# How do solar cells and LEDs work?

It’s obvious to point out that generating renewable energy is hugely important, and one way of doing that is to make electricity using solar cells. Solar cells turn the energy carried by light into an electrical current, which can directly power a device, be connected into the grid, or be stored in a battery for future use. Understanding how solar cells work depends on the principles of quantum mechanics that I’ve already written about on this blog, so the middle of this long, dark, northern winter is the perfect time to think about it and dream of all that sunlight!

It’s possible to understand how a solar works by looking at band structure. I’ve written about band structure before, so feel free to read that post for more details. To briefly recap, quantum particles can only exist in certain allowed states and the band structure is essentially a map of these states in a crystal material. Electrons fill up all these states starting from the lowest energy, but there are usually more possible states than there are electrons to fill them, meaning that some of the higher energy states are not filled. The energy of the last filled state or first empty state is called the Fermi level.

### Absorption of light

At the fundamental level, light is also made up of quantum particles, called photons. The amount of energy that is carried by a photon is directly related to the wavelength (or colour) of the light. When a photon hits an object it can be absorbed, but the energy that it carries cannot be created or destroyed, so it must be transferred into the material.

It just so happens that the amount of energy that is carried by a photon of visible light is in the same range as the energy spacing between the quantum-mechanical states in lots of crystals. This is why most materials are opaque: they are good at absorbing photons.

The energy absorbed by the material is accounted for by an electron changing its state and gaining energy in the process. I’ve tried to show this in the sketch above. A photon (green wavy line) plus an electron below the Fermi level (black circles) becomes an electron above the Fermi level. There is also a space left below the Fermi level from which the absorbing electron came (the hollow circle). This space is called a hole.

### How to capture electrons

A solar cell converts light into electrical current by capturing the excited electron and hole. But this process can be quite difficult to engineer because naturally electrons (and everything else) will tend to rearrange so that they lower their energy. For the electron, this is most obviously done by “falling” back down into the state that it left, while emitting another photon to make sure that energy is conserved. So, the solar cell must have a way of driving the electron and hole apart from each other so that they can be captured before this recombination happens.

One way that can be done is to make a structure that has different band structure in different places, shown in the sketch below. There, a solar cell device is shown on the top, and the band structure of three different regions is below. The left-hand end of the solar cell is made so that it is “p-type” meaning that it has an excess of positively charged holes. Another way of saying this is that the Fermi level is in the valence band. The right-hand end is “n-type” meaning that it has extra negative electrons, or the Fermi level is in the conduction band.

The electron-hole pairs that are formed in either the p-type or n-type regions will recombine very quickly, but those that are made in the zone in between (called a “p-n junction”, highlighted by the dashed box) might not. Another way in which the excited electron can lose energy is by moving into an unoccupied state in the n-type region (shown by the blue arrow). Simultaneously, electrons in the valence band of the p-type region can lose energy by filling in the hole in the junction region. This process is equivalent to the hole moving to the left, towards the p-type region (red arrow).

This moving charge is exactly the current that the solar cell is designed to generate, and it can be collected by attaching wires (or “contacts”) at the ends (brown areas).

### Efficiency

There are a few things that can be optimised to make solar cells more efficient. The obvious thing is to use materials which absorb a lot of photons, so finding a material that has energy transitions at lots of different energies (corresponding to a lot of different photon wave lengths) is very important. Then, the recombination time can be increased so that the electrons and holes have more time to move apart from each other. Lastly, using a material that has a good electrical conductivity will allow the electrons to move faster and so can get more separation from the holes within the recombination time. This is a massive industry, and even small gains in efficiency can be worth a lot of money!

### LEDs

As a little coda, there is another electronic device which can be understood from this kind of thinking. Instead of absorbing light and creating current, an LED does the opposite: It uses current flowing through it to emit light.

Electrons moving through the p-n junction have to lose energy to get from the n-type region to the p-type region and they can do this by emitting photons – the inverse of the absorption process. By changing the energy spacing between the levels that the electron has to move between, the colour of the emitted light can be changed. Of course, there is a bit of detail which I am leaving out here, but it’s kinda neat that an LED is like a solar cell running in reverse!

# 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.

# 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.

# How a hard disk works

This post is going to explain the fundamental part of how the hard drive in your old computer works. Modern solid state disks work completely differently, so this applies only to the older type that have been common for several decades. Specifically, when your computer writes something to the drive, it has to turn the sequence of zeroes and ones which make up the binary data into something physical on the disk. Then, when it needs to read this information later, it can go back and look at that part of the disk and recover the zeroes and ones from whatever material they were written to. But how do you tell the difference between a one and a zero? That’s the question I’ll try to answer.

