We physicists spend a lot of time talking up the weird and wacky aspects of quantum mechanics — entanglement, teleportation, many worlds, tunnelling, the philosophical ramifications of the wavefunction…, you know the drill. For a change, I wanted to make a video with Brady that highlighted just how many aspects of the quantum world can be explained in terms of phenomena and patterns we’re used to seeing in the world around us; in other words, to ground quantum principles in everyday physics. And what could be more commonplace — some might even say mundane (though not me) — than a cup of coffee?
The video describes the staggering quantum corral images which were created by Mike Crommie, Chris Lutz, and Don Eigler back in the early nineties, as discussed in this ground-breaking paper. (Unfortunately, due to some crossed wires between Brady and me, and largely because I swamped his e-mail inbox with different links to various descriptions of the quantum corral work, the video mistakenly credits Joe Stroscio and Don Eigler — rather than Crommie, Lutz, and Eigler — for the image below. Joe Stroscio has done some phenomenal scanning probe work in his time, but he’s not responsible for the corral.)
The corral is formed of 48 iron atoms which have been painstakingly put in place, one at a time, using the tip of a scanning tunnelling microscope. (Coincidentally, Joe Stroscio and colleagues have introduced autonomous atom manipulation which allows these types of atomic arrangements to be “dialled in” and fabricated directly under computer control). The ripples that can be seen both inside and outside the corral are due to the variation in electron density across the surface — electron waves scatter off (i.e. are reflected from) the Fe atoms, interfere, and we’re left with a standing wave inside the corral. Because the corral is circular, that standing wave is described mathematically by something known as a Bessel function. And that precise mathematical function also describes the standing wave that forms in a cup of coffee. Even though the diameter of the coffee cup is roughly six million times larger than that of the corral.
Physicists, and scientists in general, are very used to seeing mathematics describe very many aspects of our reality. This degree of familiarity with the ubiquity of mathematics in nature can sometimes make us — well, at least sometimes makes me — rather too blasé about just how utterly remarkable it is that precisely the same mathematical function can describe behaviour in completely different materials, spanning a huge range of length scales, and in entirely different environments. The only thing that’s common between the cup of coffee and the quantum corral is the symmetry. And yet the coffee and the electrons produce exactly the same pattern. (Well, as long as the critical “sloshing” point for the coffee isn’t reached. There was a great paper in Physical Review E back in 2012 on this topic).
What I don’t say in the video, however, is that there’s something very special about the copper sample on which the Fe atoms are sitting. It’s called a Cu(111) surface, where the numbers, known as Miller indices, describe the direction in which a copper crystal has to be cut to expose that particular plane. (Symmetry is all-important here too). At the Cu(111) surface the electrons are free to move across the plane; we call the system a 2D electron gas (although, in the video, I use the term “electron fluid” to bring out the comparison with the coffee. This isn’t such a “reach” – the term Fermi liquid is used throughout solid state physics). Not all surfaces give electrons this freedom to roam. The corral experiment would never work on a Si(111) surface, for example, because the electrons there, due to the strong covalent bonding in the crystal, simply don’t have the same leeway to explore the space around them.
I’ve written before that I’ve always been impressed that the comments under Sixty Symbols videos buck the usual trend for below the line online commentary, particularly at YouTube: the points the Sixty Symbols audience raise are very often insightful, smart, and even erudite at times. This is again true for the “quantum coffee” video. The following comment asks a particularly perceptive question related to what is causing the waves — are the electrons “driven” by the STM tip in some way?
The current from the STM tip is not responsible for “driving” the pattern. Or, to put it another way, the standing wave state of the electrons is not produced by the probe. Although STM can certainly be used in a very invasive way — this is precisely how the atoms are arranged to form the corral in the first place — it can also be used as relatively non-invasive probe of the electron density. Indeed, the same type of scattering is seen at naturally occuring defects (e.g. atomic step edges), as clearly seen in the image below (also taken from the IBM gallery). The ripples at the step edges are what are called Friedel oscillations and, again, arise from electron waves being back-reflected from the step.
As is my wont, I sneak a guitar into the video as an example of a one dimensional standing wave, in contrast to the 2D Bessel function pattern. In another key example of the pervasiveness of mathematics, there are particularly striking parallels between waves on a guitar (and other lesser musical instruments) and the quantum world. I’ve banged on about this at length before in the context of the Heisenberg uncertainty principle, so won’t hammer home the point again here. But what you might well ask is whether it’s possible to make a one-dimensional “corral” out of a line of atoms (as opposed to a 2D container).
It is. The image below shows the electron density in a 1D chain of Pd atoms, created and imaged using an STM by Nilius, Wallis and Ho ten years ago and elegantly described in this paper. By applying a different voltage to the STM tip, they can access different electron energies. The patterns of electron density, i.e. the standing waves, that they see as a function of voltage are very similar to those seen for waves on a guitar string. If you want to know more about this, including some of the not-so-gory mathematical detail, I cover it in the 1st year undergraduate Frontiers in Physics module here at Nottingham. Chapter 4 of the ebook for the nanoscience component of the module covers standing waves in the 1D atomic chain.
Another aspect of 2D standing waves we didn’t explore in the video, but which I’m hoping Brady and I will cover in the not-too-distant future, is the relationship of the quantum corral to drums and drumming. One of my all-time favourite scientific papers had that precise topic as its theme. But I’ll bang that particular drum in a future blog post.