density matrix for this regime. Then the evolution of the density ¨ matrix is given by the time-dependent Schrodinger equation @rij @t
¼
1 ½H0 qz~ F ðtÞ, rij Gij ðrrð0Þ Þij , i‘
ð5Þ
where H0 is the Hamiltonian for this system without the electromagnetic field ~ EðtÞ, rð0Þ is the unperturbed density matrix, and Gij is the relaxation rate. Here we select Gij ¼ G0 ¼ 1=T0 when ia j for simplicity. Eq. (5) is calculated by the following iterative method: X rðtÞ ¼ rðnÞ ðtÞ, ð6Þ n
with þ 1Þ @rðn ij
1 1 þ 1Þ EðtÞ: ð7Þ f½H0 , rðn þ 1Þ ij i‘ Gij rðn g ½qz, rðnÞ ij ~ ij i‘ i‘ The electric polarization of the quantum dot due to ~ EðtÞ can be ¼
@t
expressed as [15] 2
2
~ iot þ e0 wð2Þ E~ e2iot Þ þ c:c: þ e0 wð2Þ E~ , PðtÞ ¼ ðe0 wð1Þ o Ee 2o 0
ð8Þ
ð2Þ ð2Þ where wð1Þ o , w2o and w0 are the linear susceptibility, second-harmonic generation and optical rectification susceptibility, respectively. e0 is the vacuum dielectric constant. The electronic polarization of the nth order is given as
P ðnÞ ðtÞ ¼
1 TrðrðnÞ ezÞ, V
3. Results and discussions In this section, numerical calculations are carried out for GaAs/AlAs CQDs with the external electric field. The parameters adopted in our calculations are as follows [14]: mn ¼0.067m0, ˚ e0 ¼ 8:85 1012 F m1 , T0 ¼0.2 ps, and Rn ¼ 5.83 meV, an ¼98 A, r ¼ 5 1024 m3 . In Fig. 1, we investigate the SHG coefficient wð2Þ 2o as a function of the incident photon energy ‘ o for six different lengths L of the CQDs, L¼14, 15, 16, 17, 18, 20 nm, at F¼200 kV/cm. From Fig. 1, it can be obviously seen that the SHG coefficient is not a monotonous function of the incident photon energy ‘ o. As clearly shown in Fig. 1, the value of SHG coefficient can reach the magnitude of 2.5 10 5 m/V, which is two orders higher than that in the spherical quantum dots shown in Fig. 5 in the article [19]. This illustrates that quantum confinement of the electron in CQDs is much stronger than that in spherical quantum dots. The reason is that, the higher the asymmetry of the system is, the stronger the secondary polarization will be. As we all know, the symmetry of spherical quantum dot is more perfect than in cubic quantum dot. On the other hand, the applied electric field F also leads to the asymmetry of the cubic quantum dot system. Thus we can obtain the value of the SHG coefficient is much stronger than that in spherical quantum dot. More importantly, as shown in Fig. 1, we can obtain a sharper peak at L 16 nm. That is to say, doubleresonance can be realized at this condition, i.e., E10 E20 =2. When L is apart from both sides of the direction, the value of the SHG coefficient will decrease obviously, which can be seen in Fig. 1, and it will appear two resonant peaks, the major one and the minor one. The major one appears at the position ‘ o ¼ E20 =2, and the minor one occurs at the vicinity of ‘ o ¼ E10 . Furthermore, with the enlargement of length L, both the major and minor resonant peaks have an obvious redshift phenomenon. This interesting feature can be attributed to the dependence of the energy intervals E10 and E20 on length L. In order to explain this clearly, we plot Fig. 2 about the energy intervals E10 and E20/2 as the function of length L from 10 to 20 nm. From Fig. 2 we can see that with increasing of length L, the energy intervals E10 and E20 both decrease evidently, and they will meet each other at the position of length L¼16.5 nm, with E10 E20 =2 ¼ 108 meV. In other words, the double-resonant condition can be met at this location. Therefore, we can find out the double-resonant peak in Fig. 1 as it shows to us. In Fig. 3, we investigate that the external field F affects on the SHG coefficient as a function of the incident photon energy ‘ o.
