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arXiv:2302.08349 · Results in Optics

Iterative density matrix revisited: Excitonic phenomena on the InAs quantum well as a saturable absorber

Sami Ortakaya

Published Aug 27, 2023• 5 citations
arXivHTMLPDF

Abstract

By solving the Liouville equation, third-order nonlinear terms of the optical absorptive two quantum-level system are obtained through iterative density matrix calculations. In the improved model, the full frequency range is taken into account instead of just the weak absorptive limit. he process investigated in this study can be likened to saturation fitting for heavy-hole excitons in the InAs quantum well. Additionally, we demonstrate that the iterative method used for analyzing band-to-band excitonic spectra enables easy retrieval of the static dielectric constant.

DOI
10.1016/j.rio.2023.100512

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By solving the Liouville equation, third-order nonlinear terms of the optical absorptive two quantum-level system are obtained through iterative density matrix calculations. In the improved model, the full frequency range is taken into account instead of just the weak absorptive limit. he process investigated in this study can be likened to saturation fitting for heavy-hole excitons in the InAs quantum well. Additionally, we demonstrate that the iterative method used for analyzing band-to-band excitonic spectra enables easy retrieval of the static dielectric constant.
A detailed study of bandgap resonant optical nonlinearities in n-type InAs and their use in an optical bistable device is presented. The results of measurements of both nonlinear absorption and nonlinear refraction taken with an HF laser show good agreement with a band-filling model in which the contribution from the light-hole band and the effects of large initial free carrier densities are included. Evidence of the saturation of the nonlinear refraction through the carrier-density-dependent recombination rate is presented, and it is shown that this effect together with diffraction effects accounts for the critical power for bistability observed in the reflected signal from an InAs etalon at 77 K.
Saturation properties of absorption, dispersion, and mixing susceptibilities in semiconductors have been described using the density matrix approach. This model, which takes into account both intraband thermalization and interband relaxation including Auger recombination, allows for numerical computation of the susceptibilities whatever the temperature and laser intensity are. As a simplification, an equivalent two-level system is derived which represents very well absorption and mixing susceptibilities for any laser wavelength, dispersion being correctly described by this model only when the pump frequency is close to or smaller than the band-gap energy. Such an equivalence is very easy to handle and allows for great simplification when the susceptibilities are introduced into the propagation equations.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.
Taking as a starting point the Liouville equation for the one-body density matrix operator, a general expression for the third-order nonlinear current density responsible for the optical Kerr effect associated with intersubband transitions in a quantum well is derived by use of the perturbative expansion method. The nonlinear current density is related to the local field inside the quantum well via a nonlocal relation. By combining the field-induced nonlinear part of the current density with the Maxwell equation, a nonlinear integral equation for the local field is established. For a two-level quantum well, the integral equation is solved exactly. The nonlinear optical reflectivity, transmittance, and absorbance of the quantum well in turn are obtained. Numerical calculations of the light-induced change of the optical spectra are presented for a Al 0.5 Ga 0.5 As/GaAs/Al 0.5 Ga 0.5 As quantum well. The numerical results show that the optical reflectivity and absorbance decrease, and the transmittance increases as the optical intensity of the incident light is increased. In the case where the quantum-well structure is heavily doped, a notable light-induced red-shift of the location of the resonance peak in the optical spectra stemming from the combined local-field effect and field-induced population redistribution is predicted.
The bleaching of the n=1 and n=2 heavy-hole exciton absorption has been observed at room temperature in ternary alloy Al0.37Ga0.63As/AlAs multiple-quantum-well structures by nanoseconds pump-and-probe spectroscopy. The relative variation of the absorption coefficient [(−Δα)/α0] are about 15% and 8%, respectively. The nonlinear absorption coefficient (α2) and the nonlinear absorption cross section has been deduced. The value of the saturation density deduced from our spectra is in good agreement with the theoretical predictions. The influence of the staggered band alignment on the observed process is also discussed.
In the past few years advance in nanofabrication techniques have made it possible to make well-like compound semiconductor structures of low nanometer size.In these structures,the state of the electron in the direction perpendicular to the length of the well is quantized with sufficiently large separation in energy levels and the motion of the electron parallel to the length is that of a quasi-free electron.One of the most remarkable properties of these one-dimensional electron system is that the optical transitions between the size-quantized subbands are feasible.The result shows there is a large optical nonlinearities in asymmetric quantum wells.The problem of optical saturation of intersubband absorption has recently attracted considerable attention.The problem is very important to the development of superlattice devices such as fast infrared detectors and(modulators) because one needs to know the range of input optical intensities for which linear operation is(possible.) The analytical expression of the linear and the third-order nonlinear optical intersubband absorption coefficients are obtained in Pschl-Teller wells using the density matrix formalism taking into account the(intrasubband) relaxation.Since there are two adjustable parameters κ and λ in Pschl-Teller wells,the shapes of the wells will change with the change of the parameters κ and λ,accordingly the linear and the third-order nonlinear optical absorption coefficients will change with the parameters κ and λ.It is shown that the optical absorption coefficient depends on the parameter,and the optical intensity strongly. The absorption saturation will occur with I increase.Numerical results are presented for a typical AlGaAs/GaAs Pschl-Teller well.