I have recently written a paper (together with my supervisor Prof. Christoph Ortner) about understanding how materials behave when impurities are added to them. In particular, I am interested in seeing what happens at a very small scale (roughly between 10-100 nanometres). This area has been well-studied, though there is plenty more to figure out. Although my work is mostly focused on the underlying maths problem, a lot of important ideas from physics come into play. The idea of this page is to give a sense of the problem without going into too much detail.

Since this probably sounds intimidating, here’s a nice video that shows what I’m interested in, where you can see what’s happening in a material on an atomic level.

This movie was made using stop-motion, by manually positioning carbon monoxide molecules (the video calls them "atoms", but they're really molecules) on a copper surface using a scanning tunnelling microscope. What we’re looking at is an image of the charge over the surface, with positive charge being brighter and negative charge darker. This page has plenty of additional information.

While this is really cool, the idea of manipulating individual atoms or molecules is not new. In fact, back in 1990 researchers at IBM made headlines when they spelt out the company name, using the same technology; here's a link if you're interested.

First, let’s consider the background material, which is made up of copper atoms. Even though the material is made up of positive nuclei and negative electrons, the charge is constant over the surface. This is because copper is a metal, so you might remember from school that it has a “sea of free electrons”. In physics textbooks, this is often called an electron gas or the even better old-fashioned name jellium. The idea is that the electrons are free to move and spread out across the surface, which results in a uniform charge and also explains why copper is a good conductor.

What I’m interested in is the ripple patterns that form around the carbon monoxide molecules. This happens whenever you add an impurity to the surface of a metal, at very low temperatures. You can see them in the movie since they cooled down the material to less than 13 K, or -260 °C (which is not far from absolute zero). At higher temperatures, the carbon monoxide molecules would move randomly across the surface.

These ripples are called Friedel oscillations and are caused by the electrons on the surface being disturbed by the impurity. The positive charge of the carbon monoxide nuclei attracts additional electrons to the copper surface, so the charge around the nuclei is no longer uniform. This also generates an effect called screening. Using different models to study materials leads to different screening effects. You can study this effect using various models. Depending on the model used and the material being investigated, the calculations lead to different rates of screening for different models.

If you look at the carbon monoxide nuclei in a vacuum, as they are positively charged, they create a force that attracts negative charges and repels positive charges. This is described by Coulomb’s law, and satisfies an inverse-square law, meaning that if you double the distance, the force becomes four times weaker. Alternatively, when the carbon monoxide nuclei are on a copper surface, it attracts nearby electrons. As a result, the overall force generated by the nuclei is much weaker, as the surrounding electrons counteract it with their own opposite force, as they have an opposite charge. It is the screening effect that causes the oscillations to die down further from the impurity.

However, this doesn’t fully explain the ripple patterns that form on the surface, since the darker bands contain fewer electrons than in the background material. In order to explain this, we need some quantum mechanics. One fundamental and strange idea is called the wave-particle duality, which states that all matter has the properties of a particle and a wave. We can view the behaviour of the electrons on the surface as a diffraction pattern, like how dropping a stone in water generates circular ripples. I could go into more detail but it’d get pretty technical, so if you want more information, I’d recommend this and this.

While screening and Friedel oscillations aren’t the main focus of my paper, they come up in a discussion in Remark 6 on Page 8. The model I work with is called the Thomas—Fermi—von Weizsӓcker (TFW) model, named after Enrico Fermi, Llewellyn Thomas and Carl Friedrich von Weizsӓcker. It’s worth pointing out that I don’t look at impurities on the surface, but inside the metal instead. That said, screening and Friedel oscillations occur for the same reasons.

I have shown that in the TFW model, adding an impurity to a metal causes a local disturbance of the electrons. This is clear from the movie, but is not so easy to show mathematically. As a consequence, this result generalises Thomas—Fermi screening to look at impurities in metals where the background isn’t uniform. Since the TFW model considers a simpler electron-electron interaction, it is inconclusive as to whether Friedel oscillations occur in this model. To take this into account, we need to consider more advanced models, like Hartree—Fock or Kohn—Sham, which are much better at treating the electron-electron interaction. I might look into this some day.

For more information, here are some references:

• Kittel - Quantum Theory of Solids,
• Ashcroft and Mermin - Solid State Physics,
• Atkins and Friedman - Molecular Quantum Mechanics,

and our paper, Locality of the Thomas—Fermi—von Weizsӓcker equations.