ð9Þ
where V is the volume of interaction and Tr denotes the trace or summation over the diagonal elements of the matrix rðnÞ ez. In our paper, the SHG coefficient per unit volume is given as
wð2Þ 2o ¼
e3 r
M01 M12 M20
e0 ‘ 2 ðoo10 iG0 Þð2oo20 iG0 Þ
,
ð10Þ
where r is the electron density in the system and e is the electronic charge. oij ¼ ðEi Ej Þ=‘ is the transition frequency, Mij ¼ j/ci j zjcj Sj ði,j ¼ 0,1,2Þ is the off-diagonal matrix element, and M01M12M20 is matrix elements’ product. From Eq. (10), we can get a peak value of the SHG coefficient when the double-resonant condition can be met, i.e., ‘ o E10 E20 =2, and it can be obtained by
wð2Þ 2o,max ¼
e3 r M01 M12 M20
e0 ‘ 2
G20
:
ð11Þ
Fig. 1. SHG coefficient wð2Þ 2o as a function of incident photon energy ‘ o for six different lengths L, L¼ 14, 15, 16, 17, 18, 20 nm, and F ¼200 kV/cm.
S. Shao et al. / Physica B 406 (2011) 393–396
395
Fig. 2. Interval energy E10 and E20/2 as a function of length L.
Fig. 4. Interval energy E10 and E20/2 as a function of applied electric field F.
Fig. 3. SHG coefficient w2o ð2Þ as a function of incident photon energy ‘ o for four different values of applied electric field F ¼100, 150, 200, 250, 300 kV/cm with L¼ 16 nm.
Fig. 5. SHG coefficient wð2Þ 2o as a function of incident photon energy ‘ o for four different values of relaxation time T0 ¼0.08, 0.12, 0.16, 0.2 ps for length L¼16 nm and F¼ 200 kV/cm.
Here we select five different values of applied electric field F, F ¼100, 150, 200, 250, 300 kV/cm with length L¼16 nm. It can be easily shown in Fig. 3 that the SHG coefficient is not a monotonic function of the electric field F. When F is around 200 kV/cm, the SHG coefficient reaches maximum, and it will decline while increasing or decreasing F. And it can be apparently seen that, not only the minor resonant peak but also the major resonant peak shift to the right side with enlargement of F, that is to say, they both happen to suffer a prominent linear Stark shift (blueshift). In order to explain this clearly, we plot Fig. 4, about the energy intervals E10 and E20/2 as a function of electric field F from 100 to 300 kV/cm. In Fig. 4, we find that both the interval energies E10 and E20/2 increase in a crow line. But the interval energy E10 increases faster than the interval energy E20/2 and they will come across around at the position of F ¼210 kV/cm with the interval energy E10 E20 =2 ¼ 112 meV, where they meet the condition of double-resonance. Finally, in Fig. 5, we discuss the SHG coefficient wð2Þ 2o as a function of the photon energy ‘ o with L¼16 nm, and F¼ 200 kV/cm for different relaxation times: T0 ¼0.08, 0.12, 0.16, 0.2 ps, respectively. From Fig. 5, we can see that the larger the relaxation time T is, the higher the peak value of the SHG coefficient is. So, we can get the higher value of the SHG coefficient through increasing the relaxation time. As we all know,
the relaxation rate relates to several external factors, such as character of material, temperature of environment and so on. And we can correctly select these factors and get higher value of SHG coefficient.
4. Conclusion In conclusion, the second-harmonic generation coefficient has been researched in detail and it can reach the magnitude of 10 5 m/V. Meanwhile, we also find that the SHG coefficient is not a monotonic function of the applied field F, and it relates to the length L of CQDs as well. When we set the value of applied electric field F¼200 kV/cm and length L¼16 nm, with relaxation time T¼0.2 ps, we can obtain the higher value of the SHG coefficient. These theoretical results may have great influence on experimental studies for optical devices.
Acknowledgments Project supported by the National Natural Science Foundation of China (under Grant no. 60878002), the Science and Technology
396
S. Shao et al. / Physica B 406 (2011) 393–396
Committee of Guangdong Province (under Grant nos. 2009B011100007, 2010B010800031, 2010B090400130 and 8251009101000002), and the Science and Technology Bureau of Guangzhou (under Grant no. 2009J1-C421-1). References
[8] [9] [10] [11] [12] [13] [14] [15]
[1] [2] [3] [4]
A.D. Yoffe, Adv. Phys. 50 (2001) 1. D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. D. Heiss, M. Kroutvar, J.J. Finley, G. Abstreiter, Solid State Commun. 135 (2005) 591. V.A. Harutyunyan, E.M. Kazryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 36 (2007) 114. [5] A.D. Andreev, A.A. Lipovsky, Semiconductors 33 (1999) 1450. [6] M.S. Atoyan, H.A. Sarkisyan, E.M. Kazaryan, Physica E 22 (2004) 860. [7] W.F. Xie, Superlatt. Microstruct. 48 (2010) 239.
