Pick up a copper coin and a piece of window glass. Both are lumps of atoms, with electrons around nuclei, held by the same electric force. Connect either to a battery. Copper carries amps. Glass carries essentially nothing. The ratio of their resistivities is of order 1022 or more, a bigger spread than any other everyday property of matter. Where does a number that large come from?
The answer is not in any single atom. It appears only when something like 1023 atoms sit together in a lattice. Add a trace of phosphorus to silicon, a few atoms in every ten million, and its conductivity changes by a factor of thousands. Cool lead or niobium close to absolute zero and its resistance drops to exactly nothing. A physicist trained in condensed matter sees all of these as the same subject: what happens to electrons when a huge crowd of atoms cooperates.
A single atom has sharp energy levels. Put two atoms close together and each level splits in two: electrons can sit in a combination that is lower in energy than before, or one that is higher. Put N atoms in a row, and each level splits into N. The levels spread across a range set by how strongly neighbouring atoms overlap, and with 1023 atoms they are so closely spaced they form a continuous band. Atoms close together, strong overlap, wide band.
Electrons fill levels from the bottom, two per level (spin up and spin down), because of the Pauli exclusion principle. Now the whole of electrical behaviour is a question of filling. If one electron per atom fills the band halfway, the next empty level sits a hair above the last full one, and a tiny push will move electrons: that is a metal. If two electrons per atom fill it to the top, there is no empty level to move into. Try it.
Nearest-neighbour tight-binding chain, E₀ = 0, exact eigenvalues. Dark lines are filled levels, light are empty, half-tone is half-filled. Two electrons fit in each level.
Notice what the readout says as you add atoms: the gap to the next empty level shrinks like 1/N. For a real wire with 1023 atoms it is smaller than any thermal wobble, so electrons are effectively free to move.
A band is a range of allowed energies. Between bands there can be a band gap where no electron state exists. Whether a material conducts depends on two things: how many electrons it has, and whether the highest filled band is full. Copper has one outer electron per atom, so its band is half full and it is a metal. Diamond and glass have completely full bands with a large gap above: an insulator. Silicon has the same full-band structure but a gap of only 1.12 eV. That is small enough that, at room temperature, a few electrons are knocked up by heat: a semiconductor.
The metal's electrons are a strange crowd. Copper has 8.49 × 10²⁸ conduction electrons per cubic metre, packed so tightly that the Pauli principle forces them up to a Fermi energy of about 7.0 eV and a speed near 1.6 × 10⁶ m/s, even at absolute zero. Yet when 1 amp runs through a 1 mm² copper wire, the electrons' average drift is only about 0.07 mm per second. The signal moves at the speed of the field, but the carriers barely crawl.
The number of heat-freed carriers in a semiconductor depends on the gap with a ferocious exponential: it goes as e−Eg/2kT. Drag the gap.
ni = 1.7 × 10¹⁹ cm⁻³ × (T/300)3/2 × e−Eg/2kT, with the prefactor fixed at silicon's value for every material (illustrative). For silicon at 300 K it gives 6.6 × 10⁹ cm⁻³; the textbook value is about 1 × 10¹⁰. Copper's conduction electrons: 8.49 × 10²² cm⁻³.
Between silicon and diamond the gap differs by a factor of about five, but the carrier count differs by a factor of 1037. That is why one is the basis of electronics and the other is a jewel. And note the temperature slider: semiconductors improve when hot (more carriers), the opposite of metals. Silicon chips are limited to roughly 150 °C for this reason, and one reason wide-gap materials such as GaN and silicon carbide are used in hot, high-voltage power electronics.
Pure silicon has about 10¹⁰ free carriers per cubic centimetre, out of 5 × 10²² atoms. Replace one atom in five million with phosphorus (one extra outer electron) and you add 10¹⁶ electrons per cubic centimetre: a million times more. This is doping. A law of mass action holds the product of electron and hole densities fixed at room temperature, n·p = ni². So adding electrons removes holes, in step: n-type silicon has electrons as its majority carriers, p-type (boron) has holes.
