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Exciton effects on the nonlinear optical rectification in one-dimensional quantum dots
Physics Letters A 335 (2005) 175–181 www.elsevier.com/locate/pla
Exciton effects on the nonlinear optical rectification in one-dimensional quantum dots You-Bin Yu a,∗ , Shi-Ning Zhu a , Kang-Xian Guo b a National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China b School of Physics and Electronic Engineering, Guihuagang Campus, Guangzhou University, Guangzhou 510405, China
Received 2 May 2003; received in revised form 23 November 2004; accepted 1 December 2004 Available online 22 December 2004 Communicated by A.R. Bishop
Abstract Exciton effects on the nonlinear optical rectification in one-dimensional quantum dots are studied. The numerical results are presented for GaAs/AlGaAs quantum dots. The results show that the optical rectification coefficient is greatly enhanced because of the quantum confinement of exciton. It is over two times bigger than that obtained by without considering exciton effects. In addition, the optical rectification coefficient is related to the relaxation time T2 . 2004 Elsevier B.V. All rights reserved. PACS: 71.35.-y; 68.65.La; 78.20.Nv Keywords: Exciton effects; Optical rectification; Quantum dot
1. Introduction In the past few decades, the nonlinear optical properties of semiconductor quantum wells, superlattices and macrostructures have attracted much attention [1–20]. These include interband transitions with excitonic effects and intersubband transitions. One of the most important features in these structures is that excitonic spectrum persists even at the room tempera* Corresponding author.
E-mail address: [email protected] (Y.-B. Yu). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.12.013
tures. This is due to the enhancement of the excitonic binding energy caused by the quantum confinement effect in these structures. With the recent advances in nanofabrication techniques, it is possible to make wire-like compound semiconductor structures of nanometer size. In these structures, the state of the electron in the direction perpendicular to the length of the wire and dot is quantized with sufficiently large separation in energy levels and the motion of the electron parallel to the length is that of a quasi-free electron. Recently, onedimensional (1D) quantum dots have been investigated in the past few decades [21–23]. 1D quantum dot is
176
Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
different from the 1D quantum wire. The electron is free to move along the quantum wire in 1D quantum wire. However, it is confined to move along the 1D quantum dot. In fact, 1D quantum dot is the small parts of a quantum wire separated by two-wall potential. The existing theoretical understanding of 1D interacting electrons provide in the past quantitative results for transport in the presence of correlations and impurities [24,25]. One of the most remarkable properties of these 1D electron systems is that the optical transitions between the size-quantized subbands are also feasible. Many authors have interests in studying the nonlinear optical properties of quantum wells, quantum wires, and quantum dots [15–18] because of their relevance for studying practical applications and as a probe for the electronic structure of mesoscopic media. Far-infrared spectroscopy on quantum dots offers the opportunity to study the internal excitations of a electron system. It has been shown that the quantum dots display very large optical nonlinearities. In the previous studies, few works take exciton effects into account in studying the nonlinearities of the quantum wells, quantum wires, or quantum dots [1–18]. In peculiar, the exciton effects on the nonlinearities in 1D quantum dots have not been studied. In this Letter, exciton effects on the optical rectification in 1D wire-like semi-parabolic quantum dots which is strongly confined in the x- and y-directions and is confined by semi-parabolic potential in the zdirection are studied. Here, we take the semi-parabolic potential as infinite potential which is approximate the realistic potential only when considering low excited states. So it is appropriate to investigate the optical rectification which only requires one excited state in this model. In Section 2, the eigenfunctions and eigenenergies of the exciton states are obtained using the effective-mass approximation, with centerof-mass and relative coordinates. And the analytical expression for the optical rectification coefficient in 1D semi-parabolic quantum dots is derived by density matrix treatment. In Section 3, the numerical results are presented for GaAs/AlGaAs semi-parabolic quantum dots. The results show that the optical rectification coefficient is greatly enhanced when the exciton states in the quantum dots are considered and it is over two times larger than that obtained only considering electron states. Therefore, it is not appropriate to study the nonlinearities in quantum dots only considering the
electron state. Finally, brief conclusions are given in Section 4.
