band gap metal vs semiconductor
Inside one atom an electron sits only on separated energy levels. Gather a vast number of atoms into a solid and those same levels push apart into a dense crowd. That crowd is called an energy band. The top band packed with electrons is the valence band, the empty band above it is the conduction band, and the empty gap between them is the band gap. Whether this one gap is 0, narrow, or wide splits conductors, semiconductors, and insulators. One gap splits the materials.
Grow the band gap from 0 to 10. That gap is the energy width between the top of the valence band and the bottom of the conduction band, and the unit is the electron volt. It is a different quantity from an elastic modulus or an electric field that uses the same letter, so a subscript always keeps them apart. When the gap is 0 the two bands touch and electrons flow freely, a conductor. Around 1.1 it is a semiconductor such as silicon, and wider than 4 it is an insulator such as diamond. One number splits three worlds.
Where does a band come from? With exactly one atom an electron sits on one sharp level. Bring many atoms close and the same level cannot overlap, so it splits into many levels that differ only a little. Raise the count and those levels grow denser until, to the eye, they merge into one continuous band. This is the bridge from separated levels to the continuous band of a solid. Many of them become a band.
Now fill the band with electrons. Electrons fill from low energy upward, and how many seats there are at a given energy is told by the density of states S(E). Here S(E) is the number of states per unit energy, states/eV, a completely different quantity from the entropy of heat S(J/K). The top height the electrons have filled is called the Fermi level. Raise the temperature and the boundary near EF smears, and a sharp step becomes a soft threshold. That threshold sits near the Fermi level.
In a pure semiconductor the level sits in the middle of the gap, and electrons and empty seats appear only in tiny pairs. Mixing in a little impurity is doping. One kind grows a level just under the conduction band that gives extra electrons, the level rises, and the carriers that mostly move are electrons. The other kind grows a level just above the valence band that takes electrons, the level falls, and the carriers that mostly move are empty seats. This EF shift is the heart of making diodes and transistors. Moving the level makes the device.
Last, raise the temperature and measure the conductivity. A metal already has free electrons, so as the temperature rises they collide with the shaking lattice, the hindrance grows, and the conductivity falls like 1/T. A semiconductor is the opposite. As the temperature rises, thermal energy carries electrons across the gap into the conduction band, the carriers grow sharply like exp(-Eg/2kT), and the conductivity rises. These opposite directions reveal the gap at once. The curve below is a normalized sketch of the current. The direction speaks the gap.