Introduction
Solid state physics applies quantum mechanics (Chapter II) to the ordered, repeating structure of crystalline solids, explaining why some materials conduct electricity, some insulate, and some — semiconductors — do something in between and controllable, which is the entire basis of modern electronics.
1. Crystal lattice and Bravais lattices
A crystal is matter whose atoms are arranged in a repeating, ordered pattern. This pattern is described mathematically as a lattice (a set of points reproduced by translation) combined with a basis (the atom or group of atoms placed at each lattice point). There are exactly 14 Bravais lattices in three dimensions — the complete set of distinct ways points can be arranged with full translational symmetry (for example, simple cubic, body-centred cubic, and face-centred cubic). Classifying a real crystal by its Bravais lattice and basis is the starting point for predicting its mechanical, electrical, and optical properties.
2. Free-electron model and band theory of solids
The simplest model of a metal treats its valence electrons as a “gas” of free particles, ignoring the periodic lattice — this free-electron model already explains basic metallic conductivity reasonably well. Band theory improves on this by accounting for the periodic potential of the lattice (see point 3): allowed electron energies form continuous bands, separated by forbidden band gaps. Whether a material conducts, insulates, or is a semiconductor depends on how these bands are filled and how large the gaps are:
- Conductors have a partially filled band, so electrons can move into nearby empty states under a small applied field.
- Insulators have a full valence band separated from the empty conduction band by a large gap, so electrons cannot easily cross.
- Semiconductors have the same structure as insulators but a much smaller gap, small enough that thermal energy or added impurities can promote some electrons across.
3. Motion of electrons in a periodic potential
Electrons in a real crystal do not move through empty space but through the periodic electric potential created by the regularly spaced positive ion cores. Bloch’s theorem shows that in such a potential, the allowed electron wavefunctions take a specific form (a plane wave modulated by a function with the same periodicity as the lattice), and this periodicity is precisely what forces electron energies into the allowed bands and forbidden gaps described above — band theory is the direct mathematical consequence of solving the Schrödinger equation for a periodic potential.
4. Fermi energy and density of states
The Fermi energy is the energy of the highest occupied electron state in a solid at absolute zero temperature — a direct consequence of the Pauli exclusion principle (Chapter II), which forces electrons to fill successively higher energy states rather than all occupying the lowest one. The density of states describes how many available electron energy states exist per unit energy interval; combined with Fermi–Dirac statistics, it determines how electrons are actually distributed among the bands at any given temperature, and hence the material’s electrical and thermal properties.
5. n-type and p-type semiconductors
A pure (“intrinsic”) semiconductor like silicon has very few free charge carriers at room temperature. Doping — deliberately introducing controlled impurities — dramatically changes this:
- n-type doping adds atoms with one more valence electron than the host (e.g., phosphorus in silicon), contributing extra, loosely bound electrons that become the majority charge carriers (negative).
- p-type doping adds atoms with one fewer valence electron (e.g., boron in silicon), creating “holes” (absent electrons that behave like mobile positive charge carriers) as the majority carriers.
Joining n-type and p-type material forms a p–n junction, the fundamental building block of diodes and, by extension, nearly all semiconductor electronics.
6. The transistor and MOSFET
A transistor uses a small input current or voltage to control a much larger output current, functioning as both an amplifier and an electronic switch — the basic building block of digital logic. The MOSFET (metal-oxide-semiconductor field-effect transistor) controls current flow between two terminals (source and drain) using an electric field applied through an insulated gate electrode, rather than a direct current into the control terminal; its combination of low power consumption and ease of miniaturisation is why it dominates modern integrated circuits, from processors to memory chips.
7. Dielectric and magnetic properties of solids
Beyond conductivity, solids respond to electric and magnetic fields in ways rooted in their atomic and electronic structure. Dielectric behaviour in solids follows the polarization mechanisms introduced in Electricity and Magnetism, but crystal structure strongly affects the degree and direction-dependence (anisotropy) of the response. Magnetism in solids originates fundamentally from electron spin and orbital motion; whether a material is diamagnetic, paramagnetic, or ferromagnetic (Chapter I) depends on how these individual atomic moments are arranged and whether quantum-mechanical exchange interactions favour their mutual alignment — the origin of the strong, permanent magnetism seen in materials like iron.
Key Takeaways
- Crystals are described by a lattice (one of 14 Bravais types) plus a basis; band theory — the result of solving the Schrödinger equation in a periodic potential — explains why materials are conductors, insulators, or semiconductors based on band filling and gap size.
- The Fermi energy and density of states, governed by Fermi–Dirac statistics and the Pauli exclusion principle, determine how electrons populate the available bands.
- Doping a semiconductor with donor (n-type) or acceptor (p-type) impurities creates controllable majority charge carriers; joining the two types forms the p–n junction at the heart of diodes and transistors.
- The MOSFET’s field-controlled conduction, rather than direct current injection, makes it the basis of low-power, highly miniaturised modern electronics.
- Both dielectric and magnetic behaviour in solids trace back to electron structure and, for magnetism, to quantum-mechanical exchange interactions between atomic moments.