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_2O_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.

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