2. Theory The effective-mass Hamiltonian for an electron– hole pair in 1D quantum dots can be written as: ph2 e2 pe2 , + V (z ) + + V (z ) − e h 2m∗e 2m∗h |ze − zh | (ze , zh > 0), (1)
H=
where m∗e and m∗h are the effective mass of the electron and hole, respectively. is background dielectric constant, and V (zi ) is the semi-parabolic potential which is defined by 1 ∗ 2 2 V (zi ) = 2 mi ω0 zi , zi 0, (2) ∞, zi < 0 (i = e, h). The Hamiltonian can be separated into two parts that represent the center of mass and the relative motion, respectively: H=
1 1 p2 e2 P2 + Mω02 Z 2 + + µω02 z2 − , (3) 2M 2 2µ 2 |z|
where the center-of-mass coordinates m∗e ze + m∗h zh h¯ , P = ∇Z , M i and the total mass M = m∗e +m∗h , as well as the relative coordinates z = ze − zh and p = h¯i ∇z , and the reduced mass µ = m∗e m∗h /M. Then the exciton wave functions and energy levels can be written as Z=
ψ(ze , zh ) = φ(Z)ϕ(z),
(4)
E = EZ + Ez .
(5)
The center-of-mass part is the problem of onedimensional semi-parabolic oscillator which has been solved as 1 2 2 φ(Z) = NN exp − α Z H2N+1 (αZ), (6) 2 3 EZ = EN = 2N + hω ¯ 0 2
(N = 0, 1, 2, . . .), (7)
Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
where α=
where
−1/2
1 √ 2N NN = π 2 (2N + 1)! α
Mω0 , h¯
.
The relative-motion part can be solved analytically both in the strong and weak-confinement limits. In the strong-confinement limit where the Coulomb term is neglected and the ϕ(z) is solved as 1 2 2 ϕ(z) = Nm exp − β z H2m+1 (βz), (8) 2 3 Ez = 2m + hω ¯ 0 2 where µω0 , β= h¯
(m = 0, 1, 2, . . .),
(9)
−1/2
1 √ 2m π 2 (2m + 1)! Nm = β
.
In the weak-confinement limit where the confinement term is neglected, the relative-motion part is a 1D hydrogenic problem. So the ϕ(z) is also solved as 2γ z γz F −m, 2, , (10) ϕ(z) = Cm z exp − m+1 m+1 µe4
Ez = Em = −
2(m + 1)2 2 h¯ 2 (m = 0, 1, 2, . . .),
(11)
2
where γ = µe2 , F is the confluent hypergeometric h¯ function, and Cm is the normalization constant. The nonlinear optical rectification coefficient for a two-level system in 1D semi-parabolic quantum dots can be obtained by density matrix approach and perturbation expansion method [3,21,22] and it can be written as χ0(2) = 4
e3σs ε0h¯ 2
×
2 1+ ω01
2 T1 T2 + ω
(ω01 − ω)2 +
1 T22
+
1 T22
T1
−1
T2
(ω01 + ω)2 +
1 T22
, (12)
(2)
χ0,max =
2e3 T1 T2 σs ε0 h¯ 2
µ01 = ψ0 |z|ψ1 , δ01 = ψ1 |z|ψ1 − ψ0 |z|ψ0 , ρii − ρjj E1 − E0 , σs = S h¯ is the density of electron in the quantum dot, T1 is the longitudinal relaxation time and T2 is the transverse relaxation time. Only considering electron state, the electron wave functions and energy levels can be written as 1 φn (z) = Nn exp − α 2 z2 H2n+1 (αz), (14) 2 3 h¯ ω0 (n = 0, 1, 2, . . .), En = 2n + (15) 2
ω01 =
where α=
m∗e ω0 , h¯
−1/2
1 √ 2n π2 (2n + 1)! Nn = α
.
