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Journal of Applied Physics

Calculation of giant third-order nonlinear susceptibilities due to excitonic processes in rectangular GaAs quantum well wires

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

Published Jan 1, 1994• 10 citations
Publisher

Abstract

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.

DOI
10.1063/1.355803

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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.

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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.
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 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 Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$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.
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.
In recent years,the subject of optical nonlinearity in semiconductor quantum confined structures such as quantum wires and quantum dots has attracted a considerable attention because of their relevance for studying practical applications and as a probe for the electronic structure of media.It has been shown that the low dimensional quantum systems present very large nonlinear optical susceptibilities .Theoretically,most of the studies on quantum dots are focused on spherical quantum dots for simplicity.The shape of quantum dot depends on growth conditions.In fact,quantum dots are better described as thin disk or cylinder.In the present work third order nonlinear optical susceptibilities in disk shaped quantum dots due to resonant intersubband transition are analyzed by a compact density matrix approach.It has rarely been discussed in literatures.Firstly,we consider an electron moving in a quantum disk whose radius R is very large compared to its thickness d .For simplicity,we assume that the isotropic effective mass approximation is valid within the disk and that the inner potential is null while the outer confining potential is approximately infinite.Using the adiabatic approximation we solve the three dimensional Schrdinger equation and obtain the analytical expressions of wave functions and energy levels in cylindrical coordinates.Secondly,we consider the system is excited by an electromagnetic field with frequency ω. The electromagnetic field is incident with a polarization vector normal to the disk.We treat H′=ezE(t) as a perturbation term.By using the compact density matrix method,we derive a general formula for third order nonlinear optical susceptibilites employing a four level model in quantum disk.Finally,numerical results are illustrated for a typical GaAs quantum disk.We calculate the third order nonlinear optical susceptibilities as a function of the disk thickness.It is shown that the third order nonlinear optical susceptibilities in quantum disk are 10～100 times larger than that in quantum well for various disk thickness.Furthermore,if we choose an optimized thickness of quantum disk,we could obtain a quite large refractive index with a comparatively small absorption coefficient.Therefore,we suggest that the quantum dots would be an excellent system in the field of optical communications and optical switching.
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.
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.