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Third-harmonic generation in cubical quantum dots
Superlattices and Microstructures 46 (2009) 672–678
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Third-harmonic generation in cubical quantum dots Zhi-Hai Zhang, Kang-Xian Guo ∗ , Bin Chen, Rui-Zhen Wang, Min-Wu Kang Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China
article
info
Article history: Received 3 April 2009 Received in revised form 9 July 2009 Accepted 20 July 2009 Available online 5 August 2009 Keywords: Third-harmonic generation Cubical quantum dots Quantum-confinement effect
abstract Third-harmonic generation (THG) for cubical quantum dots (CQDs) with an applied electric field is theoretically investigated in the framework of the compact-density-matrix approach and an iterative method. The confined wave functions and energies of electrons in the CQDs are calculated in the effective-mass approximation. Numerical calculations are presented for typical GaAs/AlAs CQDs. The results demonstrate that the THG strongly depends on the length of the CQDs and the magnitude of the electric field. Also, the peaks shift towards the higher energy region with increasing electric field. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction In the last two decades, much attention has been focused on the nonlinear properties in semiconductor quantum dots [1]. The studies on these structures have opened a new field in fundamental physics and chemistry, and also offer a wide range of potential applications for optoelectronic devices [2,3]. Some of these applications include semiconductor lasers [4], singleelectron transistors [5], quantum computing [6], optical memories [7] and infrared photodetectors [8]. In 1975, Esaki et al. were the first to present the concept of quantum wires and dots [9,10]. With recent rapid advances of modern technology, such as molecular beam epitaxy [11] and metal–organic chemical vapour deposition [12,13], it is now possible to reduce the dimensionality of semiconductor from bulk sample to zero-dimensional semiconductor nanostructures or cubical quantum dots (CQDs) [14,15]. Among the nonlinear optical properties, more attention has been paid to the third-order ones. Several types of systems have been researched with respect to third-harmonic generation (THG), such as parabolic quantum systems [16,17], semi-parabolic quantum systems [18–20], coupled quantum systems [21,22], and so on. But CQDs have not been researched in this area, so research in this system is important both theoretically and in view of practical applications.
∗
Corresponding author. Tel.: +86 13342886687. E-mail address: [email protected] (K.-X. Guo).
0749-6036/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2009.07.023
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
673
Fig. 1. Schematic diagram for the electronic confined potential profile in a cubical quantum dot.
In this paper, THG in the GaAs/AlAs CQDs system with an applied electric field is investigated. The construction of this CQD system is shown in Fig. 1. This paper is organized as follows. In Section 2, the Hamiltonian, relevant eigenstates and eigenenergies are discussed in the GaAs/AlAs CQDs, and the analytical expression of the THG coefficient is deduced by the compact-density-matrix approach and an iterative method. In Section 3, numerical calculations for typical GaAs/AlAs CQDs are performed. We find that the dipolar matrix elements µ01 µ12 µ23 µ30 decrease with enhancement of the electric
− →
field F , and the THG coefficient as a function of photon energy is plotted and analyzed in detail. Finally, a brief conclusion is given in Section 4. 2. Theory 2.1. Electronic structure In the effective-mass approximation, the Hamiltonian of an electron in a CQD in the presence of an
− →
electric field F is H =−
h¯ 2 2m∗
− → → ∇2 + e F · − r + V (x, y, z ),
(1)
with
(
0
V ( x, y , z ) =
∞
L
|x|, |y| and |z | ≤ , 2 elsewhere,
(2)
− →
where F is the electric field vector, and m∗ 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, V is the confining potential and the spatial electric field.