Then the dipolar matrix element can be solved as 4N0 N1 2 , µ01 = φ0 (z)|z|φ1(z) = (16) =√ 2 α 6πα 2N 2 2 µ00 = φ0 (z)|z|φ0(z) = 20 = √ , (17) α πα 72N12 3 =√ . µ11 = φ1 (z)|z|φ1(z) = (18) 2 α πα Finally, from Eq. (13) to Eq. (18), the maximum optical rectification coefficient can be given by √ 8 2h¯ e3 T1 T2 σs (2) . χ0,max (ω) = (19) 3ε0 (m∗e π h¯ ω)3/2 3. Results and discussions
µ201 δ01
when ω ∼ ω01 , there is a peak value of χ0 (2)
177
µ201 δ01,
(13)
We assumed T1 = 1 ps and T2 = 0.2 ps [3] and the parameters used in our numerical work are adopted as: m∗e = 0.067m0 , m∗h = 0.09m0 (m0 is the mass of a free electron), σs = 5 × 1024 m−3 , = 12.53. From Eq. (19), we can see the peak value of optical rectifica(2) tion coefficient χ0,max (ω) is related to the photon energy hω. ¯ However, their relation depends on the potential shape. The classical polarizable sphere model of
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Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
(2)
Fig. 1. The maximum optical rectification as a function of photon energy. The results (solid line) are compared with the step quantum wells (dashed line) of Rosencher and Bois and the polarizable sphere model (dotted line) of Gurnick and DeTemple, respectively.
Gurnick and DeTemple, which yields to (h¯ ω)−2 [26]. The step quantum wells which Rosencher and Bois have studied yields to (h¯ ω)−1 [3]. Nevertheless, in our model it yields to (h¯ ω)−3/2 . We plotted he maximum optical rectification coefficient versus the photon energy according to Eq. (19) in Fig. 1. Moreover, the (2) χ0,max (ω) of the classical polarizable sphere model and step quantum wells are also plotted in Fig. 1. From Fig. 1, we can see the maximum optical rectification coefficient obtained in our model is one order of magnitude larger than that obtained in step quantum wells, and two orders of magnitude larger than that obtained in polarizable sphere model. The enhancement of optical rectification coefficient is attributed to the large asymmetry of the semi-parabolic potential. From Eq. (13), we can see the maximum optical (2) rectification coefficient χ0,max is also related to the frequency ω0 of the parabolic potential. In the follow(2) ing, we will discuss the optical rectification χ0 in semi-parabolic quantum dots for GaAs/AlGaAs in the strong-confinement (ω0 1 × 1013 s−1 ) and weakconfinement (ω0 1 × 1013 s−1 ) regimes [20]. 3.1. Strong-confinement regime
Fig. 2. The optical rectification coefficient |χ0 | with the ω0 = 2 × 1014 s−1 versus the photon energy hν, considering exciton effects (solid line) and without considering exciton effects (dashed line).
functions and energy levels can be obtained through Eqs. (4)–(9). In Fig. 2, we show the optical recti(2) fication coefficient χ0 as a function of the photon energy hν with the semi-parabolic confinement frequency ω0 = 2 × 1014 s−1 for two cases: considering exciton effects and without considering exciton effects. The case of without considering exciton effects is the single electron state which is the problem of one-dimensional semi-parabolic oscillator. The wave functions and energy levels of the single electron state are similar to Eq. (6) and Eq. (7) while the mass of the electron m∗e instead of the total mass M. From Fig. 2 we can see that the optical rectification coefficient χ0(2) obtained by considering exciton effects is over two times larger than that obtained without considering exciton effects. This enhancement is attributed to the increase of the dipolar matrix element µ201 δ01 . However, it is clear that the symmetric system do not produce second-order nonlinearities because δ01 is equal to zero. In Fig. 3, we show the peak (2) values of optical rectification coefficient |χ0,max | as a function of the frequency ω0 for two case: considering exciton effects and without considering exciton effects according to the Eq. (13). From Fig. 3, we can see that (2) | will increase when the frequency ω0 decrease |χ0,max (2)
In this case, the Coulomb energy of the electron and hole interaction is a small correction to the quantization energies of electrons and holes [27], the Coulomb term in Eq. (3) may be neglected, the exciton wave
and |χ0,max | will increase more quickly as ω0 decrease (2)
for considering exciton effects. The |χ0,max | obtained by considering exciton effects is also over two times larger than that obtained when exciton effects are not
Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
(2)
Fig. 3. The optical rectification coefficient maximum value |χ0,max | versus the semi-parabolic confinement frequency ω0 for two cases: considering exciton effects (solid line) and without considering exciton effects (dashed line).
