Excitonic effects on the secondorder nonlinear optical properties of semispherical quantum dots
 Jefferson Flórez^{1}Email author and
 Ángela Camacho^{1}
DOI: 10.1186/1556276X6268
© Flórez and Camacho; licensee Springer. 2011
Received: 25 August 2010
Accepted: 29 March 2011
Published: 29 March 2011
Abstract
We study the excitonic effects on the secondorder nonlinear optical properties of semispherical quantum dots considering, on the same footing, the confinement potential of the electronhole pair and the Coulomb interaction between them. The exciton is confined in a semispherical geometry by means of a threedimensional semiparabolic potential. We calculate the optical rectification and second harmonic generation coefficients for two different values of the confinement frequency based on the numerically computed energies and wavefunctions of the exciton. We present the results as a function of the incident photon energy for GaAs/AlGaAs quantum dots ranging from few nanometers to tens of nanometers. We find that the secondorder nonlinear coefficients exhibit not only a blueshift of the order of meV but also a change of intensity compared with the results obtained ignoring the Coulomb interaction in the socalled strongconfinement limit.
Introduction
Nonlinear optical properties of semiconductor quantum dots have attracted considerable interest due to their several potential applications [1–4]. In particular, secondorder nonlinear optical properties, such as nonlinear optical rectification (OR) and second harmonic generation (SHG), have received special theoretical [5–8] and experimental [9, 10] attention due to their magnitudes being stronger than those of highorder ones, making them the first nonlinear optical effects experimentally observed.
The confinement of carriers provided by a quantum dot is well described by a parabolic potential when only the lowest excited states of the carriers are considered. However, selfassembled quantum dots growth in the laboratory usually exhibit asymmetrical shapes that ensure the generation of nonlinear optical effects. In order to model these asymmetries, an asymmetrical potential is required.
Recently, several authors [5, 6, 8] studied the effects of an exciton on the secondorder nonlinear properties in onedimensional semiparabolic quantum dots. Using analytical approximate results, they showed that the excitonic effects enhance significantly the OR and SHG coefficients. They used the socalled strongconfinement limit, ignoring in this way the Coulomb interaction between electron and hole because of the quantum dot dimensions are smaller than the effective Bohr radius, and finding that the excitonic effect reduces itself to an effectivemass model in one particle scheme.
In this study we find eigenenergies and eigenstates of an exciton in a semispherical quantum dot solving the corresponding threedimensional Schrödinger equation using a finite elements method and taking into account both the confinement and Coulomb potentials of the electronhole pair. We present the OR and SHG coefficients as a function of the incident photon energy with and without Coulomb potential. Our results show that energy and intensity of the peaks in the secondorder nonlinear optical coefficients change when Coulomb interaction is introduced.
This article is organized as follows. In "Theory" section, we present the characteristic quantities of the harmonic and Coulomb potentials, and the definitions of the weak and strongconfinement limits in terms of these parameters. In addition, we present the analytical expressions for the optical nonlinearities, such as OR and SHG, obtained by the density matrix formalism. In "Results" section, we show the OR and SHG coefficients with and without Coulomb interaction as a function of the incident photon energy for two quantum dot sizes. We also give account of the changes presented by the secondorder nonlinear coefficients focusing in the role played by the Coulomb interaction. Conclusions are summarized in final section.
Theory
The angle θ is the usual polar angle in spherical coordinates, and ω _{0} the oscillator frequency considered in this study the same for the electron and the hole. The potential defined in Equation 2 confines the exciton in the upper half of a sphere, i.e., the quantum dot has a semispherical shape.
Hamiltonian (8) has been solved analytically in two limiting cases (strong and weak confinement) for onedimensional quantum dots. The eigenfunctions and eigenvalues are presented in references [5] and [8]. In onedimensional case, the confinement potential also imposes constraints to spatial coordinates, resulting in a hydrogenlike (asymmetricharmonic) reduced particle Hamiltonian for weak (strong) limit.
The strongconfinement limit is established when , or equivalently , and the weakconfinement limit when , or .
Results
In this study, the results are presented for a GaAs/AlGaAs structure. We have used the following parameters in the calculations: = 0.067m _{0}, = 0.09m _{0} (m _{0} is the mass of a free electron) [4], T _{1} = 1 ps, T _{2} = 0.2 ps [12], σ_{s} = 5 × 10^{24} m ^{3}[5], ε = 12.53, Γ _{0} = 1/0.14ps ^{1}, N = 3 × 10^{16} cm^{3}[8].
With the aim of exploring the nonlinear behavior at higher frequencies, i.e., when the quantum dot size is smaller than in the previous case, we choose ω _{0} = 2 × 10^{14} s^{1}, in which the quantity L is less than , or ħω _{0} is greater than , differing in both cases by one order of magnitude as can be seen in Figure 1. Because of this difference, several authors[5, 6, 8] used the strongconfinement limit as a satisfactory approximation in the case of small quantum dots. Accordingly with Equation 9, the frequencies ω _{0} = 1 × 10^{13} s^{1} and ω _{0} = 2 × 10^{14} s^{1} define a quantum dot size of L = 17.4 nm and L = 3.9 nm, respectively. This means that our results are suitable for the current quantum dot sizes that range from few nanometers to tens of nanometers.
We find numerically eigenenergies and eigenstates of Hamiltonian (8) by a finite elements method for the two frequencies mentioned above. We have used the software COMSOL Multiphysics, which offers the possibility of defining a geometry, in this case the upper half of a sphere, and to solve the timeindependent Schrödinger equation with appropriate boundary conditions.
Eigenenergies of the exciton and peak energies of the OR and SHG coefficients with and without Coulomb interaction for ω _{0} = 1 × 10^{13} s^{1}
Energy (meV)  With Coulomb  Without Coulomb  Diff. 