− →
In spherical coordinates, the external spatial electric field is F = F (sin θ cos ϕˆe1 + sin θ sin ϕˆe2 + cos θ eˆ 3 ). By using the effective Rydberg constant R∗ = m∗ e4 /2 h¯ 2 ε 2 as the unit of the energy and the effective Born radius a∗ = h¯ 2 ε/m∗ e2 as the unit of length, the Hamiltonian becomes H = −∇ 2 + η(x sin θ cos ϕ + y sin θ sin ϕ + z cos ϕ) + V (x, y, z ),
(3)
where η = ea F /R is the dimensional measure of the electric field, and ε is the static dielectric constant of the material. The numerical method for solving the Schrödinger equation is well known [23–25]. The solutions are given by ∗
∗
ψi (εi ) = C1 Ai(εi ) + C2 Bi(εi ), 2 3
(4)
where εi = (Ai xi − Ei + Vi )/Ai , and Vi is infinite barrier potential (V0 = ∞). A1 = η sin θ cos ϕ , A2 = η sin θ sin ϕ , A3 = η cos θ , Ai(εi ) and Bi(εi ) are Airy functions. C1 and C2 are the normalized
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Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
coefficients of the wave function. All of these normalized coefficients and the eigenenergy Ei can be numerically solved by the standard boundary condition of the electronic bound state. 2.2. THG coefficient Next, the formula for the THG coefficient in CQDs can be obtained by the compact-density-matrix
− →
method and an iterative procedure [26]. The system is excited by an electromagnetic field E (t ) = ˜ iωt + Ee ˜ −iωt . Let us denote ρ as the one-electron density matrix for this regime. Then the evolution Ee of the density matrix is given by the time-dependent Schrödinger equation
∂ρij 1 − → = [H0 − qz F (t ), ρ]ij − Γij (ρ − ρ (0) )ij ∂t h¯
(5)
− →
where H0 is the Hamiltonian for this system without the electromagnetic field E (t ); ρ (0) is the unperturbed density matrix; Γij is the phenomenological relaxation rate. Eq. (5) is calculated by the following iterative method [27,28].
ρ(t ) =
X
ρ (n) (t ),
(6)
n
with
∂ρij(n+1)
o 1 − → [H0 , ρ (n+1) ]ij − ih¯ Γij ρij(n+1) − [qz , ρ (n) ]ij E (t ). ∂t ih¯ ih¯ − → The electric polarization of the quantum dot due to E (t ) can be expressed as ˜ iωt + χ2(ω2) E˜ 2 e2iωt + c .c . + ε0 χ3(ω3) E˜ 3 e3iωt , P (t ) = ε0 χω(1) Ee =
1 n
χω(1) , χ2(ω2)
(7)
(8)
χ3(ω3)
where and are the linear susceptibility, optical rectification and third-harmonic generation, respectively. ε0 is the vacuum dielectric constant. The electronic polarization of the nth order is given as 1 Tr(ρ (n) ez ), (9) V where V is the volume of interaction and Tr denotes the trace or summation over the diagonal elements of the matrix ρ (n) ez. In our paper, the THG coefficient per unit volume is given as P (n) (t ) =
χ3(ω3)
=
e4 σ v
ε0 h¯ 3
µ01 µ12 µ23 µ30
1
(3ω − ω21 − iΓ21 )(2ω − ω20 − iΓ20 )(ω − ω23 − iΓ23 ) 1
+
(3ω − ω21 − iΓ21 )(2ω − ω20 − iΓ20 )(ω − ω30 − iΓ30 )
+
(3ω − ω23 − iΓ23 )(2ω − ω20 − iΓ20 )(ω − ω10 − iΓ10 )
1 1
+
(3ω − ω30 − iΓ30 )(2ω − ω20 − iΓ20 )(ω − ω21 − iΓ21 )
+
(3ω − ω30 − iΓ30 )(2ω − ω31 − iΓ31 )(ω − ω21 − iΓ21 )
+
(3ω − ω01 − iΓ01 )(2ω − ω31 − iΓ31 )(ω − ω32 − iΓ32 )
+ +
1 1 1
(3ω − ω21 − iΓ21 )(2ω − ω31 − iΓ31 )(ω − ω30 − iΓ30 ) 1
(3ω − ω21 − iΓ21 )(2ω − ω31 − iΓ31 )(ω − ω01 − iΓ01 )
,
(10)
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
675
Fig. 2. The dipolar matrix elements µ01 µ12 µ23 µ30 as a function of the applied electric field for θ = 0◦ , ϕ = 0◦ with the length of cubical quantum dots for L = 11, 12 nm, respectively.
