- Fermi Dirac Distribution Function Definition: The Fermi Dirac distribution function describes the probability that a fermion, such as an electron, will occupy a particular energy level at a given temperature.
- Material Conductivity: This function is essential in electronics for understanding how many free electrons are available to conduct electricity in a material
- Energy Band Theory: The distribution function ties into energy band theory, helping to explain the concentration of electrons in the conduction band.
- Fermi Level: The Fermi level is defined as the energy level where the probability of finding an electron is 50%.
- Temperature Effects: The Fermi-Dirac distribution function shows how electron energy states change with temperature, affecting the material’s conductive properties.
Distribution functions are nothing but the probability density functions used to describe the probability with which a particular particle can occupy a particular energy level. When we speak of Fermi-Dirac distribution function, we are particularly interested in knowing the chance by which we can find a fermion in a particular energy state of an atom (more information on this can be found in the article “Atomic Energy States”). Here, by fermions, we mean the electrons of an atom which are the particles with ½ spin, bound to Pauli exclusion principle.
Necessity of Fermi Dirac Distribution Function
In electronics, material conductivity is crucial. This property depends on the number of free electrons available to conduct electricity.
As per energy band theory (refer to the article “Energy Bands in Crystals” for more information), these are the number of electrons which constitute the conduction band of the material considered. Thus inorder to have an idea over the conduction mechanism, it is necessary to know the concentration of the carriers in the conduction band.
Fermi Dirac Distribution Expression
Mathematically the probability of finding an electron in the energy state E at the temperature T is expressed as
Where,
is the Boltzmann constant
T is the absolute temperature
Ef is the Fermi level or the Fermi energy
Now, let us try to understand the meaning of Fermi level. In order to accomplish this, put
in equation (1). By doing so, we get,
This means the Fermi level is the level at which one can expect the electron to be present exactly 50% of the time.
Fermi Level in Semiconductors
Intrinsic semiconductors are the pure semiconductors which have no impurities in them. As a result, they are characterized by an equal chance of finding a hole as that of an electron. This inturn implies that they have the Fermi-level exactly in between the conduction and the valence bands as shown by Figure 1a.
In n-type semiconductors, there are more electrons than holes. Therefore, the Fermi level is closer to the conduction band, indicating a higher probability of finding electrons near it.
Following on the same grounds, one can expect the Fermi-level in the case of p-type semiconductors to be present near the valence band (Figure 1c). This is because, these materials lack electrons i.e. they have more number of holes which makes the probability of finding a hole in the valence band more in comparison to that of finding an electron in the conduction band.
Effect of temperature on Fermi-Dirac Distribution Function
At T = 0 K, the electrons will have low energy and thus occupy lower energy states. The highest energy state among these occupied states is referred to as Fermi-level. This inturn means that no energy states which lie above the Fermi-level are occupied by electrons. Thus we have a step function defining the Fermi-Dirac distribution function as shown by the black curve in Figure 2.
As temperature rises, electrons gain energy and can move to the conduction band. At higher temperatures, the distinction between occupied and unoccupied states becomes less clear, as shown in the blue and red curves in Figure 2.
