What exactly is Crystal Momentum, $\hbar k$?

Question

The title says it all really.

Does this mean that the crystal is moving?

From my notes, I read that

The effect of an external force on an electron in the crystal is to change the crystal momentum $\hbar k$. In the absence of a force, the crystal momentum must be constant and thus conserved.

In a full band the net crystal momentum of electrons should be zero

For an electric field, $\mathcal{E_x}$, we find a change in crystal momentum, $k_x$, $$\hbar\frac{dk_x}{dt}=q\mathcal{E_x}$$

The missing ingredient, required to describe conduction, is scattering, which we assumed to be negligible. But where does the scattering come from? While the force of the electric field on electrons causes a change in crystal momentum, scattering must somehow restore the crystal momentum to their original values. Since Bloch oscillations are incredibly difficult to observe, electrons must be scattered before they can cross the FBZ boundary. The only other entities within the crystal with large crystal momentum are of course phonons.

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These quotes are not related and quite honestly I have no idea what any of the quotes mean, as I don't know what is meant by 'crystal momentum'. If anyone wants to know the context from which those quotes came please let me know.

So, my question is as follows: In layman's terms (if possible), what is crystal momentum?

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N.B

Before posting this question I looked at this question and I'm finding it hard to understand. I'm only a 2nd year undergraduate, who has begun reading solid state.

Answer 1

Crystal momentum arises when you consider the allowed energy eigenstates of the electrons which inhabit a crystal. One finds that the allowed energies in crystals are not discrete (as they are for an isolated atom), but rather form continuous bands separated by gaps.

In the plot below, I've plotted the first band in blue and the second band in red; the dotted lines on the vertical axis show the continuous intervals of allowed energies which correspond to each band.

These states are labeled by two numbers, $n$ and $k$. $n$ is the band index, which tells you what energy band the state lives in; $k$ (or $\hbar k$, I suppose) is the so-called crystal momentum, which tells you where (left to right) in that energy band you are. In the above diagram, I plotted two different possible states - $n=1$ and $k=-1$, and $n=2,k=0.7$.

You should not think of $\hbar k$ as the physical momentum of the crystal itself. Rather, it is a kind of pseudo-momentum which labels the allowed electronic energy states. Neither should you think of it as the genuine momentum of the particular energy eigenstate which it labels, for the simple reason that the energy eigenstates do not have definite momentum.

Despite not being the genuine momentum, $\hbar k$ has some momentum-like properties. Applying an external force to the electrons in the system causes the crystal momentum of a particular state to change according to $$\frac{d}{dt}(\hbar \mathbf k) = \mathbf F_{ext}$$

Furthermore, if the electronic states interact with external particles (phonons, photons, etc), the sum of the crystal momentum of the electrons and genuine momentum of the external particles is conserved$^\dagger$. This is important when analyzing how electrons can jump from one energy state to the other by interactions with such particles.

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$^\dagger$Not quite conserved - rather, conserved up to a reciprocal lattice vector. In other words, if $\mathbf k$ and $\mathbf k'$ are the crystal momenta before and after some event, then we say that $\mathbf k$ is conserved up to a reciprocal lattice vector if

$$\mathbf k' -\mathbf k = \mathbf G$$ for some reciprocal lattice vector $\mathbf G$ (possibly the zero vector).