179
(2)
Fig. 5. The peak values of optical rectification coefficient |χ0,max | as a function of the frequency ω0 .
frequency: ω0 = 1 × 1014 s−1 , ω0 = 1.5 × 1014 s−1 , ω0 = 2 × 1014 s−1 , ω0 = 2.5 × 1014 s−1 , and ω0 = 3 × 1014 s−1 , respectively. We find five peaks which start at hν = 0.132 eV, 0.198 eV, 0.263 eV, 0.329 eV, and 0.395 eV, respectively. A very important feature is ω01 will increase with the enhancement of the semiparabolic confinement frequency ω0 which makes the peaks move to the high-energy side which shows a strong-confinement-induced blue-shift of the exciton resonance in semiconductor quantum dots, which agree with the theoretical results [28]. These features make the semi-parabolic quantum dots very promising candidates for nonlinear optical materials applications.
(2)
Fig. 4. The optical rectification coefficient |χ0 | versus the photon energy hν for five different values of the semi-parabolic confinement frequency ω0 = 1 × 1014 s−1 , 1.5 × 1014 s−1 , 2 × 1014 s−1 , 2.5 × 1014 s−1 , and 3 × 1014 s−1 , respectively.
considered. In addition, exciton effects will be smaller with the frequency ω0 increase. From Eq. (13) we can see that the enhancement of µ201 δ01 result in the in(2) crease of |χ0,max |. From Fig. 2 and Fig. 3, we can see that the exciton effects on optical rectification coeffi(2) cient χ0 are very important. So it is apparently very important to take the exciton effects into account when we study the optical rectification coefficient for quantum dots. Fig. 4 shows the optical rectification coefficient (2) χ0 as a function of the photon energy hν for five different values of the semi-parabolic strong-confinement
3.2. Weak-confinement regime In the weak-confinement limit, the Coulomb interaction is more important than the quantization energies of the electrons and holes [27], so the parabolic confinement term of the relative motion part in Eq. (3) is neglected, the exciton wave functions and energy levels can be obtained through Eqs. (4)–(7) and Eqs. (10), (11). In Fig. 5, we show the peak values of optical (2) rectification coefficient |χ0,max | as a function of the (2) frequency ω0 . From Fig. 5, we can see that |χ0,max | will increase with the enhancement of the frequency ω0 , which is just opposite to that happens in the strongconfinement regime. In Fig. 6, we show the optical rectification coefficient χ0(2) as a function of the photon energy hν with the semi-parabolic confinement
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Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
tion in this model. Moreover, the bigger the transverse relaxation time T2 is, the sharper the peaks will be and the bigger the peak intensities will be. However, the transverse relaxation time T2 is related not only to the quantum dot material and confining potential, but also to other factors, such as temperature and boundary conditions. Therefore, theoretical studies may make a great contribution to experimental studies, may have profound consequences as regards improvements of practical devices such as ultrafast optical switches, and may open up new opportunities for practical exploitation of the quantum-size effect in devices. (2)
Fig. 6. The optical rectification coefficient |χ0 | versus the photon energy hν for five different transverse relaxation time: T2 = 0.16 ps, T2 = 0.18 ps, T2 = 0.2 ps, T2 = 0.22 ps, and T2 = 0.24 ps, which are illustrated by the solid line, dashed line, dotted line, dash-dotted line, and dash-dot-dotted line, respectively.
frequency ω0 = 1 × 1012 s−1 for five different transverse relaxation time: T2 = 0.16 ps, T2 = 0.18 ps, T2 = 0.2 ps, T2 = 0.22 ps, and T2 = 0.24 ps, respectively. From Fig. 6, we can see that the peak value will increase as the transverse relaxation time T2 increase. And the bigger the peak is, the sharper the peak intensity will be. In addition, the five curves have two points of intersection at hν = 0.3905 eV and hν = 0.3975 eV, respectively. Because T2 is related to the effects of outer surroundings, this feature shows that the influence of outer surroundings can be neglected when hν = 0.3905 eV and hν = 0.3975 eV. So we can apply the feature to decrease the deviations in experiment.
4. Conclusions We present a simple and straightforward study of the optical rectification coefficient for a nanometersize quantum dots. The results show that the theoretical value of the optical rectification coefficient is greatly enhanced due to the exciton effects. The optical rectification coefficient obtained by considering exciton effects is over two times larger than that obtained only considering electron states. In addition, the large asymmetry of the semi-parabolic potential is also the causes for the enhancement of the optical rectifica-
Acknowledgements This work is supported by Guangdong Provincial Natural Science Foundation of China.
