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Physical review. B, Condensed matter

Frequency-dependent optical dephasing and the nature of inhomogeneous broadening in crystals

Leslie J. Root, James L. Skinner

Published Sep 15, 1985• 25 citations
Publisher

Abstract

We present a theory of frequency-dependent optical dephasing as measured by narrow-band photon-echo experiments. We consider a substitutionally disordered crystal with inhomogeneous broadening (diagonal disorder) and electronic interactions between occupied sites. We include the possibility of site-energy correlations. We find that the frequency dependence of the dephasing rate 1/${T}_{2}$ is very sensitive to the ratio \ensuremath{\xi}/a, where \ensuremath{\xi} is the correlation length and a is the lattice spacing. When \ensuremath{\xi}/a\ensuremath{\rightarrow}\ensuremath{\infty}, and there are macroscopic domains of resonant ions, 1/${T}_{2}$ becomes frequency independent. In the other limit when \ensuremath{\xi}/a\ensuremath{\rightarrow}0, and the site energies are uncorrelated (microscopic broadening), 1/${T}_{2}$ is proportional to the inhomogeneous line shape. We have compared our results to experiments by Macfarlane and Shelby on ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$:${\mathrm{Eu}}^{3+}$ (2 at. %) and ${\mathrm{EuP}}_{5}$${\mathrm{O}}_{14}$. We find reasonable agreement with experiment when \ensuremath{\xi} is less than a few lattice sites. Thus, for these systems the model of microscopic broadening is more appropriate than that of macroscopic broadening.

