Nonlinear optical properties of an exciton in a spherical quantum dot (2011)
We study the nonlinear optical properties of an exciton confined in a quantum dot and placedin an uniform static electric field. The exciton is confined using a parabolic potential. The electric field is parallel to thez-axis of the quantum dot, providing the asymmetry needed toget nonlinear optical phenomena. Within the effective-mass approach, we write the Hamiltonian of the exciton in terms of the center-of-mass and the relative coordinate, obtaining two separated Hamiltonians. The center-of-mass motion is a harmonic oscillator, while the relative motion includes both a parabolic and a Coulomb potential from the electron-hole electrostatic interaction, so that this Hamiltonian is unsolvable analytically. In order to solve approximately the relative Hamiltonian, we treat perturbatively the Coulomb potential in the so-called strong-confinement regime, where the matrix elements of this interaction are much smaller than the unperturbed energies. Thus we analyze to first order the energy corrections to the ground state and the triply degenerate first excited states. We obtain that the corrections to the eigenvalues are only reliable in the strong regime, which in terms of the QD characteristic sizes means that they must be much smaller than 18 nm forGaAs/AlGaAs QDs. These sizes are reduced even more when the electric field is introduced. To obtain results for larger quantum dots, from a few to tens of nanometers, we solve numerically the relative Hamiltonian in the intermediate regime using a finite elements method. In this regime the Coulomb interaction and the confinement potential are taken exactly. As an interesting result, we obtain that the Eigen functions are not only shifted in the opposite direction of the electric field, but also they are elongated. Both effects, the shift and the elongation, enhance the asymmetry of the QD compared with the results obtained in the strong regime where the Eigen functions are only shifted. With the numerical results, we compute the nonlinear optical rectification susceptibility as well as the linear and nonlinear absorption coefficient and refractive index changes as a function of the incident photon energy. The main result is that the nonlinear optical proper-ties of the QD are improved in the intermediate regime as a straightforward consequence of the enhanced asymmetry. We also find that the nonlinear optical coefficients exhibit a wide spectrum of behaviors, such as a sign inversion. This fact offers the possibility of tuning the confinement energy (or the size) of a QD to obtain the desirable optical properties.