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When N atoms come together to form a solid, their N discrete atomic orbitals combine to form N molecular orbitals so closely spaced in energy that they form continuous bands. The band structure of a solid — the arrangement of these energy bands and the gaps between them — determines whether the material is a metal, semiconductor, or insulator. Metals have overlapping or partially filled bands with no gap at the Fermi level; semiconductors have a small band gap (< ~3.5 eV) between a filled valence band and an empty conduction band; insulators have a large band gap (> ~3.5 eV). The Fermi level, density of states, and band gap are the key quantities that govern electronic, optical, and thermal properties.
Band theory is the bridge between the molecular orbital theory you already know and the electronic properties of bulk solids. The conceptual extension is simple: if two atoms form a bonding and an antibonding orbital, and three atoms form three molecular orbitals, then 1023 atoms form 1023 orbitals packed so tightly in energy that they form a continuous band. The bandwidth — the total energy spread — equals the bonding-antibonding splitting for the relevant atomic orbitals and depends on the degree of orbital overlap between neighbors.
The critical question is how electrons fill these bands. Each band can hold 2N electrons (N orbitals, 2 electrons each from spin). If a band is completely filled, electrons cannot respond to an electric field because there are no empty nearby states to move into — the material is an insulator or semiconductor. If a band is partially filled, electrons near the top of the occupied states can be promoted to nearby empty states with minimal energy input, enabling conduction — the material is a metal. The Fermi level marks the boundary between filled and empty states at absolute zero.
The band gap — the energy range between the top of the valence band (highest filled) and the bottom of the conduction band (lowest empty) — is the single most important parameter in semiconductor physics and materials chemistry. It determines the minimum energy needed to excite an electron from bonding to antibonding states. For silicon (1.1 eV), visible light photons have more than enough energy to excite electrons across the gap, which is why silicon absorbs light and can generate photocurrent. For diamond (5.5 eV), only deep ultraviolet photons carry enough energy, so diamond is transparent to visible light and electrically insulating.
The distinction between direct and indirect band gaps matters for optical properties. In a direct gap semiconductor (GaAs, CdTe), the valence band maximum and conduction band minimum occur at the same crystal momentum (k-point), so photon absorption can occur without phonon assistance. In an indirect gap material (Si, Ge), the band extrema are at different k-points, requiring a phonon to conserve momentum — this makes absorption less efficient. Direct gap semiconductors are preferred for light-emitting devices and solar cells because they absorb and emit light much more efficiently. Band theory makes these distinctions quantitative and connects them to crystal structure and bonding.