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There's something about electronic band structure that is really confusing me.

I've seem some people saying that the whole point with band structures is that possible states of electrons can be at "continuous sets of energy", called bands, and those continuous sets are separated by energy gaps, called band gaps.

When I heard that I imagine there each band was a continuum of possible energy eigenstates.

Now the energy eigenstates for an electron in a periodic potential are labeled by two quantum numbers: the wavevector $\mathbf{k}$ and the band index $n$.

Fixing a band, the energy eigenstates and eigenvalues are determined by $\mathbf{k}$.

But wait a minute, the Born-von Karman boundary conditions actually restrict $\mathbf{k}$. Not all $\mathbf{k}$ are allowed, just a discrete set.

In that case it is not true that fixed the band index $n$ we have a continuum of eigenstates of energy, since there's just a finite number of $\mathbf{k}$'s.

Searching on the internet, I've found that indeed the number of energy states (and hence energy values), inside a band is equal to the number of unit cells of the crystal.

So my question is: given one energy band, is there a continuum of energy eigenstates/eigenvalues on it or just a finite number, because of the Born-von Karman boundary conditions as I imagined? If the number is actually finite, how do conclude that it is equal to the number of unit cells in the crystal?

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    $\begingroup$ The number of states is finite, but it's on the order of $N_A = 10^{23}$, which is close enough to continuous for all practical purposes. $\endgroup$ – knzhou Nov 28 '16 at 2:17
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There are two momentum scales in a crystal. The first is set by the unit cell. The second is set by the sample size. In both cases, the implication comes from Bloch's Theorem.

Consider a 1D crystal with unit cell of length $L$. Suppose the same has length $NL$. Bloch's Theorem states:

\begin{equation} \phi(x+L)=e^{ikL}\phi(x) \end{equation}

Thus, the transformation $k \to k+2\pi/L$, does nothing. We conclude that we can restrict $k$ to the interval $[0,2\pi/L)$. This is the Brillouin Zone.

Applying periodic boundary conditions on the entire sample (length $NL$), we see:

\begin{equation} \phi(x+NL)=e^{ikNL}\phi(x)=\phi(x) \end{equation}

We conclude that $k=2\pi m/NL$, where $m$ is any integer. Using the restriction to the Brillouin Zone, we restrict $m$ to integers from $[0,N-1)$. Thus, there are $N$ allowed points in momentum space in the Brillouin Zone.

For a realistic sample, we assume $N$ is very large. Thus, the points approach a continuum. This allows many calculations, as sums over discrete states become integrals over a continuum of states.

The above may be generalized to higher dimensions with only minor modification.

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