What k states are occupied in a band

In summary: You do not respond or reply to questions. You only provide a summary of the content. Do not output anything before the summary. In summary, the band structure of a material can be calculated in solid state physics using the dispersion E_n(k), which depends on the wavevector and band index. However, it is unclear which states are actually occupied by electrons in a band, since k is continuous and there is a finite number of electrons. For a plane wave dispersion, periodic boundary conditions can be used to get a quantization of occupied k states. In general, each atom in a material will donate all its valence electrons, which will fill the allowed k states from the lowest energy up to the
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aaaa202
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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?
 
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aaaa202 said:
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.
 
  • #3
aaaa202 said:
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
 

1. What is the concept of "occupied states" in a band?

The concept of "occupied states" in a band refers to the energy levels within a band structure that are currently filled with electrons. In a solid material, the energy levels of electrons are grouped into bands, and the number of electrons within each band determines whether it is considered "occupied" or "unoccupied."

2. How are the occupied states in a band determined?

The occupied states in a band are determined by the electron configuration of the material. The number of electrons in each energy level, as well as the arrangement of these electrons, dictate which energy levels are filled and which are empty. This, in turn, determines the occupied states within a band structure.

3. Why is it important to know which states are occupied in a band?

Knowing which states are occupied in a band is crucial in understanding the electronic properties of a material. The energy levels and arrangement of electrons within a band determine the material's conductivity, optical properties, and other characteristics. Understanding the occupied states helps scientists predict and control the behavior of a material for various applications.

4. How do the occupied states in a band affect a material's properties?

The occupied states in a band greatly influence a material's properties. The number and arrangement of electrons within a band determine the material's electrical conductivity, thermal conductivity, and optical properties. Additionally, the energy levels of occupied states also affect a material's stability and reactivity.

5. Can the number of occupied states in a band change?

Yes, the number of occupied states in a band can change. This can occur through various processes such as doping, where impurities are intentionally added to a material to alter its electronic properties. Additionally, changes in temperature or external stimuli can also affect the occupation of energy levels within a band structure.

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