# How do you measure the quantum states of a material?

I’ve talked a lot on this blog about how understanding the quantum states of a material can be helpful for working out its properties. But is it possible to directly measure these states in an experiment? And what sort of equipment is needed to do so? I’ll try to explain here.

First, a quick recap. The band structure is like a map of the allowed quantum states for the electrons in a material. The coordinates of the map are the momentum of the electron, and at each point there are a series of energy levels which the electron can be in. The energy states close to the “Fermi energy” largely determine things like whether the material can conduct electricity and heat, absorb light, or do interesting magnetic things.

There are various ways that the band structure can be investigated. Some of them are quite indirect, but last week, I visited an experimental facility in the UK where they can do (almost) direct measurements of the band structure using X-rays.

The technical name for this technique is “angle-resolved photoemission spectroscopy”, or ARPES for short. Let’s break that down a bit. Spectroscopy just means that it’s a way of measuring the spectrum of something. In this case, it’s the electrons in the material. I’ll come back to the “angle-resolved” part in a minute, but the crucial thing to explain here is what photoemission is.

The sketch above shows a hypothetical band structure. When light is shone on a material, the photons (green wavy arrows) that make up the beam can be absorbed by one of the electrons in the filled bands below the Fermi energy. When this happens, the energy and momentum of the photon is transferred into the electron.

This means that the electron must change its quantum state. But the band structure gives the map of the only allowed states in the material, so the electron must end up in one of the other bands. In the left-hand picture, the energy of the photon is just right for the electron at the bottom of the red arrow to jump to an unfilled state above the Fermi energy. This is called “excitation”.

But in the right-hand picture, the energy of the photon is larger (see the thicker line and bigger wiggles on the green arrow) so there is no allowed energy level for the excited electron to move to. Instead, the electron is kicked completely out of the material. To put that another way, the high-energy photons cause the material to emit electrons. This is photoemission!

The crucial part about ARPES is that the emitted electrons retain information about the quantum state that they were in before they absorbed the photons. In particular, the photons carry almost no momentum, so the momentum of the electron can’t really change during the emission process. But also, energy must be conserved, so the energy of the emitted electron must be the energy of the photon, plus the energy of the quantum state that the electron was in before emission.

So, if you can catch the emitted electrons, and measure their energy and momentum, then you can recover the band structure! The angle-resolved part in the ARPES acronym means that the momentum of the electrons is deduced from what angle they are emitted at.

But what does this look like in practise? Fortunately, a friendly guide from Diamond showed me around and let me take pictures.

The upper-left picture is an outside view of the Diamond facility. (The cover picture for this blog entry is an aerial view.) It’s a circular building, although this picture is taken from close enough that this might be hard to see. This gives a sense of scale for the place!

Inside is a machine called a synchrotron. They didn’t let us go near this, so I don’t have any pictures, but it is a circular particle accelerator which keeps bunches of electrons flowing around it very, very fast. As they go around, they release a lot of X-ray photons which can be captured and focused. (There is a really cool animation of this on their web site.) The X-rays come down a “beam line” and into one of many experimental “hutches” which stand around the outside of the accelerator.

The upper-right picture shows the ARPES machine inside the main hutch of beamline I05. Most of the stuff you can see at the front is designed for making samples under high vacuum, which can then be transferred straight into the sample chamber without exposure to air.

The lower-left picture is behind the machine, where the beam line comes in. It’s kinda hard to see the metal-coloured pipe, so I’ve drawn arrows. The lower-right picture shows where the real action happens. The sample chamber is near the bottom (there is a window above it which allows the experimentalists to visually check that the sample is okay), and you can just about see the beam line coming in from behind the rack in the foreground.

The X-rays come into the sample chamber from the beam line, strike the sample, and the emitted electrons are funnelled into the analyser which is the big metallic hemisphere towards the right of the picture. The spherical shape is important, because the momentum of the electrons is detected by how much they are deflected by a strong electric field inside the analyser. This separates the high momentum electrons from the low momentum ones in a similar way that a centrifuge separates heavy items from light ones.

And what can you get after all of this? The energy and momentum of all the electrons is recorded, and pretty graphs can be made!

Above is a picture that I stole from the Diamond web site. On the left is a theoretical calculation for the band structure of a material called tungsten diselenide (WSe2). On the right is the ARPES data. The colour scheme shows the intensity of the photoemitted electrons. As you can see, the prediction and data match very well. After all the effort of building a massive machine, it works! Hooray science!