where σv is the electron density in the system and e is the electronic charge. ωij = (Ei − Ej )/h¯ is the transition frequency, µij = |hψj |z |ψi i|(i, j = 0, 1, 2, 3) is the off-diagonal matrix element, and µ01 µ12 µ23 µ30 is a geometric factor. 3. Results and discussions (3)
In this section, we will discuss the THG coefficient χ3ω in GaAs/AlAs CQDs under external electric fields. The parameters adopted in our calculations are as follows [29,30]: m∗ = 0.067m0 , the effective Rydberg constant is R∗ = 5.83 meV, the effective Bohr radius is a∗ = 98 Å, ε0 = 8.85 × 10−12 F m−1 , h¯ Γ12 = h¯ Γ23 = h¯ Γ34 = 1 meV, h¯ Γ13 = h¯ Γ24 = 0.3 meV, h¯ Γ14 = 0.1 meV, and σv = 5 × 1024 m−3 . Fig. 2 shows the dipolar matrix elements µ01 µ12 µ23 µ30 as a function of the applied electric
− →
field F for different values of the length of the CQDs. From Fig. 2, we can observe that the dipolar
− →
matrix elements µ01 µ12 µ23 µ30 subsequently decrease with enhancement of the electric field F . It can be further noted that the larger the length of the CQDs is, the higher the value of the dipolar
− → − →
matrix elements µ01 µ12 µ23 µ30 will be. Since we know that if the electric field F is too strong, the semiconductors will be broken down, this means that there is an upper limit to F . There is also a range for L in our model: the effect of corners on the levels in systems for L = 100 Å is less than 1%, and it can be neglected for wider CQDs [27,31]. (3) In Fig. 3(a) and (b), we present the imaginary part and the real part of χ3ω as a function of the ◦ ◦ incident photon energy h¯ ω with θ = 0 , ϕ = 0 and L = 12 nm for different values of the electric field. From Fig. 3(a) and (b), we can see that the resonant peaks will move to the right-hand side of the curve with the enhancement of electric field, and that the larger the value of the electric field is, the more prominent the peak will be. This enhancement is attributed to the increase of the dipolar matrix elements µ01 µ12 µ23 µ30 (see Fig. 2). (3) In Fig. 4(a) and (b), we present the imaginary part and the real part of χ3ω as a function of the
− →
incident photon energy h¯ ω with θ = 0◦ , ϕ = 0◦ and F = 200 kV/cm for different values of the length of the CQDs. From Fig. 4(a) and (b), we can see that the resonant peaks will move to the lefthand side of the curve as the length of the CQDs increases, and that the narrower the size of the CQDs is, the smaller the value of the THG coefficient will be. This is due to the fact that the quantumconfinement effect results in a separation of energy levels, and the stronger the confinement effect is, the broader the separation will be. The resonant peaks will move to higher energy when L decreases, which causes a strong confinement-induced blueshift in the quantum dots. (3) Fig. 5(a) and (b) show the imaginary part and real part of χ3ω as a function of the photon energy
− → h¯ ω with ϕ = 0◦ , L = 12 nm and F = 200 kV/cm for different values of the angle θ . It is obvious that
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Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
a
b
Fig. 3. (a) and (b) Contribution of electric field to (a) the real part, (b) the imaginary part of third-harmonic generation as
− →
a function of the photon energy for θ = 0◦ , ϕ = 0◦ , L = 12 nm with the applied electric field for F respectively.
a
= 50, 250 kV/cm,
b
Fig. 4. (a) and (b) Contribution of L to (a) the real part, and (b) the imaginary part of third-harmonic generation as a function
− →
of the photon energy for θ = 0◦ , ϕ = 0◦ , F = 200 kV/cm with the length of cubical quantum dots for L = 11, 12 nm, respectively.
