Free Electrons
Consider the usual cubic box of side lengths \(L\) with nothing inside it (i.e. a vacuous Bravais lattice \(\Lambda=\emptyset\)). Now put an electron \(e^-\) inside this box. Its position space wavefunction will be a free plane wave of the form \(\langle\textbf x|\textbf k\rangle=L^{-3/2}e^{i\textbf k\cdot\textbf x}\) with energy \(E=\hbar^2|\textbf k|^2/2m\). However, the periodic Born-von Karmen boundary conditions imposed by the box cause the wavevector to be quantized \(\textbf k_{\textbf n}=\frac{2\pi}{L}\textbf n\) for some \(\textbf n\in\textbf Z^3\) (this defines a lattice in reciprocal \(\textbf k\)-space but it’s not the same thing as the reciprocal lattice \(\Lambda^*=\emptyset^*=\textbf R^3\)). Thus, the energies are also quantized as \(E_{\textbf n}=\frac{2\pi^2\hbar^2|\textbf n|^2}{mL^2}\). The ground state energy is \(E_{\textbf 0}=0\) with degeneracy \(\Omega(E_{\textbf 0})=1\), followed by the next lowest energy \(E_{\hat{\textbf i}}=\frac{2\pi^2\hbar^2}{mL^2}\) with degeneracy \(\Omega(E_{\hat{\textbf i}})=6\), etc. More generally there is some interesting number theory associated with such degeneracies.
Now suppose that first electron \(e^-\) naturally “burrows” its way down to the ground state \(\textbf n=\textbf k=\textbf 0\) in order to minimize its energy \(E=0\). Now put a second electron \(e^-\) into the box. In reality, the two electrons \(e^-\) would by their mutual Coulomb repulsion run away from each other on a hyperbolic orbit so to speak, raising all energy levels \(E_{\textbf n}\). However, for now we shall ignore all interactions. Then basically we have right now an ideal electron gas of two \(e^-\) that don’t talk to each other. However, actually there is one fundamental quantum mechanical “interaction” so to speak between the two \(e^-\) that cannot be ignored; this is the Pauli exclusion interaction arising from the identical fermionic nature of the spin \(s=1/2\) electrons \(e^-\). Here, as typical, we incorporate spin in an ad hoc manner as just another good quantum number whose associated operators \(\textbf S^2,S_3\) commute with the free space Hamiltonian \(H=T\) of the box. So if the first electron \(e^-\) is in state \(|\textbf n=\textbf 0\rangle\otimes|\uparrow\rangle\), then the second electron \(e^-\), if it also wants to minimize its energy, would have to occupy the state \(|\textbf n=\textbf 0\rangle\otimes|\downarrow\rangle\) (so the total state of the \(2\)-body electron system would be \(|\textbf n=\textbf 0\rangle\otimes|\textbf n=\textbf 0\rangle\otimes\frac{1}{\sqrt 2}(|\uparrow\rangle\otimes|\downarrow\rangle-|\downarrow\rangle\otimes|\uparrow\rangle)\)). The third electron \(e^-\) would then have to live in state \(|\textbf n=(1,0,0)\rangle\otimes|\uparrow\rangle\) for instance and in general we can house \(2\Omega(E_{(1,0,0)})=12\) electrons \(e^-\) in the next energy level, and so forth according to the non-zero values of the scatter plot above. In particular, if one puts in \(N\sim 10^{23}\) electrons for this ideal free electron gas, one would basically fill up a (discrete) ball of electrons in \(\textbf k\)-space (called the Fermi sea) of some radius \(k_F\approx\left(\frac{3N}{8\pi}\right)^{1/3}\), with each lattice point holding two electrons of opposite spin. The spherical boundary \(S^2\) of the Fermi sea of the ideal free electron gas would be called its Fermi surface, whose states thus have momentum \(\hbar k_F\) (called the Fermi momentum) and energy \(E_F=\frac{\hbar^2k_F^2}{2m}\) (called the Fermi energy).
Metals vs. Insulators
In practice, we’re not interested in an empty lattice \(\Lambda=\emptyset\), but rather a non-empty lattice \(\Lambda\neq\emptyset\) such as an atomic lattice in a solid! And what’s more, electrons \(e^-\) don’t just “get added” by some external agent, but rather emerge naturally as the valence electrons \(\partial e^-\) from the atoms located at the lattice sites \(\textbf x\in\Lambda\); thus, working inside a solid kills both birds with one stone.
