Gapless Electrons in Graphene
Most recent answer: 10/21/2013
- Elliot (age 17)
Manchester
This is a great, tough question. I'm sorry that the explanation here is neither easy nor complete. It would be even sketchier if I had not gotten some key help from my friend Eduardo Fradkin. You can follow up for more explanation if you wish.
First, we should explain what's meant by "massless" here. We mean that the energy E of a traveling charged particle in the material is some constant s times the magnitude of its momentum p: E=ps. That's like a rest-mass-less particle in a vacuum: E=pc, but s is some particular material property, with no particular connection to c, the speed of light.
It turns out that the key to understanding this behavior is that the bands of states, which have some dependence of energy on momentum, E(p), don't show a gap in graphene. This will take a lot more explanation than most of our answers, so bear with us.
I'll try to start with a 1-D picture. Say you had free electrons, traveling at much less than c. Then you'd have the old expression for kinetic energy: E=p2/2m. Now we have to remember that electrons are not little dots but spread-out quantum waves. The momentum is related to the wavelength λ by the universal relation p=h/λ, where h is called Planck's constant. Now let's also remember that these waves are traveling in a crystal, which means in a regular array of atoms. Waves bounce of regular patterns very strongly if the waves bouncing off each spot add up together. "in phase". That means that a wave traveling left with p=-h/2a bounces over toward the right at p=+h/2a, and vice versa. These two waves get all mixed together. Instead of two waves traveling opposite directions with energy (h/2a)2/2m there are two different standing waves, one sitting on the atoms in a way the lowers that energy and the other shifted by half the spacing between the atoms in a way that raises the energy. That makes a gap in the range of possible energies.
The lower energy standing waves form the top part part of a low-energy band of states with a range of wave patterns. The high-energy standing waves form the bottom of a higher-energy band of states with different wave patterns. Near the top and bottom of each band the energy changes as the square of the change in how much the wave phase changes from atom-to-atom. That's like the usual E=p2/2m sort of pattern, although the "m" now isn't that of bare electrons but something that depends on how the waves interact with the crystal. The left-hand picture in illustrates such behavior.
Ok, that's the background about a 1-D simple version of a crystal. What happens in graphene? First of all it's 2-D, not 1-D. So think of 2-D sheets of E(k), where k describes how much the wave phase changes along each direction in the crystal as you go from one unit cell to the next. (If there were no other complications, k would just be proportional to p, but the way the wave wiggles in each unit cell of the crystal makes each k have a mixture of different p's) Off-hand, you'd expect wavy sheets with smooth, rounded maxima and minima for the different bands of E(k). Then E would change quadratically right near any maximum or minimum, again like the usual E=p2/2m sort of pattern. Graphene, however, has a special feature. In just the right directions the scattered waves cancel. For each atom that would make the wave bounce backwards there's another shifted just enough to make a wave that cancels. In these special directions, there's no gap! The bottom band comes up and touches the top band without mixing the states. The result is a cone-like region where E depends linearly, not quadratically, on the change in wave pattern.
You can find some decent pictures of this special region , and in cruder form on the right-hand side of the link above. captures how there are six points (different k's) at which E(k) forms those cones, as you'd expect from the 6-fold symmetry of the graphene sheets.
Mike W.
(published on 10/21/2013)