the resonant peaks will move to the right of the curve as the angle θ decreases and that the smaller the value of θ is, the sharper the peak will be. 4. Conclusion (3)
We have presented an efficient study of the THG coefficient |χ3ω | for GaAs/AlAs CQDs. The calculations mainly focus on the dependence of the THG coefficient on the applied electric field, the (3) length of CQDs and the angle θ . We find that a very large |χ3ω | can be obtained compared to the other (3)
quantum models. The maximum |χ3ω | under the parameters we choose is 29.67 × 10−15 (m/V)2 , as shown in Fig. 3(b), which is estimated to be around one order higher than that in semiconductor carbon nanotubes (5 × 10−16 (m/V)2 ) [32], and one order higher than that in a quantum disk (4.5 × 10−16 (m/V)2 ) [30]. The THG coefficient in our model is greatly enhanced. Based on this CQD (3) system, we can get a much stronger peak of |χ3ω | by adjusting the electric field and the length of the CQDs in an appropriate region, as L primarily influences the energy levels of these CQDs. Therefore, it is expected that an optimum system will be achieved by choosing appropriate values of electric field
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
a
677
b
Fig. 5. (a) and (b) Contribution of θ to (a) the real part, and (b) the imaginary part of third-harmonic generation as a function
− → of the photon energy for ϕ = 0◦ , L = 12 nm F = 200 kV/cm with θ for θ = 5◦ , 70◦ , respectively. (3)
and the length of CQDs to obtain a stronger |χ3ω |. Finally, we hope these important conclusions can make a great contribution to the experimental studies, have a significant influence on improvements of optical devices, such as ultrafast optical switches, and open up new opportunities for practical exploration of the quantum-size effect on devices. Acknowledgments Project supported by the National Science Foundation of China (under Grant No. 60878002), the Science and Technology Committee of Guangdong Province (under Grant Nos. 2007B010600061, 2008B010200043, 2008B010600050, and 8251009101000002). 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]
W. Langbein, B. Patton, Opt. Lett. 31 (2006) 1151. M.A. Reed, Science 268 (1993) 118. S. Ünlü, İ Karabulut, H. Safak, Physica E 33 (2006) 319. N. Kirstaedter, O.G. Schmidt, N.N. Ledentsov, D. Bimberg, V.M. Ustinov, A.Yu. Egorov, A.E. Zhukov, Appl. Phys. Lett. 69 (1996) 1226. E. Leobandung, L. Guo, S.Y. Chou, Appl. Phys. Lett. 69 (1995) 2338. K.K. Likharev, IBM J. Res. Dev. 69 (1998) 1444. K. Imamur, Y. Sugiyama, Y. Nakata, S. Muto, N. Yokoyama, Jpn. J. Appl. Phys. 34 (1995) L1445. X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. C.S. Lent, P.D. Tougaw, W. Porod, G.H. Bernstein, Nanotechnology 4 (1993) 49. K.X. Guo, J. Guangzhou Univ. 4 (2007) 1. A.Y. Cho, J.R. Arhur, Prog, Solid. State. Chem. 10 (1975) 157. P.D. Dapkus, Annu. Rev. Mater. Sci. 12 (1982) 243. S. Hersee, J.P. Duchemim, Annu. Rev. Mater. Sci. 12 (1982) 65. J. Simon, N.T. Pelekanos, C. Adelmann, E. Martinez-Guerrero, R. Andrè, B. Daudin, Le Si Dang, H. Mariette, Phys. Rev. B 68 (2003) 035312. D. Lagarde, A. Balocchi, H. Carrère, P. Renucci, T. Amand, X. Marie, Phys. Rev. B 77 (2008) 041304. G.H. Wang, K.X. Guo, Physica E 315 (2002) 234. G.Q. Hai, F.M. Peeters, Phys. Rev. B 60 (1999) 8984. C.J. Zhang, K.X. Guo, Physica B 383 (2006) 183. L. Zhang, H.J. Xie, Physica E 22 (2004) 791. Y.B. Yu, K.X. Guo, Physica E 18 (2003) 492. S. Gangopadbyay, B.R. Nag, Nanotechnology 8 (1997) 14. R.Z. Wang, K.X. Guo, Z.L. Liu, B. Chen, Y.B. Zheng, B. Li, Supperlatt. Microstruct. 45 (2009) 8. S. Gangopadbyay, B.R. Nag, Nanotechnology 8 (1997) 14. D. Gershoni, H. Temkin, G.J. Dolan, J. Dunsmuir, S.N.G. Chu, M.B. Panish, Appl. Phys. Lett. 53 (1988) 995. J. Shertzer, L.R. Ram-Mohan, Phys. Rev. B 41 (1990) 9994. H. Yildirim, M. Tomak, Phys. Rev. B 72 (2005) 115340. M.M. Fejer, S.J.B. Yoo, R.L. Byer, A. Harwit, J.S. Harris, Phys. Rev. Lett. 62 (1989) 1041.
