Why solid state physics: from atoms to materials

Almost everything you can hold is a solid, and almost every technology that defines modern life depends on understanding how solids behave. The copper in a power cable, the silicon in a phone, the steel in a bridge, the magnet in an electric motor and the superconducting coil in a hospital scanner are all made of the same ingredients: nuclei and electrons interacting through electromagnetism and obeying quantum mechanics. Solid state physics, now usually called condensed matter physics, asks how those simple ingredients produce the enormous variety of properties seen in real materials. Why does copper conduct electricity about \(10^{31}\) times better than a plastic insulator? Why is diamond the hardest natural material while graphite, made of the same carbon atoms, is soft enough to write with? Why does iron become a permanent magnet while aluminium does not? Why does a piece of lead lose all electrical resistance below 7.2 K?

This course answers those questions with a small number of powerful ideas. The first is periodicity: in a crystal, atoms sit on a regular lattice, and that regularity simplifies the quantum mechanics enormously. The second is the idea of collective excitations: instead of following \(10^{23}\) atoms individually, we describe a solid by its normal modes of vibration, called phonons, and by electrons moving through the periodic potential, which organize themselves into energy bands. The third is the role of the Pauli exclusion principle, met in PHYS-373, which decides how electrons fill those bands and therefore whether a material is a metal, a semiconductor or an insulator. Statistical mechanics, which PHYS-361 develops in parallel, supplies the tools to turn energy levels into measurable quantities such as heat capacity and magnetization.

What a solid is

A solid is a collection of atoms held close together by chemical bonds, so close that the outer electron clouds of neighbouring atoms overlap. Typical spacings between atoms are 0.2 to 0.4 nm, which means a cubic centimetre contains around \(10^{23}\) atoms. Two broad classes exist. In a crystal, the atoms are arranged in a pattern that repeats exactly through space, like the tiles of a floor. In an amorphous solid, such as window glass, the arrangement is regular only over a few neighbouring atoms and has no long-range order. Most of this course concerns crystals, because their periodicity makes the physics tractable, but many of the ideas, such as phonons and the distinction between localized and extended electron states, carry over to amorphous materials too.

Crystals are far more common than they look. A copper wire or a steel beam is polycrystalline: it consists of many small crystal grains, each a few micrometres across, packed together with different orientations. Single crystals can also be grown. The silicon wafers used for computer chips are cut from single crystals up to 300 mm in diameter and two metres long, with fewer than one missing atom per billion. The perfection of those crystals is the foundation of the semiconductor industry.

Energy scales

A useful first step in any physics problem is to identify the important energies, because they tell you which effects matter at which temperatures. The program below estimates the main scales for copper, a typical metal, starting from its density and molar mass. It also lists the electrical resistivity of a range of materials, to show the astonishing span of behaviour that this course must explain.

import math
# Energy and length scales of a simple solid: copper
NA, kB, e, hbar, me = 6.02214076e23, 1.380649e-23, 1.602176634e-19, 1.054571817e-34, 9.1093837e-31
rho, M = 8960.0, 0.063546            # kg/m^3, kg/mol
n_atoms = rho / M * NA                # atoms per m^3
a_fcc = (4 / n_atoms) ** (1 / 3)      # fcc cubic cell holds 4 atoms
d_nn = a_fcc / math.sqrt(2)
print(f"atoms per m^3        : {n_atoms:.3e}")
print(f"cubic cell edge a    : {a_fcc * 1e9:.4f} nm (x-ray value 0.3615 nm)")
print(f"nearest-atom spacing: {d_nn * 1e9:.4f} nm")
# characteristic energies
kT300 = kB * 300 / e
theta_D = 343.0
E_phonon = kB * theta_D / e
EF = hbar ** 2 / (2 * me) * (3 * math.pi ** 2 * n_atoms) ** (2 / 3) / e
E_coh = 3.49                          # measured cohesive energy, eV per atom
print()
print("energy scale                    eV")
for name, E in (("thermal kT at 300 K", kT300), ("top phonon k*theta_D", E_phonon),
                ("cohesive energy per atom", E_coh), ("Fermi energy (free electrons)", EF)):
    print(f"  {name:30s}{E:8.3f}")
# electrical resistivity spans more than 30 orders of magnitude
print()
for name, r in (("copper", 1.7e-8), ("graphite (in plane)", 4e-6), ("germanium (pure)", 0.46),
                ("silicon (pure)", 2.3e3), ("glass", 1e12), ("polytetrafluoroethylene", 1e23)):
    print(f"  resistivity of {name:26s} {r:8.1e} ohm m")

atoms per m^3        : 8.491e+28
cubic cell edge a    : 0.3612 nm (x-ray value 0.3615 nm)
nearest-atom spacing: 0.2554 nm

energy scale                    eV
  thermal kT at 300 K              0.026
  top phonon k*theta_D             0.030
  cohesive energy per atom         3.490
  Fermi energy (free electrons)    7.045

  resistivity of copper                      1.7e-08 ohm m
  resistivity of graphite (in plane)         4.0e-06 ohm m
  resistivity of germanium (pure)            4.6e-01 ohm m
  resistivity of silicon (pure)              2.3e+03 ohm m
  resistivity of glass                       1.0e+12 ohm m
  resistivity of polytetrafluoroethylene     1.0e+23 ohm m

