Field Note: The Band Structure
Field Note · Trolla · Condensed matter physics
A crystal is a periodic potential. Solve the Schrödinger equation in a periodic potential, and you get energy bands. The band structure is the map of all possible electron energies as a function of crystal momentum. It is the single most important object in condensed matter physics. Without it, you cannot answer whether a material is a metal, an insulator, or a semiconductor. With it, you can calculate everything else.
The starting point
The Hamiltonian is:
$$H = -\frac{\hbar^2}{2m_e} \nabla^2 + V(\mathbf{r})$$
where $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$ for all lattice vectors $\mathbf{R}$. Bloch's theorem states that the eigenstates have the form:
$$\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}(\mathbf{r})$$
where $u_{n\mathbf{k}}(\mathbf{r})$ has the periodicity of the lattice. The index $n$ is the band index, and $\mathbf{k}$ is the crystal momentum, defined modulo reciprocal lattice vectors. The spectrum $E_n(\mathbf{k})$ is the band structure.
Bloch's theorem is deceptively simple. It turns an infinitely dimensional problem into a family of finite-dimensional problems, one for each $\mathbf{k}$ in the Brillouin zone. Each $\mathbf{k}$ gives you a set of eigenvalues $E_n(\mathbf{k})$ and eigenfunctions $u_{n\mathbf{k}}(\mathbf{r})$. Plot $E_n(\mathbf{k})$ as a function of $\mathbf{k}$, and you have the band structure.
What the band structure tells you
Metals: The Fermi energy crosses one or more bands. Electrons at the Fermi surface are available for transport. The material conducts.
Insulators: All bands below the Fermi energy are fully occupied. The next available band is separated by a large energy gap ($E_g > 3$ eV). Electrons cannot jump the gap at reasonable temperatures. The material does not conduct.
Semiconductors: Same as insulators, but the gap is small ($E_g \sim 0.1$–2 eV). Thermal or optical excitation can promote electrons across the gap, creating mobile carriers in both the conduction and valence bands.
Topological insulators: A special class of semiconductors where the band structure has a nontrivial topological invariant (a $\mathbb{Z}_2$ index, a Chern number, or some other discrete label). The bulk is insulating, but the surface hosts gapless conducting states protected by time-reversal symmetry.
Computational methods
There are three principal approaches to computing band structures, each with different trade-offs.
Tight-binding
The tight-binding model assumes that electrons are strongly bound to atomic sites and only weakly hop between them. The Hamiltonian is written as a matrix of hopping amplitudes $t_{ij}$ between sites $i$ and $j$. Diagonalize the matrix for each $\mathbf{k}$, and you get approximate band energies.
Tight-binding is fast, intuitive, and surprisingly accurate for many systems. It fails when the band structure is dominated by delocalized states or when the hopping amplitudes are too numerous to compute by hand.
Density functional theory (DFT)
DFT computes the band structure from first principles by solving the Kohn-Sham equations:
$$\left(-\frac{\hbar^2}{2m_e} \nabla^2 + V_{\text{eff}}(\mathbf{r})\right) \phi_i(\mathbf{r}) = \varepsilon_i \phi_i(\mathbf{r})$$
where $V_{\text{eff}}(\mathbf{r})$ includes the ionic potential, the Hartree potential, and the exchange-correlation potential. The Kohn-Sham eigenvalues $\varepsilon_i$ are not true quasiparticle energies, but they correlate remarkably well with experimental band structures for most purposes.
DFT is the workhorse of computational condensed matter. It is accurate for ground-state properties and gives band structures that are qualitatively correct for most materials. Its weaknesses — underestimating band gaps, failing for strongly correlated systems — are well known and actively researched.
GW approximation
The GW approximation corrects DFT's band-gap problem by computing the electron self-energy $\Sigma$ within a screened-interaction framework. The name "GW" comes from the approximation $\Sigma \approx iGW$, where $G$ is the one-particle Green's function and $W$ is the screened Coulomb interaction.
GW gives quantitatively accurate band gaps and quasiparticle dispersions. It is computationally expensive — $O(N^3)$ to $O(N^4)$ in the number of electrons — and is usually applied on top of a DFT calculation as a perturbative correction.
A practical observation
When you plot band structures, the high-symmetry points in the Brillouin zone — $\Gamma$, $X$, $L$, $K$, $H$ — define the standard paths along which the dispersion is sampled. The choice of path is arbitrary but conventional, and two papers plotting different paths through the same Brillouin zone will show different pictures of the same bands. The bands themselves do not change. The plot does.
This is a small, persistent annoyance in the literature. Always check the path. A band gap that appears to close along one path may be open along another.