I know that all the different carrier concentrations in the semiconductor can be confusing so let
me provide more info and some exercises that will hopefully help to clarify.
INTRINSIC CARRIER CONCENTRATION
Let’s say we have a sample of pure, crystalline Si that is [1 cm x 1 cm x 5 cm] and I have stored
it at 0K. If I apply a small voltage no current will flow because all energy states in the valence
band are full and all states in the conduction band are empty.
- - - -
Conduction Band
eeee
+ + +
hhh h
+
Valence Band
To energize electrons into the conduction band as shown above, we will heat up the sample.
[Kelvin is Celsius + 273] As the sample heats up more and more electrons get enough energy to
move to the conduction band and each of these electrons that move to the conduction band leave
behind a hole in the valence band. As you see below, between 27C and 600C the concentration
of electrons and holes increases by seven orders of magnitude.
Data for Silicon
Degrees Kelvin Degrees Celsius ni [per cm3]
300
27 [room temp] 1.45 x 1010
473
200
~1014
673
400
~5 x 1016
873
600
~5 x 1017
pi [per cm3]
1.45 x 1010
~1014
~5 x 1016
~5 x 1017
The concentration of intrinsic electrons and holes is not static at any temperature – there is
always an equilibrium between electrons moving to the conduction band and others returning to
the valence band. So the numbers in the table above are average values that result from the
steady-state thermodynamic equilibrium. These numbers are determined empirically using
conductivity measurements.
EXTRINSIC CARRIER CONCENTRATIONS
The mechanisms that create intrinsic carriers go on all the time if the sample is above 0K. To get
EXTRINSIC carriers we have to add an element to the Silicon and we call that element a dopant.
The dopants typically come from either column III of the periodic table [called acceptors] or
column V [called donors]. If we add some column III dopant [e.g. Boron] to the Silicon the
dopant will replace Si at some of the spots in the crystal. Since there are only 3 electrons in the
outer shell, the acceptor dopant will grab an electron from the valence band and become a
negatively charged ion that is stuck in the crystal. The result is an extra hole in the valence band
and one negatively charged ion in the Si crystal for every column III atom added to the Si. This
is p-type material.
If we add some column V dopant [e.g. Arsenic] to the Silicon the dopant will replace Si at some
of the spots in the crystal. Since there are 5 electrons in the outer shell, the donor dopant will
easily give away an electron from its outer shell and become a positively charged ion that is
stuck in the crystal. The result is an extra electron in the conduction band and one positively
charged ion in the Si crystal for every column V atom added to the Si. This is n-type material.
Thermodynamic equilibrium gives us the following relationship:
𝑛𝑜 𝑝𝑜 = 𝑛𝑖2
Where no and po are the equilibrium concentrations of electrons and holes after the addition of
dopant and ni is the intrinsic concentration of electrons [which is equal to the intrinsic
concentration of holes].
EXCESS CARRIER CONCENTRATION
Once the sample of Si is prepared, at a stable, known temperature and has been doped with the
correct concentration of impurity [dopant] atoms, we can calculate the concentrations of
equilibrium carriers.
Nd = concentration of donor atoms
Na = concentration of acceptor atoms
It is typical for the dopant concentration to be orders of magnitude greater than the intrinsic
concentration. So we can write:
𝑛𝑜 ≅ 𝑁𝑑 for n-type Silicon and 𝑁𝑑 𝑝𝑜 = 𝑛𝑖2
𝑝𝑜 ≅ 𝑁𝑎 for p-type Silicon and 𝑁𝑎 𝑛𝑜 = 𝑛𝑖2
The above equations represent the equilibrium conditions. When the Si is used in a device such
as a diode or a transistor, current driven by an applied energy source creates an additional
quantity of electrons and/or holes by energizing more electrons from valence to conduction band
– or – by pushing carriers from one region to another in the device. This additional quantity of
carriers, which comprises the current, will change with the increase or decrease of the applied
energy. If the applied energy is turned off, the additional carriers will diffuse throughout the
circuit until the equilibrium situation returns. Some of the sources of applied energy are: light
[photodiode], heat [usually unwanted], voltage [diode, BJT], etc.