*distribution functions*, and how many electrons are in the levels with energies between \(E\) and \(E + dE\). In thermal equilibrium (meaning our system of interest is free to exchange energy in the form of heat with some large reservoir described by a well-defined temperature \(T\)), the distribution of electrons as a function of energy is given by the Fermi-Dirac distribution.

So, what are "hot" electrons? If we have a system driven out of equilibrium, it's possible to have the electrons arranged in a non-thermal (non-FD distribution!) way. Two examples are of particular interest at the nanoscale. In a transistor, say, or other nanoelectronic device, it is possible to apply a voltage across the system so that \(eV >> k_{\mathrm{B}}T\) and inject charge carriers at energies well above the thermally distributed population. Often electron-electron scattering on the 10-100 fs timescale redistributes the energy across the electrons, restoring a thermal distribution at some higher effective temperature (and on longer timescales, that energy cascades down into the vibrations of the lattice). Electrons in a metal like Au at the top of the distribution are typically moving at speeds of \(\sim 10^{6}\) m/s (!!), so that means that near where the current is injected, on distance scales like 10-100 nm, there can be "hot" electrons well above the FD distribution.

The other key way to generate "hot" electrons is by optical absorption. A visible photon (perhaps a green one with an energy \(\hbar \omega\) of 2 eV) can be absorbed by a metal or a semiconductor, and this can excite an electron at an energy \(\hbar \omega\) above the top of the FD distribution. Often, on the 10-100 fs timescale, as above, that energy gets redistributed among many electrons, and then later into the lattice. That's heating by optical absorption. In recent years, there has been an enormous amount of interest in trying to capture and use those hot electrons or their energy before there is a chance for that energy go become converted to heat. See here, for instance, for thoughts about solar energy harvesting, or here for a discussion of hot electron photochemistry. Nanoscale systems are of great interest in this field for several reasons, including the essential fact that hot electrons generated in them can access the system surface or boundary in the crucial timespan before energy relaxation.

(Talking about this and thermoelectricity now sets the stage so I can talk about our recent paper in an upcoming post.)

*Really, the whole many-body electron wavefunction has to be antisymmetric under the exchange of any two electrons, so it's wrong to talk as if one particular electron is sitting in one particular state, but let's ignore that for now. Also, in general, the energy levels of the many-electron system actually depend on the number and arrangement of the electrons in the system (correlation effects!), but let's ignore that, too.