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Physical review. B, Condensed matter

Two-photon transitions to excitons in quantum wells

Alfredo Pasquarello, A. Quattropani

Published Nov 15, 1990• 21 citations
Publisher

Abstract

We calculate two-photon transition rates of exciton states in GaAs quantum wells for varying well widths. The exciton states are described in the effective-mass approximation and are obtained variationally assuming wave functions that are separable in the r coordinates. The two-photon transition rate is calculated taking explicitly into account the sum over all the intermediate states, using a variational procedure. We calculate transition rates of excitons of s symmetry as well as of p symmetry, which are allowed for polarization of the radiation beam along the growth axis and in the layer planes, respectively. From our calculations we find that transitions to s-type excitons have sufficiently large rates to be observable, whereas transitions to p-type excitons seem to have rates too small to be observable; these results are in agreement with experiment. The transition rate of s excitons increases significantly for decreasing well width, whereas the transition rate of p excitons is found to be almost independent of the well width. Moreover, our model allows us to calculate transition rates for photons of different energy. We have shown the resonant enhancement of the transition rate of s-type excitons when one of the photon energies approaches the energy of an intermediate state.

DOI
10.1103/physrevb.42.9073

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Physical Review Letters•Jun 26, 1989•35 citations
We calculate two-photon transition rates of exciton states in GaAs quantum wells for varying well widths. The exciton states are described in the effective-mass approximation and are obtained variationally assuming wave functions that are separable in the r coordinates. The two-photon transition rate is calculated taking explicitly into account the sum over all the intermediate states, using a variational procedure. We calculate transition rates of excitons of s symmetry as well as of p symmetry, which are allowed for polarization of the radiation beam along the growth axis and in the layer planes, respectively. From our calculations we find that transitions to s-type excitons have sufficiently large rates to be observable, whereas transitions to p-type excitons seem to have rates too small to be observable; these results are in agreement with experiment. The transition rate of s excitons increases significantly for decreasing well width, whereas the transition rate of p excitons is found to be almost independent of the well width. Moreover, our model allows us to calculate transition rates for photons of different energy. We have shown the resonant enhancement of the transition rate of s-type excitons when one of the photon energies approaches the energy of an intermediate state.
Transition rates of an electron confined in a quantum well due to interaction with photon (electromagnetic radiation) are approximated and compared with the exact results. The energy levels of the system were calculated using the Schrödinger equation within the effective mass approximation. Wave functions of an electron confined in a quantum well are approximated, which are in turn used to approximate optical transition rates when an electron interacts with electromagnetic radiation. The approximate wave functions exhibit close agreement with the exact functions at lower quantum states but become less accurate as the principal quantum number increases. The energy eigen values of the system increase with decreasing quantum well width and the accuracy of the approximate energy eigen values decreases for higher states. Transition rates are characterized by peaks which occur whenever the energy of the incident photon equals the energy difference between states. Since the accuracy of the approximations decreases for higher energy states, the positions of the peaks corresponding to transitions involving higher states become less accurate. Increase in temperature decreases peaks of absorption transition rates while it enhances peaks of emission transition rates, a behavior captured well by the approximations.
Two-photon absorption spectroscopy was performed in GaAs quantum wells with photon energy \ensuremath{\Elzxh}\ensuremath{\omega} near half the band gap Eg{\mathrm{E}}_{\mathrm{g}}Eg​. For the case of incident light polarized perpendicular to the confinement direction \ensuremath{\epsilon}^\ensuremath{\perp}z^, the onset of two-photon absorption occurs at the 2P state of the lowest-energy heavy-hole exciton and the absorption increases linearly with 2\ensuremath{\Elzxh}\ensuremath{\omega}-Eg{\mathrm{E}}_{\mathrm{g}}Eg​. For \ensuremath{\epsilon}^\ensuremath{\parallel}z^, the two-photon spectrum is stepped with superpositions of sharp 1S exciton features at odd \ensuremath{\Delta}N light-hole--conduction-band transitions. These results are used to accurately determine exciton binding energy, successive subband energy splittings, and band offsets.
Coherent Rayleigh scattering and incoherent luminescence are studied by femtosecond upconversion under resonant excitation as a function of exciton density in GaAs quantum wells. Comparing samples with different well widths allows to establish the relation between inhomogeneous broadening and the shape of resonant Rayleigh scattering. We find an excellent agreement between experiment and a classical model. In samples with moderate inhomogeneous broadening we provide evidence for the transition from coherent to incoherent emission. The comparison with excitonic dephasing times T2 obtained from four-wave-mixing experiments demonstrates that the incoherent exciton population rises within T2/2, at any exciton density when changed by more than an orders of magnitude. The experimental results confirm very clearly the concept predicting that the coherent polarization made up by localized excitons decays into an incoherent exciton population under conservation of the number of excited quanta.
We have identified several physical processes by which below-gap photons can modify excitonic absorption in GaAs quantum wells.1-3 These different phenomena can be distinguished by their temporal response times, which range from tens of nanoseconds to less than one picosecond. Our experiments also show that the relative contributions of these mechanisms to below-gap optical modulation of exciton absorption depend strongly on temperature as well as on the wavelength of the below-gap modulating beam. The size of the modulation indicates that a new class of absorptive and dispersive devices for optical signal processing is possible in which signals propagate in the transparent region of the crystal and experience negligible attenuation. In order to successfully interpret and calculate the magnitude of these exciting new effects, we show that it is necessary to take into account the special properties of two-dimensional excitons in semiconductor quantum wells.
We calculate the anisotropic two-photon absorption spectra of an AlGaAs-GaAs V-shaped quantum wire with realistic band structure, including excitonic effects. The absorption coefficient is obtained by integrating effective multiband Bloch equations accounting for the Coulomb interaction within the Hartree-Fock approximation. The various excitonic peaks are identified with respect to the involved energy subbands and the symmetry properties. Furthermore, A/sub 2/-excitons, which are dark for one-photon excitation, become bright for two-photon spectroscopy when the light is linearly polarized along a direction that is not a symmetry axis. We show that some of them may be observable.
Photoconductivity measurements revealing two-photon absorption in GaAs/AlGaAs multiple quantum wells are presented. It is shown that both the forbidden 1S and the allowed 2P excitons can be observed. The energy separation between 1S and 2P excitons is determined directly from the absorption spectra to high accuracy. The data give evidence for two different absorption mechanisms: One is the two-photon absorption and the other one is the absorption of the second-harmonic wave which is generated inside the sample by nonlinear optical processes. By means of this additional absorption mechanism, the observation of the 1S excitions in two-phonon absorption spectra can be explained qualitatively.