Theory of a Single Dye Molecule Blinking with a Diffusion-Based Power Law Distribution

Abstract

In single molecule studies of injection of an electron from a photoexcited dye into a semiconductor nanoparticle or into a film of such nanoparticles, the injection may be into the conduction band or into the band gap, depending on the system. The theory of the process and its return are discussed, in particular when a power law for the waiting time distribution may be expected and what that power might be. To this end a reaction−diffusion equation is set up and solved. When the injection is into the conduction band, a power law is predicted for the return of the electron to the dye cation but not for the injection. After a short time, the law for the waiting time distribution has a power of −1. At short times, before the slower return due to an increasing radius is recognized, the power law is −1/2. When the injection is into the band gap, a −1 power law is predicted for both the injection and the return. Available data are discussed in terms of the theory. A corollary is that single molecule studies for the injection can determine whether the injection is into the band gap or into the conduction band. The theory is tested by single molecule studies of various systems, such as comparing different dye−TiO 2 , dye−Al 2 O 3 , and dye−ZrO 2 systems and comparing specific dye−TiO 2 systems as a function of pH, and dye hole injection into p-type NiO.

Understanding Photoinduced Charge Transfer Dynamics of Single Perylenediimide Dyes in a Polymer Matrix by Bin-Time Dependence of their Fluorescence Blinking Statistics

“…For example, the statistics of organic dyes in biological system, and at semiconductor interfaces clearly differ from power-law behavior (e.g., multiexponential kinetics). Although the power-law distribution of the on-time kinetics has been explained by several models, ,, its mechanism is incompletely understood, and a unified view appears to be still lacking.…”

A Maximum Likelihood Method for Power Law Distributions That Does Not Break Down When the Slope Is Close to Unity

“…Indeed, eqs 11 and 12 do not give a close fit to the data (Figures 1and 2). For the single-molecule study of the electron injection from the dye onto a semiconductor surface, m 0 ≅ 1 is the theoretically expected value, 1 and a treatment of this particular case is particularly necessary.…”

Long-range photoinduced electron transfer dynamics in rigid media

“…[97][98][99] Power law kinetics have also been reported for electron injection and/or recombination between surface-bound dye molecules and semiconductor metal oxide surfaces. [100][101][102][103] We have modeled the power law kinetics for back electron transfer by the following procedure. The number of electron transfer products at time t, N(t), is given in eqn (16) based on the distance dependence of the electron transfer rate constant k(R) and the distribution function describing the initial distances between reduced complex and oxidized quencher, f (R).…”