2.3 Bloch's theorem

Bloch's theorem describes how electrons behave in the periodic potential of a crystal lattice. It explains why electrons in solids form energy bands separated by gaps, which is the foundation for classifying materials as metals, semiconductors, or insulators.

Bloch's theorem fundamentals

In a crystal, atoms are arranged in a regular, repeating pattern. This arrangement creates a periodic potential that every electron in the solid experiences. Bloch's theorem tells you something powerful about the solutions to the Schrödinger equation in this periodic environment: the electron wavefunctions aren't random or complicated. They take a very specific, structured form.

Electrons in periodic potentials

The potential energy felt by an electron in a crystal repeats with the same periodicity as the lattice itself. Mathematically, this means:

$V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R})$

where $\mathbf{R}$ is any lattice vector. This periodicity constrains what the electron wavefunctions can look like, and that constraint is exactly what Bloch's theorem captures. The result is that electrons can only occupy certain ranges of energy (bands), separated by forbidden regions (gaps).

Bloch wavefunctions

Bloch's theorem states that the eigenstates of an electron in a periodic potential can be written as:

$\psi_{n,\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} \, u_{n,\mathbf{k}}(\mathbf{r})$

This has two pieces:

• Plane wave factor $e^{i\mathbf{k} \cdot \mathbf{r}}$: Describes the overall propagation of the electron through the crystal. This is what gives the electron its wavelike, extended character.
• Periodic function $u_{n,\mathbf{k}}(\mathbf{r})$: Has the same periodicity as the lattice, so $u_{n,\mathbf{k}}(\mathbf{r}) = u_{n,\mathbf{k}}(\mathbf{r} + \mathbf{R})$. This part encodes how the wavefunction is modulated by the local atomic environment.

Each Bloch state is labeled by a band index $n$ (which band it belongs to) and a wavevector $\mathbf{k}$ (which lies within the first Brillouin zone).

Bloch states vs atomic orbitals

Atomic orbitals are eigenstates of isolated atoms, localized around a single nucleus. In a crystal, these orbitals on neighboring atoms overlap and hybridize. The result is Bloch states, which extend throughout the entire lattice rather than being stuck on one atom.

The tight-binding approximation makes this connection explicit: you build Bloch states as linear combinations of atomic orbitals from every atom in the crystal. This is one of the most intuitive ways to see how discrete atomic energy levels broaden into continuous energy bands.

Mathematical formulation of Bloch's theorem

The formal starting point is the time-independent Schrödinger equation for an electron in a periodic potential:

$\left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right]\psi(\mathbf{r}) = E\,\psi(\mathbf{r})$

Bloch's theorem tells you that the solutions can always be chosen in the form $\psi_{n,\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} \, u_{n,\mathbf{k}}(\mathbf{r})$.

Derivation of Bloch's theorem

The proof relies on the symmetry of the problem. Here's the logic:

1. Define the translation operator $\hat{T}_{\mathbf{R}}$, which shifts every position by a lattice vector $\mathbf{R}$: it acts as $\hat{T}_{\mathbf{R}}\psi(\mathbf{r}) = \psi(\mathbf{r} + \mathbf{R})$.
2. Because the potential is periodic, $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$, the translation operator commutes with the Hamiltonian: $[\hat{H}, \hat{T}_{\mathbf{R}}] = 0$.
3. Operators that commute share a common set of eigenstates. So you can choose energy eigenstates that are simultaneously eigenstates of all $\hat{T}_{\mathbf{R}}$.
4. The eigenvalues of $\hat{T}_{\mathbf{R}}$ must be phase factors (since translation operators are unitary), so $\hat{T}_{\mathbf{R}}\psi = e^{i\mathbf{k}\cdot\mathbf{R}}\psi$ for some wavevector $\mathbf{k}$.
5. Writing $\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} \, u(\mathbf{r})$ and imposing the eigenvalue condition shows that $u(\mathbf{r})$ must be lattice-periodic.

The key insight is that translational symmetry alone forces the wavefunctions into the Bloch form. You don't need to know the details of the potential.

Periodic boundary conditions

To make the problem tractable, you impose Born-von Karman periodic boundary conditions on a finite crystal of $N_1 \times N_2 \times N_3$ unit cells:

$\psi(\mathbf{r} + N_i \mathbf{a}_i) = \psi(\mathbf{r})$

where $\mathbf{a}_i$ are the primitive lattice vectors. This condition quantizes the allowed values of $\mathbf{k}$: each component of $\mathbf{k}$ can only take discrete values spaced by $2\pi / (N_i |\mathbf{a}_i|)$. In the thermodynamic limit (large $N_i$), these discrete values become effectively continuous, but the total number of allowed $\mathbf{k}$-points per band equals the number of unit cells in the crystal.