# What next for integrated circuits?

There is currently a big problem in the semiconductor industry. While technological progress and commercial pressure demand that electronics must be made smaller and faster, we are getting increasingly close to the fundamental limits of what can be achieved with current materials.

In the last couple of weeks, two academic papers have come out which describe ways in which we might be able to get around these limitations.

### Size matters

A quick reminder about how transistors work. (You can read more detail here.) Transistors are switches which can be either on or off. They have a short conducting channel through which electricity can flow. When they are on, electrical current is flowing through them, and when they are off it is not. They have three connections, one which supplies current (called the source), one which collects it (the drain), and one which controls whether the channel is open or closed.

There is something called the International Technology Roadmap for Semiconductors which lays out targets for improvements in transistor technology which companies such as Intel are supposed to aim for. The stages in this plan are called “nodes”, which are described by the size of the transistor. Having smaller transistors is better because you can fit more into a chip and do more computations in a given space.

At the moment, transistors at the 14 nanometre node are being produced. This means that the length of the gate/channel is 14nm (a nanometre is one millionth of a millimetre). According to the roadmap, within a decade or so, the channel length is supposed to be as short as 3nm. But, overall, transistors are rather bigger than this length, in part because of the size of the source and drain contacts. Transistors at the 3nm node will have an overall size of about 40nm.

### Carbon nanotube transistors

The first paper I want to mention, which came out in the journal Science, reports the fabrication of a transistor made out of different materials which allows the overall size to be reduced. Instead of using doped silicon for the contacts and channel,  these researchers made the channel out of a carbon nanotube, and contacts from a cobalt-molybdenum alloy.

Carbon nanotubes are pretty much graphene which has been rolled up into a cylinder which is a few nanometres wide. Depending on the details, they can have semiconducting electronic properties which are excellent for making transistors from, but they also are interesting for a whole range of other reasons.

By doing this, they could make a channel/gate region of about 11 nm long, with two contacts that were about 10nm each. Even with some small spacers, the total width of the transistor was only 40nm. This should satisfy the demands of the 3nm node of the roadmap, even though the channel is nearly four times as long as that.

### 3D chips

The second approach is completely different. At the moment, integrated circuits are mostly made in a single layer, although there are some exceptions to this in the most modern chips. This means that the various parts of the chip that do calculations and store memory can be located quite a long way away from each other. This can lead to a bottleneck as data is moved around to where it is needed.

A group of researchers, publishing in the journal Nature, designed an entirely new architecture for a chip in which the memory, computation, input, and output were all stacked on top of each other. This means that even though the transistors in their device are not particularly small, the data transfer between memory and computation can all happen at the same time. This leads to a huge increase in speed because the bottleneck is now much wider.

The prototype they designed was actually a gas sensor, and a rough idea of its construction is shown in the sketch above. Gas molecules fall on the top layer, which is made up of a large number of individual detectors that can react to single molecules. These sensors can then write the information about their state into the memory which is directly below it via vertical connections that are built into the chip itself.

The point of the sensor is to work out what type of gas has fallen on it. To do this, the information stored in the memory from the sensors must be processed by a pattern recognition algorithm which involves a lot of calculations. This is done by a layer of transistors which are placed below the memory, and are directly connected to it. In the new architecture, the transistors doing the computation have much quicker access to the data they are processing than if it were stored in another location on the chip. Finally, an interface layer allows the chip to be controlled and through which it outputs the result of the calculation are below the transistors, again connected vertically.

The paper shows results for accurate sensing of gaseous nitrogen, lemon juice, rubbing alcohol, and even beer! But that’s not really the crucial point. The big new step is the vertical integration of several components which would otherwise be spaced out on a chip. This allows for much quicker data processing, because the bottleneck of transferring data in and out of memory is drastically reduced.

So, the bottom line here is that simply finding ways to make traditional silicon transistors smaller and smaller is only one way to approach the impending problems facing the electronics industry. It will be a while before innovations like this become the norm for consumer electronics, and perhaps these specific breakthroughs will not be the eventual solution. But, in general, finding new materials to make transistors from and designing clever new architectures are very promising routes forward.

# Particle-wave duality and the two slit experiment

Particle-wave duality is the concept in quantum mechanics that small objects simultaneously behave a bit like particles and a bit like waves. This comes very naturally from the mathematics, but instead of talking about those boring details, I’m going to describe a famous experiment that proves it.

