It’s not much of an exaggeration to say that electrons form the backbone of the modern world. They are our workhorses—they bring energy from power plants to our homes and factories. They are our couriers—they carry information through circuits in our computers. Some day, they will be our providers—as the central elements in photovoltaic solar cells, they will capture the sun’s energy for our use. These current and future electronic devices rely on very precise answers to the question *What do electrons do inside materials?* Out of technological need and intellectual curiosity, condensed matter physicists have spent over a century discovering better and better answers—and they continue to do so today. In this post, I will try to give you a little glimpse into the form, beauty, and utility of such an answer.

# A Single Electron Alone in One-Dimensional Space

The question *What do electrons do in materials?* is a **big** question. There is an essentially infinite amount of different materials, and all of them behave differently in different situations. This post is just going to be an overview of some basic properties. We’ll consider a few simple examples, and I’ll explain how and when they relate to the real world. We’ll start simple and gradually add new elements—by the end of this post, you’ll be able to understand how solar cells work!

The technological applications I mentioned above use electrons to move energy from place to place. Therefore, it seems natural to study the relationship between electron motion (in the form of momentum) and energy. This is exactly what we’ll do. But before we consider electrons inside materials, let’s first look at a single electron living in an otherwise-empty one-dimensional space. (I.e., an electron confined to move along an infinitely-long line.)

We can think of this single electron as a classical point particle. In this case, the electron obeys the simple kinetic energy formula E = p^{2}/2m_{e}, where E is the energy, m_{e} is the electron’s mass, and p is the electron’s momentum. If you don’t want to read equations, you can just look at the figure below. This type of figure—where energy is plotted as a function of momentum—is often called a *dispersion plot* or an *E vs. k curve*. Negative momentum corresponds to the electron moving left in its one-dimensional world, while positive momentum corresponds to the electron moving right.

# A Single Electron in a One-Dimensional Crystal

In a real material, each electron is surrounded by ions and other electrons. So as a first step towards moving to a real material, let’s add ions to our one-dimensional space. In particular, let’s add *periodically-located* ions such as the ones below. (A periodic arrangement of ions constitutes a crystal lattice.)

The electron’s motion is now quite different because the lattice ions are exerting electromagnetic forces on it. To fully analyze this situation, we would need to treat the electron+lattice system quantum-mechanically by solving Schrödinger’s equation. This is not too difficult when the electrons only interact weakly with the lattice, but I want to focus on results so I’ll skip the math for now.

Before I show you a plot, I’d like to mention that three things—quantum-mechanical tunneling, the wave nature of electrons, and broken translational symmetry—produce some strange effects on the electron’s motion and dispersion. To be a little more specific:

- Classically, the electron would get trapped by one of the positively-charged ions and would never be able to move through the crystal. However, quantum mechanics allows the electron to tunnel past the ions and move (almost) freely.
- In quantum mechanics, all particles also behave like waves. This means that the ions in the crystal can scatter the electron, and that the electron can interfere with itself. Discontinuities in the dispersion can occur when the scattering and interference combine to create a standing wave.
- In empty space, every location is physically the same—this is known as
*continuous translational symmetry*and it is responsible for momentum conservation. In a crystal, the presence of ions breaks this symmetry—only some locations are physically the same. This causes momentum to only be partially conserved. The end result is that the electron’s dispersion in momentum space also becomes periodic.

I will write supporting posts about these ideas in the future, but I didn’t want to clog this one up with too many details.

Now, without any further ado—here is the electron’s dispersion inside a crystal. (I included the free electron dispersion as a dotted line for comparison.)

It’s quite different! The most important things to notice are that:

- Energy goes up and down when momentum gets farther from zero. (For the free electron, energy always went up when momentum got farther from zero.)
- The dispersion is periodic. (I already mentioned this.)
- There are some energies that do not correspond to any momentum value. (For the free electron, there was at least one momentum value corresponding to each energy value.)

The second point allows us to simplify our dispersion plots—we only need to show one repeated unit instead of many. The third point allows us to conceptually split the dispersion into energy bands (energy regions where there are corresponding momentum values) and energy gaps (energy regions that are forbidden to the electron).

# Many Non-Interacting Electrons in the Same 1D Crystal

For our final and most realistic model, we will now consider a whole bunch of electrons located in the same one-dimensional crystal. (To simplify things, we’ll assume that the electrons interact only with the lattice and not with each other.) Each electron in the gas will have an energy and momentum. In particular, since the electrons don’t interact with each other, their energy and momentum values will all lie on the dispersion curve discussed above. But which values will they take?

We can answer this question if we assume the electrons are at zero temperature. In this case, the electrons will arrange themselves into their lowest-energy configuration. You might guess that all of the electrons would have zero energy and zero momentum—basically, that all electronic motion would cease. However, this cannot happen because of a quantum mechanical effect called the *Pauli exclusion principle*.