Bloch wavevector

The wavevector $\mathbf{k}$ in the Bloch wavefunction is called the crystal momentum (up to a factor of $\hbar$). A few things to keep straight:

• $\mathbf{k}$ is restricted to the first Brillouin zone. Any $\mathbf{k}$ outside this zone is equivalent to a $\mathbf{k}$ inside it, shifted by a reciprocal lattice vector $\mathbf{G}$.
• Crystal momentum is not the same as true momentum. An electron in a Bloch state doesn't have a definite momentum because the periodic potential breaks continuous translational symmetry. However, $\hbar\mathbf{k}$ plays an analogous role in conservation laws (e.g., in scattering processes, crystal momentum is conserved modulo $\mathbf{G}$).
• $\mathbf{k}$ determines the phase relationship between unit cells: how the wavefunction's phase advances as you move from one cell to the next.

Brillouin zones

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. You construct it by drawing perpendicular bisecting planes to the nearest reciprocal lattice vectors from the origin. Everything enclosed is the first zone.

Higher Brillouin zones are defined by the next set of bisecting planes. Every zone has the same volume in reciprocal space, and any point in a higher zone can be mapped back into the first zone by subtracting an appropriate reciprocal lattice vector $\mathbf{G}$. This is why the first Brillouin zone contains all the unique information about the band structure.

Consequences of Bloch's theorem

Electronic band structure

Applying Bloch's theorem to a crystal yields the energy dispersion relation $E_n(\mathbf{k})$, which gives the energy of an electron in band $n$ as a function of wavevector $\mathbf{k}$. Plotting $E_n(\mathbf{k})$ along high-symmetry directions in the Brillouin zone produces the familiar band structure diagrams.

The shape of these bands depends on the crystal structure and the types of atomic orbitals involved. Bands that derive from tightly bound core orbitals tend to be narrow (flat), while bands from delocalized outer orbitals tend to be wide (dispersive).

Allowed vs forbidden energy bands

Bloch's theorem predicts that electrons can only have energies within certain allowed bands, separated by band gaps where no states exist. The distinction between material types comes down to how these bands are filled:

• Metals: The highest occupied band is only partially filled, so electrons near the Fermi level can easily move to nearby empty states. This gives high electrical conductivity.
• Insulators: The valence band is completely filled and the conduction band is completely empty, separated by a large gap (typically > 4 eV). Electrons can't easily reach the conduction band.
• Semiconductors: Same picture as insulators, but with a small gap (e.g., ~1.1 eV for Si, ~0.67 eV for Ge). Thermal or optical excitation can promote electrons across the gap.

Effective mass of electrons

Near a band extremum (a maximum or minimum in $E_n(\mathbf{k})$), the dispersion is approximately parabolic, similar to a free particle but with a modified mass. The effective mass tensor is defined as:

$\frac{1}{m_{ij}^*} = \frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k_i \partial k_j}$

• A band with strong curvature (wide, dispersive band) gives a small effective mass, meaning electrons respond readily to applied fields.
• A flat band gives a large effective mass, meaning electrons are sluggish.
• Near the top of a valence band, the curvature is negative, giving a negative effective mass. This is the origin of the hole concept: it's more convenient to describe the missing electron as a positive quasiparticle with positive effective mass.

The effective mass directly enters transport equations, so it governs quantities like electrical conductivity and the Hall coefficient.

Density of states in bands

The density of states (DOS) counts how many electronic states exist per unit energy:

$g(E) \propto \int_{E(\mathbf{k})=E} \frac{dS}{|\nabla_{\mathbf{k}} E|}$

where the integral is over the constant-energy surface in $\mathbf{k}$-space. The DOS is large wherever bands are flat (small $|\nabla_{\mathbf{k}} E|$), producing van Hove singularities at critical points.

The DOS determines many measurable properties: the position of the Fermi level, carrier concentrations, electronic specific heat, and optical absorption spectra.

Applications of Bloch's theorem

Nearly free electron model

This model starts from free electrons and treats the periodic potential as a weak perturbation. The key result: energy gaps open up at the Brillouin zone boundaries due to Bragg reflection of electron waves. At these boundaries, the electron wavelength satisfies the Bragg condition, and the forward- and backward-traveling waves mix to form standing waves. One standing wave concentrates electron density on the ions (lower energy), the other between the ions (higher energy), creating the gap.

