Condensed Matter Physics

Condensed matter physics studies the collective behavior of very large numbers of interacting particles in solids and liquids. It is the largest subfield of physics by number of practitioners, and it supplies the physics behind transistors, lasers, magnets, batteries, and superconducting qubits. This hub covers the foundations every other page builds on: how atoms arrange into crystals, how that periodicity turns electron energies into bands, and how band filling decides whether a solid is a metal, an insulator, or a semiconductor. The pages listed below extend this to lattice vibrations, magnetism, disorder, emergent and topological phases, experimental probes, and the many-body formalism.

The organizing idea of the field is P. W. Anderson’s “more is different” (1972). Knowing everything about one electron and one ion does not tell you that $10^{23}$ of them will superconduct, order magnetically, or form a rigid solid. These emergent properties belong to the organization of the particles rather than to the particles themselves, and each comes with its own low-energy description: quasiparticles, order parameters, broken symmetries, and topological invariants.

Pages in this section

The concepts on this page feed the rest of the section roughly as follows.

flowchart LR
    L["Crystal lattice<br/>(Bravais lattice + basis)"] --> R["Reciprocal lattice<br/>and Brillouin zone"]
    R --> D["Diffraction<br/>(Bragg / Laue)"]
    R --> B["Bloch theorem<br/>and energy bands"]
    L --> P["Phonons<br/>(lattice dynamics)"]
    B --> G["Density of states<br/>and Fermi level"]
    G --> C{"Band filling"}
    C --> M["Metal"]
    C --> S["Semiconductor"]
    C --> I["Insulator"]
    M --> MM["Fermi liquids<br/>and magnetism"]
    P --> SC["Superconductivity"]
    MM --> SC
    B --> T["Topological bands"]

Crystal structure

Bravais lattices and bases

A Bravais lattice is the infinite set of points

generated by three primitive vectors $\mathbf{a}_i$. Every lattice point has an identical environment. A real crystal structure is a Bravais lattice plus a basis: the group of atoms attached to each lattice point. Diamond, for example, is an FCC lattice with a two-atom basis, and NaCl is an FCC lattice with a Na-Cl basis.

Combining translations with point-group symmetries (rotations, reflections, inversion) leaves exactly 14 Bravais lattices in three dimensions, grouped into 7 crystal systems. Adding a basis and the glide and screw operations it permits gives the 230 space groups, which classify every crystal.

P = primitive, C = base-centered, I = body-centered, F = face-centered, R = rhombohedral.

FCC and HCP are the two ways of stacking close-packed planes (ABCABC and ABAB respectively) and share the maximum packing fraction $\pi/(3\sqrt{2}) \approx 0.74$.

Miller indices

A lattice plane is labeled by Miller indices $(hkl)$: take the plane’s intercepts with the axes in units of $a_1, a_2, a_3$, invert them, and clear fractions. The family of all planes with those indices is written ${hkl}$, a direction $[uvw]$, and a family of directions $\langle uvw\rangle$. The planes $(hkl)$ are perpendicular to the reciprocal lattice vector $\mathbf{G}_{hkl} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$, and their spacing is

Reciprocal lattice and Brillouin zone

The reciprocal lattice is the set of wavevectors $\mathbf{G}$ for which $e^{i\mathbf{G}\cdot\mathbf{R}} = 1$ for every lattice vector $\mathbf{R}$. Its primitive vectors satisfy $\mathbf{a}i \cdot \mathbf{b}_j = 2\pi\delta{ij}$:

Any function with the lattice periodicity, such as the crystal potential, is a Fourier series over reciprocal lattice vectors only: $V(\mathbf{r}) = \sum_{\mathbf{G}} V_{\mathbf{G}}\, e^{i\mathbf{G}\cdot\mathbf{r}}$. The reciprocal of an SC lattice is SC, the reciprocal of FCC is BCC, and vice versa. The reciprocal primitive cell has volume $(2\pi)^3/V_c$.

The first Brillouin zone (BZ) is the Wigner-Seitz cell of the reciprocal lattice. Electron and phonon states are labeled by a wavevector inside it, so band structures are plotted along paths joining its high-symmetry points. For the 2D square lattice the zone is a square of side $2\pi/a$, and the standard path is $\Gamma \to X \to M \to \Gamma$. For FCC crystals (Si, GaAs, Cu) the zone is a truncated octahedron with points $\Gamma$, X, L, W, K.

Diffraction

Periodic arrays of atoms diffract waves whose wavelength is comparable to the lattice spacing: X-rays (about 1 Å), thermal neutrons, and electrons. Diffraction is how crystal structures are determined.

