Excitonic effects on the second-order nonlinear optical properties of semi-spherical quantum dots
© Flórez and Camacho; licensee Springer. 2011
Received: 25 August 2010
Accepted: 29 March 2011
Published: 29 March 2011
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. 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 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 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]. 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.
In this study we find eigenenergies and eigenstates of an exciton in a semi-spherical quantum dot solving the corresponding three-dimensional Schrödinger equation using a finite elements method and taking into account both the confinement and Coulomb potentials of the electron-hole 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 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 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 semi-spherical shape.
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  and . 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 strong-confinement limit is established when , or equivalently , and the weak-confinement limit when , or .
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) , T 1 = 1 ps, T 2 = 0.2 ps , σs = 5 × 1024 m -3, ε = 12.53, Γ 0 = 1/0.14ps -1, N = 3 × 1016 cm-3.
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 × 1014 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 strong-confinement limit as a satisfactory approximation in the case of small quantum dots. Accordingly with Equation 9, the frequencies ω 0 = 1 × 1013 s-1 and ω 0 = 2 × 1014 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 COM-SOL Multiphysics, which offers the possibility of defining a geometry, in this case the upper half of a sphere, and to solve the time-independent 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 × 1013 s-1
OR peak energy
SHG peak energy
Eigenenergies of the exciton and peak energies of the OR and SHG coefficients with and without Coulomb interaction for ω 0 = 2 × 1014 s-1
OR peak energy
SHG peak energy
We can see from Tables 1 and 2 that the eigenenergies obtained with Coulomb inter-action are smaller than those obtained without that interaction. The explanation to this fact is that there is an attractive Coulomb potential between the electron-hole 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 × 1013 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 × 1014 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 blue-shift of the OR and SHG peaks of the order of meV for both ω 0 = 1 × 1013 s-1 and ω 0 = 2 × 1014 s-1.
Dipole matrix element products of the OR and SHG coefficients with and without Coulomb interaction for ω 0 = 1 × 1013 s-1
M 01 M 12 M 20
Dipole matrix element products of the OR and SHG coefficients with and without Coulomb interaction for ω 0 = 2 × 1014 s-1
μ 01 μ 12 μ 20
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 second-order nonlinear optical properties of semi-spherical quantum dots. We find that Coulomb interaction manifests itself in a blue-shift 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 second-order optical nonlinearities in asymmetrical quantum dots with the aim of to detect the magnitude of this effect.
second harmonic generation.
This work was partially supported by Facultad de Ciencias of Universidad de los Andes.
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