[16] [17] [18] [19] [20] [21]
W.F. Xie, Physica B 405 (2010) 3436. C. Dane, H. Akbas, N. Talip, K. Kasapoglu, Physica E 39 (2007) 95. Sh.W. Feng, Thin Solid Films 516 (2008) 7695. O. Husberga, A. Khartchenkoa, Physica E 13 (2002) 1090. C.S. Yang, Microelectron. J. 39 (2008) 1469. C. Dane, H. Akbas, N. Talipb, K. Kasapoglua, Physica E 39 (2007) 95. Z.H. Zhang, K.X. Guo, B. Chen, R.Z. Wang, M.W. Kang, Superlatt. Microstruct. 46 (2009) 672. Z.H. Zhang, K.X. Guo, B. Chen, R.Z. Wang, M.W. Kang, Physica B 404 (2009) 2332. G.W. Bryant, Phys. Rev. B 37 (1988) 8763. S. Gangopadbyay, B.R. Nag, Nanotechnology 8 (1997) 14. J. Shertzer, L.R. Ram-Mohan, Phys. Rev. B 41 (1990) 9994. B. Li, K.X. Guo, Ch.J. Zhang, Y.B. Zheng, Phys. Lett. A 367 (2007) 493. E. Rosencher, P. Bois, Phys. Rev. B 44 (1991) 11315. B. Chen, K.X. Guo, Z.L. Liu, R.Z. Wang, Y.B. Zheng, B. Li, J. Phys. Condens. Matter 20 (2008) 255214.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Studies on the second-harmonic generations in cubical quantum dots with applied electric field Shuai Shao, Kang-Xian Guo , Zhi-Hai Zhang, Ning Li, Chao Peng Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China
a r t i c l e in f o
abstract
Article history: Received 9 August 2010 Received in revised form 28 October 2010 Accepted 29 October 2010
The second-harmonic generation (SHG) coefficient for cubical quantum dots (CQDs) with the applied electric field is theoretically investigated. Using the compact density-matrix approach and the iterative method, we get the analytical expression of the SHG coefficient. And the numerical calculations for the typical GaAs/AlAs CQDs are presented. The results show that the SHG coefficient can reach the magnitude of 10 5 m/V, about two orders higher than that in spherical quantum dot system. More importantly, the SHG coefficient is not a monotonic function of the length L of CQDs as well as the applied field F. If we select suitable values of F and L, we will get a higher value of the SHG coefficient. In addition, the relaxation rate also affects the SHG coefficient obviously. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quantum dots Second-harmonic generation
1. Introduction With the rapid development of semiconductor nanoelectronics in the past two decades, it became possible to produce quantum dots (QDs). These structures are very useful because of the fact that charge carrier motions are restricted in all three directions, which gives the possibility for physical characteristics effective control of those structures. The controlling quantum dots physical properties are attractive not only from the fundamental science point of view, but also because of its potential application in the development of semiconductor optoelectronics devices [1–4]. Modern methods for semiconductor nanostructures growth make it possible to obtain QDs of different geometrical shapes and sizes, such as spherical QDs, cubical QDs and cylindrical QDs. In the previous studies, considerable attentions had been paid to the spherical QDs, because they are easier to investigate taking into account their symmetry, which allows to obtain analytical solutions for the energy spectrum, coefficient of absorption, charge carriers mobility, etc. [5–8]. Recently, different authors discussed the electronic and optical properties in the CQDs [9–15]. For example, Feng [10] studied size-dependent emission properties and intersubband (ISB) transitions in cubic InN (c-InN) quantum dots (QDs). In 2008, Yang [12] calculated quantum states of a hydrogenic donor impurity in a cubic quantum dot by the infinite difference method. Dane et al. [13] discussed the effect of spatial electric field on the sub-band energy in a cubic GaAs/AlAs quantum dot. The result showed that the electron sub-band energy E decreased with the spatial electric field strength F increasing. What is more, the value of the electron sub-band energy E also
Corresponding author. Tel.: +86 13 342886687; fax: + 86 20 39366871.
