3.1 Bloch theorem

Bloch theorem is a cornerstone of condensed matter physics that explains how electrons behave in crystalline solids. It describes wave functions in periodic potentials, connecting free-electron behavior with the influence of the crystal lattice. The theorem leads directly to the concepts of energy bands and crystal momentum, which are essential for understanding why metals conduct, why semiconductors have gaps, and why insulators don't conduct at all.

Fundamentals of Bloch theorem

Bloch theorem provides the framework for understanding how electrons move through the periodic arrangement of ions in a crystal. Rather than treating each atom individually, the theorem exploits the translational symmetry of the lattice to dramatically simplify the problem. Instead of solving for electron states across an entire macroscopic crystal, you only need to solve within a single unit cell.

Periodic potentials in crystals

The regular arrangement of atoms in a crystal creates a repeating pattern of electrostatic potentials. This periodicity is expressed as:

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

where $\mathbf{R}$ is any Bravais lattice vector. The potential an electron "sees" at position $\mathbf{r}$ is identical to what it sees at $\mathbf{r} + \mathbf{R}$. This periodicity is what makes Bloch theorem possible, and it's also what gives rise to allowed energy bands separated by forbidden gaps.

Bloch's wave function

Bloch theorem states that the eigenstates of an electron in a periodic potential take the form:

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

This has two parts:

• $e^{i\mathbf{k} \cdot \mathbf{r}}$ is a plane wave, representing the electron's propagation through the crystal (like a free electron).
• $u_{\mathbf{k}}(\mathbf{r})$ is a function with the same periodicity as the lattice, meaning $u_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r} + \mathbf{R})$. This part encodes how the wave function is modulated by the ions.

The physical picture: electrons can propagate through a perfect crystal despite the presence of ions. The periodic part $u_{\mathbf{k}}$ adjusts the wave function near each ion, but the plane-wave envelope allows coherent transport across the lattice.

Crystal momentum

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

• $\hbar \mathbf{k}$ is not the true momentum of the electron. In a periodic potential, true momentum isn't conserved because the potential breaks continuous translational symmetry.
• Crystal momentum is conserved modulo a reciprocal lattice vector $\mathbf{G}$. This means $\mathbf{k}$ and $\mathbf{k} + \mathbf{G}$ label the same physical state.
• Conservation of crystal momentum governs selection rules in electron-phonon scattering, optical transitions, and other processes in solids.

Mathematical formulation

Bloch's theorem equation

An equivalent (and often more useful) statement of the theorem is:

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

This says that translating the wave function by a lattice vector $\mathbf{R}$ simply multiplies it by a phase factor $e^{i\mathbf{k} \cdot \mathbf{R}}$. The proof follows from the fact that the translation operator $T_{\mathbf{R}}$ commutes with the Hamiltonian (because $V$ is periodic), so they share simultaneous eigenstates. The eigenvalues of $T_{\mathbf{R}}$ must have unit modulus (since $T_{\mathbf{R}}$ is unitary), which forces the $e^{i\mathbf{k} \cdot \mathbf{R}}$ form.

Periodic boundary conditions

To get a countable set of allowed $\mathbf{k}$ values, you apply Born-von Kármán periodic boundary conditions to a finite crystal of $N$ unit cells:

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

where $\mathbf{a}_i$ are the primitive lattice vectors and $N_i$ is the number of unit cells along direction $i$. Combining this with Bloch's theorem requires $e^{i k_i N_i a_i} = 1$, which quantizes the allowed wave vectors:

$k_i = \frac{2\pi n_i}{N_i a_i}, \quad n_i = 0, 1, 2, \ldots, N_i - 1$

This gives exactly $N$ allowed $\mathbf{k}$ points per band in each dimension, so the total number of $\mathbf{k}$ states per band equals the number of unit cells in the crystal.

Reciprocal lattice vectors

The reciprocal lattice vectors $\mathbf{G}$ satisfy $e^{i\mathbf{G} \cdot \mathbf{R}} = 1$ for all lattice vectors $\mathbf{R}$. In one dimension, $G = 2\pi n / a$ where $n$ is an integer. These vectors define the periodicity in $\mathbf{k}$-space:

• Since $\mathbf{k}$ and $\mathbf{k} + \mathbf{G}$ label the same state, all unique information is contained within a single primitive cell of the reciprocal lattice.
• That primitive cell is the first Brillouin zone.
• Reciprocal lattice vectors also appear in Fourier expansions of the periodic potential and the periodic part $u_{\mathbf{k}}(\mathbf{r})$.

