Fraction of occupied states (Fermi-Dirac distribution + DOS)

In summary, the Fermi-Dirac distribution and density of states of a semiconductor are important concepts in understanding the behavior of electrons in a material. To compute the total number of states in a given energy range, we can simply integrate the density of states function over that range. To determine the percentage of these states that have electrons in them, we need to consider both the density of states and the Fermi-Dirac distribution. By combining these two factors, we can calculate the number of electrons per m3 that have energies in a specific range.
  • #1
su3liminal1
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Homework Statement
Find the percentage of these states that have electrons in them, assuming the number of electrons above Ec+2kT is negligible.
Relevant Equations
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I just want to clear some confusion I am having with the Fermi-Dirac distribution & density of states (DOS) of a semiconductor, which are given by

1.png


Say we have a piece of Silicon in equilibrium and its Fermi level lies 0.25 eV below the conduction band edge, i.e. Ec - EF = 0.25 eV. Let us say we want to compute two things:
(1) Total number of states in the range Ec ≤ E ≤ E+ 2kBT.
(2) The percentage of these states that have electrons in them, assuming the number of electrons above Ec+2kBT is negligible.

For (1), it is straight forward: we just integrate the density of states function in the conduction band, gc(E) over the indicated range:
1568765673022.png

At room temperature and using an effective mass of silicon is, say, mn*=1.09m0. This yields
1568765694868.png


For (2), I know that the Fermi-Dirac distribution in this context represents the the probability of an electron occupying a state at energy E, which can also be interpreted as the ratio of filled state to total states at the energy. But I am really not sure what do here. Do I compute the difference of Fermi-Dirac distributions in that range, or do I integrate, or both are wrong?
 

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  • #2
Welcome to PF.

From part (1) you know the number of states (per m3) in the energy range ##E_c## to ##E_c + 2k_BT##. I think you can answer part (2) if you know how many electrons (per m3) have energies in this range.

Suppose you consider a small range of energies ##E## to ##E+dE##. Can you see how to get the number of electrons per m3 that have energies in this range? This will involve both ##g_c(E)## and ##f(E)##.
 

FAQ: Fraction of occupied states (Fermi-Dirac distribution + DOS)

1. What is the Fermi-Dirac distribution?

The Fermi-Dirac distribution is a statistical distribution that describes the probability of a given energy level being occupied by a fermion (particle with half-integer spin) in a system at thermal equilibrium. It takes into account the Pauli exclusion principle, which states that no two fermions can occupy the same energy state simultaneously.

2. How is the fraction of occupied states calculated using the Fermi-Dirac distribution?

The fraction of occupied states, also known as the Fermi level, is calculated by integrating the Fermi-Dirac distribution function over all energy levels. This gives the total number of occupied states divided by the total number of available states, which represents the probability of a state being occupied by a fermion at a given temperature.

3. What is the relationship between the fraction of occupied states and the density of states (DOS)?

The density of states (DOS) is a measure of the number of energy states per unit energy interval in a system. The fraction of occupied states is directly related to the DOS, as it represents the proportion of all available states that are occupied by fermions at a given energy level. In other words, the DOS provides information about the distribution of occupied states in a system.

4. How does temperature affect the fraction of occupied states?

As temperature increases, the Fermi-Dirac distribution function becomes more spread out, resulting in a higher fraction of occupied states. This is because at higher temperatures, fermions have more thermal energy and are more likely to occupy higher energy states. At absolute zero temperature, the fraction of occupied states is zero, as all fermions are in their lowest available energy states.

5. What are some real-world applications of the Fermi-Dirac distribution and DOS?

The Fermi-Dirac distribution and DOS are widely used in condensed matter physics, materials science, and electronic device engineering. They are essential in understanding the behavior of electrons in semiconductors, metals, and insulators, and play a crucial role in the design of electronic devices such as transistors and solar cells. They are also used in quantum mechanics and statistical mechanics to describe the behavior of fermions in various systems.

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