Electrooptical functions and ellipsometric parameters of excitons in cylindrical quantum dots

Abstract

We show how to compute the optical functions (the complex electrosusceptibility tensor, dielectric function, electroreflection spectra and ellipsometric parameters) for semiconductor quantum dots (QD) exposed to a uniform electric field in the growth direction, including the excitonic effects. The method uses the microscopic calculation of the QD excitonic wave functions and energy levels, and the macroscopic real density matrix approach (RDMA) to compute the electromagnetic fields and susceptibilities. The electron-hole screened Coulomb potential is adapted and the valence band structure is taken into account in the cylindrical approximation. In the microscopic calculations we solve the 6-dimensional two-particles Schrödinger equation by transforming it into an infinite set of coupled second order 2-dimensional differential equations with the appropriate boundary conditions. These differential equations are solved numerically giving the eigenfunctions and the energy eigenvalues. Then we used the RDMA and computed the frequency- and electric field strength dependent complex excitonic susceptibility tensor. The above approach enables us to determine the relative oscillator strength connected with excitonic resonances and to find the averaged susceptibilities for light- and heavy-holes excitons. Having the frequency dependent complex susceptibility tensor, we calculate the electrooptical functions for a QD. Numerical calculations have been performed for a InGaAs QD with a constant electric field applied in the growth direction. The optical Stokes parameters and ellipsometric parameters ψ and Δ as functions of the frequency and the angle of incidence are also determined. A good agreement with experiment is obtained.