# 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!

# How does a transistor work?

The world would be a very different place if the transistor had never been invented. They are everywhere. They underpin all digital technology, they are the workhorses of consumer electronics, and they can be bewilderingly small. For example, the latest Core i7 microchips from Intel have over two billion transistors packed onto them.

But what are they, and how do they work?

In some ways, they are beguilingly simple. Transistors are tiny switches: they can be on or off. When they are on, electric current can flow through them, but when they are off it can’t.

The most common way this is achieved is in a device called called a “field effect transistor”, or FET. It gets this name because a small electric field is used to change the device from its conducting ‘on’ state to it’s non-conducting ‘off’ state.

At the bottom of the transistor is the semiconductor substrate, which is usually made out of silicon. (This is why silicon is such a big deal in computing.) Silicon is a fantastic crystal, because by adding a few atoms of another type of element to its crystal, it can become positively or negatively charged. To explain why, we need to turn to chemistry! A silicon atom has 14 electrons in it, but ten of these are bound tightly to the atomic nucleus and are very difficult to move. The other four are much more loosely bound and are what determines how it bonds to other atoms.

When a silicon crystal forms, the four loose electrons from each atom form bonds with the electrons from nearby atoms, and the geometry of these bonds is what makes the regular crystal structure. However, it is possible to take out a small number of the silicon atoms and replace them with some other type of atom. If this is done with an atom like phosphorus or nitrogen which has five loose electrons, then four of them are used to make the chemical bonds and one is left over. This left-over electron is free to move around the crystal easily, and it gives the crystal an overall negative charge. In the physics language, the silicon has become “n-doped”.

But, if some silicon atoms are replaced by something like boron or aluminium which has only three loose electrons, the atom has to ‘borrow’ an extra electron from the rest of the crystal, meaning that this electron is ‘lost’ and the crystal becomes positively charged. This is called “p-doped”.

Okay, so much for the chemistry, now back to the transistor itself. Transistors have three connections to the outside world, which are usually called the source, drain, and gate. The source is the input for electric current, the drain is the output, and the gate is the control which determines if current can flow or not.

The source and drain both connect to a small area of n-doped silicon (i.e. they have extra electrons) which can provide or collect the electric current which will flow through the switch. The central part of the device, called the “channel” is p-doped which means that there are not enough electrons in it.

Now, here’s where the quantum mechanics comes in!

A while back, I described the band structure of a material. Essentially, it is a map of the quantum mechanical states of the material. If there are no states in a particular region, then electrons cannot go there. The “Fermi energy” is the energy at which states stop being filled. I’ve drawn a rough version of the band structure of the three regions in the diagram below. In the n-doped regions, the states made by the extra electrons are below the Fermi surface and so they are filled. But in the p-doped channel, the unfilled extra states are above the Fermi energy. This makes a barrier between the source and drain and stops electrons from moving between the two.

Now for the big trick. When a voltage is applied to the gate, it makes an electric field in the channel region. This extra energy that the electrons get because they are in this field has the effect of moving energy of the quantum states in the channel region to a different energy. This is shown on the right hand side of the band diagrams. Now, the extra states are moved below the Fermi energy, but the silicon can’t create more electrons so these unfilled states make a path through which the extra electrons in the source can move to the drain. This removes the barrier meaning that applying the electric field to the channel region opens up the device to carrying current.

In the schematic of the device above, the left-hand sketch shows the transistor in the off state with no conducting channel in the p-doped region. The right-hand sketch shows the on-state, where the gate voltage has induced a conducting channel near the gate.

So, that’s how a transistor can turn on and off. But it’s a long leap from there to the integrated circuits that power your phone or laptop. Exactly how those microchips work is another subject, but briefly, the output from the drain of one transistor can be linked to the source or the gate of another one. This means that the state of a transistor can be used to control the state of another transistor. If they are put together in the right way, they can process information.

# Topology and the Nobel Prize

You may have seen that the Nobel Prize for Physics was awarded this week. The Prize was given “for theoretical discoveries of topological phase transitions and topological phases of matter”, which is a bit of a mouthful. Since this is an area that I have done a small amount of work in, I thought I would try to explain what it means.

You might have seen a video where a slightly nutty Swede talks about muffins, donuts, and pretzels. (He’s my boss, by the way!) The number of holes in each type of pastry defined a different “topology” of the lunch item. But what does that have to do with electrons? This is the bit that I want to flesh out. Then I’ll give an example of how it might be a useful concept.

### What is topology?

In a previous post, I talked about band structure of crystal materials. This is the starting point of explaining these topological phases, so I recommend you read that post before trying this one. There, I talked about the band structure being a kind of map of the allowed quantum states for electrons in a particular crystal. The coordinates of the map are the momentum of the electron.

