Condensed Matter Physics

Exploring the Quantum World of Materials

Condensed matter physics studies the physical properties of matter in its condensed phases, primarily solids and liquids. It is the largest field of contemporary physics, encompassing phenomena from superconductivity to topological insulators.

The big idea: more is different. You could in principle know everything about a single electron and a single proton and still have no way to predict that $10^{23}$ of them, packed into a crystal, will conduct electricity, turn magnetic, or carry current with zero resistance. Condensed matter physics is built on this insight — Philip Anderson’s “more is different”: when enormous numbers of simple parts interact, qualitatively new emergent phenomena appear that exist only collectively. A superconductor’s resistanceless flow, a magnet’s spontaneous alignment, the rigidity of a solid — none of these are properties of the constituent particles; they are properties of the organization. This hub follows the foundational thread: start from the periodic arrangement of atoms (the crystal lattice), see how that periodicity reshapes the allowed electron energies into bands separated by gaps, and read off — from a single fact, where the Fermi level falls relative to a gap — whether a material is a metal, an insulator, or the technologically pivotal in-between case, the semiconductor. The deeper machinery, and the genuinely new states of matter that emerge when interactions and topology take over, live on the dedicated pages linked below.

Explore Condensed Matter

Lattice Dynamics & Phonons

How a crystal vibrates: the harmonic crystal, normal modes and phonon dispersion, acoustic vs. optical branches, the Debye and Einstein models, phonon heat capacity, and electron-phonon coupling.

Metals & Magnetism

The electron sea and magnetic order: Drude, Sommerfeld and Fermi-liquid theory, the Fermi surface, screening, the exchange interaction, Heisenberg/Ising models, mean-field magnetism, and spin waves.

Disorder & Localization

How randomness turns a metal into an insulator: Anderson localization, weak localization, the mobility edge and metal-insulator transition, scaling theory, and many-body localization.

Experimental Techniques

How we actually see band structure, order, and excitations: ARPES, STM/STS, neutron and Raman scattering, quantum oscillations, transport, and thermodynamic measurements.

Superconductivity, Quantum Hall & Topological Phases

Superconductivity (Ginzburg-Landau, BCS, Josephson), the integer and fractional quantum Hall effects, topological insulators and Chern insulators, strongly correlated systems, and soft condensed matter.

Graduate-Level Formalism

The full mathematical machinery: second quantization, Green's functions, advanced band theory, Bogoliubov-de Gennes, Chern-Simons, DMFT, and tensor networks.

Crystal Structure

Bravais Lattices

14 distinct lattice types in 3D, characterized by lattice vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$

Position vector:

$$\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3$$

Reciprocal Lattice

Defined by vectors satisfying $\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi\delta_{ij}$:

$\mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}$

The reciprocal lattice is the natural arena for everything that follows: Bloch's theorem labels electron states by a wavevector $\mathbf{k}$ living here, and the Brillouin zone is the unit cell over which the entire band structure is defined.

First Brillouin zone: Wigner-Seitz cell of reciprocal lattice

X-ray Diffraction

Bragg's law:

$2d\sin\theta = n\lambda$

Structure factor:

$F_{\mathbf{G}} = \sum_j f_j e^{i\mathbf{G} \cdot \mathbf{r}_j}$

The lattice does more than hold atoms still. The same periodic lattice that diffracts X-rays can also vibrate, and its quantized vibrations — phonons — carry heat, scatter electrons, and ultimately glue together the Cooper pairs of conventional superconductors. That story has its own page: Lattice Dynamics & Phonons. Here we follow the other consequence of periodicity — what it does to the electrons.

Electronic Band Theory

A free electron can have any energy. Put it in a perfectly periodic crystal and something remarkable happens: its allowed energies collapse into continuous bands separated by forbidden gaps. This single fact — that periodicity carves the energy axis into allowed bands and forbidden gaps — is the foundation of all electronic structure, and the next section will show that it is also the entire reason a semiconductor exists.

Bloch's Theorem

Wavefunctions in periodic potential:

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

Where $u_{n\mathbf{k}}(\mathbf{r})$ has lattice periodicity

Bloch's theorem is the master key: because the potential repeats with the lattice, every electron eigenstate can be written as a plane wave $e^{i\mathbf{k}\cdot\mathbf{r}}$ modulated by a cell-periodic function. The label $\mathbf{k}$ (the crystal momentum, defined only within the first Brillouin zone) and the band index $n$ together replace the continuum of free-electron momenta. Each band $E_n(\mathbf{k})$ is a smooth surface over the Brillouin zone, and the gaps between bands are exactly the energies no $\mathbf{k}$ can reach.

