Solid-State Physics

Quantum mechanics resolved both problems. In 1927 Arnold Sommerfeld kept Drude's gas but made the electrons obey the exclusion principle, using the statistics of Fermi and Dirac. No two electrons can share a state, so they fill the energy levels from the bottom up to a high energy, the Fermi energy. Warming the metal can only excite the few electrons near the top. The missing heat capacity was explained.

A year later Felix Bloch asked how an electron wave moves through the regular lattice of ions that crystallography had mapped. The answer was surprising. In a perfect crystal it moves forever without scattering. Resistance comes only from flaws and from the vibrations of the atoms, which is why it falls as a metal is cooled.

The allowed energies in a crystal form bands separated by gaps. In 1931 Alan Herries Wilson saw what this meant. If the highest occupied band is only partly full, electrons can shift to nearby empty states and carry a current: the solid is a metal. If the band is completely full, there is nowhere to go, and the solid is an insulator. A semiconductor is an insulator with a gap small enough that heat or impurities can put a few electrons across it. Wolfgang Pauli wrote that year that one should not work on semiconductors, which were a filthy mess. Their behaviour depended on traces of impurity no one could yet control.

The Second World War changed that. Radar needed crystal detectors, and wartime work produced germanium and silicon of unprecedented purity. After the war Bell Laboratories set up a group to build a solid replacement for the vacuum tube. In December 1947 John Bardeen and Walter Brattain pressed two gold contacts close together on a germanium crystal and found that a small current into one controlled a larger current through the other. Their leader, William Shockley, who had not been present, spent the following weeks designing the junction transistor, which was easier to make and became the standard.

Shockley later left to found his own company in California. His management was so difficult that eight of his staff left in 1957 to form Fairchild Semiconductor, the seed of Silicon Valley. He spent his later years promoting discredited theories about race and intelligence. At Texas Instruments in 1958 Jack Kilby built a whole circuit on one piece of germanium. At Fairchild in 1959 Robert Noyce designed an integrated circuit on flat silicon with its connections printed on the surface, the form every chip has taken since.

Drude's model gives the conductivity of a metal as $\sigma = n e^2 \tau / m$, where $n$ is the number of free electrons per cubic metre, $e$ and $m$ are the electron's charge and mass, and $\tau$ is the average time between collisions. Copper has one free electron per atom. Its density is 8.96 g/cm³ and a mole weighs 63.55 g, so

Copper's resistivity at room temperature is $1.68 \times 10^{-8}$ Ω m. Turning the formula round,

Drift speed. A current $I$ in a wire of cross-section $A$ needs the electrons to drift at $v = I/(nAe)$. For a current of 10 A in a household cable of 2.5 mm², this gives about 0.29 mm/s. For 1 A in a 1 mm² wire it is 0.074 mm/s, and an electron would take nearly four hours to travel one metre. The lamp lights at once because the electric field travels at nearly the speed of light and pushes all the electrons together.

How far between collisions? Here Drude and Sommerfeld disagree.

Drude's distance is about 13 times the spacing between copper atoms, 0.23 nm, which fitted the idea of electrons bouncing off ions. Sommerfeld's quantum electrons move fourteen times faster, and travel about 170 atomic spacings between collisions. Something must let them pass most atoms untouched. Bloch's theorem supplied it: a perfect lattice does not scatter an electron wave at all.

The same numbers explain the heat capacity. At room temperature the thermal energy $k_B T$ is 0.025 eV, which is 0.36% of the Fermi energy. Only electrons within that sliver of the top can be excited, so they add almost nothing to the heat a metal absorbs.

Solid-state physics, renamed condensed matter physics in the 1970s, is now the largest branch of physics. Its devices are everywhere: transistors, lasers made of semiconductors, light-emitting diodes and solar cells. It also branched into stranger territory. Some metals lose their resistance entirely, a problem that defeated band theory for decades and became superconductivity. The magnetism of iron needed quantum mechanics of its own, in magnetism. And a closer look at electrons in two dimensions revealed a new kind of order, topological matter.