In our situation, the Pauli exclusion principle says that *no more than two electrons can have the same energy and momentum*. So only two electrons can have zero energy and momentum. Two other electrons will take the next-best thing and situate themselves slightly higher up the energy-vs-momentum curve. Two more will take the next-next-lowest energy. So on and so forth. In the end, all the allowed energies below a certain special energy (called the Fermi energy or E_{F}) will be “filled” by electrons, and all energies above the Fermi energy will have no electrons. The exact location of the Fermi level will depend on the number of electrons in the material.

The existence of band gaps and the Fermi energy are two of the most unique and interesting properties of crystalline materials. In the next (and final) section, we’ll see how these two ideas work together inside photovoltaic solar cells.

# Photovoltaic Solar Cells

The goal of a solar cell is to directly convert solar energy into electricity. Conceptually, the simplest way for this to happen is for a solar photon to transfer all its energy to an electron in a collision. If this happens many times, people could collect all the energetic electrons and send them down power lines. You could receive them at your home and use them to power all of your electronic devices.

However, this process can’t happen for a free electron. The problem is that photons have very little momentum—in fact, for all practical purposes, they have no momentum at all. Imagine an electron at rest getting hit by a photon. If it absorbed all of the photon’s energy, its energy would change from zero to E_{photon}. However, because E = p^{2}/2m_{e} for a free electron, its momentum would also need to increase. A photon is unable to supply this momentum, so this process is impossible!

On the other hand, this *can* happen inside of a crystal. Imagine a crystal where the Fermi energy lies at the very top of a band, as shown in the left side of the figure below. If E_{photon} = E_{band gap}, an incoming photon can transfer an electron from the occupied band to the unoccupied band *because momentum is not conserved inside a crystal*. The right hand side of the figure shows the end result—one of the electrons has gained energy (but not momentum). This energetic electron can now be harvested—its energy can be used to power your home!

Of course, this description is overly simplified, and it’s quite difficult to make a good solar cell. The three most pressing technological issues facing solar cells today are collection efficiency, extraction efficiency, and cost. In more detail:

- The size of the band gap determines which photons get absorbed by the solar cell. In reality, the best solar cells use several stacked crystals that have different band gaps—this ensures that photons of many different energies get absorbed. However, these multi-layered solar cells are expensive and are currently unable to compete with fossil fuels in terms of immediate cost.
- Once an electron is excited by a photon, we need to remove it from the solar cell. This requires low-resistance solar cell materials with built-in electric fields. This is quite the engineering challenge! As with collection efficiency, complicated schemes are more effective but also more expensive.

Solar cells need to strike a balance between cost and efficiency in order to become a viable energy source. When that balance is found, people everywhere will be able to drastically decrease their impact on the environment without making any major changes to their lives!

# Conclusions

To sum this all up, we have seen that:

- Electrons inside crystalline materials are affected by:
- Quantum mechanical tunneling
- Wave-particle duality
- Discretely-broken translational symmetry
- The Pauli exclusion principle

- And that these effects lead to:
- The existence of band gaps
- The existence of the Fermi level

- These two properties—band gaps and the Fermi level—allow solar cells to exist (among other things).

If you want to know more, you can read textbooks, take classes, read my blog, or become a condensed matter physicist!

## Further Topics

Here are some other things you can think about. I plan on turning several of these into full-fledged posts in the future.

- What happens in non-crystalline materials?
- What happens in 2D and 3D?
- What happens if there are defects in the crystal structure?
- What happens at nonzero temperature?
- What happens if the electrons interact with each other?
- Why do some crystals conduct electricity while others don’t?
- Why is silicon so useful?
- How can electrons tunnel past so many ions?
- How is an electron in a crystal like a wave on a string? What does this have to do with band gaps?
- What is the connection between the crystal lattice and the dispersion’s periodicity?
- Can we accurately predict band structures? (
**Mostly yes**.) - How can we experimentally measure band structures?
- How do plants (and other phototrophs) capture the sun’s energy?
- Are solar cells the only way for humans to use the sun’s energy? (
**No**.) - Do we really need quantum mechanics to explain the electronic properties of crystals? (
**Yes**.)

## Further Reading

Most of what I know about this topic comes from these three books:

- Solid State Physics, by Neil Ashcroft and David Mermin
- Introduction to Solid State Physics, by Charles Kittel
- Elementary Solid State Physics: Principles and Applications, by M. Ali Omar

(Solid state physics is plagued by being too big—there is no definitive text. However, I like the first two books the best. They may not be totally complete, but they provide a good background that will help you understand more modern concepts in the field. Kittel is more appropriate for younger undergraduates, while Ashcroft & Mermin is for graduate students or upper-level undergrads.)