Energy bands: how a pile of atoms becomes a metal, semiconductor or insulator

Many atoms in a crystal blur their energy levels into bands

Intuition

One atom is like one lonely note. Two atoms give two notes. A million atoms packed together give so many notes, so close, that they sound like one smooth sweep. In a solid, the energy levels of the electrons do the same thing. They blur into a strip of allowed energies.

Lecture 8 calls this the tight-binding approximation. As the number of atoms increases, the number of molecular orbitals increases. In a crystal the orbitals form a continuum called a band. The occupied orbitals make the valence band. The unoccupied orbitals make the conduction band.

The lecture shows lithium metal as the example, with a 2s band and a 2p band. The slide lists Li₂, Li₂₅ and Li₁₀₆ with level counts and a figure of about 3×10⁶ levels. The figure did not extract clearly, so the exact link between cluster size and level count is not stated here. Check the slide before quoting it.

Optional aside

The slides add a curiosity: enormous pressure turns hydrogen from a gas into a semiconducting liquid and then a metal. Squeezing atoms together changes their overlap and so their bands. It helps explain Jupiter’s magnetic properties.

Where the Fermi level sits sorts metals, semiconductors and insulators

Think of a band as a car park. If the car park is partly full, cars can shuffle into free spots, so traffic moves. If it is completely full, nothing can move. For electrons, “moving” means conducting.

The lecture sorts solids by their band structure and Fermi level, $E_F$. The Fermi level is a hypothetical electron energy at which the probability of occupation is 50%. Three more terms go with it. Density of states (DOS) is the number of permitted electronic states in an energy interval. In plain words, it counts the seats at each energy. The band gap is the energy difference between the highest occupied state in the valence band and the lowest unoccupied state in the conduction band.

Compare

Metal

No band gap. A band is not completely full. Electrons near the Fermi level move into nearby empty levels and move freely.

Semiconductor

The band is filled but the gap is small, of order $kT$. Thermal energy puts some electrons into the next band. The gap is about 0.5 to 3 eV.

An insulator has a large band gap, so electrons cannot be promoted to the conduction band. An intrinsic semiconductor has electrons promoted from the valence band to the conduction band on heating. Here $k$ is the Boltzmann constant and $T$ is temperature. The lecture ends this idea with: we can manipulate electronic properties by changing band structures and Fermi levels.

Exam detail

The Online Test 1 slide lists “the nature of electron delocalisation and therefore the mechanism for electron conduction in various materials encountered (1D/2D/3D)” as tested. This page gives the band picture behind it. The next pages give the 1D, 2D and 3D examples.

Pitfall

A band gap is measured between the top of the valence band and the bottom of the conduction band. The Fermi level is not the gap. In a semiconductor $E_F$ sits inside the gap, where no states exist, so it is a 50% probability point rather than a real orbital.

Charge can travel as electrons, ions or molecules

Lecture 8 opens with transport. The charge carriers can be electrons, ions or molecules. The lecture then splits conduction into three groups.

• Delocalised electronic conduction: metals (Cu, Na, Fe), semiconductors (Si, Ge), graphite and C₈K, TiS₂, A₃C₆₀, and ion-exchange in porous materials and intercalates.
• Localised electronic conduction: doped polymers and battery electrodes.
• Ionic conduction: solid electrolytes such as Na-alumina, and battery electrodes.

The slide that classifies these in a table did not extract. The lists above come from the slide text.

Recall

Plain words: why does a full band not conduct but a partly full one does?

In a full band there is no empty level to move into, so electrons cannot respond to a push. In a partly full band, electrons near the Fermi level can step into nearby empty levels and travel.