Band Structure Fundamentals

Learning Objectives

By the end of this section you should be able to:

1. Explain in plain language why isolated atomic levels turn into continuous bands when the atoms are packed into a crystal — and tell the story two different ways: bottom-up (tight-binding) and top-down (nearly-free electrons).
2. Recognise the band structure as the function $E_n(\mathbf{k})$ — a finite catalogue of energies indexed by a band number $n$ and a Brillouin-zone wavevector $\mathbf{k}$.
3. Read a band structure plot fluently: identify the Fermi level, the valence band maximum (VBM), the conduction band minimum (CBM), and distinguish a direct gap from an indirect one.
4. Connect band slope to group velocity $\mathbf{v} = \hbar^{-1} \nabla_{\mathbf{k}} E_n$ and band curvature to effective mass $(m^{*})^{-1} = \hbar^{-2} \partial^2 E / \partial k^2$ — the two quantities that determine how electrons in a band actually behave.
5. Trace a band structure all the way back to the VASP files that produced it: KPOINTS (line mode), EIGENVAL, vasprun.xml, and the INCAR tags $\texttt{ICHARG}=11$, $\texttt{LORBIT}=11$.

One-line preview: a band structure is the answer to the question "if I drop one electron into this crystal, what energies are available to it, and which directions can it flow?" Every electronic property a materials scientist cares about — colour, conductivity, mobility, magnetism — is a sentence written in the vocabulary of $E_n(\mathbf{k})$.

From a Single Atom to a Crystal

Take a hydrogen atom. Solve the Schrödinger equation. You get a familiar ladder of discrete energies: $-13.6\,\text{eV}$ for 1s, $-3.4\,\text{eV}$ for 2s, and so on. Now bring a second hydrogen atom close. Each atomic level splits into two — bonding and anti-bonding — exactly like the lab experiment of coupling two pendulums of identical frequency. Bring a third atom and you get three slightly displaced levels. Bring $N$ atoms and you get $N$ levels. Bring an Avogadro's number of atoms — and the levels are so dense that no instrument can distinguish them. They have become a band.

That is the bottom-up picture: atoms → molecules → solids → bands. It is a story chemists tell, and it is exactly right. Where it gets interesting is when you realise that the relabelling $n \to k$ at the end is not cosmetic. The crystal's translation symmetry hands you back something better than discrete eigenvalues — a continuous function $E(k)$ on the first Brillouin zone, with a beautiful and physical interpretation. Before we get there, let's watch the levels close up with our own eyes.

Interactive — Levels Become Bands

Drag the slider through $N$. Each cyan tick is an allowed energy for a chain of $N$ hydrogenic atoms with hopping $t = 1$. The dashed amber lines $E = \pm 2t$ are the band edges in the infinite limit.

Two observations to lock in. First, the band edges $\pm 2t$ do not depend on $N$ — they are set by the hopping alone. The band width ($4t$) is a local property of the chemistry; only the filling of the band depends on how many atoms (and electrons) you have. Second, the levels never escape outside $[-2t,\,+2t]$ — the band has a hard floor and a hard ceiling. Outside that interval is a gap, the forbidden region between this band and the next one.

The Bloch Catalog: E as a Function of (n, k)

From Section 3.5 we already know the punch line: in a periodic potential, every electronic eigenstate carries two labels — a band index $n$ (which discrete "solution" we are on) and a wavevector $\mathbf{k}$ (which point of the Brillouin zone). The corresponding eigenenergy

is a function of two variables: a discrete index $n \in \{1, 2, 3, \dots\}$ and a continuous vector $\mathbf{k}$ running over the first Brillouin zone. This function is the band structure.

Why "catalogue"? Because for a fixed $\mathbf{k}$, the Schrödinger equation in the plane-wave basis becomes a finite (but large) matrix eigenvalue problem, and matrices have a finite, ordered list of eigenvalues. Numbering those eigenvalues from the bottom gives the band index $n$. Doing this at every $\mathbf{k}$ in the Brillouin zone and stacking the results gives the bands.

The Anatomy of a Band Structure Plot

A band structure plot is a one-dimensional cross-section of the function $E_n(\mathbf{k})$. The horizontal axis is a path through the Brillouin zone — a sequence of straight-line segments joining special points like $\Gamma, X, L, K, M$. The vertical axis is energy, almost always shifted so that the Fermi level $E_F$ sits at zero. Each curve in the plot is one band $E_n$.

Six features every materials scientist looks for in the first three seconds:

Interactive — Folding Parabolas, Opening Gaps

The bottom-up tight-binding story is one way to derive bands. The complementary top-down story starts from a free electron gas — the bare-bones quantum mechanics $E = \hbar^2 k^2 / 2m$ — and asks what happens when you slowly turn on a periodic potential. The widget below is the cleanest visualisation of that picture.

