Exciton/biexciton energies in rectangular GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum-well wires including finite Al-graded band offsets with application to third-order optical susceptibilities

Exciton and biexciton binding energies and wave functions are calculated with a three-parameter variational model in an effective-mass approximation for a rectangular GaAs quantum-well wire surrounded by an ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As cladding. Moreover, the Al interdiffusion into the wire and the finite band offsets between the wire and the cladding have been included. For the range of dimensions studied, the inclusion of the Al interdiffusion had a pronounced effect on the binding energies when compared to those obtained from the infinite barrier model [Frank L. Madarasz, Frank Szmulowicz, F. Kenneth Hopkins, and Donald L. Dorsey, Phys. Rev. B 49, 13 528 (1994); J. Appl. Phys. 75, 639 (1994); Phys. Rev B 51, 4633 (1995)]. Using the results of the exciton and biexciton calculation, we calculate the third-order nonlinear optical susceptibility as a function of pump-probe frequencies in a small range about the exciton absorption resonance. We have found, depending upon wire dimensions, broadening parameter(s) size, and the amount of pump detuning, values of the imaginary parts of the susceptibilities to be on the order of -${10}^{\mathrm{-}1}$ esu and a fairly large off-resonance absorption due to biexciton formation.