Bragg and Laue conditions

Waves reflected from successive planes of spacing $d$ interfere constructively when the extra path length is a whole number of wavelengths:

The equivalent Laue condition states that the scattering vector must be a reciprocal lattice vector:

Structure factor and systematic absences

When a cubic crystal is described with its conventional cell, the extra lattice points make some $F_{hkl}$ vanish identically. These systematic absences identify the lattice type:

Thermal vibrations reduce Bragg intensities by the Debye-Waller factor $e^{-2W}$ with $2W = \langle(\mathbf{G}\cdot\mathbf{u})^2\rangle$, without broadening the peaks. Modern structure determination relies on synchrotron X-ray and neutron sources; Experimental Techniques covers the probes that go beyond static structure.

The same lattice also vibrates. Its quantized normal modes, phonons, carry heat, scatter electrons, and provide the pairing interaction in conventional superconductors. They are treated on Lattice Dynamics & Phonons. The rest of this page covers what periodicity does to the electrons.

Electronic band theory

Bloch’s theorem

For an electron in a potential with the lattice periodicity, $V(\mathbf{r}+\mathbf{R}) = V(\mathbf{r})$, every energy eigenstate can be chosen in the form

The eigenstate is a plane wave modulated by a function with the periodicity of the lattice. The crystal momentum $\hbar\mathbf{k}$ is conserved modulo $\hbar\mathbf{G}$ and can be restricted to the first Brillouin zone. For each $\mathbf{k}$ the cell-periodic problem has a discrete spectrum labeled by the band index $n$. The resulting band structure $E_n(\mathbf{k})$ consists of smooth surfaces over the zone with $E_n(\mathbf{k}+\mathbf{G}) = E_n(\mathbf{k})$. Energies that no band reaches for any $\mathbf{k}$ form band gaps.

With periodic boundary conditions on a crystal of $N$ unit cells, $\mathbf{k}$ takes exactly $N$ values in the zone. Each band therefore holds $2N$ electrons once spin is counted. This counting argument drives the metal/insulator classification below.

Two complementary approximations explain where bands and gaps come from.

Nearly-free-electron model

Tight-binding model

The opposite limit starts from localized atomic orbitals $\phi(\mathbf{r})$ and lets electrons hop between neighboring sites with amplitude $t$. The Bloch combination

broadens each atomic level $\epsilon_0$ into a band. With nearest-neighbor hopping on a simple cubic lattice,

The bandwidth is $W = 2zt$ for coordination number $z$: $4t$ in 1D, $8t$ on the square lattice, $12t$ on the cubic lattice. Narrow bands (small $t$), as in transition-metal $d$ and rare-earth $f$ states, are where electron-electron interactions compete with kinetic energy. That competition leads to the Hubbard model and Mott insulators covered on Superconductivity, Quantum Hall & Topological Phases. Well-separated orbitals give isolated bands; overlapping orbitals of different character hybridize, and the gaps between the resulting bands are what make covalent semiconductors such as Si insulating at $T=0$.

In practice, band structures of real materials are computed with density functional theory (DFT) and refined with many-body corrections such as GW or DMFT. Electronic Structure Beyond DFT describes these methods.

Semiclassical dynamics, effective mass, and holes

A wavepacket built from states near $\mathbf{k}$ in band $n$ moves with the group velocity, and its crystal momentum responds to external forces:

Near a band extremum the dispersion is quadratic and defines an effective mass tensor:

The effective mass absorbs the effect of the periodic potential, so carriers near band edges behave as free particles with mass $m^*$. Typical values are $0.067\,m_e$ for electrons in GaAs, and up to hundreds of $m_e$ in heavy-fermion compounds. A full band carries no current. A nearly full band is described as a small number of holes: quasiparticles with positive charge and positive effective mass at the top of the band. This is why the Hall coefficient of some metals and all p-type semiconductors has the “wrong” sign. The full semiclassical equations also include an anomalous velocity from the Berry curvature, $-\dot{\mathbf{k}}\times\boldsymbol{\Omega}_n(\mathbf{k})$, which is the starting point for the anomalous Hall effect and topological band theory.

Metals, insulators, and semiconductors

Because each band holds $2N$ electrons, the ground state follows from counting. If the valence electrons fill some bands completely and leave the rest empty, there is a gap at the Fermi level and the solid is an insulator. If a band is partly filled, states exist just above the Fermi energy $E_F$ and the solid is a metal. An odd number of electrons per primitive cell therefore forces a metal in band theory, unless strong interactions intervene (a Mott insulator). An even number permits either outcome, depending on whether bands overlap: Mg and Ca are metals despite having two valence electrons per atom.

