Band Theory of Solids

From a single atom to a solid

An isolated atom has sharp, discrete energy levels — the familiar 1s, 2s, 2p orbitals, each holding electrons at precise energies set by the Schrödinger equation. Bring two identical atoms close enough that their orbitals overlap and the Pauli exclusion principle forbids two electrons from occupying the same quantum state, so each level splits into two: a bonding (lower) and an antibonding (higher) combination, exactly as in molecular orbital theory. Add a third atom and you get three levels; add N atoms and you get N levels.

In a real crystal N is enormous — on the order of 10²³ atoms per cubic centimetre. Splitting one atomic level into 10²³ sub-levels spread over a few electron-volts means the spacing between adjacent levels is around 10⁻²³ eV, far smaller than any energy an electron could resolve. For all practical purposes the levels merge into a continuous energy band. Each atomic orbital generates its own band: the 1s atomic states form a 1s band, the 3s/3p states form the bands that matter for conduction, and so on. Between some bands lie ranges of energy that no electron can occupy — the band gaps, or forbidden zones.

The two bands that decide a material's electrical fate are the highest band that is filled with electrons at absolute zero — the valence band — and the next band above it, normally empty — the conduction band. Electrons in a completely full band cannot carry current: every available state is occupied, so an applied field has nowhere to push them. Only when there are empty states immediately accessible can electrons accelerate and conduct.

The Fermi level and who sits where

Electrons fill bands from the bottom up, two per state (spin up and spin down), following the Fermi–Dirac distribution. The Fermi level (E_F) is the energy at which the occupation probability is exactly ½ at equilibrium. Its location relative to the bands is the single most important fact about a solid:

• Metal. E_F falls inside a band. Either the valence band is only partly filled (sodium, with one 3s electron per atom half-fills its band), or a full band overlaps an empty one so there is no gap at all (magnesium, where the 3s and 3p bands overlap). Empty states sit a hair above the filled ones, so even a tiny field produces current. Copper reaches σ ≈ 5.96×10⁷ S/m.
• Semiconductor. The valence band is full and E_F sits near the middle of a small gap (≈ 0.1–3 eV). At 0 K the material is an insulator, but at room temperature thermal energy (k_BT ≈ 0.025 eV) excites a measurable number of electrons across the gap, leaving mobile holes behind. Conductivity rises with temperature — the opposite of a metal.
• Insulator. The valence band is full and the gap is large (> ~4 eV). The Boltzmann factor e^−E_g/2k_BT for crossing a 5.5 eV gap at 300 K is around 10⁻⁴⁸, so essentially no electrons make it across and the material refuses to conduct.

The numbers that separate the three classes

The exponential sensitivity of carrier concentration to the gap is what produces the staggering range of conductivities. Doubling the gap from 1 eV to 2 eV cuts the intrinsic carrier density by roughly e^{−(1 eV)/(2·0.025 eV)} ≈ 10⁻⁹. That single exponential explains why silicon and diamond — chemically both group-14 elements with the same diamond crystal structure — differ in conductivity by more than 25 orders of magnitude.

Material	Class	Band gap Eg (eV)	Gap type	Conductivity σ (S/m, 300 K)
Copper	Metal	0 (overlap)	—	~6.0×10⁷
Germanium	Semiconductor	0.66	Indirect	~2.2 (intrinsic)
Silicon	Semiconductor	1.12	Indirect	~4×10⁻⁴ (intrinsic)
Gallium arsenide	Semiconductor	1.42	Direct	~10⁻⁶ (intrinsic)
Gallium nitride	Wide-gap semic.	3.4	Direct	very low (intrinsic)
Diamond	Insulator	5.5	Indirect	<10⁻¹³
Quartz (SiO₂)	Insulator	~9	—	~10⁻¹⁸

Note the temperature behaviour, which is diagnostic. A metal's conductivity drops as it heats because lattice vibrations scatter the already-mobile electrons. A semiconductor's conductivity rises roughly as e^−E_g/2k_BT because heat is creating the very carriers it relies on. Measuring dσ/dT therefore tells you which class you are holding.

Direct vs. indirect gaps: why silicon can't glow

A band is not a single energy but a curve of energy versus crystal momentum (the E–k dispersion). In a direct-gap material such as GaAs or GaN, the bottom of the conduction band lies directly above the top of the valence band at the same momentum. An electron can fall across the gap by emitting one photon — momentum is conserved automatically. This is why GaAs and GaN make efficient LEDs, laser diodes, and the blue/UV emitters that won the 2014 Nobel Prize in Physics.

In an indirect-gap material such as silicon or germanium, the conduction-band minimum is shifted in momentum from the valence-band maximum. To recombine, an electron must change both its energy and its momentum, which requires emitting or absorbing a phonon (a quantum of lattice vibration) at the same instant as the photon — a far less probable three-body event. Silicon therefore makes a poor light emitter even though it is the ideal logic material, which is why a smartphone uses silicon for its processor but compound semiconductors for its camera flash and laser autofocus.

Doping: tuning the gap from the inside

Pure (intrinsic) silicon is nearly an insulator at room temperature. Its power comes from doping — deliberately adding parts-per-million of impurity atoms that place new energy levels inside the gap, close to one band edge.

• n-type. A group-15 atom (phosphorus, arsenic, antimony) has one more valence electron than silicon. It sits ~0.045 eV below the conduction band, easily ionised at room temperature, donating a free electron. The Fermi level shifts upward, toward the conduction band.
• p-type. A group-13 atom (boron, gallium) has one fewer electron, creating an acceptor level ~0.045 eV above the valence band. It captures a valence electron and leaves a mobile hole. The Fermi level shifts downward.

The leverage is extraordinary: a doping level of just 10¹⁶ atoms/cm³ — about one boron atom per 5 million silicon atoms — raises silicon's conductivity by several orders of magnitude. Joining an n-type region to a p-type region creates a p-n junction: electrons and holes diffuse across, Fermi levels equalise, and a built-in potential of ~0.7 eV forms. That junction is the diode, the solar cell, and the building block of the bipolar and field-effect transistors that make up the ~10¹⁰ devices on a modern chip.

Where band theory shows up

• Microelectronics. Every transistor exploits doping-controlled Fermi levels and junctions; CMOS logic, memory, and power devices are band engineering in action.
• Photovoltaics. A solar cell absorbs photons with energy above E_g; the ~1.1 eV gap of silicon is near the theoretical Shockley–Queisser optimum (~1.34 eV) for the solar spectrum.
• Lighting and lasers. LED colour is set directly by the gap — GaN (3.4 eV) for blue, InGaN alloys for green, AlGaAs for red.
• Photocatalysis. TiO₂ (E_g ≈ 3.2 eV) absorbs UV to drive water splitting and self-cleaning surfaces.
• Thermoelectrics and sensors. Narrow-gap materials (Bi₂Te₃, PbTe) convert heat to electricity; gap-tuned detectors sense specific wavelengths.

Where the simple picture bends

Band theory assumes electrons move almost independently in a periodic potential. That works beautifully for silicon, but breaks where electron–electron interactions dominate. Some materials that band theory predicts to be metals — nickel oxide, for instance — are actually insulators (Mott insulators) because strong on-site repulsion freezes the electrons in place. Disordered and amorphous solids blur the sharp band edges into band tails. And in nanostructures such as quantum dots, the gap itself becomes size-dependent: shrink a CdSe dot from 6 nm to 2 nm and quantum confinement widens its effective gap, tuning its colour from red to blue. The bands are an emergent property of order and scale, and reshaping either reshapes the electronics.