Three things to do with the slider, in order:

1. Set V = 0 (free electron limit). The dashed grey parabolas are $E = (k + nG)^2$ for $n = -1, 0, 1$. Inside the first BZ, the coloured lines are exactly the same parabolas folded back into $[-\pi/a, \pi/a]$ — this is the empty-lattice band structure. Notice that curves cross at $k = \pm \pi/a$ and at $k = 0$: those crossings are accidental degeneracies of the empty lattice.
2. Drag V up to ~0.4 (nearly free electrons). The crossings turn into avoided crossings. A gap opens at every point where two plane waves were degenerate. At the zone boundary the gap is exactly $2|V|$ (read it off the red marker). This is the textbook nearly-free-electron result, and it is the origin of every band gap in a covalent or ionic insulator.
3. Push V toward 1.5 (strong potential). Watch the lowest band flatten. The lower it goes, the more it looks like a cosine — the tight-binding band of the previous interactive. The two pictures are the same picture, viewed from opposite ends.

Why We Plot Along High-Symmetry Paths

A band structure in three dimensions is a function on a 3-D Brillouin zone — that means each band is a 3-D iso-energy hypersurface. We cannot plot that on paper. So we cheat: we pick a 1-D path through the most physically interesting points and plot $E_n$ along the path.

The points are not arbitrary — they are the high-symmetry points of the Brillouin zone introduced in Section 3.4: $\Gamma$ (the centre), $X$, $L$, $K$, $M$, etc. They matter because:

• The Hamiltonian has extra symmetry there, so eigenstates carry extra quantum numbers and degeneracies show up.
• Band extrema (the VBM and CBM) almost always sit at high-symmetry points or along high-symmetry lines — physics is conservative about where it puts important things.
• Different conventions agree on the labelling, so two papers using different DFT codes are describing the same path.

For an FCC Brillouin zone (Si, Ge, GaAs, CdSe in zincblende), the canonical path is $L \to \Gamma \to X \to W \to K \to \Gamma$. For a simple cubic BZ it is $\Gamma \to X \to M \to \Gamma \to R$. You don't need to memorise these — VASP's band-structure tools and SeeK-path will hand them to you for any space group. What you do need is to understand that the wiggles you see on a band structure plot are 1-D cross sections of a higher-dimensional landscape.

Direct vs. Indirect Band Gaps

For semiconductors the single most decisive feature of the band structure is the relative location of the VBM and the CBM in the Brillouin zone.

The reason a direct gap matters for optics is conservation of crystal momentum. A photon at visible frequency carries $k_\text{photon} \approx 2\pi/\lambda \sim 10^{-3}\,\text{Å}^{-1}$, which on a Brillouin-zone scale ($\sim 1\,\text{Å}^{-1}$) is essentially zero. A photon-driven transition is therefore a vertical arrow on the band-structure plot. If your VBM and CBM sit at the same $\mathbf{k}$, the arrow finds them; if not, you need a phonon to supply the missing $\Delta \mathbf{k}$ — a slow, three-body process. This is why silicon makes lousy LEDs and GaAs makes the laser in your CD player.

Group Velocity and Effective Mass

A band structure is more than a list of allowed energies. The shape of each band — its slopes and curvatures — encodes how electrons in that band actually move and respond to forces. Two derivatives of $E_n(\mathbf{k})$ do all the work.

Group velocity (first derivative)

A Bloch electron in band $n$ at wavevector $\mathbf{k}$ is a wave packet that travels at the group velocity

Same formula as a free particle ($v = \hbar k / m$ becomes $v = (1/\hbar)\,dE/dk$ when you differentiate $E = \hbar^2 k^2/2m$), but now the function $E_n(\mathbf{k})$ is set by the crystal, not the vacuum. The slope of a band is the velocity of an electron in that state. Flat band → near-zero velocity → localised electron. Steep band → fast electron. At a band extremum, the slope is zero: an electron exactly at the VBM or CBM does not move on average.

Effective mass (second derivative)

Newton's second law for a Bloch electron in an external force $\mathbf{F}$ reads

The matrix $(m^*)^{-1}_{ij} = \hbar^{-2}\,\partial^2 E_n / \partial k_i \partial k_j$ plays the role of an inverse mass tensor. In an isotropic minimum,

which is just the curvature of the band, expressed as a mass. Sharp parabolic minima → light, mobile carriers (GaAs has $m^*_e \approx 0.07\,m_0$). Flat minima → heavy, sluggish carriers. We will return to this in Section 5.4 when we compute carrier mobilities.