Put p-type and n-type side by side and you get a p–n junction, the diode. Electrons and holes diffuse across and annihilate, leaving a barrier. Pushing the voltage one way lowers the barrier and current rises exponentially; the other way it raises the barrier and almost nothing flows. The Shockley formula is the single equation to keep.
Doping: n = Nd (fully ionised), p = ni²/Nd with ni = 10¹⁰ cm⁻³ for silicon at 300 K, 5 × 10²² Si atoms/cm³. Diode: ideal Shockley equation with Is = 1 pA (illustrative) at 300 K; real diodes add series resistance and an ideality factor between 1 and 2.
On the log plot the forward branch is a straight line: every 60 mV more gives about ten times the current. A transistor is two such junctions, or a gate that moves a barrier, so one signal controls a much larger current, and billions of them form a processor. Condensed matter is where that sentence comes from.
A crystal is a repeating arrangement, and waves with a wavelength comparable to the spacing, about an ångström (10⁻¹⁰ m), bounce off it like light off a diffraction grating. In 1912 Max von Laue's group showed that X-rays diffract from crystals, proving both that X-rays are waves and that crystals are regular. In 1913 William Henry Bragg and his son William Lawrence Bragg turned it into a method for finding where atoms sit. Their rule is one line:
A worked number: silicon's (111) planes are 3.136 Å apart. Copper-Kα X-rays have λ = 1.5406 Å. The first strong reflection comes at sin θ = 1.5406 ÷ (2 × 3.136), so θ = 14.2°, a detector at 28.4° from the beam. Measure the angles and intensities for many planes and you can reconstruct the whole structure: this is how salts, metals, DNA and proteins were first seen. Slide through the angles. More planes (a bigger crystal) means sharper peaks.
Intensity from N planes: [sin(Nφ) / (N sin φ)]² with φ = 2π d sin θ / λ. Spacings from lattice constants (Si 5.431 Å, NaCl 5.640 Å); Cu Kα 1.5406 Å, Mo Kα1 0.7093 Å. Idealised: ignores atomic scattering strength and absorption.
In 1911 Heike Kamerlingh Onnes in Leiden cooled mercury to about 4 K and watched its resistance fall to an immeasurable value. Below a critical temperature Tc many metals carry current with no loss at all. Two decades later Meissner and Ochsenfeld found the second property: a superconductor also pushes magnetic fields out of itself, which is why a magnet floats over one. It is not just a perfect wire.
The explanation, by Bardeen, Cooper and Schrieffer in 1957 (Nobel Prize 1972), is that electrons near the Fermi surface pair up through tiny lattice distortions. Pairs behave like bosons and fall into one shared quantum state, separated from the next excitation by an energy gap, so there is nothing for a bump to scatter them into. The weak-coupling estimate for the gap is 3.52 kBTc.
The 1986 discovery of copper-oxide superconductors by Bednorz and Müller (Nobel Prize 1987) pushed Tc beyond liquid nitrogen's 77 K within months: YBa₂Cu₃O₇ at about 92 K. These are not explained by simple BCS theory, and no one yet agrees on the mechanism. In 2015 hydrogen sulfide at about 155 GPa (1.5 million atmospheres) reached 203 K, and a few hydrides since. No ambient-pressure room-temperature superconductor has been confirmed: the 2023 claim of "LK-99" was not reproduced by groups that tried.
Tc values: Hg 4.15 K (Onnes 1911), Nb 9.25 K, MgB₂ 39 K (2001), YBCO about 92 K (1987), H₃S 203 K (2015). *H₃S needs about 155 GPa. The resistance curve is schematic (a straight line above Tc); the gap is the weak-coupling estimate 3.52 kBTc, and real values differ by material.