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Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Third-harmonic generation in cubical quantum dots Zhi-Hai Zhang, Kang-Xian Guo ∗ , Bin Chen, Rui-Zhen Wang, Min-Wu Kang Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China
article
info
Article history: Received 3 April 2009 Received in revised form 9 July 2009 Accepted 20 July 2009 Available online 5 August 2009 Keywords: Third-harmonic generation Cubical quantum dots Quantum-confinement effect
abstract Third-harmonic generation (THG) for cubical quantum dots (CQDs) with an applied electric field is theoretically investigated in the framework of the compact-density-matrix approach and an iterative method. The confined wave functions and energies of electrons in the CQDs are calculated in the effective-mass approximation. Numerical calculations are presented for typical GaAs/AlAs CQDs. The results demonstrate that the THG strongly depends on the length of the CQDs and the magnitude of the electric field. Also, the peaks shift towards the higher energy region with increasing electric field. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction In the last two decades, much attention has been focused on the nonlinear properties in semiconductor quantum dots [1]. The studies on these structures have opened a new field in fundamental physics and chemistry, and also offer a wide range of potential applications for optoelectronic devices [2,3]. Some of these applications include semiconductor lasers [4], singleelectron transistors [5], quantum computing [6], optical memories [7] and infrared photodetectors [8]. In 1975, Esaki et al. were the first to present the concept of quantum wires and dots [9,10]. With recent rapid advances of modern technology, such as molecular beam epitaxy [11] and metal–organic chemical vapour deposition [12,13], it is now possible to reduce the dimensionality of semiconductor from bulk sample to zero-dimensional semiconductor nanostructures or cubical quantum dots (CQDs) [14,15]. Among the nonlinear optical properties, more attention has been paid to the third-order ones. Several types of systems have been researched with respect to third-harmonic generation (THG), such as parabolic quantum systems [16,17], semi-parabolic quantum systems [18–20], coupled quantum systems [21,22], and so on. But CQDs have not been researched in this area, so research in this system is important both theoretically and in view of practical applications.
∗
Corresponding author. Tel.: +86 13342886687. E-mail address: [email protected] (K.-X. Guo).
0749-6036/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2009.07.023
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
673
Fig. 1. Schematic diagram for the electronic confined potential profile in a cubical quantum dot.
In this paper, THG in the GaAs/AlAs CQDs system with an applied electric field is investigated. The construction of this CQD system is shown in Fig. 1. This paper is organized as follows. In Section 2, the Hamiltonian, relevant eigenstates and eigenenergies are discussed in the GaAs/AlAs CQDs, and the analytical expression of the THG coefficient is deduced by the compact-density-matrix approach and an iterative method. In Section 3, numerical calculations for typical GaAs/AlAs CQDs are performed. We find that the dipolar matrix elements µ01 µ12 µ23 µ30 decrease with enhancement of the electric
− →
field F , and the THG coefficient as a function of photon energy is plotted and analyzed in detail. Finally, a brief conclusion is given in Section 4. 2. Theory 2.1. Electronic structure In the effective-mass approximation, the Hamiltonian of an electron in a CQD in the presence of an
− →
electric field F is H =−
h¯ 2 2m∗
− → → ∇2 + e F · − r + V (x, y, z ),
(1)
with
(
0
V ( x, y , z ) =
∞
L
|x|, |y| and |z | ≤ , 2 elsewhere,
(2)
− →
where F is the electric field vector, and m∗ 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, V is the confining potential and the spatial electric field.
− →
In spherical coordinates, the external spatial electric field is F = F (sin θ cos ϕˆe1 + sin θ sin ϕˆe2 + cos θ eˆ 3 ). By using the effective Rydberg constant R∗ = m∗ e4 /2 h¯ 2 ε 2 as the unit of the energy and the effective Born radius a∗ = h¯ 2 ε/m∗ e2 as the unit of length, the Hamiltonian becomes H = −∇ 2 + η(x sin θ cos ϕ + y sin θ sin ϕ + z cos ϕ) + V (x, y, z ),
(3)
where η = ea F /R is the dimensional measure of the electric field, and ε is the static dielectric constant of the material. The numerical method for solving the Schrödinger equation is well known [23–25]. The solutions are given by ∗
∗
ψi (εi ) = C1 Ai(εi ) + C2 Bi(εi ), 2 3
(4)
where εi = (Ai xi − Ei + Vi )/Ai , and Vi is infinite barrier potential (V0 = ∞). A1 = η sin θ cos ϕ , A2 = η sin θ sin ϕ , A3 = η cos θ , Ai(εi ) and Bi(εi ) are Airy functions. C1 and C2 are the normalized
674
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
coefficients of the wave function. All of these normalized coefficients and the eigenenergy Ei can be numerically solved by the standard boundary condition of the electronic bound state. 2.2. THG coefficient Next, the formula for the THG coefficient in CQDs can be obtained by the compact-density-matrix
− →
method and an iterative procedure [26]. The system is excited by an electromagnetic field E (t ) = ˜ iωt + Ee ˜ −iωt . Let us denote ρ as the one-electron density matrix for this regime. Then the evolution Ee of the density matrix is given by the time-dependent Schrödinger equation
∂ρij 1 − → = [H0 − qz F (t ), ρ]ij − Γij (ρ − ρ (0) )ij ∂t h¯
(5)
− →
where H0 is the Hamiltonian for this system without the electromagnetic field E (t ); ρ (0) is the unperturbed density matrix; Γij is the phenomenological relaxation rate. Eq. (5) is calculated by the following iterative method [27,28].