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[23] A. Braggio, M. Grifoni, M. Sassetti, F. Napoli, Europhys. Lett. 50 (2000) 236. [24] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [25] J. Voit, Rep. Prog. Phys. 57 (1995) 977. [26] M.K. Gurnick, T.A. DeTemple, IEEE J. Quantum Electron. QE-19 (1983) 791.
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Exciton effects on the nonlinear optical rectification in one-dimensional quantum dots You-Bin Yu a,∗ , Shi-Ning Zhu a , Kang-Xian Guo b a National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China b School of Physics and Electronic Engineering, Guihuagang Campus, Guangzhou University, Guangzhou 510405, China
Received 2 May 2003; received in revised form 23 November 2004; accepted 1 December 2004 Available online 22 December 2004 Communicated by A.R. Bishop
Abstract Exciton effects on the nonlinear optical rectification in one-dimensional quantum dots are studied. The numerical results are presented for GaAs/AlGaAs quantum dots. The results show that the optical rectification coefficient is greatly enhanced because of the quantum confinement of exciton. It is over two times bigger than that obtained by without considering exciton effects. In addition, the optical rectification coefficient is related to the relaxation time T2 . 2004 Elsevier B.V. All rights reserved. PACS: 71.35.-y; 68.65.La; 78.20.Nv Keywords: Exciton effects; Optical rectification; Quantum dot
1. Introduction In the past few decades, the nonlinear optical properties of semiconductor quantum wells, superlattices and macrostructures have attracted much attention [1–20]. These include interband transitions with excitonic effects and intersubband transitions. One of the most important features in these structures is that excitonic spectrum persists even at the room tempera* Corresponding author.
E-mail address: [email protected] (Y.-B. Yu). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.12.013
tures. This is due to the enhancement of the excitonic binding energy caused by the quantum confinement effect in these structures. With the recent advances in nanofabrication techniques, it is possible to make wire-like compound semiconductor structures of nanometer size. In these structures, the state of the electron in the direction perpendicular to the length of the wire and dot is quantized with sufficiently large separation in energy levels and the motion of the electron parallel to the length is that of a quasi-free electron. Recently, onedimensional (1D) quantum dots have been investigated in the past few decades [21–23]. 1D quantum dot is
176
Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
different from the 1D quantum wire. The electron is free to move along the quantum wire in 1D quantum wire. However, it is confined to move along the 1D quantum dot. In fact, 1D quantum dot is the small parts of a quantum wire separated by two-wall potential. The existing theoretical understanding of 1D interacting electrons provide in the past quantitative results for transport in the presence of correlations and impurities [24,25]. One of the most remarkable properties of these 1D electron systems is that the optical transitions between the size-quantized subbands are also feasible. Many authors have interests in studying the nonlinear optical properties of quantum wells, quantum wires, and quantum dots [15–18] because of their relevance for studying practical applications and as a probe for the electronic structure of mesoscopic media. Far-infrared spectroscopy on quantum dots offers the opportunity to study the internal excitations of a electron system. It has been shown that the quantum dots display very large optical nonlinearities. In the previous studies, few works take exciton effects into account in studying the nonlinearities of the quantum wells, quantum wires, or quantum dots [1–18]. In peculiar, the exciton effects on the nonlinearities in 1D quantum dots have not been studied. In this Letter, exciton effects on the optical rectification in 1D wire-like semi-parabolic quantum dots which is strongly confined in the x- and y-directions and is confined by semi-parabolic potential in the zdirection are studied. Here, we take the semi-parabolic potential as infinite potential which is approximate the realistic potential only when considering low excited states. So it is appropriate to investigate the optical rectification which only requires one excited state in this model. In Section 2, the eigenfunctions and eigenenergies of the exciton states are obtained using the effective-mass approximation, with centerof-mass and relative coordinates. And the analytical expression for the optical rectification coefficient in 1D semi-parabolic quantum dots is derived by density matrix treatment. In Section 3, the numerical results are presented for GaAs/AlGaAs semi-parabolic quantum dots. The results show that the optical rectification coefficient is greatly enhanced when the exciton states in the quantum dots are considered and it is over two times larger than that obtained only considering electron states. Therefore, it is not appropriate to study the nonlinearities in quantum dots only considering the
electron state. Finally, brief conclusions are given in Section 4.
