Fermi Level in Semiconductors:
The concept of Fermi level is of cardinal importance in semiconductor physics.
The topic is not so easy to understand and explain. After reading again and again from different books and other resources I was able to understand and write… Now I aim to share my knowledge. I tried to make it simple. The concepts of Fermi Dirac's Function will be useful in deriving the equation for carrier concentration. Which is explained later. First I will explain all the terms used in this context.
Forbidden gap/Forbidden energy band: In insulators and semiconductors, the energy bands within which no electrons may not be located. In metals, there are no such energy bands.
Valence band: It is simply the outermost orbit of an atom of any element.
Conduction band: The band of orbitals that is higher in energy.
Fermi level: It is the imaginary energy level that lies at the top of the available electrons energy levels at absolute zero. The probability of finding an occupied space at the fermi level is ½.
Fermi energy: The energy at the position of the Fermi level.
Bandgap: The energy difference between the top of the valence band and the bottom of the conduction band.
Fermi Dirac statistics: Electrons in solids follows Fermi Dirac Distribution function f(E). It gives the probability that an energy state with energy E will be occupied by the electron.
f(E)=1/(1+e(E-Ef)/kT)
The density of states N(E): The number of electrons state per unit energy per unit volume.
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| Simplified energy band diagram |
A Closer examination of Fermi Distribution Function:
The Fermi distribution function can be used to calculate the concentration of electrons and holes in a semiconductor if the density of states in the valence and conduction band is known.
f(E)=1/(1+e(E-Ef)/kT)
Where,
Ef is fermi energy or Fermi level where the probability of occupancy of an electron is 50%
T is the temperature in Kelvin
K is a Boltzmann constant 8.62*10-5 eV/K
At T=0 … f(E)=1
At E<Ef … exponent becomes negative f(E)=1
At E>Ef … exponent becomes positive f(E)=0
It explains that T=0 all the electrons are below Fermi level as the temperature increases the probability of finding an electron at a higher energy level also increases.
At higher temperatures, the probability of occupation of energy states near the conduction band is higher and the energy states near the valence band are most likely to be emptied.
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| Graphical representation of Fermi Derac's Function |
The symmetry of f(E) around Ef
f(E)=Probability of filled states above Ef
1-f(E)=Probability of filled states below Ef
f(Ef+∆E)=1-f(Ef-∆E)
∆E any state above or below Fermi level has the same probability to be filled.
Where N(E)dE is the density of states
According to the above equation, total electron concentration is the integral over the entire conduction band. The probability values (Ev and Ec) for intrinsic material are quite small. In Si at 300K n=p=1010cm-3. The density of available states is much higher at this temperature like 1019. The probability of occupancy f(E) for an individual state in the conduction band and the hole probability (1-f(E)) for a state in the valence band is quite small. Because of the large density of states in each band, small changes in f(E) can result in significant changes in carrier concentration.
Fermi Level In Semiconductors:
Intrinsic Semiconductors: As we know that the concentration of holes in the valence band (p) is equal to the concentration of electrons in the conduction band (n). n=p it implies that the chances of finding an electron near the conduction band edge are equal to the chances of finding a hole near the valence band edge. (i.e. symmetry property which is explained above). f(Ec)=1-f(Ev)
Thus Fermi level lies at the middle of the bandgap. Intrinsic Fermi level With is shown by:
Ei=Ec-Eg/2=Ev+Eg/2
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Fermi level position in intrinsic semiconductors |
Fermi Level In Extrinsic/ Doped Semiconductors:
How does the Fermi Level shift as doping concentration increases?
n-type Semiconductors: There are more electrons in the conduction band than there are holes in the valence band. Also, the probability of finding an electron near the conduction band edge is larger than the probability of finding a hole near the valence band edge. Thus Fermi level for n-type Semiconductors lies near the conduction band.
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| Fermi level position in n-type semiconductors |
p-type semiconductors:
There are more holes in the valence band than there are electrons in the conduction band. Also, the probability of finding a hole near the valence band edge is larger than the probability of finding an electron near the conduction band edge.
There are more holes in the valence band than there are electrons in the conduction band. Also, the probability of finding a hole near the valence band edge is larger than the probability of finding an electron near the conduction band edge.
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| Fermi level position in p-type material |
I hope you find this post helpful. The Fermi Dirac's function and Fermi level concept will helpful for the evaluation of the equation for carriers concentration in semiconductors. The link to the post is given below.
to evaluate equation for carriers concentration using Fermi Dirac's function
Reference: Solid State Electronic Devices 6th edition Ben G Streetman and Sanjay Kumar Banerjee
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