Properties of Bloch states

Band structure formation

When you solve the Schrödinger equation with a periodic potential, the energy eigenvalues $E_n(\mathbf{k})$ form continuous functions of $\mathbf{k}$ within each Brillouin zone. These are the energy bands, indexed by the band index $n$. The relationship $E_n(\mathbf{k})$ is called the dispersion relation.

Think of it this way: a free electron has a single parabolic dispersion $E = \hbar^2 k^2 / 2m$. Turning on a weak periodic potential opens up gaps at the Brillouin zone boundaries (where Bragg reflection occurs), splitting the single parabola into distinct bands.

Allowed vs. forbidden energy bands

• Allowed bands are energy ranges where electron states exist. Electrons can occupy these states and move through the crystal.
• Band gaps (forbidden bands) are energy ranges with no available states. No electron can have an energy within the gap in a perfect, infinite crystal.

The size of the band gap depends on the strength of the periodic potential. Stronger potentials produce wider gaps. This is why:

• Metals have partially filled bands (or overlapping bands), so electrons easily find empty states to move into.
• Semiconductors have modest gaps (on the order of 1 eV, e.g., 1.12 eV for Si).
• Insulators have large gaps (several eV or more, e.g., ~9 eV for diamond).

Brillouin zones

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It contains all the unique $\mathbf{k}$ values needed to fully describe the electronic structure.

• Higher-order Brillouin zones can be mapped back into the first zone by translation through reciprocal lattice vectors. This is the reduced zone scheme.
• The extended zone scheme unfolds the bands across multiple zones, which is useful for seeing how bands connect to the free-electron parabola.
• High-symmetry points in the Brillouin zone (like $\Gamma$, $X$, $L$ in an FCC lattice) are where band extrema and degeneracies often occur.

Applications in solid state physics

Electronic structure calculations

Bloch theorem reduces the problem of solving for electrons in a macroscopic crystal ($\sim 10^{23}$ atoms) to solving within a single unit cell for each $\mathbf{k}$. This makes band structure calculations tractable. From the band structure, you can extract the density of states, Fermi energy, and optical transition energies, which together determine a material's electrical, optical, and magnetic properties.

Conductivity in metals

Metals have partially filled bands, meaning there are empty states just above the Fermi level that electrons can scatter into when an electric field is applied. Bloch theorem explains why a perfect crystal at zero temperature would have infinite conductivity: Bloch states are exact eigenstates, so electrons propagate without scattering. Finite resistivity arises from deviations from perfect periodicity (phonons, impurities, defects).

The temperature dependence of resistivity in metals (roughly linear at high $T$, proportional to $T^5$ at low $T$ via the Bloch-Grüneisen formula) reflects how phonon scattering disrupts the Bloch states.

Semiconductors and insulators

The band gap predicted by Bloch theorem is the central concept in semiconductor physics:

• Intrinsic semiconductors have a gap small enough that thermal excitation promotes some electrons from the valence band to the conduction band.
• Extrinsic semiconductors are doped with impurities that introduce states within the gap, providing carriers at lower temperatures.
• Devices like p-n junctions, transistors, solar cells, and LEDs all rely on controlling electron transitions across band gaps.

Experimental evidence

X-ray diffraction patterns

X-ray diffraction confirms the periodic crystal structure that Bloch theorem requires. Bragg peaks reveal the lattice parameters and symmetry group of the crystal. The measured structure factors also give information about electron density distributions, which can be compared with band structure calculations.

Angle-resolved photoemission spectroscopy (ARPES)

ARPES is the most direct experimental probe of band structure. By measuring the kinetic energy and emission angle of photoelectrons, you can map out $E(\mathbf{k})$ for occupied states. ARPES data routinely confirms the band dispersions, Fermi surfaces, and gap structures predicted by Bloch-theorem-based calculations.