2. Theory The effective-mass Hamiltonian for an electron– hole pair in 1D quantum dots can be written as: ph2 e2 pe2 , + V (z ) + + V (z ) − e h 2m∗e 2m∗h |ze − zh | (ze , zh > 0), (1)
H=
where m∗e and m∗h are the effective mass of the electron and hole, respectively. is background dielectric constant, and V (zi ) is the semi-parabolic potential which is defined by 1 ∗ 2 2 V (zi ) = 2 mi ω0 zi , zi 0, (2) ∞, zi < 0 (i = e, h). The Hamiltonian can be separated into two parts that represent the center of mass and the relative motion, respectively: H=
1 1 p2 e2 P2 + Mω02 Z 2 + + µω02 z2 − , (3) 2M 2 2µ 2 |z|
where the center-of-mass coordinates m∗e ze + m∗h zh h¯ , P = ∇Z , M i and the total mass M = m∗e +m∗h , as well as the relative coordinates z = ze − zh and p = h¯i ∇z , and the reduced mass µ = m∗e m∗h /M. Then the exciton wave functions and energy levels can be written as Z=
ψ(ze , zh ) = φ(Z)ϕ(z),
(4)
E = EZ + Ez .
(5)
The center-of-mass part is the problem of onedimensional semi-parabolic oscillator which has been solved as 1 2 2 φ(Z) = NN exp − α Z H2N+1 (αZ), (6) 2 3 EZ = EN = 2N + hω ¯ 0 2
(N = 0, 1, 2, . . .), (7)
Y.-B. Yu et al. / Physics Letters A 335 (2005) 175–181
where α=
where
−1/2
1 √ 2N NN = π 2 (2N + 1)! α
Mω0 , h¯
.
The relative-motion part can be solved analytically both in the strong and weak-confinement limits. In the strong-confinement limit where the Coulomb term is neglected and the ϕ(z) is solved as 1 2 2 ϕ(z) = Nm exp − β z H2m+1 (βz), (8) 2 3 Ez = 2m + hω ¯ 0 2 where µω0 , β= h¯
(m = 0, 1, 2, . . .),
(9)
−1/2
1 √ 2m π 2 (2m + 1)! Nm = β
.
In the weak-confinement limit where the confinement term is neglected, the relative-motion part is a 1D hydrogenic problem. So the ϕ(z) is also solved as 2γ z γz F −m, 2, , (10) ϕ(z) = Cm z exp − m+1 m+1 µe4
Ez = Em = −
2(m + 1)2 2 h¯ 2 (m = 0, 1, 2, . . .),
(11)
2
where γ = µe2 , F is the confluent hypergeometric h¯ function, and Cm is the normalization constant. The nonlinear optical rectification coefficient for a two-level system in 1D semi-parabolic quantum dots can be obtained by density matrix approach and perturbation expansion method [3,21,22] and it can be written as χ0(2) = 4
e3σs ε0h¯ 2
×
2 1+ ω01
2 T1 T2 + ω
(ω01 − ω)2 +
1 T22
+
1 T22
T1
−1
T2
(ω01 + ω)2 +
1 T22
, (12)
(2)
χ0,max =
2e3 T1 T2 σs ε0 h¯ 2
µ01 = ψ0 |z|ψ1 , δ01 = ψ1 |z|ψ1 − ψ0 |z|ψ0 , ρii − ρjj E1 − E0 , σs = S h¯ is the density of electron in the quantum dot, T1 is the longitudinal relaxation time and T2 is the transverse relaxation time. Only considering electron state, the electron wave functions and energy levels can be written as 1 φn (z) = Nn exp − α 2 z2 H2n+1 (αz), (14) 2 3 h¯ ω0 (n = 0, 1, 2, . . .), En = 2n + (15) 2
ω01 =
where α=
m∗e ω0 , h¯
−1/2
1 √ 2n π2 (2n + 1)! Nn = α
.
