Electric field effect on the second-order nonlinear optical properties of parabolic and semiparabolic quantum wells

By using the compact-density-matrix approach and iterative procedure, a detailed procedure for the calculation of the second-harmonic generation (SHG) susceptibility tensor is given in the electric-field-biased parabolic and semiparabolic quantum wells (QW's). The simple analytical formula for the SHG susceptibility in the systems is also deduced. By adopting the methods of envelope wave function and displacement harmonic oscillation, the electronic states in parabolic and semi parabolic QW's with applied electric fields are exactly solved. Numerical results on typical ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{Al}/\mathrm{GaAs}$ materials show that, for the same effective widths, the SHG susceptibility in semiparabolic QW is larger than that in parabolic QW due to the self-asymmetry of the semiparabolic QW, and the applied electric field can make the SHG susceptibilities in both systems enhance remarkably. Moreover, the SHG susceptibility also sensitively depends on the relaxation rate of the systems.