I’m With the (Valence) Band: Band Structure and the Science of Conduction

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

Iron Maiden
Iron Maiden is one type of band. As we will discover, another type of band lives in most materials and is much more exciting (source).

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.

A schematic of a band structure
In a large crystalline material, the number of allowed quantum states becomes very large–so large that we can pretend there are infinitely many. If we plot the energy of each allowed state as a function of the allowed momentum, they form sets of connected states, called “bands.” The energies that no electron can have form holes in these bands, called “bandgaps.”

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.”

The valence and conduction bands inside a material
In principle, there could be infinite bands. However, there are usually only two. They’re called the “valence band” and 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.

A half full band
Because of the Pauli exclusion principle, bands fill up from lowest energy to highest energy, much like buckets of water.

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.

The different "conduction phases" of a crystalline material are analogous to traffic on a road.
The different “conduction phases” of a crystalline material are analogous to traffic on a road. (A) If the Fermi level is somewhere in the conduction band, then there are many states for electrons to occupy and they can flow smoothly, like traffic on an empty road. (B) If all of the states in the valence band are full and the conduction band is empty, there is no room for motion, just as in a traffic jam. (C) If the Fermi level is in the conduction band, then there are just a few empty states for electrons to move into. These empty states (or “holes”) can travel through the material carrying positive charge in the opposite direction of the electron flow, just as a gap in traffic can move backwards through a line of cars on a busy road. Road images are from here, here, and here.

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.

A conductor, an insulator, and a semiconductor
In metals, the valence and conduction bands overlap, so they always conduct. Materials we usually think of as insulators, like glass and rubber, have full valence bands and very large bandgaps; electrons can rarely enter the conduction band. Semiconductors also have full valence bands, but they also have small band gaps. This means that they can conduct or insulate depending on the exact circumstances.

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.)

The band structure of graphene
The band structure of graphene, a very exciting new semiconductor.

Further Reading

Unfortunately, most resources you can find about semiconductors deal with microchip design or with rather technical physics. Here’s the best I could find.

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 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.

About jonah

Jonah is the author of The Physics Mill.
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14 Responses to I’m With the (Valence) Band: Band Structure and the Science of Conduction

  1. Hamilton says:

    What does the band structure for a superconductor look like?

    • jonah says:

      Thanks for reading, Hamilton! I’m not an expert on superconductors, so take everything I say with a grain of salt, but here’s my understanding:

      The current most popular theory of superconductors is BCS theory (http://en.wikipedia.org/wiki/BCS_theory), which is incomplete. BCS theory says that at extremely low temperatures, electrons in a superconductor pair-bond into what are called “cooper pairs,” which are bosonic. As bosons, the cooper pairs aren’t bound by the Pauli exclusion principle, and they ignore band stracture. Instead, they obey Bose-Einstein statistics, meaning that almost all of them are in the lowest energy state.

      In this extremely low energy state, each cooper pair has a very large wavefunction, and these wavefunctions overlap to form a large superposition state. We can sort of think of them as becoming an electron superluid, which can travel through the material with no resistance. Alternatively, we can sort of think of htem as forming one single electron that permeates the entire material. In either case, the result is superconductivity.

      This effect does not fully explain high temperature superconductors, which are largely still a mystery.

      tl;dr, band theory can’t explain superconductivity. We need additional mathematical machinery.

      Hope this helped! I’ll try and look into BCS theory so that I can maybe wirte a post about it.

  2. jonah says:

    I’ve been informed by one of my friends who works on superconductors that I’m somewhat incorrect that band theory doesn’t play a role at all. Only electrons at the Fermi level can form cooper pairs. When they do, they “reserve” their place in the band structure so that no other electrons can inhabit those states. Over time, more and more states become forbidden as electrons form cooper pairs and go to the lowest energy state. This creates a large band gap. Eventually, the band gap is so large that if an electron changes its energy, there is absolutely nowhere to go.

    Materials electrically resist flow because electrons hit the atomic nuclei inside the material. However, this changes an electron’s energy. If the electrons don’t have energy states available to them, they’re simply forbidden from hitting anything. This causes electrons to flow without resistance.

    • Hamilton says:

      Thanks for the explanation of the superconductor band gap! I’ve always heard the term used, but I’ve never found a clear explanation of what it its. It’s nice that the explanation not only describes how it works, but links it to the mechanism of superconductivity as well. Cool!

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  5. Jason says:

    Hi jonah,

    Thanks for an excellent outline of the solid band theory… is there any chance you could help me interpret the following image from my physics textbook?

    http://i.imgur.com/JQpDqxf.jpg

    The main thing I don’t understand is why the conduction band and valence bands merge on the right side of the graph. Also I’m not sure why the conduction band from the lower level intersects with the valence band of the higher level or why the conduction bands merge on the left, as do the valence bands. Any insight would be greatly appreciated. Thanks again for an excellent site!

    • Jonah Miller says:

      Hi Jason,

      Thanks for reading! I’m very rusty on solid state physics, so I’m not sure I feel comfortable answering an actual technical question on the topic. However, I’ve passed your question along to some of my friends who work in the field. So I’ll get back to you. Sorry about that!

      • Jason says:

        Hi jonah,

        That would be awesome. Thank you… No worries if you can’t figure it out though. The graph is kind of bizarre. My theory was maybe that it is possible to have two valence subshells at once, so perhaps the same applies to the bands?

        Thanks again… please let me know if you do happen to find anything out though. :)

        • Matthew says:

          So, if you have an isolated atom
          then the electrons have discrete energy levels
          if you have atoms in a close lattice, you get energy bands

          thus if you slowly increase the interatomic separation in a (very artificial lattice) you would approach the free atom limit
          so on one side of the graph you have the band structure
          on the other side you have the free atom structure
          and you just sort of demand continuity in between to get the figure
          now, I don’t know that the energy levels of the carbon would be the same as the energy levels of the silicon at large interatomic spacing
          in fact I rather think they wouldn’t be, and suspect that the figure might just be wrong about that
          Either way, calling this the band structure is misleading at best, as that term is generally reserved for an energy vs momentum diagram rather than an energy vs interatomic spacing diagram (which frankly is a bit bizarre, I’ve never seen one before)

          Hope that helps,

          • jonah says:

            Thanks for your help, Matthew! Matthew is one of my condensed matter friends.

          • Jason says:

            Thanks a lot Matthew and jonah. Yeah, I was suspecting that the graph was just bizarre to begin with. The diagram was from a very introductory (high school level) textbook and rather than make the topic more accessible via the graph, I think the authors made it more confusing. Your input was greatly appreciated. :)

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