We study the excitonic effects on the second-order nonlinear optical properties of semi-spherical quantum dots considering, on the same footing, the confinement potential of the electron-hole pair and the Coulomb interaction between them.
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The exciton is confined in a semi-spherical geometry by means of a three-dimensional semi-parabolic 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 find that the second-order nonlinear coefficients exhibit not only a blue-shift of the order of meV but also a change of intensity compared with the results obtained ignoring the Coulomb interaction in the so-called strong-confinement limit. Nonlinear optical properties of semiconductor quantum dots have attracted considerable interest due to their several potential applications [ 1 — 4 ].
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In particular, second-order 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 high-order 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, self-assembled 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 second-order nonlinear properties in one-dimensional semi-parabolic quantum dots. Using analytical approximate results, they showed that the excitonic effects enhance significantly the OR and SHG coefficients.
They used the so-called strong-confinement 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 effective-mass model in one particle scheme.
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 second-order 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 strong-confinement 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 second-order nonlinear coefficients focusing in the role played by the Coulomb interaction. Conclusions are summarized in final section. The effective-mass Hamiltonian for the electron-hole pair in the three-dimensional quantum dot reads [ 11 ].
The potential defined in Equation 2 confines the exciton in the upper half of a sphere, i.
Hamiltonian 1 can be separated in terms of center-of-mass and relative coordinates, respectively,. The center-of-mass and relative position coordinates are defined as usual,.
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The explicit separability of the center-of-mass and relative coordinates in Equation 3 lead to the following total envelope wave function and total energy for the system:. The center-of-mass part of Hamiltonian 3 is a three-dimensional semi-parabolic oscillator that can be solved analytically. Therefore, the problem is now reduced to solve the relative motion Hamiltonian:.
Hamiltonian 8 has been solved analytically in two limiting cases strong and weak confinement for one-dimensional quantum dots. The eigenfunctions and eigenvalues are presented in references [ 5 ] and [ 8 ]. In one-dimensional case, the confinement potential also imposes constraints to spatial coordinates, resulting in a hydrogen-like asymmetric-harmonic reduced particle Hamiltonian for weak strong limit.
The harmonic potential in Equation 8 defines both the size L of the quantum dot,. On the other hand, the Coulomb potential defines the effective Bohr radius and the effective Rydberg energy of the electron-hole interaction,. The strong-confinement limit is established when , or equivalently , and the weak-confinement limit when , or. The second-order nonlinear optical coefficients can be obtained by density matrix approach and perturbation expansion method [ 12 , 13 ]. The expression for the OR coefficient, within a two-level system approach, is given by [ 5 , 6 ].
The SHG coefficient in a three level system is [ 8 ]. Under the double resonance condition, i. For this reason, we conclude that, in this frequency range, neither the strong-confinement limit nor the weak limit can be assumed because both interactions, harmonic and Coulomb, are important. Therefore, we propose a numerical technique to calculate eigenenergies and eigenstates of Hamiltonian 8 , considering the harmonic and Coulomb potentials. With the aim of exploring the nonlinear behavior at higher frequencies, i. Because of this difference, several authors[ 5 , 6 , 8 ] used the strong-confinement limit as a satisfactory approximation in the case of small quantum dots.
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.
The terms involving quantum states and energies in Equations 13 and 19 are found using the eigenstates and eigenenergies previously calculated. In each figure, we present the corresponding nonlinear optical coefficient considering excitonic effects with and without Coulomb interaction. For comparative purposes, we also present the case without excitonic effects, i. The blue line corresponds to the case without excitonic effects. Figures 2 and 3 reproduce the reported results quite well [ 5 , 8 ] regarding the enhancement of the nonlinear optical coefficients due to the quantum confinement of the exciton.
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