### Spin

But before we can get to that point, I have to explain a really important concept in quantum mechanics called “spin”. This is a quantity which is carried by all quantum mechanical particles, and is linked in a loose way to the rotational symmetry of the particle. Look at the right-pointing arrow in the picture. Hopefully it’s easy to see that the only way you can rotate the arrow so that it looks exactly the same as it does when you start (this is called a symmetry operation) is to rotate it through 360°. A particle that has this rotational symmetry is said to have a spin of 1. Now look at this double-headed arrow. If you rotate around the axis indicated by the red dot, you only have to rotate it by 180° to get back to where you started. This has a spin of 2 because you have to rotate half a turn to get the first symmetry operation. The other pictures show a few different spins.

But what about electrons? Well, they have spin of ½. Just to be clear about what that means, using the same analogy it implies that you have to rotate by 720° before the electron “looks” like it did when you started. There isn’t a good way to draw that so I can’t give you a picture of a spin-½ particle, so this is one of those places where quantum mechanics is weird and counter-intuitive and we just have to get on with it. The other building blocks of atoms (protons and neutrons) also have spin-½ so in this post I’ll focus on that strange case. The crucial thing about spin-½ particles is that their spin can exist in one of two states, usually called ‘up’ and ‘down’, and typically are represented by arrows pointing in those two directions.

But why does this matter? Well, individual spins generate a magnetic field. The reason that iron is a magnetic material is that the interaction between the spins in the iron atoms makes their spins all line up in the same direction. Therefore, the tiny magnetic fields associated with each of the spins all add up to make a large field. Non-magnetic materials don’t have this alignment (in fact, their spins are all randomly aligned) and so the tiny magnetic fields all cancel each other out because they are pointing in opposite directions. Materials like iron which have this alignment are called ‘ferromagnetic’.

### Reading and writing in a hard disk

But, what does this have to do with your laptop? Well, in a hard disk, the part where the zeroes and ones are stored is made from two small pieces of ferromagnetic material. Then, the difference between a one and a zero is made by manipulating the spins of the atoms in one of the ferromagnetic layers. When an electric current is passed through this region, the electrons behave differently depending on the spins. Specifically, if the electrons have the same spin as the atoms, then they don’t interact very strongly and the electrical resistance is quite low. But if they have opposite spins, the electrons interact strongly with the atoms so they bounce off the atoms (or “scatter” in the technical language), their progress is impeded, and the electrical resistance is high.

The way to encode a one or a zero is shown in the picture below. A one is encoded by aligning the ferromagnets (the pink layers) so that their spins point in the same direction. In the left-hand picture, I show this with both layers having up-spins. A current of electrons (shown by the red arrows) has a half-and-half mix of electrons with up-spin and down-spin. When it is passed through the stack, the up-spin electrons interact weakly with the ferromagnet up-spins in both layers (black arrows) and encounter low resistance. This means that some of the current put in at the top of the stack emerges from the bottom and this characterises the one state. Note that the down-spin electrons are blocked from getting to the bottom of the stack because they scatter strongly off the up-spin atoms in the first ferromagnet layer and so the resistance for them is high.

For the zero state, one of the ferromagnetic layers has its spins reversed. In the right-hand picture, this is shown by the lower layer now having a down-spin black arrow. For electric current, the down-spin electrons still scatter strongly from the up-spin atoms in the top layer. The up-spin electrons still pass through this layer, but then they encounter the down-spin atoms in the lower layer where the electrons and the atoms have opposite spin, so they scatter strongly. This means that no current emerges at the bottom of the device, and so this defines the zero state.

This means that, for the hard disk to work, it needs to be able to do two things. Firstly, the “write head”, which is the part that encodes the zeroes and ones when data is written to the disk, needs to be able to flip the spins of one of the ferromagnetic layers. Then, to recover the information at a later time, the “read head” tries to pass current through a specific piece of the disk material. If current flows (because the ferromagnet spins are the same) then this is a one. If current does not flow (because the ferromagnet spins are opposite) then it is a zero.

And this works entirely because of the quantum-mechanical property of particles called spin: aligned spins is a one, opposite spins is a zero. And as a bonus, it also explains why you have to be careful with hard drives and strong magnetic fields, because a magnet can change the alignment of all the ferromagnetic areas in the hard disk and destroy the encoded ones and zeros. Don’t say you weren’t warned!

# 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.