Fermi-Dirac Distribution Function:
This formula gives the probability that an energy state with energy E is occupied by an electron.
Where:
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E: energy level
EF: Fermi energy (Fermi level)
kB: Boltzmann constant
T: absolute temperature (in Kelvin)
1) At Absolute 0 (0 Kelvins):
 All energy states below EF are filled with electrons.
 All energy states above EF are empty.
 So, EF is the highest occupied energy level at 0 K.
2) At Temperatures above 0K:
 Some electrons gain thermal energy and jump to higher energy levels (above EF)
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Thus, f(E) gradually decreases instead of a sharp drop.
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At E=EF, f(E)=0.5, meaning that there is a 50% chance that the level is occupied.
Note:
f(E) = 1, 100% full, below EF (Fermi Level)
f(E) = 0, 0 empty, above EF (Fermi Level)
Carrier Concentration:
The number of charge carriers (electrons and holes) in intrinsic semiconductor is determined by
multiplying two things:
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N(E): DOS, number of available energy states at each energy level.
f(E): Fermi-Dirac distribution, probability that a given state is occupied by an electron.
Note: Because it’s intrinsic, the number of electrons n in the conduction band equals the
number of holes p in the valence band. n = p = ni
What is a hole:
The hole is the absence of an electron in a particular place in an atom.
When an electron is freed, it leaves an empty spot with a positive charge in the crystal lattice.
Intrinsic semi-conductors:
Undoped, thermal energy can free some electrons to form electron hole pairs some
conductivity. (carrier concentration e.g. 1.45 x 10^10/𝑐�𝑚�3)
ni = 1.45 x 10^10/𝑐�𝑚�3 means that, at 300 K(room temperature), in each cm^3 of silicon,
there are roughly 14.5 billions of free electrons and the same number of holes.
Doping (n-type); extrinsic semiconductors
An extrinsic semiconductor is one where impurity atoms are deliberately added to a pure
semiconductor (like silicon) to change its electrical properties.
In n-type doping, extra electrons are introduced by using donor atoms.
Fermi-Level
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A phosphorus atom (pink) is introduced into the silicon lattice.
Phosphorus belongs to group V, so it has 5 valence electrons.
Four of those electrons bond with surrounding silicon atoms, just like silicon would.
The 5th electron has no bond to fill — it becomes a free electron.
This free electron is easily available for conduction, greatly increasing conductivity
compared to intrinsic silicon.
The Fermi level moves closer to the conduction band. Conductivity increases dramatically.
p-type Doping:
In p-type doping, impurity atoms are added that create holes (missing electrons) in the crystal
lattice. These holes behave like positive charge carriers.
Fermi-Level
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A boron atom (blue) is introduced into the silicon lattice.
Boron is a group III element, so it has only 3 valence electrons.
It forms 3 covalent bonds with surrounding silicon atoms.
But the 4th bond is incomplete — there is a missing electron, called a hole (shown as an
empty circle).
This hole can move through the lattice: when a neighboring electron jumps to fill it, the
hole effectively shifts position.
Thus, the hole behaves as a mobile positive charge carrier.
The Fermi level moves closer to the valence band. Conductivity increases because holes allow
current to flow.
Useful equations for doping:
For n-type doping:
Rearranging:
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n = electron concentration.
𝑛�𝑖�= intrinsic carrier concentration.
kT = thermal energy.
EF−Ei= how far the Fermi level has shifted upward relative to the intrinsic level.
For p-type doping:
Rearranging:
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p = hole concentration.
Law of Mass Action:
Even when doping heavily, the product of electron (n) and hole (p) concentrations always equals
ni^2.
Example: In n-type, n≫p, but the tiny hole concentration is still determined by this law.
In summary:
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N-type: EFE_FEF shifts upwards → more electrons in conduction band.
P-type: EFE_FEF shifts downwards → more holes in valence band.
Carrier concentrations obey the exponential relation with EF−Ei.
Optoelectronic devices: