Number of electrons in conduction band

In summary, the density of states for an atom is a function that takes in the energy of the atom and gives the number of states that are available between that energy and the next higher energy. If there are multiple states at an energy, the Fermi probability distribution is used to determine which state is most likely to be occupied.
  • #1
EmilyRuck
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Hello!
In order to obtain the number of actual electrons in the conduction band or in a range of energies, two functions are needed:
1) the density of states for electrons in conduction band, that is the function [itex]g_c(E)[/itex];
2) the Fermi probability distribution [itex]f(E)[/itex] for the material at its temperature [itex]T[/itex].
(as a reference, http://ecee.colorado.edu/~bart/book/carriers.htm can be used). So, the number of electrons between the energy level [itex]E[/itex] and the energy level [itex]E + dE[/itex] is given by

[itex]n(E)dE = g_c(E) f(E) dE[/itex]

where [itex]g_c(E) dE[/itex] is the number of states between energy level [itex]E[/itex] and energy level [itex]E + dE[/itex] and [itex]f(E)[/itex] is the probability that they are occupied.
This would work if the number of states in that range is just 1 or 0. But what if there are multiple available states?

Fermi probability distribution gives the probability that a state at a certain energy [itex]E[/itex] is occupied: it is just one state. If we need to handle multiple states (like two electrons with opposite spin) at the same energy (or in the same infinitesimal interval of energies, between [itex]E[/itex] and [itex]E + dE[/itex]), how should we use the value of the Fermi probability for each of them?
 
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  • #2
Shouldn't degenerate states be taken care of in the density of states? For spin up and spin down, a factor of 2 is included in the density of states, and doesn't affect the Fermi-Dirac distribution.
 
  • #3
Vagn said:
Shouldn't degenerate states be taken care of in the density of states? For spin up and spin down, a factor of 2 is included in the density of states, and doesn't affect the Fermi-Dirac distribution.

Yes, a factor of 2 is included in the density of states. My question is: if an energy level [itex]E_1[/itex] is such that (for example) [itex]f(E_1) = 0.1[/itex] and there are two possible states with opposite spins for the electrons at [itex]E = E_1[/itex]: will both the states have the same [itex]0.1[/itex] probability of being occupied? Or what else?
 
  • #4
EmilyRuck said:
Yes, a factor of 2 is included in the density of states. My question is: if an energy level [itex]E_1[/itex] is such that (for example) [itex]f(E_1) = 0.1[/itex] and there are two possible states with opposite spins for the electrons at [itex]E = E_1[/itex]: will both the states have the same [itex]0.1[/itex] probability of being occupied? Or what else?
All degenerate states with energy [itex]E[/itex] have the same probability of being occupied, namely [itex]f(E)[/itex], no matter what the spin is.
 
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  • #5
Fermi-Dirac distribution f(E) gives the probability that a quantum state with energy E is occupied. So, as long as there is no symmetry breaking that lifts the degeneracy of spin up and spin down electrons, they have the same energy, thereore f(E) for spin up and spin down electrons are the same. That's why we have a factor of two. Furthermore, if there are more available states with the same energy, f(E) for all those states would be the same.
 
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  • #6
Ok, that's right now, so thank you both!
 

1. How are the number of electrons in the conduction band determined?

The number of electrons in the conduction band is determined by the number of available energy states for electrons to occupy. These energy states are dependent on factors such as the material's band structure and temperature.

2. What is the relationship between the number of electrons in the conduction band and electrical conductivity?

The number of electrons in the conduction band is directly proportional to the electrical conductivity of a material. A higher number of electrons in the conduction band means a greater number of free charge carriers, leading to higher electrical conductivity.

3. Can the number of electrons in the conduction band be changed?

Yes, the number of electrons in the conduction band can be changed through the application of external factors such as electric fields, magnetic fields, and temperature. This can alter the band structure and change the number of available energy states for electrons to occupy.

4. How does the number of electrons in the conduction band affect a material's properties?

The number of electrons in the conduction band plays a crucial role in determining a material's properties such as electrical conductivity, thermal conductivity, and optical properties. It also influences the material's behavior in different environments and under different conditions.

5. What is the difference between the number of electrons in the conduction band and the number of valence electrons?

The number of electrons in the conduction band refers to the electrons that are able to move freely and contribute to electrical conductivity. On the other hand, the number of valence electrons refers to the outermost electrons in an atom that are involved in chemical bonding. These two numbers may not necessarily be the same for a given material.

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