What k states are occupied in a band

[aaaa202]

In solid state physics you can calculate the band structure of a material, which is effectively the dispersion E_n(k), which depends on the wavevector as well as the band index. What I don't understand is this: Which states are occupied in a band? With this I mean: Which k values correspond to states actually occupied by electrons. Since k is continuous it cannot be all k values since there is not an infinite number of electrons. For a plane wave dispersion I remember that you can use periodic boundary conditions to get a quantization of k in states that are actually occupied. How do you do this for a general dispersion, i.e. a general band structure?

 

[ZapperZ]

In solid state physics you can calculate the band structure of a material, which is effectively the dispersion E_n(k), which depends on the wavevector as well as the band index. What I don't understand is this: Which states are occupied in a band? With this I mean: Which k values correspond to states actually occupied by electrons. Since k is continuous it cannot be all k values since there is not an infinite number of electrons. For a plane wave dispersion I remember that you can use periodic boundary conditions to get a quantization of k in states that are actually occupied. How do you do this for a general dispersion, i.e. a general band structure?

But your E(k) also contains the location of the Fermi energy AND the Fermi wavevector. For a metal, this tells you the maximum energy and wavevector that are occupied.

Zz.

 

[Henryk]

For a plane wave dispersion I remember that you can use periodic boundary conditions to get a quantization of k in states that are actually occupied.

Actually, you get quantization of allowed k values regardless if they are occupied or not.
The answer to your question is quite simple. Each atom will donate all its valence electrons. At absolute zero temperature, you start filling the allowed k states with electrons starting from the lowest energy until you use up all the electrons. Remember, each k state can accept two electrons: one with spin 'up' the second with spin 'down'.
The energy of the highest occupied state at zero temperature is Fermi energy. If the Fermi energy is within an allowed energy band you have a metal. If you fill up some bands completely and all the other bands are empty you have an insulator. Semiconductor is essentially an insulator whose energy band is small enough so you can get a 'reasonable' number of electrons thermally excited to the lowest energy band at ambient temperature.

Henryk