Q.Explain how solids are classified on the basis of band theory of solids.

[!TLDR] Band theory classifies solids by how their valence and conduction bands relate: overlapping bands give a metal, a small gap gives a semiconductor, and a wide gap gives an insulator. [!ANSWER] Metals have overlapping valence/conduction bands (no gap); semiconductors have a small gap (~1 eV or less); insulators have a wide gap (diamond ~5.0 eV) -- classification is entirely by band-gap size.

In an isolated atom, electrons occupy sharply discrete energy levels. When a very large number of atoms come together to form a solid, the Pauli exclusion principle forces each of these levels to split into an enormous number of extremely closely-spaced sub-levels -- so closely spaced that they merge into what looks like a continuous ENERGY BAND. The topmost band that is occupied by electrons is called the VALENCE BAND, and the next band above it, into which electrons can be excited, is called the CONDUCTION BAND.

Band theory classifies solids according to the relationship between these two bands. In a METAL, the valence band and the conduction band actually OVERLAP -- there is no gap between them at all -- so electrons need almost no extra energy to move into conduction levels, and metals conduct electricity very easily; sodium is a worked example of this, where the valence band itself is only half filled and directly overlaps the conduction band.

In a SEMICONDUCTOR, there is a genuine gap between the top of the valence band and the bottom of the conduction band -- the BAND GAP or ENERGY GAP -- but it is fairly small, of the order of one electron-volt or less (1.12 eV for silicon, 0.66 eV for germanium). At absolute zero, all levels up to and including the valence band are completely full and no conduction is possible; but at ordinary (room) temperatures, a reasonable number of valence electrons gain enough thermal energy (of order $kT$) to cross this small gap into the conduction band, giving the material a moderate, and usefully tuneable, conductivity that rises further as temperature increases.

In an INSULATOR, the band gap is WIDE -- diamond's gap, for example, is about 5.0 eV. Electrons essentially never gain enough thermal energy at ordinary temperatures to cross a gap this large, so almost no electrons reach the conduction band, and the material conducts extremely poorly, if at all.

So the entire metal/semiconductor/insulator classification comes down to one single quantity: the size of the band gap between the valence and conduction bands -- zero for a metal, small for a semiconductor, and wide for an insulator. [!ANSWER] Metals have overlapping valence/conduction bands (no gap); semiconductors have a small gap (~1 eV or less); insulators have a wide gap (diamond ~5.0 eV) -- classification is entirely by band-gap size.

Build up from discrete atomic levels to bands (via the Pauli exclusion principle), define the valence band, conduction band and band gap, then classify the three solid types purely by the size of that gap.

Describing the band gap qualitatively without giving the concrete numeric contrast (roughly 0 eV for a metal, ~1 eV or less for a semiconductor, ~5 eV for an insulator like diamond) that makes the classification precise rather than just descriptive.