ρ(t ) =
X
ρ (n) (t ),
(6)
n
with
∂ρij(n+1)
o 1 − → [H0 , ρ (n+1) ]ij − ih¯ Γij ρij(n+1) − [qz , ρ (n) ]ij E (t ). ∂t ih¯ ih¯ − → The electric polarization of the quantum dot due to E (t ) can be expressed as ˜ iωt + χ2(ω2) E˜ 2 e2iωt + c .c . + ε0 χ3(ω3) E˜ 3 e3iωt , P (t ) = ε0 χω(1) Ee =
1 n
χω(1) , χ2(ω2)
(7)
(8)
χ3(ω3)
where and are the linear susceptibility, optical rectification and third-harmonic generation, respectively. ε0 is the vacuum dielectric constant. The electronic polarization of the nth order is given as 1 Tr(ρ (n) ez ), (9) V where V is the volume of interaction and Tr denotes the trace or summation over the diagonal elements of the matrix ρ (n) ez. In our paper, the THG coefficient per unit volume is given as P (n) (t ) =
χ3(ω3)
=
e4 σ v
ε0 h¯ 3
µ01 µ12 µ23 µ30
1
(3ω − ω21 − iΓ21 )(2ω − ω20 − iΓ20 )(ω − ω23 − iΓ23 ) 1
+
(3ω − ω21 − iΓ21 )(2ω − ω20 − iΓ20 )(ω − ω30 − iΓ30 )
+
(3ω − ω23 − iΓ23 )(2ω − ω20 − iΓ20 )(ω − ω10 − iΓ10 )
1 1
+
(3ω − ω30 − iΓ30 )(2ω − ω20 − iΓ20 )(ω − ω21 − iΓ21 )
+
(3ω − ω30 − iΓ30 )(2ω − ω31 − iΓ31 )(ω − ω21 − iΓ21 )
+
(3ω − ω01 − iΓ01 )(2ω − ω31 − iΓ31 )(ω − ω32 − iΓ32 )
+ +
1 1 1
(3ω − ω21 − iΓ21 )(2ω − ω31 − iΓ31 )(ω − ω30 − iΓ30 ) 1
(3ω − ω21 − iΓ21 )(2ω − ω31 − iΓ31 )(ω − ω01 − iΓ01 )
,
(10)
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
675
Fig. 2. The dipolar matrix elements µ01 µ12 µ23 µ30 as a function of the applied electric field for θ = 0◦ , ϕ = 0◦ with the length of cubical quantum dots for L = 11, 12 nm, respectively.
where σv is the electron density in the system and e is the electronic charge. ωij = (Ei − Ej )/h¯ is the transition frequency, µij = |hψj |z |ψi i|(i, j = 0, 1, 2, 3) is the off-diagonal matrix element, and µ01 µ12 µ23 µ30 is a geometric factor. 3. Results and discussions (3)
In this section, we will discuss the THG coefficient χ3ω in GaAs/AlAs CQDs under external electric fields. The parameters adopted in our calculations are as follows [29,30]: m∗ = 0.067m0 , the effective Rydberg constant is R∗ = 5.83 meV, the effective Bohr radius is a∗ = 98 Å, ε0 = 8.85 × 10−12 F m−1 , h¯ Γ12 = h¯ Γ23 = h¯ Γ34 = 1 meV, h¯ Γ13 = h¯ Γ24 = 0.3 meV, h¯ Γ14 = 0.1 meV, and σv = 5 × 1024 m−3 . Fig. 2 shows the dipolar matrix elements µ01 µ12 µ23 µ30 as a function of the applied electric
− →
field F for different values of the length of the CQDs. From Fig. 2, we can observe that the dipolar
− →
matrix elements µ01 µ12 µ23 µ30 subsequently decrease with enhancement of the electric field F . It can be further noted that the larger the length of the CQDs is, the higher the value of the dipolar
− → − →
matrix elements µ01 µ12 µ23 µ30 will be. Since we know that if the electric field F is too strong, the semiconductors will be broken down, this means that there is an upper limit to F . There is also a range for L in our model: the effect of corners on the levels in systems for L = 100 Å is less than 1%, and it can be neglected for wider CQDs [27,31]. (3) In Fig. 3(a) and (b), we present the imaginary part and the real part of χ3ω as a function of the ◦ ◦ incident photon energy h¯ ω with θ = 0 , ϕ = 0 and L = 12 nm for different values of the electric field. From Fig. 3(a) and (b), we can see that the resonant peaks will move to the right-hand side of the curve with the enhancement of electric field, and that the larger the value of the electric field is, the more prominent the peak will be. This enhancement is attributed to the increase of the dipolar matrix elements µ01 µ12 µ23 µ30 (see Fig. 2). (3) In Fig. 4(a) and (b), we present the imaginary part and the real part of χ3ω as a function of the
− →
