Questions: Electronic Band Theory of Solids

Question 1

Silicon has a band gap of 1.1 eV and diamond has a band gap of 5.5 eV. Both are group 14 elements with the same crystal structure. Why is silicon a semiconductor while diamond is an insulator?

Model answer: The distinction is purely quantitative — band gap size relative to thermal energy (kT ~ 0.026 eV at 300 K). Silicon's 1.1 eV gap allows thermal excitation of a measurable number of electrons across the gap at room temperature (about 10^10 per cm^3), giving intrinsic conductivity. Diamond's 5.5 eV gap means the probability of thermal excitation is negligibly small (exp(-5.5/0.052) is essentially zero), so diamond has no mobile carriers at room temperature. The larger gap in diamond arises from stronger C-C sigma bonds and greater orbital overlap in the smaller carbon lattice.

The semiconductor/insulator boundary is conventional, not fundamental — it is typically placed around 3-4 eV. Materials with gaps just above this threshold can become semiconducting at high temperatures or with doping. The key insight from band theory is that the same physical picture (filled valence band, empty conduction band, gap between them) applies to both; only the magnitude of the gap differs.

Question 2

In band theory, a half-filled band always produces metallic behavior because electrons can be promoted to empty states with infinitesimally small energy input.

• T. True
• F. False

Answer: True

A half-filled band has the Fermi level in the middle of the band, with occupied states immediately below and empty states immediately above. An arbitrarily small electric field can promote electrons to these nearby empty states, giving them net momentum in the field direction — this is electrical conduction. Metals like sodium (one 3s electron per atom, half-filling the 3s band) exemplify this. The only exception is when electron-electron correlations are strong enough to split the half-filled band into upper and lower Hubbard bands (Mott insulators), but this is beyond standard band theory.

Question 3

What does the density of states (DOS) at the Fermi level tell you about a metal's properties?

Model answer: The DOS at the Fermi level, g(E_F), determines how many electronic states are available for thermal excitation and for response to external perturbations. A high g(E_F) means more electrons can participate in conduction (higher electronic specific heat), more states are available for Cooper pairing (higher superconducting T_c in BCS theory), and stronger magnetic susceptibility (Pauli paramagnetism). It is one of the most important single numbers characterizing a metal.

The DOS is the number of electronic states per unit energy per unit volume. At absolute zero, all states below E_F are filled and all above are empty. At finite temperature, only electrons within ~kT of E_F can be thermally excited, so only g(E_F) x kT electrons participate in thermal properties. This explains why the electronic specific heat of metals is small (only a thin shell of electrons near E_F contributes) and proportional to temperature.

Question 4

Band theory is simply molecular orbital theory extended to an infinite number of atoms.

• T. True
• F. False

Answer: True

This is exactly the conceptual link. Two hydrogen atoms form bonding and antibonding MOs. Three atoms form three MOs. N atoms form N MOs. As N approaches Avogadro's number, the spacing between adjacent energy levels becomes negligibly small (~10^-23 eV) and the discrete levels merge into a continuous band. The bandwidth (energy spread) equals the bonding-antibonding splitting and depends on orbital overlap. Band theory adds the periodicity of the crystal lattice (Bloch's theorem) to make the problem tractable, but the physical origin is MO theory applied to ~10^23 atoms.