Nearly Free Electron Model

Weak periodic potential creates band gaps at Brillouin zone boundaries

Gap size:

$\Delta E = 2|V_{\mathbf{G}}|$

where $V_{\mathbf{G}}$ is Fourier component of potential

Start from free electrons and switch on a faint periodic potential. At a Brillouin-zone boundary two plane waves $\mathbf{k}$ and $\mathbf{k}-\mathbf{G}$ are degenerate, so even a weak potential mixes them strongly, splitting the energy by $2|V_{\mathbf{G}}|$. That split is the band gap — the gap is not an accident of any particular material but a generic consequence of periodicity meeting degeneracy.

Tight-Binding Model

Start from atomic orbitals:

$\psi_{\mathbf{k}}(\mathbf{r}) = \sum_{\mathbf{R}} e^{i\mathbf{k} \cdot \mathbf{R}} \phi(\mathbf{r} - \mathbf{R})$

Dispersion relation:

$E(\mathbf{k}) = \epsilon_0 - 2t[\cos(k_xa) + \cos(k_ya) + \cos(k_za)]$

The nearly-free-electron and tight-binding pictures are the two complementary limits of the same physics. Where the nearly-free model starts from delocalized waves and lets the lattice carve gaps into them, tight-binding starts from sharp atomic levels and lets electrons hop between neighbors (amplitude $t$), broadening each isolated level into a band of width $\sim 4t$ (in 1D). Both routes arrive at the same conclusion: discrete allowed bands with forbidden gaps in between.

With the band-gap concept in hand, we can immediately read off the most consequential application of band theory before returning to the finer machinery. The next section pairs the theory with its single most important product — the semiconductor.

Semiconductors

Band theory delivers a strikingly simple classification of solids: it is not how many electrons a material has that decides whether it conducts, but where the Fermi level sits relative to the band gap. A partly filled band conducts (a metal); a filled band separated from the next empty band by a large gap insulates; a filled band separated by a small gap is a semiconductor — an insulator that thermal energy or doping can switch on. This single distinction underpins the entire electronics industry.

Class	Band filling	Gap $E_g$	Conductivity vs. $T$
Metal	Partly filled band	none (bands overlap)	decreases with $T$
Semiconductor	Filled valence band	small ($\sim 0.1$–$2$ eV)	increases with $T$
Insulator	Filled valence band	large ($\gtrsim 4$ eV)	negligible

Why semiconductors heat up into conductors. In a metal, conductivity falls as temperature rises because lattice vibrations scatter the already-mobile electrons. A semiconductor does the opposite: its valence band is full and the conduction band empty, so it can only conduct once electrons are thermally promoted across the gap. The number of carriers grows exponentially as $e^{-E_g/2k_BT}$, swamping the scattering effect. That exponential sensitivity is exactly what makes a semiconductor a controllable switch.

Band Structure

The two band edges that matter are the valence band maximum (VBM) and the conduction band minimum (CBM). Their relative position in momentum space sets the optical behavior:

Direct gap — VBM and CBM lie at the same $\mathbf{k}$. An electron can cross the gap by absorbing or emitting a single photon, so direct-gap materials (e.g. GaAs) make efficient LEDs and lasers.
Indirect gap — VBM and CBM lie at different $\mathbf{k}$. A photon alone cannot conserve momentum, so a phonon must assist; this makes silicon a poor light emitter despite being the workhorse of electronics.

Carrier Statistics

For an intrinsic (undoped) semiconductor, electrons and holes are created in pairs, and their equilibrium concentration is set by the Boltzmann factor for crossing the gap:

\[n_i = \sqrt{N_c N_v}\, e^{-E_g/2k_BT}\]

where $N_c$ and $N_v$ are the effective densities of states in the conduction and valence bands. The factor of $2$ in the exponent reflects that each promoted electron leaves a hole behind, so the carriers are shared between the two bands.

Doping

Pure semiconductors carry too few intrinsic carriers to be useful. Doping — substituting a few-parts-per-million of a foreign atom — overwhelms the intrinsic population with carriers of one chosen sign:

n-type: donor atoms (e.g. phosphorus in silicon) contribute extra electrons to the conduction band.
p-type: acceptor atoms (e.g. boron) accept electrons, leaving mobile holes in the valence band.

