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Physical review. B, Condensed matter

Theoretical studies of the effects of a magnetic field on excitonic nonlinear optical properties of quantum wires

Alexander A. Balandin, Supriyo Bandyopadhyay

Published Aug 15, 1996• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.54.5721

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The magnetic-field dependence of the third-order excitonic nonlinear susceptibility \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3) in a quantum wire is explored within the rotating wave approximation. Both the real and imaginary parts of \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(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 \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(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 magnetic-field dependence of the nonlinear differential refractive index Δnand absorption Δα in quantum wires—measured by non-degenerate pump and probe spectroscopy—is investigated theoretically. The nonlinearities arise from population saturation of the excitonic state under optical pumping and the formation of biexcitons (excitonic molecules). Both Δnand Δ α exhibit positive and negative peaks at certain pump and probe detuning frequencies associated with the formation of biexcitons and bleaching of excitons, respectively. The amplitude, lineshape and the frequency at which these peaks occur can be modulated by a magnetic field which opens up the possibility of realizing novel magneto-optical devices. Additionally, the magnetic field may allow us to realize a relatively large variation in the differential refractive index over a range of frequencies without significant accompanying absorption, thereby allowing the observation of optical bistability.
Transient optical nonlinearities of semiconductor quantum dots in the strongly confined regime are studied, in which the exciton, biexciton and magnetic field effects are taken into account. The photo induced band-to-band electronic transitions are considered from both the heavy-hole (hh) and light-hole (lh) valence bands to the lowest (1s) exciton state below the fundamental absorption edge. The time dependent perturbation technique for a three-level system, under the near band gap resonant excitation regime, has been employed; with the incorporation of the relaxation and dephasing mechanisms phenomenologically. The induced polarization is calculated from where the real and imaginary parts of the third-order nonlinear optical susceptibility, responsible for the occurrence of nonlinear refraction and nonlinear absorption respectively, have been derived. The temporal and detuning spectra of nonlinear refraction and absorption; and their dependence on density of biexcitons and a moderately strong magnetic field, have been analyzed. It is observed that the nonlinear refraction and nonlinear absorption enhance with increasing magnetic field in small CdS quantum dots.
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.
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.
In this work, the influence of an external magnetic field on the refractive index changes and optical absorption coefficients of a typical GaAs/AlxGa1−xAs coaxial cylindrical quantum well wire is studied. Energy eigenvalues and eigenvectors in the presence of a magnetic field are calculated using the fourth-order Runge-Kutta method and optical properties are obtained using the compact density matrix approach. It is found that the resonant peak values of the linear and the third-order nonlinear refractive index changes as well as the optical absorption coefficients of a cylindrical quantum wire decrease with increasing the magnetic field. However, these optical properties of a coaxial cylindrical quantum well wire are nonmonotonic functions of the magnetic field, due to the anticrossing of the subbands. The anticrossing effect leads to a large increase of the resonant peak values of these optical properties. Results also indicate that the resonant peak values of these optical properties decrease with increasing the barrier width.
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.