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

Question

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?

Answer 1

1. How do you model quantum-confinement effects in optically active (e.g. plasmonic) mesoscopic structures?
   • There should be a Comsol module that corresponds to this work, since it's mentioned that they implement the theory using that.

2. Is it possible to use FDTD for simulating mesoscopic structures (nanostructures) where quantum size effects matter?

   • 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).

3. What is the limit in size that can accurately be described by classical electrodynamics for a gold nanoparticle?

   • Sizes above 5-20 nanometers, generally, depending on your required accuracy. This number is a bit contested, but roughly around here is when you'll notice simple Mie theory using step-function-dielectric models of spheres and core-shells start to break down compared to experiment. I recommend this, this, and this for an intuition into the length scales where these nonlocal quantum confinement/electron spill-out effects become important.

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

   • This is a bit open-ended, since the question arises: What is your intended observable? There are beautiful applications of simple models to model excitons in core-shell quantum dots in the weak confinement limit (small nanoparticles, ~10-30 nm in diameter). In these regimes, your exciton ends up behaving like a particle in a finite well, a finite amount of energy greater than the optical gap, since the first available state is raised due to the finite extent of the dot.
   • For plasmonics, you can try LCAO-based TDDFT where you can make gold nanoparticles, etc (careful of the number of atoms, this is LCAO, but it's still DFT, don't get too crazy unless you have a supercomputer at your disposal) or other finite structures and attempt to have a go. The (many) subtleties of LCAO-TDDFT apply to this, so if you're not familiar with this method, it may take a bit to get an intuition for whether a calculation's results are physically meaningful or not.