Even when doped, the product of electron and hole concentrations is pinned by the mass-action law, $np = n_i^2$ — adding majority carriers necessarily suppresses minority carriers.

p-n Junction

Built-in potential: $V_{bi} = \frac{k_BT}{e} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

Depletion width: $W = \sqrt{\frac{2\epsilon_s V_{bi}}{e}\left(\frac{N_A + N_D}{N_A N_D}\right)}$

Bring an n-type and a p-type region into contact and electrons diffuse across, leaving behind a charged depletion region with a built-in field. That field rectifies — current flows easily one way and barely the other — and the p-n junction is the elementary building block of diodes, transistors, solar cells, and LEDs.

Recent Advances in 2D Semiconductors (2023-2024)

Moiré Engineering: Twisted bilayer TMDs showing correlated insulator states
Valleytronics: Valley-selective optical excitation in monolayer WSe₂
Exciton Condensates: Room-temperature exciton-polariton BEC in perovskites
Quantum Emitters: Single-photon sources in hBN defects

Counting States: The Density of States

Bands tell us which energies are allowed; the density of states $g(E)$ tells us how many states sit at each energy — the quantity that ultimately controls heat capacity, magnetic susceptibility, optical absorption, and the carrier densities $N_c, N_v$ that appeared above. It is the first piece of the deeper band-theory machinery, and the bridge to the metals, magnetism, and transport developed on the dedicated pages.

Density of States

$g(E) = \sum_n \int \frac{d^3k}{(2\pi)^3} \delta(E - E_n(\mathbf{k}))$

Van Hove singularities occur where $\nabla_k E_n(\mathbf{k}) = 0$

The shape of $g(E)$ depends sharply on dimensionality, falling as $1/\sqrt{E}$ in 1D, flat in 2D, and rising as $\sqrt{E}$ in 3D near a band edge. Wherever a band flattens ($\nabla_k E_n = 0$) the density of states spikes into a Van Hove singularity — a feature that shows up directly in optical and tunneling spectra.

Where the Foundations Lead

Crystal structure, band theory, and the density of states are the common foundation for everything else in condensed matter. The remaining themes each take this groundwork in a different direction, and each has its own page.

Metals & magnetism — the electron sea and spontaneous order. What happens once a band is only partly filled? The conduction electrons form a degenerate quantum fluid whose understanding evolved through three pictures — a classical gas (Drude), a Pauli-blocked Fermi gas (Sommerfeld), and an interacting fluid of dressed quasiparticles (Landau’s Fermi liquid) — and whose low-energy states live on a Fermi surface. When the exchange interaction between spins becomes important, the same electrons can lock into spontaneous magnetic order: paramagnetism, ferromagnetism, antiferromagnetism, and the quantized spin waves (magnons) that excite them. The full treatment — Drude through Fermi liquids, screening, the Heisenberg and Ising models, and mean-field magnetism — is on Metals & Magnetism.

Lattice dynamics & phonons — the crystal in motion. Bands describe electrons in a static lattice, but the lattice itself vibrates. Its quantized normal modes — phonons — split into acoustic and optical branches, carry most of a solid’s heat, set the low-temperature heat capacity (Debye and Einstein models), and scatter electrons. Electron-phonon coupling is also the pairing glue of conventional superconductivity. The full story is on Lattice Dynamics & Phonons.

And beyond the perfect crystal. Real materials are neither perfectly periodic nor weakly interacting. Disorder & Localization shows how randomness can turn a metal into an insulator; Superconductivity, Quantum Hall & Topological Phases covers the genuinely new states of matter; Experimental Techniques explains how all of this is measured; and Graduate-Level Formalism develops the many-body machinery behind it.

Key Takeaways

More is different. Collective behavior of $\sim 10^{23}$ particles produces emergent phenomena absent at the single-particle level.
Periodicity makes bands. Bloch’s theorem turns a periodic potential into continuous energy bands separated by forbidden gaps.
Band structure governs solids. Whether a material is a metal, insulator, or semiconductor follows from how electron bands fill.
The gap is the switch. A small band gap plus doping makes a semiconductor — a tunable insulator that powers all of electronics.
Quasiparticles simplify the many-body problem. Phonons, holes, and Cooper pairs let us treat strongly interacting systems with effective single-particle pictures.
Topology classifies new phases. Topological insulators and the quantum Hall effect are robust against disorder because they are protected by topology.