J. Phys. Chem. 1991, 95,8453-8461
a453
FEATURE ARTICLE Calculation of Natural Radiative Lifetimes from the Absorption and Fluorescence Properties of Semiconductors and Molecules James R. Bolton*It and Mary D.Archer*'* Photochemistry Unit, Department of Chemistry, The University of Western Ontario, London, Ontario, Canada N6A 5B7, and Department of Biochemistry, Imperial College, London SW7 ZAZ, United Kingdom (Received: May 8, 1991; In Final Form: August 9, 1991)
Using a detailed balance approach to treat the upper and lower thermalized electronic states of a broadband absorber, equations have been derived from which radiative lifetimes T, can be calculated from the known absorption and fluorescence properties of semiconductors and molecular chromophores. Three methods for calculating these lifetimes are outlined. Method 1 utilizes the absorption coefficients and fluorescence distribution in the frequency range where they overlap. These lifetimes T,(ov~) are very sensitive to small errors in the choice of the bandgap energy U,. Method 2 utilizes the absorption coefficients through the entire fluorescence range and is applicable only to systems in which the fluorescence lies within the absorption spectrum. For semiconductors, method 2 results in the previously derived van Roosbroeck-Shockley equation. These r,(abs) lifetimes are also very sensitive to the choice of U8,but for semiconductors this is usually known to sufficient accuracy. Method 3 assumes a mirror image relation between the absorption and fluorescence spectra and is particularly useful for molecular chromophores, for which method 2 and (usually) method 1 are not applicable. s,(mir) values obtained by method 3 will give a good account of r,(exp) values only when the mirror image assumptions hold. Quantum mechanical arguments, based on the relation between upward and downward transition moments, have also been used to calculate radiative lifetimes for molecules. We comment on the Strickler-Berg (SB) equation for molecules, derived from this approach, and show that it reduces to the Farster equation when the mirror image assumptions are imposed. Finally we treat GaAs and rhodamine 6G as examples and compare T , values calculated using the methods we describe.
I. Introduction Over 70 years ago Einstein' showed that, for the simple system of two nondegenerate energy levels shown in Figure 1, there is a fundamental relation between the probability of excitation by the absorption of a photon from the lower level 1 to the upper level u and the probability of emission of a photon by spontaneous radiative decay from the upper level. This relation is usually written in the form
AUI= (8rhn,)ulu3/P)BI,
(1)
where 4 and po ( w 3 ) are the equilibrium concentrations of conduction-band electrons and valence-band holes, a, (m-') is the absorption coefficient at frequency u, and the integral is taken over the entire absorption band. Our eq 2 is a combination of van Roosbroeck and Shockley's eqs 4, 11, and 15. NeporentSin 1958 and independently McCumbe# in 1964 used a similar argument based on detailed balance a t thermal equilibrium to relate absorption and emission cross sections
