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Physical review. B, Condensed matter

Two-level approach to saturation properties in semiconductor materials

F. de Rougemont, Robert Frey

Published Jan 15, 1988• 31 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.37.1237

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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.
Optical-intensity-dependent gain and mixing susceptibilities of amplifying semiconductor materials have been calculated using the density-matrix approach. The model, which takes intraband relaxation and carrier-injection rate into account, allows for numerical computation of these susceptibilities, whatever the medium saturation is. As a simplification, an equivalent two-level system is derived which represents the gain and nearly degenerate four-wave mixing susceptibilities. Such an equivalence, which leads to the same field-induced polarization as the conventional rate equation model, also connects the two-level (or rate equation) parameters to the real characteristics of the semiconductor material.>
The nonlinear absorption factors in a direct-gap semiconductor are calculated both for the pump and probe waves using the density matrix formalism. The key feature of the calculation is the consideration of the carrier dispersion. It is shown for the first time that the dispersion law defines the saturation behavior of the absorption factor. The intensity and frequency dependences of the absorption factors calculated differ significantly from those obtained previously. It is also shown that the conventional theory of nonlinear interband absorption based on the kinetic equations for carriers leads to the correct results only in the limit of small pump intensities.
We develop a simple and general framework for understanding the relationships between absorption and emission in multilevel systems (molecules, semiconductors, condensed matter, etc.) in and out of equilibrium. Conditions under which the absorption and emission spectra display mirror image symmetry are determined. The relationship between the absorption and emission cross sections at frequency \ensuremath{\omega} and temperature T, previously derived by Neporent and McCumber for thermally equilibrated systems, is generalized and a new derivation is presented which extends its range of application to systems out of equilibrium. This derivation requires only microscopic reversibility and that the optically coupled manifold of levels belonging to both initial and final electronic states are each in internal thermal equilibrium at the same temperature T. The initial and final (e.g., ground and excited electronic) states need not be in equilibrium. If more than one excited-electronic-state manifold participates in the dynamics and these excited-electronic-state manifolds are not themselves in equilibrium, a somewhat more complicated relationship between absorption and emission can be obtained, provided that the internal degrees of freedom within the electronic-state manifolds are each in equilibrium at the same temperature.Effects of inhomogeneous broadening are examined. Good agreement is found between the Neporent-McCumber relationship and experimental absorption and emission spectra for the laser dyes rhodamine-6G and 2,5-biphenyloxazol PPO, and the solid-state laser crystal alexandrite. A new relationship between absorption and emission in direct-band-gap semiconductor materials is derived. This relationship allows us to calculate the emission from a direct-band-gap semiconductor provided the dependence of the band-gap energy on excitation level is not large. These relationships hold even when pumping conditions differ for emission and absorption experiments; the dependence of emission on excitation level can thereby be calculated. This permits the gain spectra of semiconductor diode lasers to be determined from their absorption spectra. In the limit that both the absorption and emission are measured for a thermally equilibrated nondegenerate direct-band-gap semiconductor system with identical pumping conditions for the absorption and emission experiments, our new relationship reduces to previous results.
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.
We have developed a model for calculating saturated absorption spectra for dipole transitions in multi-level atoms. Using a semiclassical density matrix formalism, we derive a set of coupled differential equations for the internal state of the atom in a standing wave light field. The equations are solved using standard integration techniques. The absorption at each laser detuning is found from an average of the absorption for a number of velocities along the laser field, thermally weighted. The method is relatively efficient computationally yet quantitatively predicts important details of saturated absorption spectra including saturation, crossover resonances, merging of absorption lines at high intensity and optical pumping between hyperfine levels. We have measured saturated absorption and fluorescence spectra of 85 Rb, and compare to our computational results for a 36-level model.
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.