Band Structure in VASP — From INCAR to a Plot

Computing a band structure in VASP is a two-step exercise. Step 1: a normal self-consistent calculation on a uniform Monkhorst–Pack k-grid, which produces a converged charge density $n(\mathbf{r})$ stored in CHGCAR. Step 2: a non-self-consistent calculation on a densely sampled high-symmetry path, which uses that charge density unchanged and only diagonalises the Kohn–Sham Hamiltonian at the new k-points. The two-step trick saves cost and guarantees that the path and the SCF use the same self-consistent potential.

Step 1 — the SCF calculation

1# INCAR (step 1: self-consistent)
2SYSTEM = CdSe zincblende SCF
3PREC   = Accurate
4ENCUT  = 400        # plane-wave cutoff (eV) — set by convergence test
5EDIFF  = 1E-6       # SCF tolerance (eV)
6ISMEAR = 0          # Gaussian smearing for semiconductor
7SIGMA  = 0.05
8LCHARG = .TRUE.     # write CHGCAR for step 2
9LWAVE  = .FALSE.    # WAVECAR not needed for non-SCF

1# KPOINTS (step 1: uniform grid)
2Monkhorst-Pack
30
4Gamma
58 8 8
60 0 0

Step 2 — the band structure run

1# INCAR (step 2: non-SCF along high-symmetry path)
2SYSTEM = CdSe zincblende bands
3PREC   = Accurate
4ENCUT  = 400
5ICHARG = 11         # read CHGCAR, do not update density (non-SCF)
6LORBIT = 11         # write per-atom, per-orbital projections (PROCAR)
7ISMEAR = 0
8SIGMA  = 0.05
9NBANDS = 32         # at least valence + a healthy conduction stack

1# KPOINTS (step 2: line mode through L-Γ-X-W-K-Γ for FCC)
2Bands along high-symmetry lines
340                  ! 40 points per segment
4Line
5Reciprocal
6
70.5  0.5  0.5  L
80.0  0.0  0.0  Gamma
9
100.0  0.0  0.0  Gamma
110.5  0.0  0.5  X
12
130.5  0.0  0.5  X
140.5  0.25 0.75 W
15
160.5  0.25 0.75 W
170.375 0.375 0.75 K
18
190.375 0.375 0.75 K
200.0  0.0  0.0  Gamma

Plotting in five lines of pymatgen

1from pymatgen.io.vasp import Vasprun, BSVasprun
2from pymatgen.electronic_structure.plotter import BSPlotter
3
4vr  = BSVasprun("vasprun.xml", parse_projected_eigen=True)
5bs  = vr.get_band_structure(line_mode=True)
6BSPlotter(bs).get_plot(ylim=(-6, 6)).savefig("bands.png", dpi=200)

The result is a publication-quality plot with bands coloured by spin (if spin-polarised), the Fermi level dashed at zero, and the high-symmetry labels along the bottom axis. From here, every analysis in this chapter — DOS, effective masses, optical absorption — is a few-line extension of the same code.

Looking Ahead — The Map of Chapter 5

With $E_n(\mathbf{k})$ in hand, the rest of this chapter unpacks every physical observable you can squeeze out of it.

Summary

• A band is what happens to a discrete atomic level when atoms are coupled into an infinite periodic chain: the discrete eigenvalues fill in to a continuous strip of width set by the hopping integral.
• The band structure $E_n(\mathbf{k})$ is a finite catalogue of energy surfaces, indexed by band number, defined over the first Brillouin zone. It is the central object of solid-state physics.
• Two derivations of bands meet in the middle: bottom-up tight-binding (chemistry, atomic levels) and top-down nearly-free-electrons (parabolas folded into the BZ with avoided crossings of size $2|V|$).
• A band structure plot is a 1-D slice of $E_n(\mathbf{k})$ along a high-symmetry path. Every paper showing electronic structure of a crystal is showing one of these.
• Slope $\nabla_{\mathbf{k}} E_n$ is group velocity; curvature $\partial^2 E_n / \partial k^2$ is inverse effective mass. Together they govern transport, mobility, and optical-matrix elements.
• Direct vs indirect gap is the first thing to read off when judging a semiconductor for optoelectronics: only direct-gap materials emit light efficiently.
• In VASP, a band structure is a two-step recipe: an SCF run on a uniform grid, then a non-SCF run ($\texttt{ICHARG}=11$) on a high-symmetry path with line-mode KPOINTS. The eigenvalues land in EIGENVAL and vasprun.xml; pymatgen plots them in five lines.

Coming next: Section 5.2 — Density of States — where we integrate $E_n(\mathbf{k})$ over the Brillouin zone to compute $g(E)$, the function that thermodynamics and optical experiments actually observe.