Discretized Euler-Lagrange variational study of nonlinear optical rectification coefficients

This paper is concerned with developing a discretized Euler-Lagrange variational method in order to study the nonlinear optical rectification coefficients of cosine-shaped quantum wells under the influence of an external electric field. The proposed approach is employed to solve a nonlinear Schrodinger equation, in which the nonlinear term is due to the electron-electron interaction effect. The accuracy of the proposed numerical method is first demonstrated by comparing our results for ground and first excited state energies with three other numerical methods. Two different semiconducting nanostructures GaAs/AlGaAs (weak confinement regime) and GaN/AlN (strong confinement regime) are then compared, and it is shown that GaAs/AlGaAs quantum wells have larger optical rectification coefficients. We show that the electron-electron interaction effect can increase the optical rectification coefficient in some situations up to three times. Hence, we conclude that it is necessary to include the nonlinear term due to this effect in the Schrodinger equation. This nonlinear term has been neglected in relevant past works.