Monday, May 25, 2015

What is band theory? (car analogy)

One of the commenters on my previous post asked how I could explain band theory to a nonscientist, an artist in particular.  Here's a shot.  By necessity, when trying to give an explanation that avoids math almost completely, I'm forced to lean heavily on analogy, which means sacrificing accuracy to some degree.  Still, I think this is a useful exercise - it certainly makes me think hard about what I consider to be the important elements of a concept.  (This must be what Randall Munroe had to do many times for his upcoming book!  If you haven't read his first one, what have you been waiting for?)

The electronic properties of many crystalline materials are well described by "band theory".  At its heart, band theory comes down to three important ideas that I'll explain more in a minute:  Electrons in solids can only have certain states (to be defined below); those states are determined by the arrangement of the atoms in the solid; and each state can only hold two electrons, no more.   To describe this, I'm going to have to mix metaphors a bit, but bear with me.

In a very American mode, we're going to picture the electronic states as individual lanes in a verrrrrry wide, multi-lane highway.  Each lane has a different speed limit (each state has a particular kinetic energy), with the slowest traffic off to the driver's right (in this US-centric analogy) and speed limits increasing progressively to the driver's left.   Each lane can only hold at most two cars (each state can only hold two electrons, one of each kind of "spin").   Here's where the analogy becomes more of a reach:  Not all speed limits (electron kinetic energies) are allowed.  Speeds in adjacent lanes are separated by a small amount (energy level spacings are set by the size of the crystal), and some lanes are missing altogether (some energies are outright forbidden, determined by the type and arrangement of atoms in the crystal).  So, there are "bands" of lanes, separated from each other by "gaps".

Now we start adding cars to the highway with the restriction that cars can only drive at the speed limit of their lane, and (in an un-American twist) the drivers want to go the minimum possible speed.  This is going to tell us the "ground state", the slowest/lowest energy configuration of the system.  The first two cars (electrons) go into the slowest lanes waaaaay over on the driver's right.  The next two cars go into the second-slowest lane, and so forth.  We keep adding in cars (electrons) until we run out of inventory (until we have kept track of all of the electrons).  The more cars we put in, the faster the top speed of the fastest cars!

Cars can only merge into lanes that are open (or only partly occupied).  If the last car added ends up in a lane in the middle of a band of lanes, so that it can easily merge into an adjacent unoccupied lane, this situation corresponds to a material that is a metal.  If the last car ends up right against the guard rail of a band of lanes, so that there just is no adjacent lane to the driver's left available, then this situation corresponds to a "band insulator".   (If the gap to the next band of lanes is large, we call such materials "insulators"; if it's not too big, we call those materials "semiconductors".)

One point that even this very imperfect analogy can highlight:  The speed of the fastest cars (electrons) in a block of copper is actually about 0.5% of the speed of light (!), or more than 6,000,000 kph.  For metals with even more electrons, the fastest movers can be going so quickly that relativistic effects become important!

This was a very rough cut.  I'll try to return to this later, with other ways of thinking about it.

Anonymous said...

Nice!

Purely focused on electronic properties and not on the crystal (but remarks about gap size and allowed states can be added easily) I always use the cinema seating analogy: have a cinema with seats divided in two groups (i.e. a corridor/gap in between).
People can sit 2 in a chair (at least, when we were young...). They can easily move to neighboring seats.
You fill up the seats from the far end (like a bucket fills with water). If you run out of people right at the open corridor, then you'd need to give the people a large "kick" over the seatless corridor to reach an open seat - hence conductivity is low or nonexistent. (Temperature feeds in here.)

Explaining a Mott insulator is also possible: suppose the people really dislike each other... they won't sit in each others lap and everything is stuck even if you have only one person per seat right up to the coridor. (No M-H bands in this analogy...). Now take out one person (hole doping) - stuff can move again.

Anonymous said...

Thanks so much! This is the original Anonymous. Will try it on my brother.
I started off with handwaving through the Bloch theorem, but the highway analogy is definitely better.

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