The boundary between semiconductor and insulator is a convention, not a sharp line. Wide-gap materials such as GaN (3.4 eV), SiC (about 3.3 eV), Ga$_2$O$_3$ (about 4.8 eV), and diamond (5.5 eV) are now used as semiconductors in power electronics and UV optoelectronics.

Semiconductors

Band edges and common materials

Semiconductor behavior is set by the valence-band maximum (VBM) and the conduction-band minimum (CBM). In a direct-gap material both lie at the same $\mathbf{k}$, so a photon alone can create or annihilate an electron-hole pair; these materials make efficient LEDs and lasers. In an indirect-gap material they lie at different $\mathbf{k}$, so optical transitions need a phonon to supply the missing momentum. This is why silicon, the dominant electronic material, emits light poorly.

Carrier statistics

In a nondegenerate semiconductor the electron and hole densities are Boltzmann tails of the Fermi-Dirac distribution:

with $N_v$ defined in the same way using the hole mass. Their product is independent of $E_F$. This is the law of mass action:

The factor of 2 in the intrinsic exponent arises because thermal excitation creates electrons and holes in pairs, which puts $E_F$ near midgap. For silicon at 300 K, $n_i \approx 10^{10}\ \text{cm}^{-3}$, which is far too few carriers for useful devices.

Doping

Substituting a small fraction of dopant atoms (typically $10^{14}$ to $10^{20}\ \text{cm}^{-3}$, against about $5\times10^{22}$ Si atoms per cm$^3$) sets the carrier density:

• n-type: a group-V donor (P, As in Si) has one extra electron, bound by only about 45 meV. At room temperature it is ionized into the conduction band.
• p-type: a group-III acceptor (B in Si) captures an electron from the valence band and leaves a mobile hole.

The donor binding energy follows from a hydrogen model rescaled by the effective mass and dielectric constant, $E_d \approx 13.6\ \text{eV} \times (m^*/m_e)/\epsilon_r^2$. Because $np = n_i^2$ still holds, raising the majority-carrier density suppresses the minority-carrier density. Minority carriers matter because they carry the current in bipolar devices and solar cells.

The p-n junction

When p-type and n-type regions meet, carriers diffuse across the interface and recombine. This leaves a depletion region of ionized dopants whose field opposes further diffusion. In equilibrium the Fermi level is flat across the junction and the bands bend by the built-in potential:

where $W$ is the depletion width under an applied bias $V$ (forward bias positive). The asymmetric barrier gives the Shockley diode characteristic:

The p-n junction underlies diodes, bipolar transistors, LEDs, laser diodes, photodetectors, and solar cells. The MOS field-effect transistor instead uses a gate field to invert the surface carrier type. Its nanosheet (gate-all-around) form is the transistor architecture in leading-edge logic processes of the mid-2020s.

Density of states

The density of states (DOS) $g(E)$ counts single-particle states per unit energy and volume:

where the factor 2 counts spin. The DOS at the Fermi level controls the electronic heat capacity ($C = \tfrac{\pi^2}{3}k_B^2 g(E_F)\,T$), the Pauli susceptibility, the screening length, and the superconducting coupling. The DOS near band edges fixes the carrier densities $N_c$ and $N_v$ above.

For a parabolic band $E = \hbar^2k^2/2m$ the DOS depends strongly on dimension:

Current research directions

Band theory plus interactions and topology still produces surprises. Several threads that were active in 2023–2026 build directly on the concepts above. They are treated in more depth on Superconductivity, Quantum Hall & Topological Phases and Graduate-Level Formalism.

See also

• Lattice Dynamics & Phonons: quantized vibrations, heat capacity, and electron-phonon coupling.
• Metals & Magnetism: the electron sea, Fermi surfaces, and magnetic order.
• Disorder & Localization: how randomness drives metal-insulator transitions.
• Superconductivity, Quantum Hall & Topological Phases: emergent and topological states of matter.
• Experimental Techniques: how band structure, order, and excitations are measured.
• Graduate-Level Formalism: many-body theory and field-theoretic methods.
• Quantum Mechanics: the single-particle quantum mechanics underlying band theory.
• Statistical Mechanics: Fermi-Dirac statistics and the theory of phase transitions.
• Quantum Field Theory: field-theoretic methods for collective excitations.
• Thermodynamics: macroscopic properties and phase diagrams.
• Computational Physics: DFT, Monte Carlo, and molecular dynamics.
• Physics Hub: all physics topics.