At a high level, the act of superimposing a Bravais lattice \(\Lambda\) within what used to be an empty cube of sides \(L\) can be treated perturbatively exactly as one does in the nearly free electron model. Specifically, we’re still keeping the ideality/non-interacting assumption between the electrons (with the caveat of the Pauli interaction already mentioned), but now we go from being free\(\rightarrow\)nearly free (used in a technical sense). From that analysis, we know that the presence of the \(\Lambda\)-periodic potential butchers the previous free electron dispersion relation \(E(\textbf k)=\hbar^2|\textbf k|^2/2m\) into an energy band structure \(E_{\text{bands}}(\textbf k)\) where each Brillouin zone \(\Gamma_{\Lambda^*}^{(0)},\Gamma_{\Lambda^*}^{(1)}\), etc. (or bijectively, each energy band \(E_{\text{bands}}(\Gamma_{\Lambda^*}^{(0)}),E_{\text{bands}}(\Gamma_{\Lambda^*}^{(1)})\)) can accommodate precisely \(N\) momentum \(\textbf k\)-states (and thus \(2N\) electrons \(e^-\)) where \(N\) is the number of atoms in the solid. Suppose each atom contributes \(Z\) valence electrons (called its valency). Then the total number of free electrons roaming the solid will be \(ZN\), corresponding to the occupation of \(Z/2\) Brillouin zones or equivalently energy bands. In the ideal free electron case, we saw that the Fermi sea was a ball and its Fermi surface boundary \(\partial=S^2\) a sphere. Now, in the ideal nearly free electron case, we know that the energy is lowered at the boundary \(\partial\Gamma_{\Lambda^*}\) of the Brillouin zones (viewed from “within” otherwise how would the energy band gap form?) so this will distort the Fermi sea (and by extension the Fermi surface \(E_F\)-“equipotential”) of electrons towards it (but conserving in this case the area or in \(3\)D the volume of the Fermi sea since that’s the number of occupied \(\textbf P\)-eigenstates). For \(Z=1\) alkali metal solids or other metals (e.g. \(\text{Li},\text{Cu}\)), the act of “turning” on the perturbation due to the presence of the lattice \(\Lambda\) would look (for a \(2D\) material) something like (because \(Z=1\), the area of the initial Fermi sea/disk must be half the area of the square Brillouin zone):
It is clear in this \(2\)-dimensional case that, if the potential \(V\) induces a sufficiently large energy band gap (as it turns out it does for \(\text{Cu}\)), the Fermi surface can cross the Brillouin zone boundary \(\partial\Gamma_{\Lambda^*}\), though it must do so orthogonally in the reduced scheme to maintain its smoothness by virtue of the toroidal topology \(\Gamma_{\Lambda^*}\cong S^1\times S^1\) of the Brillouin zone.
Now, why do we care about the Fermi surface so much? Short answer: because materials with Fermi surfaces are metals. Qualitatively, the idea is that only the electrons with momentum \(\textbf k\) living on the Fermi surface of the system can actually do anything (having access to a bunch of unoccupied states slightly higher in energy to be able to respond to e.g. an \(\textbf E_{\text{ext}}\) to minimize their energy and form a current \(\textbf J=\sigma\textbf E_{\text{ext}}\)) just as only the valence electrons \(\partial e^-\) could delocalize (except that the former is a “meta-layer” of valency above the latter!). Any electrons \(e^-\) deep in the Fermi sea are pretty much trapped there by the Pauli exclusion principle since there’s no room for them to climb up into nearby energy levels above because they’re already occupied by other electrons (it would take a lot of energy for them to escape)!
For a sense of scale, most metals typically have Fermi temperatures on the order of \(T_F=E_F/k\sim 10^4\text{ K}\) (which is about twice as hot as the surface of the Sun \(\odot\)). This is why Fermi surfaces are also strictly defined at absolute zero \(T=0\), since most real materials in room temperature environments are nowhere near their Fermi temperature \(T_F\).
Another point is that, of course, the number of low-energy excitations available in a metal is proportional to the surface area \(\sim k_F^2\sim E_F\) of its Fermi surface (because each point on the Fermi surface corresponds to an excitable electron \(e^-\)).
Now consider \(Z=2\) atoms (e.g. alkaline earth atoms like \(\text{Be}\)). Now the initial Fermi sea/disk for the ideal free electron system has area equal to the square, but geometrically this implies that it must leak out the boundary of the Brillouin zone a little bit. If one now superimposes the perturbing lattice \(\Lambda\), there are \(2\) possibilities depending on the strength of the periodic lattice potential \(V_{\Lambda}\):
One might think the Fermi surface in the case of band insulators is just the boundary \(\partial\Gamma_{\Lambda^*}\) of the Brillouin zone but that’s not right because it’s not an equipotential with respect to any well-defined Fermi energy \(E_F\).
From here onwards \(Z=3,4,5,…\), the qualitative classification essentially repeats. Metals for instance may have several fully-occupied core bands, a fully-occupied valence band, and then right above that a partially-occupied conduction band. Note though that the Fermi surface need not lie solely in a single Brillouin zone, but can have sections distributed through several Brillouin zones. For example, if we consider the ideal, free electron gas with \(Z=3\) this time (so the area of the circle is three times that of the square \(1\)-st Brillouin zone that it now contains), then:
Although the Fermi surface in the \(3\)-rd Brillouin zone square looks disconnected, in fact it is connected (as it was in the extended scheme) because opposite sides of that square in the reduced scheme are topologically glued together (put another way, within the \(3\)-rd Brillouin zone reduced square, if one shifts the “origin” from where it is right now to the top right corner for instance (although all corners are equivalent), then the Fermi surface clearly becomes connected (see this either by tessellating the square or thinking of it as a torus \(S^1\times S^1\) and wrapping the square on itself):
This concludes a qualitative overview of band theory (classification of materials based on whether they are gapped or gapless). While the framework in general is fairly robust, there are some situations where its predictions fail. And as one might suspect, the origin of such deviations are due to one of the assumptions of band theory not being satisfied, notably the assumption of ideality. Examples of these deviations include semiconductors, Mott insulators (cf. band insulators), and topological insulators.