The number density of copper atoms, \(8.5 \times 10^{28}\) per cubic metre, and its face-centred cubic structure (lesson 2) give a cell edge of 0.361 nm, within about one part in a thousand of the value measured by x-ray diffraction (lesson 3), and a spacing of 0.255 nm between closest atoms. The energies sort themselves into a clear hierarchy. Thermal energy at room temperature, \(k_BT = 0.026\) eV, is comparable with the largest phonon energy, \(k_B\theta_D = 0.030\) eV, where the Debye temperature \(\theta_D = 343\) K marks the frequency of the fastest lattice vibration. So at room temperature essentially all vibrational modes of copper are excited, and its heat capacity is close to the classical value, a fact lesson 6 explains. The cohesive energy, 3.5 eV per atom, is the energy needed to pull the solid apart into free atoms; it is more than 100 times \(k_BT\), which is why copper melts only at 1358 K. Finally, the Fermi energy of the conduction electrons, computed with the free-electron formula of PHYS-373, is 7.0 eV, nearly 300 times \(k_BT\). This last comparison is the key to the electronic properties of metals: the electron gas is deeply degenerate, and only a thin shell of electrons near the Fermi energy can respond to heat or to fields.

The resistivities span more than 30 orders of magnitude, from \(1.7 \times 10^{-8}\ \Omega\,\text{m}\) for copper to around \(10^{23}\ \Omega\,\text{m}\) for a fluoropolymer insulator. No other common physical property varies over such a range. Semiconductors such as germanium and silicon lie in between, and their resistivity can be changed by many orders of magnitude by adding tiny amounts of impurity atoms or by raising the temperature. Explaining this enormous range, with a single theory that also predicts which materials fall where, is one of the great achievements of quantum mechanics. The explanation, band theory, is the heart of lessons 8 to 10.

A short history

People have studied crystals since antiquity, but the modern subject began with the idea that crystals are built from repeated units. René Just Haüy proposed in the 1780s, after accidentally breaking a calcite crystal and noticing that the fragments had the same shape, that crystals are stacks of identical tiny blocks. Auguste Bravais showed in 1848 that there are only 14 distinct ways to arrange points periodically in three dimensions, the Bravais lattices of lesson 2. The internal structure of crystals remained invisible until 1912, when Max von Laue and his collaborators diffracted x-rays from a crystal of copper sulfate, proving both that x-rays are waves and that crystals are periodic. William Henry Bragg and his son William Lawrence Bragg quickly turned the method into a tool for determining crystal structures, beginning with sodium chloride in 1913.

The electronic theory of metals began with Paul Drude in 1900, three years after J. J. Thomson discovered the electron. Drude treated the conduction electrons as a classical gas bouncing off the ions. His model explained Ohm's law and the ratio of thermal to electrical conductivity, but it predicted far too large an electronic heat capacity. Arnold Sommerfeld fixed that in 1927 by applying Fermi–Dirac statistics. A year later Felix Bloch showed how electron waves move through a periodic lattice, and by 1931 Alan Wilson had used Bloch's ideas to explain the difference between metals, insulators and semiconductors. Peter Debye had already explained the low-temperature heat capacity of solids in 1912 using quantized lattice vibrations.

Those foundations made possible the transistor, invented at Bell Laboratories by John Bardeen, Walter Brattain and William Shockley in 1947. Superconductivity, discovered by Heike Kamerlingh Onnes in 1911, was finally explained by Bardeen, Leon Cooper and Robert Schrieffer in 1957. Canada has a notable place in this history. Bertram Brockhouse developed neutron scattering methods for measuring phonons and magnetic excitations at the NRX reactor at Chalk River, Ontario, in the 1950s and shared the 1994 Nobel Prize in Physics with Clifford Shull for that work. Today, the subject continues to produce surprises: graphene, isolated in 2004, and topological insulators, predicted in 2005, both feature in the capstone lesson.

How we probe solids

Much of what we know about solids comes from scattering experiments, in which waves or particles are sent into a sample and their deflection is measured. X-rays, with wavelengths around 0.1 nm, scatter from electrons and reveal the positions of atoms. Neutrons, which are uncharged and have wavelengths comparable to atomic spacings at thermal energies, scatter from nuclei and from magnetic moments and can also exchange energy with phonons, so they measure lattice vibrations and magnetic structures directly. Electrons scatter strongly and are used to study surfaces and, in electron microscopes, to image individual columns of atoms. Synchrotron light sources produce intense, tunable x-rays for these experiments: the Canadian Light Source in Saskatoon is one such facility, and neutron beams for materials research are available at the McMaster Nuclear Reactor in Hamilton, Ontario.

Other probes measure how a solid responds to external influences. Electrical and thermal transport measurements reveal how electrons and phonons carry charge and heat. Heat capacity measurements count the low-energy excitations. Optical spectroscopy probes electronic transitions and band gaps. Magnetic measurements reveal how electron spins order. Photoemission, in which light ejects electrons from a sample, maps the electron energy bands directly. In each case, the experiment measures some average over the excitations of the solid, and the theory of this course is designed to predict such averages.

The plan of the course

The course moves from the arrangement of atoms to the behaviour of electrons. Lessons 2 to 4 describe crystal structure, the reciprocal lattice and diffraction, and the forces that bind solids together. Lessons 5 and 6 treat lattice vibrations, their quantization as phonons, and their contribution to heat capacity and thermal conductivity. Lesson 7 revisits the free-electron model of metals, adding transport properties. Lessons 8 and 9 introduce electrons in a periodic potential: Bloch's theorem, the nearly-free-electron and tight-binding models, and energy bands. Lesson 10 applies band theory to semiconductors and the p–n junction, the building block of electronics. Lesson 11 treats magnetism, and lesson 12 closes with superconductivity and modern quantum materials.

Throughout, short Python programs compute the quantities discussed, using only the standard library. They are meant to be read, run and modified. Many of the numbers quoted in the lessons come directly from them, so you can check every step of each calculation.