E-mail address: [email protected] (K.-X. Guo). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.10.078
related to the orientation of the spatial electric field. Zhang et al. [14,15] discussed the nonlinear optical rectification and third harmonic generation in cubical quantum dot with applied electric field. Bryant et al. [16] focused on the correlation effects and quantum confinement of excitons in cubic quantum dot. The purpose of this paper is to discuss the second-harmonic generation in cubical quantum dot with applied electric field. In Section 2, the eigenfunctions and eigenenergies of electron states are obtained using the effective mass approximation and the analytical expression for SHG coefficient is derived by the compact density matrix approach and an iterative method. In Section 3, the numerical results and discussions are presented for GaAs/AlAs CQDs. And in Section 4, we lay out conclusion of this paper.
2. Model and theory Under the effective mass approximation, the Hamiltonian of an electron in the CQD system at the presence of an electric field ~ F can be described as [14] 2
^ ¼ ‘ r2 þ e~ H F ~ r þVðx,y,zÞ, 2m
ð1Þ
where Vðx,y,zÞ ¼
8 <0 :
1
L jxj,jyj, and jzj o , 2 elsewhere,
ð2Þ
and ~ F is the electric field vector, mn and e are the electron effective mass and charge, respectively. ~ r is the position vector of the electron, L is the length of the CQDs, and V(x,y,z) is the confining potential. In spherical coordinates, the external spatial electric field can be written ~ F ¼ Fðsinycosje^1 þ sinysinje^2 þ cosye^3 Þ. We take the effective
394
S. Shao et al. / Physica B 406 (2011) 393–396 2
Rydberg constant R ¼ m e4 =2‘ e2 as the unit of the energy and the 2 effective Born radius a ¼ ‘ e=m e2 as the unit of length. And e is the static dielectric constant of the material. Since we choose an infinite potential, the electron will be limited to movement within the cubic quantum dot. Therefore, the Hamiltonian can be rewritten as 2
H ¼ r þ Zðxsinycosj þysinysinj þ zcosyÞ,
ð3Þ
where Z ¼ ea F=R is the dimensional measure of the electric field. The ¨ numerical method of solving the Schrodinger equation is well given in Refs. [17,18]. Using the same method, we can give the corresponding solution
Ci ðei Þ ¼ C1 Aiðei Þ þ C2 Biðei Þ,
ð4Þ
2=3 , i ¼ ðAi xi Ei Þ=Ai
A1 ¼ Zsinycosj, A2 ¼ Zsinysinj, A3 ¼ where e Zcosy, Aiðei Þ and Biðei Þ are Airy functions. C1 and C2 are the normalized coefficients of the wave functions. In this paper, we set y ¼ 03 , j ¼ 03 . All of these normalized coefficients and eigenenergies can be numerically solved by the standard boundary condition of the electronic bound state. And then, we can get the formula of the SHG coefficient in CQDs through the compact-density-matrix method and an iterative procedure [20,21]. The system is excited by an electronmagnetic ~ iot þ Ee ~ iot . Let us denote r as the one-electron field ~ EðtÞ ¼ Ee density matrix for this regime. Then the evolution of the density ¨ matrix is given by the time-dependent Schrodinger equation @rij @t
¼
1 ½H0 qz~ F ðtÞ, rij Gij ðrrð0Þ Þij , i‘
ð5Þ
where H0 is the Hamiltonian for this system without the electromagnetic field ~ EðtÞ, rð0Þ is the unperturbed density matrix, and Gij is the relaxation rate. Here we select Gij ¼ G0 ¼ 1=T0 when ia j for simplicity. Eq. (5) is calculated by the following iterative method: X rðtÞ ¼ rðnÞ ðtÞ, ð6Þ n
with þ 1Þ @rðn ij
1 1 þ 1Þ EðtÞ: ð7Þ f½H0 , rðn þ 1Þ ij i‘ Gij rðn g ½qz, rðnÞ ij ~ ij i‘ i‘ The electric polarization of the quantum dot due to ~ EðtÞ can be ¼
@t
expressed as [15] 2
2
~ iot þ e0 wð2Þ E~ e2iot Þ þ c:c: þ e0 wð2Þ E~ , PðtÞ ¼ ðe0 wð1Þ o Ee 2o 0
ð8Þ
ð2Þ ð2Þ where wð1Þ o , w2o and w0 are the linear susceptibility, second-harmonic generation and optical rectification susceptibility, respectively. e0 is the vacuum dielectric constant. The electronic polarization of the nth order is given as
P ðnÞ ðtÞ ¼
1 TrðrðnÞ ezÞ, V
