Fermi energy definition and Fermi-Dirac distribution

In summary: For T>0 the Fermi-Dirac function gives a smooth decrease from 1 to 0. In summary, Fermi energy is defined as the topmost filled level in the ground state of an N electron system and this definition only holds for absolute zero. For non-absolute zero temperatures, Fermi energy is the energy at which the probability of a state being occupied is 50 percent. The Fermi-Dirac function shows a sharp step from 1 to 0 at low temperatures, indicating that levels below the Fermi level are filled and all above are empty. However, at higher temperatures, the Fermi-Dirac function decreases smoothly. This is due to the fluctuation of occupation levels at finite temperatures, and
  • #1
chikchok
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Fermi energy definition and fermi-dirac distribution
1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent. Please, tell me if I am understanding this correctly.
2)I was wondering why at low temperatures Fermi-Dirac function goes sharply from 1 to 0 and for higher temperature it goes down smoothly. Is it reasonable to assume that for low temperature, levels below the fermi level are filled and all above are empty? But why it does not happen with higher temperatures. Thank you in advance.
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  • #2
1) Stating "In the ground state" obviates mention of temperature.
2) Temperature is a measure of average internal energy. An isolated system at lowest energy will be in the ground state. If connected to a thermal reservoir it will have a finite probability of being in an excited state with a probability that is roughly negative exponential in energy e.g. the Boltzmann factor. Careful consideration of quantum effects gives rise to Fermi-Dirac statistics for half integer spin which gives a "step" function for T=0
 
  • #3
hutchphd said:
1) Stating "In the ground state" obviates mention of temperature.
2) Temperature is a measure of average internal energy. An isolated system at lowest energy will be in the ground state. If connected to a thermal reservoir it will have a finite probability of being in an excited state with a probability that is roughly negative exponential in energy e.g. the Boltzmann factor. Careful consideration of quantum effects gives rise to Fermi-Dirac statistics for half integer spin which gives a "step" function for T=0
so for non-absolute temperature fermi energy is not the energy of topmost filled level anymore?
 
  • #4
The Fermi Energy is defined as the zero T result. It is "5" for your graphed system. With finite temperature occupation of levels is fluctuating.
 

FAQ: Fermi energy definition and Fermi-Dirac distribution

What is Fermi energy?

Fermi energy, also known as Fermi level, is the highest energy level occupied by an electron at absolute zero temperature in a solid material. It represents the energy at which the probability of finding an electron is 50%.

How is Fermi energy related to the Fermi-Dirac distribution?

The Fermi-Dirac distribution is a probability distribution that describes the distribution of electrons in a solid material at a given temperature. Fermi energy is used as a reference point in this distribution, as it represents the highest energy level occupied by electrons at absolute zero temperature.

What is the significance of Fermi energy in materials science?

Fermi energy plays a crucial role in understanding the electronic properties of materials. It determines the electrical and thermal conductivity, as well as the optical and magnetic properties of a material. It also helps in predicting the behavior of electrons in different materials and their interactions with external fields.

How is Fermi energy different from band gap energy?

Band gap energy is the difference in energy between the top of the valence band and the bottom of the conduction band in a material. It represents the energy required to promote an electron from the valence band to the conduction band. On the other hand, Fermi energy represents the energy of the highest occupied electron level at absolute zero temperature.

Can Fermi energy be measured experimentally?

Yes, Fermi energy can be measured experimentally using various techniques such as photoemission spectroscopy, tunneling spectroscopy, and Hall effect measurements. These methods involve measuring the energy of electrons in a material and determining the energy level at which the electron density is 50%.

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