Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Application of Degenerately Doped Metal Oxides in the Study of Photoinduced Interfacial Electron Transfer

Degenerately doped In₂O₃:Sn semiconductor nanoparticles (<i>nano</i>ITO) have been used to study the photoinduced interfacial electron-transfer reactivity of surface-bound [Ru^II(bpy)₂(4,4′-(PO₃H₂)₂-bpy)]²⁺ (RuP²⁺) molecules as a function of driving force over a range of 1.8 eV. The metallic properties of the ITO nanoparticles, present within an interconnected mesoporous film, allowed for the driving force to be tuned by controlling their Fermi level with an external bias while their optical transparency allowed for transient absorption spectroscopy to be used to monitor electron-transfer kinetics. Photoinduced electron transfer from excited-state -RuP^2+* molecules to <i>nano</i>ITO was found to be dependent on applied bias and competitive with nonradiative energy transfer to <i>nano</i>ITO. Back electron transfer from <i>nano</i>ITO to oxidized -RuP³⁺ was also dependent on the applied bias but without complication from inter- or intraparticle electron diffusion in the oxide nanoparticles. Analysis of the electron injection kinetics as a function of driving force using Marcus–Gerischer theory resulted in an experimental estimate of the reorganization energy for the excited-state -RuP^3+/2+* redox couple of λ* = 0.83 eV and an electronic coupling matrix element, arising from electronic wave function overlap between the donor orbital in the molecule and the acceptor orbital(s) in the <i>nano</i>ITO electrode, of <i>H</i>_ab = 20–45 cm^–1. Similar analysis of the back electron-transfer kinetics yielded λ = 0.56 eV for the ground-state -RuP^3+/2+ redox couple and <i>H</i>_ab = 2–4 cm^–1. The use of these wide band gap, degenerately doped materials provides a unique experimental approach for investigating single-site electron transfer at the surface of oxide nanoparticles

Driving Force Dependent, Photoinduced Electron Transfer at Degenerately Doped, Optically Transparent Semiconductor Nanoparticle Interfaces

Photoinduced, interfacial electron injection and back electron transfer between surface-bound [Ru^II(bpy)₂(4,4′-(PO₃H₂)₂-bpy)]²⁺ and degenerately doped In₂O₃:Sn nanoparticles, present in mesoporous thin films (nanoITO), have been studied as a function of applied external bias. Due to the metallic behavior of the nanoITO films, application of an external bias was used to vary the Fermi level in the oxide and, with it, the driving force for electron transfer (Δ<i>G</i>^o′). By controlling the external bias, Δ<i>G</i>^o′ was varied from 0 to −1.8 eV for electron injection and from −0.3 to −1.3 eV for back electron transfer. Analysis of the back electron-transfer data, obtained from transient absorption measurements, using Marcus–Gerischer theory gave an experimental estimate of λ = 0.56 eV for the reorganization energy of the surface-bound Ru^III/II couple in acetonitrile with 0.1 M LiClO₄ electrolyte

Photoinduced Interfacial Electron Transfer within a Mesoporous Transparent Conducting Oxide Film

Interfacial electron transfer to and from conductive Sn-doped In₂O₃ (ITO) nanoparticles (NPs) in mesoporous thin films has been investigated by transient absorption measurements using surface-bound [Ru^II(bpy)₂(dcb)]²⁺ (bpy is 2,2′-bipyridyl and dcb is 4,4′-(COOH)₂-2,2′-bipyridyl). Metal-to-ligand charge transfer excitation in 0.1 M LiClO₄ MeCN results in efficient electron injection into the ITO NPs on the picosecond time scale followed by back electron transfer on the nanosecond time scale. Rates of back electron transfer are dependent on thermal annealing conditions with the rate constant increasing from 1.8 × 10⁸ s^–1 for oxidizing annealing conditions to 8.0 × 10⁸ s^–1 for reducing conditions, presumably due to an enhanced electron concentration in the latter