### Diffraction

It’s called the two slit experiment, and I’ve sketched how it works in the picture on the right. Before going into the full details, let’s look at the upper part of the picture. This shows a light wave shining on a barrier with a small slit in it. The thin black lines show the position of the peaks of the wave that describes the traveling light. Some of the light can get through that slit, but in doing so, it changes its form to become a circular wave with the slit at its source. This is called diffraction, and leads to a distinctive pattern when the light hits a screen placed some way behind the barrier. The red line behind the barrier shows the intensity of the light hitting the screen. This demonstrates that light can behave in a wave-like way because if the light was just particles you would not see the diffraction pattern, but there would be a small spot of light on the screen in line with the slit.

Now look at the lower part of the picture. Now the screen has been replaced with a second barrier that has two slits in it. Both of these slits act like the first one: they diffract the light that is coming through. So behind the second barrier, there are now two waves of light, one coming from each slit. These two waves interfere with each other, so that the pattern of light seen on the screen (the red line) looks very different from that made by just one slit. (I did actually calculate what the light should look like before I drew these pictures, so I hope both of the red lines are actually correct!) Interference is the process of these wave adding together to form one single pattern. The value of a light wave at a particular position can be either positive or negative. In the picture, the thin black lines show where the waves are at their maximum – so where they are their most positive. Exactly half-way between a pair of lines they are at their most negative. If the two waves are both positive at a particular position (like exactly at the center of the screen) then they add together to give intense light. But if one is positive and one is negative then they will cancel each other out and leave almost no light.

### Electrons

That’s not very controversial. But it starts to get a bit more weird when you repeat the same experiment but using a beam of electrons instead of a beam of light. Electrons are one of the three types of “particle” which make up an atom: The protons and neutrons bind together to form the nucleus, and then electrons “orbit” around it. Until this experiment was done for the first time, most physicists thought that electrons were particles. But the result of the experiment was the same kind of two-slit diffraction pattern that they got when they used light. The electrons that went through each of the slits were interfering with each other just like the light waves did. The only possible conclusion: these electrons were also wave-like.

Then, they pushed the experiment a bit further. They had the same barriers, but instead of using a beam of electrons, they fired them through one at a time. Astonishingly, even though there was only one electron, the result was still a two-slit diffraction pattern. Somehow, the electron was going through both slits and interfering with itself. Conclusion: Electrons are not just wave-like when there are lots of them, they are wave-like on their own!

### Now it gets weird

To try and verify this, they modified their apparatus to include detectors at both of the slits so they could tell which slit the electron was going though. Expecting to find a signal from both detectors, they were surprised to find that only one of the detectors sensed an electron going though, and instead of the two-slit diffraction pattern, they now saw a one-slit pattern on the screen. If they did the experiment with the detectors turned off, the two-slit diffraction pattern reappeared. It seemed like asking the electron which slit it had gone through forced it to choose one or the other. But get this: The experimentalists got sneaky. They took the electron detectors away and instead made slits that could be opened and closed very quickly. Starting with both slits open, they fired one electron from the gun. After it had passed the barrier with the two slits, but before it reached the screen, they closed one of the slits. Any guesses as to what pattern was measured on the screen?

They saw a single-slit diffraction pattern! Somehow, the electron knew that one of the slits had been closed after it went through, and behaved like only the other one had been open the whole time. This hints at many deep issues about quantum measurement and (gulp!) the nature of reality itself. But I’ll save that discussion for another time.

This experiment has been repeated with many different objects used instead of the light or electrons. Protons, whole atoms, and buckyballs all show the same behavior, so this is without doubt a general feature in quantum mechanics and not something oddly specific to light and electrons. In fact, once you allow for the possibility of wave-like particles, you start to see the effects of them in many places, including in the behavior of electrons in the materials which make computer chips and all the rest of information technology. So it’s a pretty big deal.

### And finally…

One final point of detail which I think is worth pointing out. In the first paragraph, I mentioned that “small objects” are needed to do this experiment. But what does “small” mean in this context? It turns out, this can be written down in a really simple equation. The de Broglie wavelength, referred to by the symbol $\lambda$, is the wavelength associated with the quantum object. It turns out, that to see the wave-like properties, the size of the slits has to be similar to $\lambda$.

The formula is $\lambda = h / mv$. Here, $h$ is just a number that comes from quantum mechanics and can be forgotten about. The $m$ and $v$ are the mass and speed associated with the particle-like properties of the object. So, the heavier the “particle”, the smaller the associated wavelength is. This explains why you don’t see any wave-like effects for people or cars or golf balls. Just to illustrate the kind of size that we talking about, light has a $\lambda$ of half a micron or so. For electrons, it’s a few nanometers, and for buckyballs, it’s a few thousandths of a nanometer.