Superconductors are in daily use: the magnets of MRI scanners are niobium–titanium wires cooled by liquid helium at 4 K, and the Large Hadron Collider's bending magnets run at 1.9 K.
In 1972 Philip Anderson wrote an essay called "More Is Different". Its point: a crowd of particles has properties that no single particle has, and that cannot be read off the single particle's rules. Water molecules are not wet, and one iron atom is not a magnet. A magnet appears when billions of spins agree.
The simplest model of that agreement is the Ising model: a grid of tiny arrows, each pointing up or down. Each arrow likes to match its four neighbours, and heat shakes it randomly. At low temperature agreement wins and the grid is mostly one direction: a magnet. At high temperature randomness wins. In between is a sharp critical temperature: Onsager solved the two-dimensional case exactly in 1944, with Tc = 2J ÷ (kB ln(1+√2)) = 2.269 J/kB, where J is the neighbour coupling. The same maths describes boiling and many other transitions.
A 24 × 24 grid with periodic edges, Metropolis updates, 150 sweeps each time you change temperature. Red is up, blue is down. A small grid blurs the transition: the real one is sharp only for infinitely many spins. Exact result (Onsager): magnetisation (1 − sinh−4(2J/kT))1/8 below Tc.
Condensed matter has found phases that do not fit the "ordered or disordered" picture too: the quantum Hall states discovered in 1980 (Nobel Prize 1985), and the topological phases of Thouless, Haldane and Kosterlitz (Nobel Prize 2016), whose properties are protected by the shape of the electrons' band structure rather than by a symmetry. The field has never stopped producing surprises.
When many atoms sit together their levels spread into bands, and electrons fill those bands from the bottom. Conductors have a band half-full; insulators and semiconductors have a full band below a gap, and the gap's size, through an exponential, decides how many carriers there are. Doping and the p–n junction turn that into devices, X-ray diffraction reads the structure, and in the cold, electrons can pair into a state with no resistance at all. The crowd does things the parts cannot.
Try this: this week, look at any LED. Its colour is its band gap: red is about 1.9 eV, blue about 2.7 eV. The same physics that made the copper coin conduct is choosing the colour of the light.
Sources. Copper electron density 8.49 × 10²⁸ m⁻³ and Fermi energy about 7.0 eV from Kittel's tables. Band gaps (Ge 0.67, Si 1.12, GaAs 1.42, GaN 3.4, diamond 5.47 eV) are standard room-temperature values. Bragg and Laue: Laue's X-ray diffraction experiment 1912; W. H. and W. L. Bragg 1913 (Nobel 1915). Superconductivity: Onnes 1911; Meissner and Ochsenfeld 1933; BCS 1957 (Nobel 1972); Bednorz and Müller 1986 (Nobel 1987); YBCO about 92 K, 1987; H₃S 203 K at about 155 GPa (Drozdov et al., Nature, 2015); LK-99 failed replication (South Korean Society of Superconductivity and Cryogenics review, December 2023). Ising: Onsager 1944. Anderson, "More Is Different", Science 177 (1972). Topological phases: Nobel Prize in Physics 2016. Quantum Hall effect: von Klitzing 1980 (Nobel 1985). Course contents: MIT OCW 8.231. Every number an interactive shows was computed in node: Fermi energy of copper 7.04 eV, Fermi speed 1.57 × 10⁶ m/s, drift speed at 1 A in 1 mm² = 7.35 × 10⁻⁵ m/s; ni (model) Ge 4.0 × 10¹³, Si 6.6 × 10⁹, GaAs 2.0 × 10⁷, diamond 1.9 × 10⁻²⁷ cm⁻³; Si/diamond carrier ratio e−84 ≈ 3 × 10⁻³⁷; 60 mV/decade (59.5); Si (111) first Bragg angle 14.22°; Ising Tc = 2.269 J/kB; magnetisation at T = 1.5 is 0.986.