incident photon energy h¯ ω with θ = 0◦ , ϕ = 0◦ and F = 200 kV/cm for different values of the length of the CQDs. From Fig. 4(a) and (b), we can see that the resonant peaks will move to the lefthand side of the curve as the length of the CQDs increases, and that the narrower the size of the CQDs is, the smaller the value of the THG coefficient will be. This is due to the fact that the quantumconfinement effect results in a separation of energy levels, and the stronger the confinement effect is, the broader the separation will be. The resonant peaks will move to higher energy when L decreases, which causes a strong confinement-induced blueshift in the quantum dots. (3) Fig. 5(a) and (b) show the imaginary part and real part of χ3ω as a function of the photon energy
− → h¯ ω with ϕ = 0◦ , L = 12 nm and F = 200 kV/cm for different values of the angle θ . It is obvious that
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a
b
Fig. 3. (a) and (b) Contribution of electric field to (a) the real part, (b) the imaginary part of third-harmonic generation as
− →
a function of the photon energy for θ = 0◦ , ϕ = 0◦ , L = 12 nm with the applied electric field for F respectively.
a
= 50, 250 kV/cm,
b
Fig. 4. (a) and (b) Contribution of L to (a) the real part, and (b) the imaginary part of third-harmonic generation as a function
− →
of the photon energy for θ = 0◦ , ϕ = 0◦ , F = 200 kV/cm with the length of cubical quantum dots for L = 11, 12 nm, respectively.
the resonant peaks will move to the right of the curve as the angle θ decreases and that the smaller the value of θ is, the sharper the peak will be. 4. Conclusion (3)
We have presented an efficient study of the THG coefficient |χ3ω | for GaAs/AlAs CQDs. The calculations mainly focus on the dependence of the THG coefficient on the applied electric field, the (3) length of CQDs and the angle θ . We find that a very large |χ3ω | can be obtained compared to the other (3)
quantum models. The maximum |χ3ω | under the parameters we choose is 29.67 × 10−15 (m/V)2 , as shown in Fig. 3(b), which is estimated to be around one order higher than that in semiconductor carbon nanotubes (5 × 10−16 (m/V)2 ) [32], and one order higher than that in a quantum disk (4.5 × 10−16 (m/V)2 ) [30]. The THG coefficient in our model is greatly enhanced. Based on this CQD (3) system, we can get a much stronger peak of |χ3ω | by adjusting the electric field and the length of the CQDs in an appropriate region, as L primarily influences the energy levels of these CQDs. Therefore, it is expected that an optimum system will be achieved by choosing appropriate values of electric field
Z.-H. Zhang et al. / Superlattices and Microstructures 46 (2009) 672–678
a
677
b
Fig. 5. (a) and (b) Contribution of θ to (a) the real part, and (b) the imaginary part of third-harmonic generation as a function
− → of the photon energy for ϕ = 0◦ , L = 12 nm F = 200 kV/cm with θ for θ = 5◦ , 70◦ , respectively. (3)
and the length of CQDs to obtain a stronger |χ3ω |. Finally, we hope these important conclusions can make a great contribution to the experimental studies, have a significant influence on improvements of optical devices, such as ultrafast optical switches, and open up new opportunities for practical exploration of the quantum-size effect on devices. Acknowledgments Project supported by the National Science Foundation of China (under Grant No. 60878002), the Science and Technology Committee of Guangdong Province (under Grant Nos. 2007B010600061, 2008B010200043, 2008B010600050, and 8251009101000002). 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]
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