3. Results and discussions In this section, numerical calculations are carried out for GaAs/AlAs CQDs with the external electric field. The parameters adopted in our calculations are as follows [14]: mn ¼0.067m0, ˚ e0 ¼ 8:85 1012 F m1 , T0 ¼0.2 ps, and Rn ¼ 5.83 meV, an ¼98 A, r ¼ 5 1024 m3 . In Fig. 1, we investigate the SHG coefficient wð2Þ 2o as a function of the incident photon energy ‘ o for six different lengths L of the CQDs, L¼14, 15, 16, 17, 18, 20 nm, at F¼200 kV/cm. From Fig. 1, it can be obviously seen that the SHG coefficient is not a monotonous function of the incident photon energy ‘ o. As clearly shown in Fig. 1, the value of SHG coefficient can reach the magnitude of 2.5 10 5 m/V, which is two orders higher than that in the spherical quantum dots shown in Fig. 5 in the article [19]. This illustrates that quantum confinement of the electron in CQDs is much stronger than that in spherical quantum dots. The reason is that, the higher the asymmetry of the system is, the stronger the secondary polarization will be. As we all know, the symmetry of spherical quantum dot is more perfect than in cubic quantum dot. On the other hand, the applied electric field F also leads to the asymmetry of the cubic quantum dot system. Thus we can obtain the value of the SHG coefficient is much stronger than that in spherical quantum dot. More importantly, as shown in Fig. 1, we can obtain a sharper peak at L 16 nm. That is to say, doubleresonance can be realized at this condition, i.e., E10 E20 =2. When L is apart from both sides of the direction, the value of the SHG coefficient will decrease obviously, which can be seen in Fig. 1, and it will appear two resonant peaks, the major one and the minor one. The major one appears at the position ‘ o ¼ E20 =2, and the minor one occurs at the vicinity of ‘ o ¼ E10 . Furthermore, with the enlargement of length L, both the major and minor resonant peaks have an obvious redshift phenomenon. This interesting feature can be attributed to the dependence of the energy intervals E10 and E20 on length L. In order to explain this clearly, we plot Fig. 2 about the energy intervals E10 and E20/2 as the function of length L from 10 to 20 nm. From Fig. 2 we can see that with increasing of length L, the energy intervals E10 and E20 both decrease evidently, and they will meet each other at the position of length L¼16.5 nm, with E10 E20 =2 ¼ 108 meV. In other words, the double-resonant condition can be met at this location. Therefore, we can find out the double-resonant peak in Fig. 1 as it shows to us. In Fig. 3, we investigate that the external field F affects on the SHG coefficient as a function of the incident photon energy ‘ o.
ð9Þ
where V is the volume of interaction and Tr denotes the trace or summation over the diagonal elements of the matrix rðnÞ ez. In our paper, the SHG coefficient per unit volume is given as
wð2Þ 2o ¼
e3 r
M01 M12 M20
e0 ‘ 2 ðoo10 iG0 Þð2oo20 iG0 Þ
,
ð10Þ
where r is the electron density in the system and e is the electronic charge. oij ¼ ðEi Ej Þ=‘ is the transition frequency, Mij ¼ j/ci j zjcj Sj ði,j ¼ 0,1,2Þ is the off-diagonal matrix element, and M01M12M20 is matrix elements’ product. From Eq. (10), we can get a peak value of the SHG coefficient when the double-resonant condition can be met, i.e., ‘ o E10 E20 =2, and it can be obtained by
wð2Þ 2o,max ¼
e3 r M01 M12 M20
e0 ‘ 2
G20
:
ð11Þ
Fig. 1. SHG coefficient wð2Þ 2o as a function of incident photon energy ‘ o for six different lengths L, L¼ 14, 15, 16, 17, 18, 20 nm, and F ¼200 kV/cm.
S. Shao et al. / Physica B 406 (2011) 393–396
395
Fig. 2. Interval energy E10 and E20/2 as a function of length L.
Fig. 4. Interval energy E10 and E20/2 as a function of applied electric field F.
Fig. 3. SHG coefficient w2o ð2Þ as a function of incident photon energy ‘ o for four different values of applied electric field F ¼100, 150, 200, 250, 300 kV/cm with L¼ 16 nm.
Fig. 5. SHG coefficient wð2Þ 2o as a function of incident photon energy ‘ o for four different values of relaxation time T0 ¼0.08, 0.12, 0.16, 0.2 ps for length L¼16 nm and F¼ 200 kV/cm.
