How to calculate carriers concentration in a semiconductor using Fermi Dirac's distribution function:
We have calculated the carrier concentration using the mass action law (Electrons and Holes Concentration In Semiconductors).
Before reading it is better to read about the Fermi Level concept, which is here.
Explanation of Fermi Level In Semiconductors
Key Questions
Now there is another way of calculating carrier concentration by using Fermi Dirac's function and density of states.
Before reading it is better to read about the Fermi Level concept, which is here.
Explanation of Fermi Level In Semiconductors
Key Questions
- Evaluate equation for carrier concentration in semiconductors using Fermi Dirac's function
Now there is another way of calculating carrier concentration by using Fermi Dirac's function and density of states.
Where N(E)dE is the density of states in the energy range dE. And can be calculated by quantum mechanics and the Pauli exclusion rule.
A closer examination of Fermi Dirac's function f(E) shows that Fermi function becomes extremely small for larger energies E. That is energy bands that are present far above the conduction band. and hence the product of (E)N(E) decreases for energy levels above Ec. So mostly electrons occupy energy states at the edge of the conduction band at equilibrium. And only a few electrons occupy energy states far above the conduction band. Or the probability of occupation of energy states far above the conduction band is very low.
The result of integration is obvious. Neglecting the energy states far above conduction band (because the probability of finding electrons above Ec is very low, as discussed above). For calculation of electrons concentration at equilibrium we only consider energy states at the bottom of the conduction band where the probability of finding an electron is maximum. Thus the electrons concentration at equilibrium is simply
no=Ncf(Ec)
f(E)=1/(1+e(E-Ef)/kT)
f(E)=e-(Ec-Ef)/kT… By approximation
no=Nce-(Ec-Ef)/kT
Similarly, most holes occupy energy states near the top of the valence band. Or the probability of finding an empty state below the valence band decreases. Or (1-f(E)) decreases rapidly below Ev. The product (1-f(E))NE decreases for energy levels below Ev. We discussed the symmetry property in between f(E) and 1-f(E), and Ev and Ec. So the holes concentration in the valence band at equilibrium is
po=Nv(1-f(E))
1-f(Ev)=1- 1/[1+e(Ev-Ef)/kT]
1-f(Ev)=e-(Ef-Ev)/kT… By approximation
po=Nve-(Ef-Ev)/kT
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