E _{0}  11.218  16.455  5.237 
E _{1}  26.147  29.619  3.472 
E _{2}  40.000  42.784  2.784 
OR peak energy  14.929  13.164  1.765 
SHG peak energy  14.660  13.164  1.496 
Eigenenergies of the exciton and peak energies of the OR and SHG coefficients with and without Coulomb interaction for ω _{0} = 2 × 10^{14} s^{1}
Energy (meV)  With Coulomb  Without Coulomb  Diff. 

E _{0}  306.59  329.10  22.51 
E _{1}  577.06  592.39  15.33 
E _{2}  843.31  855.67  12.36 
OR peak energy  270.47  263.28  7.19 
SHG peak energy  269.41  263.28  6.13 
We can see from Tables 1 and 2 that the eigenenergies obtained with Coulomb interaction are smaller than those obtained without that interaction. The explanation to this fact is that there is an attractive Coulomb potential between the electronhole pair that implies a reduction of the eigenenergies for the exciton. However, the eigenenergies are affected in different ways depending on the quantum state. For example, for the ground state ω _{0} = 1 × 10^{13} s^{1}, Table 1, we have an energy difference of 5.237 meV, while for the first and second excited states the differences are of 3.472 and 2.784 meV, respectively. We have a similar situation for ω _{0} = 2 × 10^{14} s^{1}, Table 2. This is because the mean spatial separation between the electron and the hole increases, and therefore the Coulomb interaction decreases, as the energy of the quantum state increases. The final result is a blueshift of the OR and SHG peaks of the order of meV for both ω _{0} = 1 × 10^{13} s^{1} and ω _{0} = 2 × 10^{14} s^{1}.
Dipole matrix element products of the OR and SHG coefficients with and without Coulomb interaction for ω _{0} = 1 × 10^{13} s^{1}
Coefficient  [nm^{3}]  With Coulomb  Without Coulomb 

OR 
 1365  1222 
SHG  M _{01} M _{12} M _{20}  1237  1635 
Dipole matrix element products of the OR and SHG coefficients with and without Coulomb interaction for ω _{0} = 2 × 10^{14} s^{1}
Coefficient  [nm^{3}]  With Coulomb  Without Coulomb 

