Band Theory of Solids
Band theory extends molecular orbital theory from individual molecules to macroscopic solids. When trillions of atoms come together in a crystal, their discrete energy levels merge into continuous energy bands that determine whether a material conducts electricity, acts as a semiconductor, or serves as an insulator. Band theory is the foundation of modern electronics and the physics behind every transistor, LED, and solar cell.
From Molecular Orbitals to Energy Bands
Consider a single sodium atom with its 3s valence electron in a discrete energy level. Bring two sodium atoms together, and the two 3s orbitals split into one bonding and one antibonding molecular orbital. Bring four atoms together, and four levels appear. Extend this to a mole of atoms — roughly 6 × 10²³ — and the levels are so closely spaced that they form a continuous band of allowed energies.
Each atomic orbital in the isolated atom gives rise to one band in the solid. The 1s, 2s, 2p, and 3s orbitals of sodium each produce their own band. The bands may overlap or be separated by energy gaps (band gaps) where no allowed electronic states exist.
Valence Band and Conduction Band
The valence band is the highest-energy band that is fully or partially occupied by electrons at absolute zero. These electrons participate in bonding and hold the crystal together.
The conduction band is the next higher band, typically empty or partially filled. Electrons in the conduction band are delocalized throughout the crystal and free to move in response to an electric field — they carry electrical current.
The band gap (E_g) is the energy difference between the top of the valence band and the bottom of the conduction band. This single parameter largely determines the electrical behavior of a solid.
Conductors, Semiconductors, and Insulators
Conductors (metals) have either an overlapping valence and conduction band (no band gap) or a partially filled valence band. In either case, electrons can move freely into empty states with negligible energy input. Copper, silver, and aluminum are excellent conductors. Their resistivity increases with temperature because lattice vibrations scatter the mobile electrons.
Insulators have a large band gap, typically greater than 4 eV. At room temperature, virtually no electrons have enough thermal energy to jump from the valence band to the conduction band. Diamond (E_g ≈ 5.5 eV) and glass are classic insulators.
Semiconductors have a moderate band gap, roughly 0.1 to 4 eV. At absolute zero they behave as insulators, but at room temperature a small fraction of electrons can be thermally excited across the gap. Silicon (E_g ≈ 1.1 eV) and germanium (E_g ≈ 0.67 eV) are the most important semiconductors. Their conductivity increases with temperature — the opposite of metals — because more electrons gain enough energy to cross the gap.
Intrinsic Semiconductors
A pure semiconductor is called intrinsic. When an electron in silicon absorbs enough energy to jump to the conduction band, it leaves behind a "hole" — a vacant state in the valence band that behaves as a positive charge carrier. Both the excited electron and the hole contribute to conductivity. In intrinsic silicon at room temperature, the carrier concentration is about 1.5 × 10¹⁰ per cm³ — tiny compared to copper's 8.5 × 10²² free electrons per cm³, which is why pure silicon is a poor conductor.
Doping: Engineering the Band Gap
The power of semiconductors lies in doping — intentionally introducing impurity atoms to dramatically alter electrical properties.
n-type doping adds atoms with one more valence electron than the host. Phosphorus (5 valence electrons) substituted into silicon's lattice donates its extra electron to the conduction band. The donor level sits just below the conduction band edge (about 0.045 eV for P in Si), so electrons are easily promoted at room temperature. The majority carriers are electrons.
p-type doping adds atoms with one fewer valence electron. Boron (3 valence electrons) in silicon creates an empty state (acceptor level) just above the valence band. Electrons from the valence band fill this level easily, creating mobile holes. The majority carriers are holes.
The p-n Junction
When p-type and n-type semiconductors are joined, electrons diffuse from the n-side into the p-side and holes diffuse the other way, creating a depletion region with an internal electric field. This p-n junction is the basis of the diode — it allows current to flow in one direction but blocks it in the other. Nearly all electronic devices — transistors, LEDs, solar cells, laser diodes — are built from combinations of p-n junctions.
Applications in Modern Technology
Transistors — The metal-oxide-semiconductor field-effect transistor (MOSFET) uses a voltage applied to a gate electrode to control current flow through a semiconductor channel. Billions of these switches on a single chip power every modern computer.
Solar cells — Photons with energy exceeding the band gap excite electrons across the gap. The internal field of a p-n junction separates the electron-hole pairs, generating electric current.
LEDs — When electrons and holes recombine across a p-n junction, they release energy as photons. The band gap determines the photon wavelength and thus the color of light emitted. Gallium nitride (GaN) LEDs produce blue and white light, revolutionizing energy-efficient lighting.
Band theory transforms our understanding of matter from a collection of atoms into a quantum mechanical framework that explains and predicts the electronic properties of every solid material.