Then the dipolar matrix element can be solved as 4N0 N1 2 , µ01 = φ0 (z)|z|φ1(z) = (16) =√ 2 α 6πα 2N 2 2 µ00 = φ0 (z)|z|φ0(z) = 20 = √ , (17) α πα 72N12 3 =√ . µ11 = φ1 (z)|z|φ1(z) = (18) 2 α πα Finally, from Eq. (13) to Eq. (18), the maximum optical rectification coefficient can be given by √ 8 2h¯ e3 T1 T2 σs (2) . χ0,max (ω) = (19) 3ε0 (m∗e π h¯ ω)3/2 3. Results and discussions
µ201 δ01
when ω ∼ ω01 , there is a peak value of χ0 (2)
177
µ201 δ01,
(13)
We assumed T1 = 1 ps and T2 = 0.2 ps [3] and the parameters used in our numerical work are adopted as: m∗e = 0.067m0 , m∗h = 0.09m0 (m0 is the mass of a free electron), σs = 5 × 1024 m−3 , = 12.53. From Eq. (19), we can see the peak value of optical rectifica(2) tion coefficient χ0,max (ω) is related to the photon energy hω. ¯ However, their relation depends on the potential shape. The classical polarizable sphere model of
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(2)
Fig. 1. The maximum optical rectification as a function of photon energy. The results (solid line) are compared with the step quantum wells (dashed line) of Rosencher and Bois and the polarizable sphere model (dotted line) of Gurnick and DeTemple, respectively.
Gurnick and DeTemple, which yields to (h¯ ω)−2 [26]. The step quantum wells which Rosencher and Bois have studied yields to (h¯ ω)−1 [3]. Nevertheless, in our model it yields to (h¯ ω)−3/2 . We plotted he maximum optical rectification coefficient versus the photon energy according to Eq. (19) in Fig. 1. Moreover, the (2) χ0,max (ω) of the classical polarizable sphere model and step quantum wells are also plotted in Fig. 1. From Fig. 1, we can see the maximum optical rectification coefficient obtained in our model is one order of magnitude larger than that obtained in step quantum wells, and two orders of magnitude larger than that obtained in polarizable sphere model. The enhancement of optical rectification coefficient is attributed to the large asymmetry of the semi-parabolic potential. From Eq. (13), we can see the maximum optical (2) rectification coefficient χ0,max is also related to the frequency ω0 of the parabolic potential. In the follow(2) ing, we will discuss the optical rectification χ0 in semi-parabolic quantum dots for GaAs/AlGaAs in the strong-confinement (ω0 1 × 1013 s−1 ) and weakconfinement (ω0 1 × 1013 s−1 ) regimes [20]. 3.1. Strong-confinement regime
Fig. 2. The optical rectification coefficient |χ0 | with the ω0 = 2 × 1014 s−1 versus the photon energy hν, considering exciton effects (solid line) and without considering exciton effects (dashed line).
functions and energy levels can be obtained through Eqs. (4)–(9). In Fig. 2, we show the optical recti(2) fication coefficient χ0 as a function of the photon energy hν with the semi-parabolic confinement frequency ω0 = 2 × 1014 s−1 for two cases: considering exciton effects and without considering exciton effects. The case of without considering exciton effects is the single electron state which is the problem of one-dimensional semi-parabolic oscillator. The wave functions and energy levels of the single electron state are similar to Eq. (6) and Eq. (7) while the mass of the electron m∗e instead of the total mass M. From Fig. 2 we can see that the optical rectification coefficient χ0(2) obtained by considering exciton effects is over two times larger than that obtained without considering exciton effects. This enhancement is attributed to the increase of the dipolar matrix element µ201 δ01 . However, it is clear that the symmetric system do not produce second-order nonlinearities because δ01 is equal to zero. In Fig. 3, we show the peak (2) values of optical rectification coefficient |χ0,max | as a function of the frequency ω0 for two case: considering exciton effects and without considering exciton effects according to the Eq. (13). From Fig. 3, we can see that (2) | will increase when the frequency ω0 decrease |χ0,max (2)
In this case, the Coulomb energy of the electron and hole interaction is a small correction to the quantization energies of electrons and holes [27], the Coulomb term in Eq. (3) may be neglected, the exciton wave
and |χ0,max | will increase more quickly as ω0 decrease (2)
for considering exciton effects. The |χ0,max | obtained by considering exciton effects is also over two times larger than that obtained when exciton effects are not
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(2)
Fig. 3. The optical rectification coefficient maximum value |χ0,max | versus the semi-parabolic confinement frequency ω0 for two cases: considering exciton effects (solid line) and without considering exciton effects (dashed line).
