# Fermi Energy, number of electrons and holes

• bcjack
At 290 K, there will be a larger number of vacancies, while at 174 K there will be a smaller number of vacancies and more electrons in level W3. This is due to the temperature dependence of the Fermi-Dirac distribution. In summary, the Fermi energy and number of electrons and vacancies in a system can be calculated using the Fermi-Dirac distribution, which takes into account the temperature and energy levels of the system.
bcjack
Fermi Energy, Quantum mechanics

Electron levels and degeneracies thereof in a system are:
W1 = 0 eV, 10^23 /cm3 Valence band
W2 = 0.9 eV, 5x10^20 /cm3 Donor level
W3 = 1 eV, 2x10^23 /cm3 Conduction level

Total number, n, of electrons in the system is (10^23 + 5 x 10^23) /cm3.

Determine Fermi energies, number of electrons in level W3, and number of vacancies (holes) in levels W1 and W2 at i) 290 K and ii) 174 K.

My first thoughts of doing this problem is by adding the number of electrons in the 3 states together and solve for Ef with the fermi distribution.

f(e)=1 / [1 +e^ (E-Ef)/kT ]
n = g(e) * f(e) density of states times the fermi distribution

n_total = n1 + n2 + n3 = 10^23 / [1 +e^ (0-Ef)/kT ] + 5x10^20/ [1 +e^ (0.9-Ef)/kT ] + 2x10^23 / [1 +e^ (1-Ef)/kT ] = (10^23 + 5 x 10^23) /cm3

and solve for Ef.

Is this a reasonable approach?

To find the number of electrons and holes, i would then plug the Ef back into the equation for each energy level.

any thoughts would be appreciated. thanks!

Last edited:
Yes, that is a reasonable approach. You can use the Fermi-Dirac distribution to calculate the Fermi energy. To find the number of electrons and holes, you can then plug the Fermi energy back into the equation for each energy level to find the number of electrons and vacancies.

## 1. What is Fermi Energy?

Fermi Energy is the highest energy state that an electron can occupy at absolute zero temperature in a solid material. It is a measure of the energy required for an electron to move from the lowest energy state to the highest energy state in a solid.

## 2. How is Fermi Energy calculated?

Fermi Energy can be calculated using the formula EF = (h2/8m)(3π2N/V)2/3, where h is the Planck constant, m is the mass of an electron, N is the total number of electrons, and V is the volume of the solid material.

## 3. What is the significance of the number of electrons and holes in a material?

The number of electrons and holes in a material determines its electrical conductivity and optical properties. The movement and interaction of electrons and holes play a crucial role in the behavior of semiconductors and other electronic devices.

## 4. How do the number of electrons and holes affect the Fermi Energy level?

The number of electrons and holes in a material can affect the Fermi Energy level by shifting it up or down. An increase in the number of electrons will shift the Fermi Energy level up, while an increase in the number of holes will shift it down.

## 5. Can the number of electrons and holes be controlled in a material?

Yes, the number of electrons and holes can be controlled in a material through various methods such as doping or applying an external electric field. By controlling the number of electrons and holes, the Fermi Energy level can also be manipulated, leading to changes in the material's properties.

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