Each of those quantum states has a wave function associated with it, which describes among other things, the probability of the electron in that state being at a particular point in space. To make a link with topology, we have to look at how the wave function changes in different parts of the map. To use a real map of a landscape as the analogy, you can associate the height of the ground with each point on the map, then by looking at how the height changes you can redraw the map to show how steep the slope of the ground is at each point.

We can do something like that in the mathematics of the wave functions. For example, in the sketches below, the arrows represent how the slope of the wave function looks for different momenta. You can get vortices (left picture) where the arrows form whirlpools, or you can get a source (right picture) where the arrows form a hedgehog shape. A sink is similar except that the arrows are pointing inwards, not outwards.

Now for the crucial part. There is a theorem in mathematics that says that if you multiply the slope of the wave function with the wave function itself at the same point, and add up all of these for every arrow on the map, then the result has to be a whole number. This isn’t obvious just by looking at the pictures but that’s why mathematics is great!

That whole number (which I’m going to call n from now on) is like the number of holes in the cinnamon bun or pretzel: It defines the topology of the electron states in the material. If n is zero then we say that the material is “topologically trivial”. If n is not zero then the material is “topologically non-trivial”. In many cases, n counts difference between the number of sources and the number of sinks of the arrows.

### What topology does

Okay, so that explains how topology enters into the understanding of electron states. But what impact does it have on the properties of a material? There are a number of things, but one of the most cool is about quantum states that can appear on the surface of topologically non-trivial materials. This is because of another theorem from mathematics, called the “bulk-boundary correspondence” which says that when a topologically non-trivial material meets a topologically trivial one, there must be quantum states localized at the interface.

Now, the air outside of a crystal is topologically trivial. (In fact, it has no arrows at all, so that when you take the sum there is no option but to get zero for the result.) So, at the edges of any topologically non-trivial material there must be quantum states at the edges. In some materials, like bismuth selenide for example, these quantum states have weird spin properties that might be used to encode information in the future.

And the best part is that because these quantum states at the edge are there because of the topology of the underlying material, they are really robust against things like impurities or roughness of the edge or other types of disorder which might destroy quantum states that don’t have this “topological protection”.

### An application

Now, finally, I want to give one more example of this type of consideration because it’s something I’ve been working on this year. But let me start at the beginning and explain the practical problem that I’m trying to solve. Let’s say that graphene, the wonder material is finally made into something useful that you can put on a computer chip. Then, you want to find a way to make these useful devices talk to each other by exchanging electric current. To do that, you need a conducting wire that is only a few nanometers thick which allows current to flow along it.

The obvious choice is to use a wire of graphene because then they can be fabricated at the same time as the graphene device itself. But the snag is that to make this work, the edges of that graphene wire have to be absolutely perfect. Essentially, any single atom out of place will make it very hard for the graphene wire to conduct electricity. That’s not good, because it’s very difficult to keep every atom in the right place!

The picture above shows a sketch of a narrow strip of graphene surrounded by boron nitride. Graphene is topologically trivial, but boron nitride is (in a certain sense) non-trivial and can have n equal to either plus or minus one, depending on details. So, remembering the bulk-boundary correspondence, the graphene in this construction works like an interface between two different topologically non-trivial regions, and therefore there must be quantum states in the graphene. These states are robust, and protected by the topology. I’ve tried to show these states by the black curved lines which illustrate that the electrons are located in the middle of the graphene strip.

Now, it is possible to use these topologically protected states to conduct current from left to right in the picture (or vice versa) and so this construction will work as a nanometer size wire, which is just what is needed. And the kicker is that because of the topological protection, there is no longer any requirement for the atoms of the graphene to be perfectly arranged: The topology beats the disorder!

Maybe this, and the example of the bismuth selenide I gave before show that the analysis of topology of quantum materials is a really useful way to think about their properties and helps us understand what’s going on at a deeper level.

(If you’re really masochistic and want to see the paper I just wrote on this, you can find it here.)

# Why do some materials conduct electricity while others don’t?

Can you tell at a glance how the electrons in a material behave? Amazingly, the answer is “yes”, and in this post I’ll explain how.

I want to introduce the concept of something called ‘band structure’ because it is an idea that underpins a lot of the quantum mechanics of electrons in real materials. In particular, the band structure of material can make it really easy to know if a material is a good conductor of electricity or not. So, here goes.

