I don’t know how to do this on a small scale in a practical way, but I do know that computing machines are very large; they fill rooms. Why can’t we make them very small, make them of little wires, little elements – and by little, I mean little.
~Richard Feynman (1959)

As of 2012, the highest transistor count in a commercially available CPU
is over 2.5 billion transistors.
~Wikipedia

In my article on quantum tunneling, I mistakenly claimed that diodes and transistors made use of this phenomenon. In an effort to correct my mistake, I’m going to explain how one type of transistor—the field effect transistor (FET)—works. Transistors are the building blocks of modern computers and they’re what make miniaturization possible. Diodes and transistors actually take advantage of a different quantum phenomenon that I’ve discussed before: band structure.

Conduction and the Electron Sea

The band theory of solids accepts that particles are waves and generalizes the Bohr Model of the atom. As we recently learned from my article on band structure, an electron in a material can exist in one of two energy bands—the valence band and the conduction band—which consists of allowed energies as a function of momentum. The bands are separated by a band gap that contains the energies electrons can’t have. The highest energy level containing an electron is called the Fermi level. You can think of the bands like buckets of water, with the Fermi level as the water line. In fact, scientists even call the filled states the electron sea or the Fermi sea.

The Fermi level determines the conduction properties of the material. If the Fermi level is near the top or bottom of the band gap, then most electrons are trapped by their neighbors and the Pauli exclusion principle. If the Fermi level is somewhere in the conduction or valence bands, electrons can move freely because they have more space to do so. Imagine a road where electrons are cars; if the Fermi level is near the band gap, you have a traffic jam.

Most insulating materials start with the Fermi level very close to the bandgap. By raising or lowering the Fermi level, we can make an insulator a conductor. This is why materials with small bandgaps are called semiconductors. 

Doping

So how do you change the Fermi level? This amounts to adding or removing electrons from the band structure. At first, you might think that you need to charge the material—stick enough electrons on it, or pull enough off, and you give the material an overall positive or negative charge. This will definitely change the number of electrons in the band structure. Actually, though, we only need to change the number of mobile electrons.

This is a subtle distinction, so I’ll try and explain. (I borrowed this metaphor from William Beaty’s excellent article on bipolar junction transistors. These aren’t the same as FETs!) Imagine that a semiconductor is like a cold pipe full of water. In its natural state, most of the water is frozen solid—only a little can flow—and the material is an insulator. However, if we raise the Fermi level, we melt the water so that it can flow.

In other words, there are many electrons already in the material—they’re just trapped and immobile. If we can free some up and make them mobile, we can add electrons to the band structure and raise the Fermi level. Similarly, we can lower the Fermi level by somehow trapping more electrons and immobilizing them.

One way to do this is chemically. For example, we could pour some nitric acid on our semiconductor, as shown below.  The acid breaks into positively charged hydrogen ions (protons) and negatively charged anions (the rest of the acid). The hydrogen ions fall onto the surface of the material and attract electrons to the surface, effectively immobilizing them. With the electrons missing, the Fermi level drops and the material conducts. In this case, the charge carrier is an empty “hole” in the electron sea. Since wherever there isn’t an electron, there’s an atom missing its electron, this hole has a net positive charge. This corresponds to part (c) in the figure above.

Because electrons are negative, we call adding mobile electrons to the band structure—which raises the Fermi level—n-doping or electron-doping. Similarly, we call removing electrons from the band structure—which lowers the Fermi level—p-doping or hole-doping.

The Transistor

We can also dope a material in a more clever way by taking advantage of the polarizability of certain materials. These materials, called dielectrics, are polarizable insulators with huge bandgaps (as opposed to semiconductors). When we apply an electric field to a dielectric, all of the negatively charged electrons and positively charged atomic nuclei in the material feel an electric force. Nothing can move very much because it’s stuck in the crystalline structure of the material. However, it does the best it can, and every charge moves just a little bit, as shown below.

In the inner bulk of the material, the charges of the electrons and nuclei mostly cancel each other out, so everything remains charge neutral. However, each of the material’s surfaces only has one kind of charge on it, so the top of the material becomes positively charged and the bottom becomes negatively charged. This surface charge can’t leave the material, though, or even move around. It just produces its own little electric field.

So what happens when we put one of these materials next to a semiconductor and apply an electric field? The dielectric polarizes and the surface charge is brought close to the semiconductor. Then the surface charge (say it’s positive, just for simplicity) acts like the acid from the example above: It attracts electrons in the semiconductor and immobilizes them. By turning on the electric field, we dope the material and it starts conducting.

Since the doping effect relies on that external electric field, it stops as soon as the field is gone. This means we can use an electric field to control whether or not our semiconductor conducts electricity! Since electric fields can be created by an electric voltage, we’ve created an electrically controlled switch that turns current on and off.

Furthemore, we can now work in binary. When current is travelling through our semiconductor, we say “1,” and when no current flows, we say “0.” Now we can use the language of computers to construct Windows, Mac OS, word processors, and the Internet!

Warnings

I’ve glossed over a lot of how a transistor works. First and foremost,the field effect transistor I described relies on the semiconductor being super thin so that the polarization of the dielectric can substantially change percentage of mobile charges in the material. This is most notably true in graphene, which is only one atom thick.

However, usually the semiconductor we use is silicon. And silicon is thicker. The process for electronically doping silicon is much more intricate. In fact, there are two common types of field effect transistors, MOSFETS and JFETS. There are also bipolar junction transistors, which work somewhat differently.

Secondly, constructing a field effect transistor is complicated and difficult; there’s a whole story of nanofabrication I didn’t get into.

Anyway, I think that the picture I’ve given you is the simplest possible model that still gives you some idea of what’s going on. But take it with a grain of salt.

Further Reading

Unfortunately, a lot of descriptions of transistors are just plain wrong, so there aren’t that many good resources I can recommend. Here’s what I could find, though.

• Wikipedia is almost always a good place to start. Here are the articles on the field effect transistor, the bipolar junction transistor, and the p-n junction, which is an important idea behind the bipolar junction transistor.
• PBS, the United States’ national public broadcasting network, has some nice stuff on transistors.
• Hyperphysics has an article on JFETS.
• William Beaty has a beautiful and simple article on how bipolar junction transistors work. He doesn’t invoke band structure explicitly, but instead uses some very nice metaphors to explain what’s going on.

Questions? Comments? Hatemail?

As always, if you have any questions, comments, corrections, or insults, please don’t hesitate to let us know in the comments!