# How do you model quantum-confinement effects in optically active (e.g. plasmonic) mesoscopic structures?

Optically active materials, and particularly metals can be described fairly well with Classical Electrodynamics. For example, interfaces can be modeled using the Drude model, and for spherical particles we have Mie theory. This is valid for large systems (for example microns in depth or diameter). However, as systems become smaller, microscopic quantum effects can play an important role in the description of the electrodynamics. These effects are commonly known as "Quantum Confinement Effects".

A classic and illustrative example of change in optical activity as a function of size is in silver and gold nanoparticles.

In the completely classical regime, Finite Difference Time Domain (FDTD) simulations are used for electrodynamic simulations. Is it possible to use FDTD for simulating mesoscopic structures (nanostructures) where quantum size effects matter? What is the limit in size that can accurately be described by classical electrodynamics for a gold nanoparticle?

What methods are there available to model quantum-confinement effects in nanostructures of different sizes?

• I can only talk about the quantum confinement effects in nanostructures and how it affects the energetic and electronic properties. Typical size ranges of the nanostructure (say a quantum dot (0D), nanoribbon (1D) , nanowire (1D) i.e. structures with finite dimension) are around 1nm in the finite direction for confinement effects to show up. These effects manifests by say increasing the band gap of materials or by affecting the potential energy surface of a reaction. And all this can be easily calculated by first principles quantum mechanical methods such as DFT.
– gogo
May 22 '20 at 3:16

• I'm not 100% sure, but it would likely be an immense pain. Modeling spatial dispersion (nonlocal dielectric response, $$q$$-dependent dielectric function, it has many names) involves different wavevectors of light experiencing different dielectric functions. I'm not sure how this can be implemented in a real-space, method such as FDTD since it involves a full convolution of your nonlocal (but still short-ranged) involved of the dielectric function. At the very least, this suggests to be at that every time step, you would have to Fourier transform your fields, use the convolution theorem to model the action of a nonlocal dielectric function on the incoming E fields, then FT back to get your real-space grid so you can then time-step to the next (there may be subtleties here too that I'm completely missing out).