DOI
10.1103/physrevb.32.4111

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Physical review. B, Condensed matter•May 15, 1989•124 citations
We present a theory of frequency-dependent optical dephasing as measured by narrow-band photon-echo experiments. We consider a substitutionally disordered crystal with inhomogeneous broadening (diagonal disorder) and electronic interactions between occupied sites. We include the possibility of site-energy correlations. We find that the frequency dependence of the dephasing rate 1/T2{T}_{2}T2​ is very sensitive to the ratio \ensuremath{\xi}/a, where \ensuremath{\xi} is the correlation length and a is the lattice spacing. When \ensuremath{\xi}/a\ensuremath{\rightarrow}\ensuremath{\infty}, and there are macroscopic domains of resonant ions, 1/T2{T}_{2}T2​ becomes frequency independent. In the other limit when \ensuremath{\xi}/a\ensuremath{\rightarrow}0, and the site energies are uncorrelated (microscopic broadening), 1/T2{T}_{2}T2​ is proportional to the inhomogeneous line shape. We have compared our results to experiments by Macfarlane and Shelby on {\mathrm{Y}}_{2}$${\mathrm{O}}_{3}:Eu3+{\mathrm{Eu}}^{3+}Eu3+ (2 at. %) and {\mathrm{EuP}}_{5}$${\mathrm{O}}_{14}. We find reasonable agreement with experiment when \ensuremath{\xi} is less than a few lattice sites. Thus, for these systems the model of microscopic broadening is more appropriate than that of macroscopic broadening.
For a crystal with both diagonal disorder (inhomogeneous broadening) and substitutional disorder we have calculated the contribution to the photon echo decay due to purely electronic intersite interactions. We have considered the two important limiting cases of macroscopic and microscopic inhomogeneous broadening. In the case of macroscopic broadening we find that the optical dephasing rate constant 1/T2∼f J, where f is the fractional concentration of impurities and J is the nearest-neighbor intersite interaction in frequency units. This result is valid for f≪1. In the opposite limit of microscopic inhomogeneous broadening, we find that 1/T2∼f J2/δ̄, where δ̄ is the width in frequency units of the inhomogeneous distribution. This result is valid for arbitrary f, including f=1, and for J≪δ̄. In the case that f≪1 and J≪δ̄, 1/T2 is much smaller for microscopic broadening than for macroscopic broadening. We discuss our results in relation to several experiments on mixed molecular crystals and to an experiment on the stoichiometric material EuP5O14.
We present a nonperturbative theory of the thermal line broadening of optical transitions due to pure dephasing by phonons. The theory extends previous perturbative results of McCumber and Sturge. The theory is applied to experimental results on mixed organic crystals and ruby. We also present a theory of dephasing at T=0 K due to interacting impurities. Our results predict a frequency dependence to 1/T2. We analyze experiments on Y2O3:Eu3+ by Macfarlane and Shelby with this theory, and are able to comment on the nature of inhomogeneous broadening in this system.
Abstract Inhomogeneous broadening is often considered a nuisance in solid state spectroscopy, and the usual treatment of it is phenomenological. In fact, the inhomogeneous broadening of absorption spectra, coupled with a microscopic theory, can provide one with interesting information about the nature of disorder in different systems. More sophisticated experiments can provide more detailed information about disorder by probing certain correlations between energy levels. For example, flourescence line narrowing spectra of isolated impurities involve correlations among three different vibronic levels. Pressure effects on hole burning spectra probe correlations of impurity transition energies before and after the pressure change. For interacting impurity systems, photon echo experiments involve transition energy correlations between different impurities. Efforts to understand these effects within a microscopic framework are discussed.
In recent years the microscopic description of inhomogeneous broadening of electronic transitions of chromophores in crystals and glasses has attracted much interest [1–3]. In the phenomenological model proposed by Selzer [1], it is assumed that the energies of the initial and final states involved in the optical transition exhibit a one-to-one correlation. In an alternative approach by Lee et al [2], the two states are considered as completely uncorrelated. Very recently, a more general microscopic theory has been presented by Laird and Skinner [3]. In the latter theory the aforementioned models are obtained as limiting cases. So far relatively few experiments have been reported which focus on the origin of inhomogeneous broadening.
We present a theory of photon echoes from impurity molecules in mixed crystals that takes into account electronic interactions among impurity molecules and inhomogeneous broadening of the electronic absorption spectrum. The system is modeled as a collection of randomly placed interacting two level systems with a distribution of transition energies. The observable of the echo experiment is expanded in powers of the density of two level systems, and the first two terms of this expansion are calculated. A scaling argument suggests that the density expansion is rapidly convergent for concentrations of experimental interest, and the density expansion is used to construct a Padé approximant. At times comparable to or less than the inverse of the width of the inhomogeneous distribution of resonance frequencies, the time dependence of the signal is determined primarily by the interactions between the impurities and is independent of the inhomogeneous width. At longer times the inhomogeneity becomes significant and the echo signal displays a much weaker time dependence. We examine the effect of different microscopic models of inhomogeneous broadening on the echo observable. We demonstrate that the photon echo experiment can provide a sensitive probe of interactions between impurities in mixed crystals.
A detailed theoretical analysis of optical dephasing measurements performed on complex systems, e.g., glasses, proteins, or complex crystals, is presented. Unlike simple crystals, such systems undergo dynamical processes which have a very broad rate distribution. Dynamics can occur on a variety of time scales ranging from subpicoseconds to seconds, or longer. The formalism is based on a four-point correlation-function description of line-narrowing experiments. The results of optical dephasing measurements (time domain) or linewidth measurements (frequency domain) depend on a time TW{T}_{W}TW​ which defines the time scale associated with the particular experimental technique. Detailed information about the rate distribution of the system's dynamics is obtained from the change in the optical dephasing rate as the experimental time scale is changed, not from the dephasing rate measured in any individual experiment. A fundamental result, which is independent of the nature of the rate distribution or the coupling to the optical center, is proven; i.e., the derivative of the optical dephasing rate with respect to TW{T}_{W}TW​, the experimental time scale, is directly proportional to the Laplace transform of the fluctuation rate distribution. As examples, the formal results are applied to two types of systems, optical centers in a glass and in a complex crystal. For the crystalline system Pr3+{\mathrm{Pr}}^{3+}Pr3+:CaF2{\mathrm{CaF}}_{2}CaF2​, TW{T}_{W}TW​-dependent optical dephasing data from the literature are analyzed quantitatively, where previously only a qualitative description was possible.