Quantum oscillations

Effects like the de Haas-van Alphen (magnetic susceptibility) and Shubnikov-de Haas (resistivity) oscillations occur when a magnetic field quantizes electron orbits into Landau levels. The oscillation period in $1/B$ is directly related to extremal cross-sectional areas of the Fermi surface. These measurements confirm that electrons in crystals occupy well-defined $\mathbf{k}$-states and provide precise data on Fermi surface geometry and effective masses.

Limitations and extensions

Disordered systems

Bloch theorem assumes perfect translational periodicity. It breaks down in:

• Amorphous materials (glasses, disordered alloys) where there's no long-range order.
• Systems with strong disorder, where Anderson localization can occur: electron wave functions become exponentially localized rather than extended Bloch states.
• The concept of a mobility edge separates localized states from extended states in disordered systems.

Many-body effects

Bloch theorem is a single-particle result. It doesn't account for electron-electron interactions. Extensions include:

• Fermi liquid theory, where interacting electrons are described as weakly interacting quasiparticles that still have well-defined $\mathbf{k}$ quantum numbers.
• Many-body perturbation theory (GW approximation, etc.) for more accurate band gaps and quasiparticle lifetimes.
• In strongly correlated systems (e.g., Mott insulators, heavy fermion materials, high-$T_c$ cuprates), the single-particle Bloch picture can fail qualitatively.

Topological materials

Bloch theorem provides the starting point for topological band theory. The key addition is that Bloch states can carry topological invariants (Chern numbers, $\mathbb{Z}_2$ indices) that characterize the global structure of the wave functions across the Brillouin zone. These invariants predict robust surface states in topological insulators and exotic bulk properties in Weyl and Dirac semimetals.

Computational methods

Tight-binding approximation

The tight-binding model starts from localized atomic orbitals and builds Bloch states by summing over lattice sites with the appropriate phase factors:

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

This approach gives an intuitive picture of how atomic levels broaden into bands as atoms are brought together. The bandwidth is controlled by the overlap (hopping) integrals between neighboring orbitals. Tight-binding is widely used for modeling graphene, carbon nanotubes, and other systems where a few orbitals dominate the physics near the Fermi level.

Density functional theory (DFT)

DFT combines Bloch theorem with the Kohn-Sham framework, replacing the many-electron problem with an effective single-particle problem in a self-consistent potential. Bloch's theorem is applied to the Kohn-Sham orbitals, making the calculation tractable for real materials. DFT is the workhorse of modern materials science, though it can underestimate band gaps (a well-known limitation of standard exchange-correlation functionals).

Pseudopotential methods

Core electrons are tightly bound and don't participate much in bonding or transport. Pseudopotential methods replace the strong ionic potential and core electrons with a smoother effective potential that reproduces the correct behavior of valence electrons outside the core region. This dramatically reduces the computational cost (fewer plane waves needed to expand the Bloch states) while maintaining accuracy for the properties that matter most.

Connection to other concepts

Fermi surface

The Fermi surface is the constant-energy surface in $\mathbf{k}$-space at the Fermi energy $E_F$. Its shape is entirely determined by the band structure $E_n(\mathbf{k})$ derived from Bloch theorem. The geometry of the Fermi surface controls electronic transport, heat capacity, magnetic response, and superconducting pairing. Metals with complex Fermi surfaces (like copper or the transition metals) show rich anisotropic behavior.

Effective mass

Near a band extremum, the dispersion can be approximated as parabolic, and the electron responds to external forces as if it had an effective mass:

$\frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2 E}{dk^2}$

This effective mass can differ greatly from the free electron mass. In GaAs, for example, the conduction band effective mass is about $0.067 \, m_e$, making electrons very mobile. Near the top of a valence band, the curvature is negative, giving rise to the concept of holes with positive effective mass. The effective mass can also be anisotropic, described by a tensor in general.

$\mathbf{k}$-space vs. real space

Bloch theorem naturally frames the problem in reciprocal ($\mathbf{k}$) space, where translational symmetry makes the Hamiltonian block-diagonal. Many properties (band structure, Fermi surface, optical selection rules) are most naturally described in $\mathbf{k}$-space. Real-space descriptions become important for localized phenomena like defect states, surface effects, and disorder. The two pictures are connected by Fourier transforms, and a complete understanding of solids requires fluency in both.