To describe how electrons behave in a particular material, a good place to start is by working out what quantum states they are allowed to be in. In essence, the band structure is simply a map of these allowed quantum states. One place where things can be a bit confusing is the coordinates that are used to draw this map. Band structure uses the momentum of the quantum state as its coordinate, and gives the energy of that state at each point.

The reason for this is that the momentum and energy of the quantum states are linked to each other so it just makes sense to draw things this way. But why not use the position of the quantum state? This is because position and momentum cannot both be known at the same time due to Heisenberg’s Uncertainty Principle. If the momentum is known very accurately then the position must be completely unknown.

In fact, there’s even more to it than that. Most solids have a periodic lattice structure and this periodicity means that only certain momentum values are important. Roughly speaking, if the size of the repeating pattern in the lattice has length a, then there is a repeating pattern of allowed energy states in momentum with length 1/a. This means that we can draw the map of the allowed quantum states in only the first of these zones. This zone has a finite size, which is very helpful when trying to draw it!

The band structure of silicon. (Picture credit: Dissertation by Wilfried Wessner, TU Wien.)

Let’s take silicon as an example because it’s a really important material since a lot of electronics are made from it. The picture above shows the band structure (left) and the shape of the first repeating zone of allowed momenta (right) of silicon. The zone of allowed momenta has quite a complicated shape which is related to the crystal structure of the silicon. Some of the important points in that zone are labeled, for example, the center of the zone is called the Γ point (pronounced “gamma point”), while the center of the square face at the edge of the zone is the X point. It’s impossible to draw all the allowed states at every momentum point in a 3D zone, so what is usually done is to draw the allowed quantum states along certain lines between these important points, and that is what is on the left of the picture. You can probably see that these allowed states form bands, which is where the name ‘band structure’ comes from.

There’s one more concept that is really important, called the “Fermi surface”. Electrons are fermions, and so they are allowed to occupy these quantum states so that there is at most one electron in each state. In nature, the overwhelming tendency is for the total energy of a system to be minimized as this is the most efficient arrangement. This is done by filling up all the quantum states, starting from the bottom, until all the electrons are in their own state. There are never enough electrons to fill all the allowed quantum states, and so the energy of the last filled (or first empty) states is called the Fermi surface. In a three dimensional material, the cutoff between filled and empty states is a two-dimensional surface.

So, how does knowing the band structure help us to understand the electronic properties of a material? As an example, let’s think about whether the material conducts electricity well or not. It turns out that for electrical conduction, most of the quantum states of the electrons play no role at all. The important ones are those near the Fermi surface.

To conduct electricity, an electron has to jump from its state below the Fermi surface to one above it, where it is free to move around the material. To do this, it has to absorb some energy from somewhere. This usually either comes from an electric field that is driving the electrical current (like a battery or a plug socket), or from the thermal energy of the material itself.

Take a look at the sketches below. They are cartoons of band structures near the Fermi surface (which is shown by the green dotted line). The filled bands are shown by thick blue lines while the empty bands are shown by thin blue lines. In the left-hand cartoon there is a big gap between the filled and empty bands so it’s very difficult for an electron to gain enough energy to make the jump from the filled band to the empty band. That means that a material with a large band gap at the Fermi surface is an insulator – it can’t conduct electricity easily. The middle cartoon shows a material with only a small band gap. That means it’s possible, but kinda difficult for an electron to make the jump and become conducting. Materials with narrow gaps are semiconductors.

The right-hand cartoon shows a material where the Fermi surface goes through one of the bands, so there are both empty states and filled states right at the Fermi surface. This means it’s really easy for an electron to jump above the Fermi surface and become conducting because it takes only a tiny amount of energy to do this. These materials are conductors.

Going back to silicon, we can look at the band structure above and see that there is a gap of about 1 electron volt at the Fermi energy. (The Fermi energy is zero on the y axis). One electron volt is too large an energy for an electron to become conducting by absorbing thermal energy, but small enough that it can be done by an electric field. This means that silicon is a semiconductor – it has a narrow gap.

One final question: How do you find the band structure of your favorite material? There is an experimental technique called ARPES where you shine high energy light at a material, and the photons hitting it cause electrons to be ejected from the surface. These electrons can be caught and the energy and momentum that they have reflect the energy and momentum of the quantum states they were filling in the material. So by careful measurement you can reconstruct the map of these states.

Another way is to use mathematics to theoretically predict the band structure. There has been a huge amount of work done to come up with accurate ways to go from the spatial definition of a crystal to its band structure with no extra information. In some cases, these work very well, but the calculations which do this are often very complicated and require supercomputers to run!

So, that is band structure. An easy way to make a link between complicated quantum mechanics and everyday properties like conduction of electricity.