Question

With a plane wave, I always took the direction of the wavevector, k, as the direction of propogation (magnitude proportional to the inverse wavelength). Alternatively, it could represent the momentum (minus a factor ?) of a particle.

However inside a crystal, the electron wavevector and the electron velocity are not necessarily in the same direction. I'm thinking here of a 2D material with a cylindrical Fermi surface where the momentum may have a z component, but the Fermi velocity does not. In everyday cases you would expect momentum and velocity to be in the same direction, moreover I considered the propogation of the wave to be in the same direction as its particle analogue.

I realise that inside a crystal the electrons are no longer simple plane waves, but what then does the k vector mean?

Answer 1

k points in the direction of phase velocity (i.e., normal to the surfaces of constant phase, also called wavefronts). The electron moves in the direction of group velocity (i.e., a localized electron would be made of a wavepacket, and the electron moves as the wavepacket center moves). (Formula for electron velocity is the same as the usual group velocity formula: )

Your question can be rephrased: "Why is the phase velocity not parallel to the group velocity?" Or more generally: "Why is the phase velocity different from the group velocity?" The answer to the latter question (both mathematically and intuitively) can be found in pretty much any introductory physics book that defines and discusses the concept of group velocity.

In one dimension, of course the group velocity and phase velocity have to be parallel, but in 2D or 3D, whenever waves are propagating with a non-spherically-symmetric dispersion relation, the group velocity and phase velocity will usually point in different directions. So this also happens with light waves in crystals, with sound waves in crystals, with sound waves in sedimentary rock, etc.