## Sunday, July 08, 2012

### What is effective mass?

Given my previous post and the Higgs excitement, it's worth thinking a bit about what we mean by "effective mass" for charge carriers in solids.  At the root of the concept is the idea that it is meaningful to describe the low energy (compared with the bandwidth, which turns out to be on the order of electron-volts) electronic excitations of (the many electrons in) solids as electron-like quasiparticles - quantum objects that are well described as occupying particular states that are described as having definite energy and momentum (for the experts, these states are approximate eigenstates of energy and momentum).  One can look at those allowed states, and ask how energy $E$ varies as a function of momentum $\mathbf{p}$.  If the leading variation is quadratic, then we can define an effective mass by $E \approx p^{2}/2m*$.  Note that this doesn't have to be the situation.  Near the "Dirac point" in graphene, where the occupied $\pi$ electron band has its maximum energy and the unoccupied $\pi$ band has its minimum energy, the energy of the quasiparticles goes linearly in momentum, analogous to what one would expect for ultrarelativistic particles in free space.

The actual situation is more rich than this.  In real space, we believe the universe to be invariant under continuous translational symmetry - that is, the properties of the universe don't depend on where we are.  Translating ourselves a little to the right doesn't change the laws of nature.  That invariance is what actually implies strict conservation of momentum.  In the case of a periodic solid, we have a lower symmetry situation, with discrete translational symmetry - move over one lattice spacing, and you get back to the same physics.  In that case, while true momentum is still conserved (the universe is what it is), the parameter that acts like momentum in describing the electronic excitations in the solid is only conserved if one allows the solid as a whole the chance to pick up momentum in certain amounts (proportional to 1/the lattice spacing).

More complicated still, when electron-electron interactions are important, say between the mobile electrons and others localized to the lattice, the spectrum of low energy states can be modified quite a bit.  This can lead to the appearance of "heavy fermions", with effective masses hundreds of times larger than the free electron mass.   Note that this doesn't mean that the real electrons are actually more massive.  Pull one out of the solid and it's like any other electron.  Rather, it means that the relationship between the energy of the electronic states and their momentum in the solid differs quite a bit from what you'd see in a free electron.

So, knowing this, how fundamental is mass?  Could there be some underlying degrees of freedom of the universe, such that our standard model of particle physics is really an effective low-energy theory, and what we think of as mass really comes from the energy and momentum spectrum of that theory?  In a sense that's one aspect of what something like string theory is supposed to do.

On a more nano-related note, this discussion highlights why certain abuses of the term effective mass annoy me.  For example, it doesn't really make sense to talk about the effective mass of an electron tunneling through a dodecane molecule - 12 carbons do not make an infinite periodic system.  You can use models where effective mass is a parameter in this sort of problem, but you shouldn't attach deep physical meaning to the number at the end of the day.  