The simplest situation is just what happens in an old-school incandescent light bulb, or the heating element in a toaster. An applied voltage \(V\) drives a current \(I\) in a wire, and as we learn in freshman physics, power \(IV\) is dissipated in the metal - energy is transferred into the electrons (spreading them out up to higher energy levels within the metal than in the undriven situation, with energy transfer between the electrons due to electron-electron interactions) and the disorganized vibrational jiggling of the atoms (as the electrons also couple to lattice vibrations, the phonons). The scattering electrons and jiggling ions emit light (even classically, that's what accelerating charges do). If we look on time scales and distance scales long compared to the various e-e and e-lattice scattering processes, we can describe the vibrations and electron populations as having some local temperature. Light is just electromagnetic waves. Light in thermal equilibrium with a system (on average, no net energy transfer between the light and the system) is distributed in a particular way generically called a black body spectrum. The short version: current heat metal structures, and hot structures glow. My own group found an example of this with very short platinum wires.
In nanostructures, things can get more complicated. Metal nanostructures can support collective electronic modes called plasmons. Plasmons can "decay" in different ways, including emitting photons (just like an atom in an excited state can emit a photon and end up in the ground state, if appropriate rules are followed). It was found more than 40 years ago that a metal/insulator/metal tunnel junction can emit light when driven electrically. The idea is, a tunneling electron picks up energy \(eV\) when going from one side of the junction to the other. Some fraction of tunneling electrons deposit that energy into plasmon modes, and some of those plasmon modes decay radiatively, spitting out light with energy \(\hbar \omega \le eV\).
This same thing can happen in scanning tunneling microscopy. There is a "tip mode" plasmon where the STM tip is above the conducting sample, and this can be excited electrically. That tip plasmon can decay optically and spit out photons, as discussed in some detail here back in 1990.
The situation is tricky, though. When it comes down to atomic-scale tunneling and all the details, there are deep connections between light emission and shot noise. Light emission is often seen at energies larger than \(eV\), implying that there can be multi-electron processes at work. In planar tunneling structures, light emission can happen at considerably higher energies, and it really looks there like there is radiation due to the nonequilibrium electronic distribution. It's a fascinating area - lots of rich physics.
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