band theory of solids
Why is copper a metal that conducts electricity, diamond a transparent insulator, and silicon something in between? They are all just atoms in a crystal, so the difference must lie in how their electrons are arranged. Band theory is the answer, and it is really molecular orbital theory scaled up from two atoms to a trillion trillion of them packed in a solid.
Recall that when two atomic orbitals combine they make two molecular orbitals, one lower (bonding) and one higher (antibonding) in energy. Combine three atoms and you get three levels; combine N atoms and you get N levels. In a real crystal N is astronomically large, so these levels are spaced so unbelievably close together that they merge into a continuous smear of allowed energies — a band. Crucially, bands do not always touch: there can be a band gap, a range of energies in which no orbital exists and no electron is allowed. Two bands matter most: the valence band (the highest band the electrons fill at low temperature) and the conduction band (the next band up). Electrons can only carry current if they can move into empty nearby states, so what matters is whether the top filled band is partly empty or whether a gap blocks the way.
This single picture sorts all solids into three classes. In a metal (copper) the valence band is only partly filled, or it overlaps the conduction band, so electrons flow freely with no barrier — and conductivity actually falls as you heat it. In an insulator (diamond) the valence band is completely full and a large band gap (several electron-volts) keeps electrons from reaching the empty conduction band, so no current flows. A semiconductor (silicon) is just an insulator with a small gap (around one electron-volt): at absolute zero it would be an insulator, but at room temperature a few electrons borrow enough thermal energy to jump the gap, so it conducts a little, and unlike a metal it conducts better when heated. That small, tunable gap is exactly what makes semiconductors the foundation of all electronics.
Diamond and silicon have the same structure (the diamond lattice) but different band gaps: about 5.5 electron-volts for diamond, far too wide to jump, so diamond is a clear insulator; about 1.1 electron-volts for silicon, small enough that thermal energy lets some electrons cross, so silicon is a semiconductor. Same arrangement, one number apart.
Diamond and silicon share a structure; their very different band gaps make one an insulator and the other a semiconductor.
A semiconductor is not a separate kind of matter — it is an insulator with a small gap. The metal-versus-nonmetal distinction here is about band filling and gap size, not about which side of the periodic table an element sits on.