The nearly free electron model works well for simple metals like alkali metals (Na, K), where the valence electrons are loosely bound and the ionic potential is relatively weak.

Tight binding approximation

The tight-binding approach starts from the opposite limit: strongly localized atomic orbitals. You construct Bloch states as linear combinations of atomic orbitals (LCAO) centered on each lattice site:

$\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \, \phi(\mathbf{r} - \mathbf{R})$

The bandwidth depends on the overlap integrals between neighboring orbitals. More overlap means wider bands. This model is especially useful for transition metals (where $d$-electrons are fairly localized) and organic semiconductors.

It also naturally captures the distinction between direct band gaps (conduction band minimum and valence band maximum at the same $\mathbf{k}$) and indirect band gaps (at different $\mathbf{k}$-points), which matters for optical properties.

Kronig-Penney model

The Kronig-Penney model is a one-dimensional, exactly solvable toy model: a periodic array of rectangular potential barriers. Despite its simplicity, it reproduces the essential features predicted by Bloch's theorem:

• Allowed energy bands separated by forbidden gaps
• Band widths that depend on barrier height and width
• The emergence of Brillouin zone structure

It's a great model for building intuition. As you increase the barrier strength, the bands narrow and the gaps widen, approaching the limit of isolated atoms. As you decrease the barrier strength, the bands widen and gaps shrink, approaching the free electron limit.

Semiconductors vs insulators

Bloch's theorem provides the framework, but the practical distinction between semiconductors and insulators comes down to the size of the band gap:

• Semiconductors (Si: ~1.1 eV, GaAs: ~1.4 eV): The gap is small enough that thermal energy at room temperature ($k_BT \approx 0.026$ eV) can excite a meaningful number of electrons into the conduction band. Doping with impurities can further control the carrier concentration.
• Insulators (diamond: ~5.5 eV, $\text{SiO}_2$: ~9 eV): The gap is so large that virtually no electrons are thermally excited across it under normal conditions.

The position of the Fermi level within the gap (influenced by doping and temperature) determines the material's conductivity.

Limitations of Bloch's theorem

Applicability to real crystals

Bloch's theorem assumes perfect translational symmetry, which no real crystal has. Real materials contain defects (vacancies, interstitials), impurities, grain boundaries, and surfaces, all of which break periodicity. These imperfections can introduce localized states within the band gap that don't have the Bloch form.

To handle deviations from perfect periodicity, physicists use tools like Wannier functions (localized counterparts of Bloch states) and treat imperfections as perturbations to the periodic system.

Effects of lattice imperfections

Lattice defects affect electronic properties in several ways:

• Gap states: Vacancies, interstitials, and substitutional impurities can create energy levels inside the band gap. These act as traps or recombination centers for carriers, which is critical in semiconductor device physics.
• Carrier scattering: Electrons scatter off defects, limiting their mean free path and reducing carrier mobility. This is one of the main factors controlling electrical resistivity at low temperatures (where phonon scattering is reduced).
• Anderson localization: In highly disordered systems, scattering can be so strong that electron states become localized rather than extended, fundamentally changing the transport behavior.

Electron-electron interactions

Bloch's theorem is built on the independent electron (single-particle) approximation: each electron moves in the average potential created by the ions and all other electrons, but direct electron-electron interactions are neglected. This works surprisingly well for many materials, but it fails for strongly correlated systems where electron-electron repulsion is comparable to the kinetic energy.

Examples include transition metal oxides (like the cuprate superconductors) and heavy fermion compounds. For these materials, methods like the Hubbard model, dynamical mean-field theory (DMFT), and the GW approximation are needed to capture the physics that Bloch's theorem misses.

Beyond the single-electron picture

Several important phenomena in solids arise from collective electron behavior that the single-particle Bloch picture can't describe:

• Superconductivity: Electrons form Cooper pairs via phonon-mediated attraction (BCS theory). The paired state has fundamentally different properties from individual Bloch states.
• Magnetism: Ordered magnetic states (ferromagnetism, antiferromagnetism) arise from exchange interactions between electrons, described by models like the Heisenberg and RKKY frameworks.
• Charge density waves: Collective instabilities where the electron density spontaneously develops a periodic modulation different from the lattice periodicity.

These phenomena require many-body theoretical approaches that go well beyond the single-electron band theory built on Bloch's theorem.