OR 
 16.95  14.03 
SHG  μ _{01} μ _{12} μ _{20}  17.65  18.55 
Conclusions
Contrary to the assumption that Coulomb interaction can be neglected when the quantum dot dimensions are smaller than the effective Bohr radius, we show that this interaction affects the excitonic effects of the secondorder nonlinear optical properties of semispherical quantum dots. We find that Coulomb interaction manifests itself in a blueshift of the energy peaks of the order of several meV in the studied spectra. These results were found for two quantum dot sizes, in the first one the characteristic quantities of the harmonic and Coulomb potentials are equals, and in the second one they differ by one order of magnitude. This means that the Coulomb interaction plays an important role even when the quantum dot sizes are smaller than the effective Bohr radius.
Therefore, we encourage experimentalists to carry out measurements of secondorder optical nonlinearities in asymmetrical quantum dots with the aim of to detect the magnitude of this effect.
Abbreviations
 OR:

optical rectification
 SHG:

second harmonic generation.
Declarations
Acknowledgements
This work was partially supported by Facultad de Ciencias of Universidad de los Andes.
Authors’ Affiliations
References
 Fu Y, Hellström S, Ågren H: Nonlinear optical properties of quantum dots: excitons in nanostructures. J Nonlinear Opt Phys Mater 2009, 18: 195. 10.1142/S0218863509004579View Article
 Gotoh H, Kamada H: Coherent nonlinear optical properties in quantum dots. NTT Tech Rev 2005, 3: 19.
 Bimberg D, Meuer C, Lämmlin M, Liebich S, Kim J, Kovsh A, Krestnikov I, Eisenstein G: Nonlinear properties of quantum dot semiconductor optical amplifiers at 1.3 μm. Chin Opt Lett 2008, 6: 724. 10.3788/COL20080610.0724View Article
 Rosencher E, Vinter B: Optoelectronics. Cambridge: Cambridge University Press; 2003.
 Yu YB, Zhu SN, Guo KX: Exciton effects on the nonlinear optical rectification in onedimesional quantum dots. Phys Lett A 2005, 335: 175. 10.1016/j.physleta.2004.12.013View Article
 Baskoutas S, Paspalakis E, Terzis AF: Effects of excitons in nonlinear optical rectification in semiparabolic quantum dots. Phys Rev B 2006, 74: 153306. 10.1103/PhysRevB.74.153306View Article
 Zhang CJ, Guo KX, Lu ZE: Exciton effects on the optical absorptions in onedimensional quantum dots. Phys E 2007, 36: 92. 10.1016/j.physe.2006.08.009View Article
 Karabulut İ, Şafak H, Tomak M: Excitonic effects on the nonlinear optical properties of small quantum dots. J Phys D: Appl Phys 2008, 41: 155104. 10.1088/00223727/41/15/155104View Article
 Brunhes T, Boucaud P, Sauvage S, Lemaître A, Gérard JM, Glotin F, Prazeres R, Ortega JM: Infrared secondorder optical susceptibility in InAs/GaAs selfassembled quantum dots. Phys Rev B 2000, 61: 5662. 10.1103/PhysRevB.61.5562View Article
 Sauvage S, Boucaud P, Brunhes T, Glotin F, Prazeres R, Ortega JM, Gérard JM: Secondharmonic generation resonant with sp transition in InAs/GaAs selfassembled quantum dots. Phys Rev B 2001, 63: 113312. 10.1103/PhysRevB.63.113312View Article
 Que W: Excitons in quantum dots with parabolic confinement. Phys Rev B 1992, 45: 11036. 10.1103/PhysRevB.45.11036View Article
 Rosencher E, Bois P: Model System for optical nonlinearities: Asymmetric quantum wells. Phys Rev B 1991, 44: 11315. 10.1103/PhysRevB.44.11315View Article
 Boyd R: Nonlinear Optics. 3rd edition. New York: Elsevier; 2008.
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.