The Law of Mass Action

For n-type materials, the number of mobile electrons in the conduction band due to doping is approximately equal to the number of donor atoms. Similarly, in p-type materials, the number of mobile holes in the valence band due to doping is approximately equal to the number of acceptor atoms.

If we call n then number of mobile electrons in the conduction band and p the number of mobile holes in the valence band. In any semiconductor (doped or pure):

This is called the law of mass action. It states that the number of mobile holes, p, and the number of mobile electrons, n, multiplied together is equal to the charge carriers in an intrinsic semiconductor squared. In this way, the number of holes and electrons in the doped material will not exceed the number of holes and electrons in the intrinsic material. The reason  is used is because in an intrinsic semiconductor, the number of holes will be equal to the number of electrons:

Therefore:

Intrinsic semiconductors have a low number of mobile holes and mobile electrons, n-type semiconductors have a large number of mobile electrons but low number of mobile holes and p-type semiconductors have a large number of mobile holes but low number of mobile electrons. This can be visualised in the table below:

Charge Carrier Density in:

Intrinsic Semiconductor

N-Type Semiconductor

P-Type Semiconductor

Density of mobile electrons in Conduction Band, 

Low

High

Very Low

Density of mobile holes in Valence Band, 

Low

Very Low

High

Hence we can say that:

  • In an N-Type Semiconductor:
    • Mobile electrons in the Conduction Band are the Majority charge carriers.
    • Mobile holes in the Valence Band are the Minority charger carriers.
      •  n > p but 
  • And similarly, in a P-Type Semiconductor:
    • Mobile holes in the Valence Band are the Majority charge carriers.
    • Mobile electrons in the Conduction Band are the Minority charger carriers.
      • n < p but 

Note that  is different for each material but remains the same for everything in this tutorial. The most important thing to remember is that  and since n and p are the same, we just call it .

Knowing all this we can now move on to calculate the densities of the mobile holes and mobiles electrons in extrinsic semiconductors.

Majority and Minority Charge Carriers

N-Type Semiconductors

The total density of mobile electrons in the conduction band is defined by:

(The overall charge is Neutral)

  •  is the concentration of mobile electrons
  •  is the concentration of Donor impurities
  •  is the concentration of mobile holes – this is the minority carrier density

The Law of Mass Action tells us that

However  is usually a lot bigger than  for an n-type semiconductor and if the donors are assumed to be fully ionised [1] then:


and so

P-Type Semiconductors

The total density of mobile holes in the valence band is defined by:

  (The overall charge is Neutral)

  •  is the concentration of mobile holes
  •  is the concentration of Acceptor impurities
  •  is the concentration of mobile electrons – this is the minority carrier density

The Law of Mass Action tells us that

Again  is usually a lot bigger than  for a p-type semiconductor and if the acceptors are assumed to be fully ionised [1] then:


and so

[1] If temperature is not too low (below 100K for Silicon) and  or  is not too high (less than )

Summary

So to summarise these equations:

Material

Majority Carrier Density

Minority Carrier Density

N-Type

 [2]

 
P-Type

 [2]

 

[2] Since the minority carrier density is much smaller than the majority carrier density we can ignore it.

Move onto the next tutorial to learn about the Electric Field that is setup in the semiconductor.