The Band Gap Splits Metal from Semiconductor
Inside a single atom an electron can only sit at discrete energy levels. Bring billions of atoms together into a solid and those levels push apart into dense clusters called energy bands. The topmost band filled with electrons is the valence band, the empty band just above it is the conduction band, and the empty span between them is the band gap Eg. Whether that one gap is zero, small, or wide is what splits metals from semiconductors from insulators.
Use the slider to open the band gap Eg from 0 up to 10 eV. Here Eg is the energy width between the top of the valence band and the bottom of the conduction band, measured in eV. It shares the letter E with the elastic modulus and the electric field E, but it is a different quantity, so it always carries the subscript g. When Eg is 0 the two bands touch and electrons flow freely, giving a metal; near 1.1 eV you get a semiconductor like silicon; wider than about 4 eV you get an insulator like diamond.
So where do the bands come from? With just one atom an electron sits on one sharp level. Push N atoms close together and, because the Pauli exclusion principle forbids identical levels from overlapping, that one level splits into N slightly different levels. Slide N upward and those N levels crowd closer until, to the eye, they merge into one continuous band. This is the bridge from the discrete levels of quantum mechanics to the continuous bands of a solid.
Now fill the band with electrons. Electrons stack in from the lowest energy upward, and how many seats exist at each energy is told by the density of states S(E). Here S(E) is the number of states per unit energy (states/eV), an entirely different quantity from the thermodynamic entropy S (J/K). The top height that electrons fill up to is the Fermi level EF. Raise the temperature T and the edge near EF blurs, so the sharp step softens into a gentle threshold.
In a pure semiconductor (intrinsic) EF sits in the middle of the gap and only a few electron-hole pairs appear. Mixing in a trace of impurity is called doping. In n-type a donor level appears just below the conduction band, giving up spare electrons, so EF shifts upward and the majority carriers are electrons. In p-type an acceptor level appears just above the valence band, catching electrons, so EF shifts downward and the majority carriers are holes. This shift of EF is the heart of the electronics that build diodes and transistors.
Finally raise the temperature T and measure conductivity. A metal already has free electrons, so as T rises they collide more with lattice vibrations, scattering grows, and the conductivity falls roughly like 1/T. A semiconductor instead uses the extra thermal energy to lift electrons across the gap Eg into the conduction band, so carriers climb sharply like exp(-Eg/2kT) and the conductivity rises. These opposite directions reveal the presence of the gap at once. The curves below are normalized shapes that show the trend.