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(2)
Fig. 5. The peak values of optical rectification coefficient |χ0,max | as a function of the frequency ω0 .
frequency: ω0 = 1 × 1014 s−1 , ω0 = 1.5 × 1014 s−1 , ω0 = 2 × 1014 s−1 , ω0 = 2.5 × 1014 s−1 , and ω0 = 3 × 1014 s−1 , respectively. We find five peaks which start at hν = 0.132 eV, 0.198 eV, 0.263 eV, 0.329 eV, and 0.395 eV, respectively. A very important feature is ω01 will increase with the enhancement of the semiparabolic confinement frequency ω0 which makes the peaks move to the high-energy side which shows a strong-confinement-induced blue-shift of the exciton resonance in semiconductor quantum dots, which agree with the theoretical results [28]. These features make the semi-parabolic quantum dots very promising candidates for nonlinear optical materials applications.
(2)
Fig. 4. The optical rectification coefficient |χ0 | versus the photon energy hν for five different values of the semi-parabolic confinement frequency ω0 = 1 × 1014 s−1 , 1.5 × 1014 s−1 , 2 × 1014 s−1 , 2.5 × 1014 s−1 , and 3 × 1014 s−1 , respectively.
considered. In addition, exciton effects will be smaller with the frequency ω0 increase. From Eq. (13) we can see that the enhancement of µ201 δ01 result in the in(2) crease of |χ0,max |. From Fig. 2 and Fig. 3, we can see that the exciton effects on optical rectification coeffi(2) cient χ0 are very important. So it is apparently very important to take the exciton effects into account when we study the optical rectification coefficient for quantum dots. Fig. 4 shows the optical rectification coefficient (2) χ0 as a function of the photon energy hν for five different values of the semi-parabolic strong-confinement
3.2. Weak-confinement regime In the weak-confinement limit, the Coulomb interaction is more important than the quantization energies of the electrons and holes [27], so the parabolic confinement term of the relative motion part in Eq. (3) is neglected, the exciton wave functions and energy levels can be obtained through Eqs. (4)–(7) and Eqs. (10), (11). In Fig. 5, we show the peak values of optical (2) rectification coefficient |χ0,max | as a function of the (2) frequency ω0 . From Fig. 5, we can see that |χ0,max | will increase with the enhancement of the frequency ω0 , which is just opposite to that happens in the strongconfinement regime. In Fig. 6, we show the optical rectification coefficient χ0(2) as a function of the photon energy hν with the semi-parabolic confinement
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tion in this model. Moreover, the bigger the transverse relaxation time T2 is, the sharper the peaks will be and the bigger the peak intensities will be. However, the transverse relaxation time T2 is related not only to the quantum dot material and confining potential, but also to other factors, such as temperature and boundary conditions. Therefore, theoretical studies may make a great contribution to experimental studies, may have profound consequences as regards improvements of practical devices such as ultrafast optical switches, and may open up new opportunities for practical exploitation of the quantum-size effect in devices. (2)
Fig. 6. The optical rectification coefficient |χ0 | versus the photon energy hν for five different transverse relaxation time: T2 = 0.16 ps, T2 = 0.18 ps, T2 = 0.2 ps, T2 = 0.22 ps, and T2 = 0.24 ps, which are illustrated by the solid line, dashed line, dotted line, dash-dotted line, and dash-dot-dotted line, respectively.
frequency ω0 = 1 × 1012 s−1 for five different transverse relaxation time: T2 = 0.16 ps, T2 = 0.18 ps, T2 = 0.2 ps, T2 = 0.22 ps, and T2 = 0.24 ps, respectively. From Fig. 6, we can see that the peak value will increase as the transverse relaxation time T2 increase. And the bigger the peak is, the sharper the peak intensity will be. In addition, the five curves have two points of intersection at hν = 0.3905 eV and hν = 0.3975 eV, respectively. Because T2 is related to the effects of outer surroundings, this feature shows that the influence of outer surroundings can be neglected when hν = 0.3905 eV and hν = 0.3975 eV. So we can apply the feature to decrease the deviations in experiment.
4. Conclusions We present a simple and straightforward study of the optical rectification coefficient for a nanometersize quantum dots. The results show that the theoretical value of the optical rectification coefficient is greatly enhanced due to the exciton effects. The optical rectification coefficient obtained by considering exciton effects is over two times larger than that obtained only considering electron states. In addition, the large asymmetry of the semi-parabolic potential is also the causes for the enhancement of the optical rectifica-
Acknowledgements This work is supported by Guangdong Provincial Natural Science Foundation of China.
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