It was not so very long ago that people thought that
semiconductors were part-time orchestra leaders
and microchips were very, very small snack foods.
~Geraldine A. Ferraro
More is different.
~Philip Warren Anderson
Metals conduct electricity. Nonmetals don’t. That’s the conventional wisdom, anyway. In truth, there is a third class of material, called semiconductors. A semiconductor sometimes conducts electricity and sometimes doesn’t. This week, we’ll learn precisely what a semiconductor is and how the forces of quantum mechanics determine whether a material is a conductor, an insulator, or a semiconductor.
More is Different
Nobel laureate Philip Warren Anderson said that “more is different.” He meant that a large number of particles will behave very differently than a small number of particles. This principle is called emergence, and it holds with the electrons in an atom.
As we learned from Niels Bohr, because of their wave nature, electrons in atoms are restricted to specific combinations of energy and momentum. Roughly speaking, momentum measures a combination of how massive a particle is and how fast it’s going, while energy measures a combination of the momentum of the particle and the external forces that act on it. And, as we learned from Wolfgang Pauli, those electrons can’t occupy the exact same combination of energy and momentum (and spin). As atoms bond together to form molecules and crystal structures, these rules continue to apply. However, more is different, and new emergent behavior appears.
As the number of atoms grows, the electrons in each atom have more physical places that they can go and more energy and momentum states that they can occupy. (A quick aside for experts: Position and momentum for quantum particles are related by a Fourier transform. Energy and time are similarly related.) For macroscopic crystals like metals, the number of allowed energies is so large that there appear to be infinite allowed states. However, each allowed energy state corresponds to a given allowed momentum, and often some energies are still forbidden.
Join the Band
To see how electrons behave in a material, we can plot the energy of each allowed state as a function of momentum. Sets of connected allowed states are called “bands” and the overall distribution of allowed states in a material is called its “band structure.” The energy-momentum states where electrons are forbidden are called “bandgaps” because they break a band in two, forming a gap.
In principle, there can be infinitely many bands, but in practice there are usually only two. For reasons that will (hopefully) become clear by the end of this post, the lower-energy band is called the “valence band” and the higher energy band is called the “conduction band.”
Because of the Pauli exclusion principle, each allowed state can only be occupied by one electron (two if you include spin). Electrons want to be in the lowest energy state possible, so bands fill up like buckets of water—first the valence band and then the conduction band. The “waterline” of a given material at a given moment is called its “Fermi level,” after Enrico Fermi.
For electrons to enter the conduction band from the valence band, they have to have enough energy to jump the band gap. So if the valence band is full and we want to add more electrons to the material, we have to give each electron a lot more energy than we would if we wanted to add electrons to a non-full band. This basically how electrons work on the atomic scale. Within the bands, we can sort of ignore the fact that there are discrete energy states–but the band gap reminds us that we still live in a quantum world.
Because of the Pauli exclusion principle, the number of electrons in each band dramatically affect how the material as a whole behaves. If a band is full, this means that each electron in the band is stuck in its own state, because there’s no state to move to. This means that if some external force (like, say, a voltage) is applied, the electrons can’t respond to it. You can think of it like a traffic jam: The road is too full, so each car has nowhere to go and traffic slows to a stop.
This means that a material can only conduct if its band structure contains a band with some electrons that can move. If the conduction band is empty but the valence band is full, then we have the traffic jam scenario for the entire material—i.e., it’s an insulator. If there are enough electrons in the material to completely fill the valence band and partially fill the conduction band, then those electrons have many more states they can enter. In our analogy, this is an empty road; the material conducts. If the valence band is only partially full, then there are just a few states for electrons to move into. But in this case, these empty states (or “holes“) carry positive charge in the opposite direction of the electron flow, rather than the electrons themselves carrying negative charge through the material. In our analogy, the gaps between cars on a busy road would seem to move backwards through traffic.
In most metals, the valence band and the conduction band overlap—i.e., there’s no bandgap—and the conduction band is always partly full. This is why most metals are conductors. In materials that we traditionally consider insulators, like glass and rubber, the valence band is completely full and the bandgap is too large for electrons to enter the conduction band easily. Thus, the material almost never enters the conduction phase. Materials where the valence band and the conduction band are close are called semiconductors because they can change between conducting and insulating.
Computers, Lasers, and More
The beauty of band theory is that no one would have predicted it from studying macroscopic objects or from studying just a few quantum particles. Band theory uses quantum mechanics to explain properties of matter with which we are very familiar. It is one of the greatest triumphs of quantum mechanics and allowed our modern Information Age to begin.
By adding or removing electrons to a semiconductor, we can empty out the valence band or put electrons into the conduction band, thus controlling whether or not a semiconductor conducts. If we can figure out a way to electronically control the number of electrons in the material, we can make a very fast electronic switch. This is how transistors work–and transistors make up computers. Thus, band structure powers your laptop.
Though this is perhaps a discussion for another day, the band structure of a material also controls how it interacts with light. My very first research project used graphene, a very exciting semiconductor material composed of a single atomic layer of graphite, to make ultrafast laser pulses. (I definitely plan to talk about this at some point.)
Unfortunately, most resources you can find about semiconductors deal with microchip design or with rather technical physics. Here’s the best I could find.
- Wikipedia is always a good place to look.
- Skeptic’s Play has a deeper description of how the two main bands form.
- Hyperphysics has a fairly brief explanation.
- Evolving portrait of the electron
- Laser pulse makes insulator conduct like a metal
- The Real Story Of The Uncertainty Principle
- How Do You Make Negative Temperatures, Anyway? [Uncertain Principles]
- Straightening Out Angular Momentum
- How does graphene convert light into electricity?
Questions? Comments? Hatemail?
As always, if you have any questions, comments, insults, or corrections, please don’t hesitate to let us know in the comments! This article had a lot of jargon and a lot of pictures; I hope the latter made the former easier to deal with. Let me know if it worked for you.
EDIT: I accidentally stated that energy and momentum are related by a Fourier relation. I apologize, this is false. I meant to say that position and momentum are related. Sorry about that.