Abstract
Understanding the interplay between illumination and the electron distribution in metallic nanostructures is a crucial step towards developing applications such as plasmonic photocatalysis for green fuels, nanoscale photodetection and more. Elucidating this interplay is challenging, as it requires taking into account all channels of energy flow in the electronic system. Here, we develop such a theory, which is based on a coupled Boltzmann-heat equations and requires only energy conservation and basic thermodynamics, where the electron distribution, and the electron and phonon (lattice) temperatures are determined uniquely. Applying this theory to realistic illuminated nanoparticle systems, we find that the electron and phonon temperatures are similar, thus justifying the (classical) single-temperature models. We show that while the fraction of high-energy “hot” carriers compared to thermalized carriers grows substantially with illumination intensity, it remains extremely small (on the order of 10−8). Importantly, most of the absorbed illumination power goes into heating rather than generating hot carriers, thus rendering plasmonic hot carrier generation extremely inefficient. Our formulation allows for the first time a unique quantitative comparison of theory and measurements of steady-state electron distributions in metallic nanostructures.
Highlights
Understanding the interplay between illumination and the electron distribution in metallic nanostructures is a crucial step towards developing applications such as plasmonic photocatalysis for green fuels, nanoscale photodetection and more
The model should be used for finding the steady-state non-equilibrium electron distribution that is established under continuous wave (CW) illumination, as appropriate for technologicallyimportant applications such as photodetection and photocatalysis
We show that the population of nonequilibrium energetic electrons and holes (i.e., negative values of the deviation from thermal equilibrium (f − fT, see below) see e.g.,36). can increase dramatically under illumination, yet this process is extremely inefficient, as almost all the absorbed energy leads to heating; the electron and phonon temperatures are found to be essentially similar, justifying the use of the classical single-temperate heat model[31]
Summary
Electron-phonon (e − ph) collisions cause energy transfer between the electrons and lattice; they occur within a (narrow) energy window (the width of which is comparable to the Debye energy) near the Fermi energy, see Fig. 1b and Fig. S1b. Recent studies of the steady-state nonequilibrium in metals (e.g.,26–28) relied on a fixed value for Tph (choosing it to be either identical to the electron temperature, or to the environment temperature, see discussion above) and/or treated the rate of e − ph energy transfer using the relaxation time approximation with an e − ph collision rate that is independent of the field and particle shape While these approaches ensure that energy is conserved in the electron subsystem, they ignore the dependence of the energy transfer to the environment on the nanoparticle shape, the thermal properties of the host material, the electric field strength and the temperature difference between electrons and phonons. This generic case leads to several surprising qualitative new insights, as well as to quantitative predictions of non-equilibrium carrier distributions
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