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Physical review. B, Condensed matter

Exciton/biexciton energies in rectangular GaAs/ Alx Ga1−x As quantum-well wires including finite Al-graded band offsets with application to third-order optical susceptibilities

Frank L. Madarasz, Frank Szmulowicz, F. Kenneth Hopkins

Published Sep 15, 1995• 15 citations
Publisher

Abstract

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{\ensuremath{-}}\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{\ensuremath{-}}1}$ esu and a fairly large off-resonance absorption due to biexciton formation.

DOI
10.1103/physrevb.52.8964

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Physical review. B, Condensed matter•Aug 15, 1996•4 citations
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{\ensuremath{-}}\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\ensuremath−1{10}^{\mathrm{\ensuremath{-}}1}10\ensuremath−1 esu and a fairly large off-resonance absorption due to biexciton formation.
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 AlGaAs cladding. Moreover, the Al interdiffusion into the wire and the finite band offsets between the wire and the cladding have been included. The Coulomb interaction terms are treated exactly in their full 3D form throughout the calculation, especially in the case of the biexciton, a more physically realistic procedure then used in previous calculations which employed an effective 1D potential. Our treatment is unique in the use of a 2D Fourier expansion in the Coulomb potential terms. For the range of dimensions studied, the inclusion of the Al interdiffusion had a pronounced affect on the binding energies when compared to those obtained from the infinite barrier model. 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 and the amount of pump detuning, values of the susceptibilities to be on the order of 10-1 esu and a large off-resonance optical gain due to biexciton formation.
Exciton and biexciton biding energies and wave functions are calculated variationally for rectangular GaAs quantum well wires in an effective mass approximation. Coulomb interaction terms are treated exactly in their full three-dimensional form throughout the calculation, a more physically realistic procedure than employed in previous calculations that used effective one-dimensional potentials. Our treatment is unique in the use of a two-dimensional Fourier expansion of the Coulomb potential, that removes the difficulty of dealing with the 1/r singularity and considerably reduces the computational effort. Using the results of exciton and biexciton calculations and following the approach of T. Ishihara [Phys. Status Solidi 159, 371 (1990)] in order to eliminate the dimensional dependence of the third-order nonlinear optical susceptibility χ(3), we estimate its magnitude for near-resonant excitonic absorption. We obtain χ(3)’s on the order of 10−1–1 esu for various wire sizes.
In a recent calculation, we obtained exciton and biexciton binding energies and wave functions for a rectangular quantum wire in order to determine its third-order nonlinear optical susceptibility ${\mathrm{\ensuremath{\chi}}}^{(3)}$. Using a variational calculation involving two parameters---exciton radius and exciton-exciton separation within the biexciton---we computed the binding energies of the exciton and biexciton. In the present paper, we have included a third parameter to allow for the variation of exciton radius within the biexciton. As the result of this added variational freedom, the biexciton binding energies changed by 7--10 % and the magnitudes of all variational parameters and of ${\mathrm{\ensuremath{\chi}}}^{(3)}$ changed by approximately 20% for wire dimensions between 25\ifmmode\times\else\texttimes\fi{}25 and 300\ifmmode\times\else\texttimes\fi{}300 \AA{} 2^{2}2.
The recently reported high excitonic and biexcitonic binding energies in GaAs/GaxAl1-xAs quantum wells and quantum wells wires suggests the possibility of strong optical nonlinearities around the biexcitonic resonance. We calculate using a simple boson model the magnitude of the absorptive and dispersive nonlinearities in GaAs, around the biexciton resonance comparing bulk, quantum well and quantum well wire structures. An assessment of inhomogeneous line broadening due to a Gaussian distribution of quantum well wire radii is also presented.
We report measurements of two-photon transitions to the biexciton state and exciton-biexciton transitions (M band) in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As multiple quantum wells by spectrally resolving the self-diffraction in a four-wave-mixing experiment. Furthermore, nonlinear quantum beats between a coherently prepared population of excitons and biexcitons are observed. The analysis yields a binding energy of 1.8 meV for the biexciton in 116-A\r{} quantum wells.
The magnetic-field dependence of the third-order excitonic nonlinear susceptibility ${\mathrm{\ensuremath{\chi}}}^{(3)}$ in a quantum wire is explored within the rotating wave approximation. Both the real and imaginary parts of ${\mathrm{\ensuremath{\chi}}}^{(3)}$, arising from population saturation of the excitonic state under optical pumping, are calculated for a GaAs wire as a function of magnetic field and pump-probe detuning frequencies. The imaginary part of ${\mathrm{\ensuremath{\chi}}}^{(3)}$ exhibits a negative peak associated with the bleaching of the excitonic resonance and a positive, broad, off-resonance absorption peak associated with biexciton formation. The amplitude, line shape, and spectral frequency of both these peaks can be modulated by a magnetic field which indicates the possibility of using such a field to probe the mechanism underlying optical nonlinearity in a quantum wire. Furthermore, the field can also be used to tune the optical nonlinearity over a range of frequencies which has device applications. \textcopyright{} 1996 The American Physical Society.