Here we select five different values of applied electric field F, F ¼100, 150, 200, 250, 300 kV/cm with length L¼16 nm. It can be easily shown in Fig. 3 that the SHG coefficient is not a monotonic function of the electric field F. When F is around 200 kV/cm, the SHG coefficient reaches maximum, and it will decline while increasing or decreasing F. And it can be apparently seen that, not only the minor resonant peak but also the major resonant peak shift to the right side with enlargement of F, that is to say, they both happen to suffer a prominent linear Stark shift (blueshift). In order to explain this clearly, we plot Fig. 4, about the energy intervals E10 and E20/2 as a function of electric field F from 100 to 300 kV/cm. In Fig. 4, we find that both the interval energies E10 and E20/2 increase in a crow line. But the interval energy E10 increases faster than the interval energy E20/2 and they will come across around at the position of F ¼210 kV/cm with the interval energy E10 E20 =2 ¼ 112 meV, where they meet the condition of double-resonance. Finally, in Fig. 5, we discuss the SHG coefficient wð2Þ 2o as a function of the photon energy ‘ o with L¼16 nm, and F¼ 200 kV/cm for different relaxation times: T0 ¼0.08, 0.12, 0.16, 0.2 ps, respectively. From Fig. 5, we can see that the larger the relaxation time T is, the higher the peak value of the SHG coefficient is. So, we can get the higher value of the SHG coefficient through increasing the relaxation time. As we all know,
the relaxation rate relates to several external factors, such as character of material, temperature of environment and so on. And we can correctly select these factors and get higher value of SHG coefficient.
4. Conclusion In conclusion, the second-harmonic generation coefficient has been researched in detail and it can reach the magnitude of 10 5 m/V. Meanwhile, we also find that the SHG coefficient is not a monotonic function of the applied field F, and it relates to the length L of CQDs as well. When we set the value of applied electric field F¼200 kV/cm and length L¼16 nm, with relaxation time T¼0.2 ps, we can obtain the higher value of the SHG coefficient. These theoretical results may have great influence on experimental studies for optical devices.
Acknowledgments Project supported by the National Natural Science Foundation of China (under Grant no. 60878002), the Science and Technology
396
S. Shao et al. / Physica B 406 (2011) 393–396
Committee of Guangdong Province (under Grant nos. 2009B011100007, 2010B010800031, 2010B090400130 and 8251009101000002), and the Science and Technology Bureau of Guangzhou (under Grant no. 2009J1-C421-1). References
[8] [9] [10] [11] [12] [13] [14] [15]
[1] [2] [3] [4]
A.D. Yoffe, Adv. Phys. 50 (2001) 1. D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. D. Heiss, M. Kroutvar, J.J. Finley, G. Abstreiter, Solid State Commun. 135 (2005) 591. V.A. Harutyunyan, E.M. Kazryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 36 (2007) 114. [5] A.D. Andreev, A.A. Lipovsky, Semiconductors 33 (1999) 1450. [6] M.S. Atoyan, H.A. Sarkisyan, E.M. Kazaryan, Physica E 22 (2004) 860. [7] W.F. Xie, Superlatt. Microstruct. 48 (2010) 239.
[16] [17] [18] [19] [20] [21]
W.F. Xie, Physica B 405 (2010) 3436. C. Dane, H. Akbas, N. Talip, K. Kasapoglu, Physica E 39 (2007) 95. Sh.W. Feng, Thin Solid Films 516 (2008) 7695. O. Husberga, A. Khartchenkoa, Physica E 13 (2002) 1090. C.S. Yang, Microelectron. J. 39 (2008) 1469. C. Dane, H. Akbas, N. Talipb, K. Kasapoglua, Physica E 39 (2007) 95. Z.H. Zhang, K.X. Guo, B. Chen, R.Z. Wang, M.W. Kang, Superlatt. Microstruct. 46 (2009) 672. Z.H. Zhang, K.X. Guo, B. Chen, R.Z. Wang, M.W. Kang, Physica B 404 (2009) 2332. G.W. Bryant, Phys. Rev. B 37 (1988) 8763. S. Gangopadbyay, B.R. Nag, Nanotechnology 8 (1997) 14. J. Shertzer, L.R. Ram-Mohan, Phys. Rev. B 41 (1990) 9994. B. Li, K.X. Guo, Ch.J. Zhang, Y.B. Zheng, Phys. Lett. A 367 (2007) 493. E. Rosencher, P. Bois, Phys. Rev. B 44 (1991) 11315. B. Chen, K.X. Guo, Z.L. Liu, R.Z. Wang, Y.B. Zheng, B. Li, J. Phys. Condens. Matter 20 (2008) 255214.