# Equilibrium Carrier Concentration

### Overview

1. Semiconductors contain majority and minority carriers. The more abundant charge carriers are the majority carriers; the less abundant are the minority carriers.
2. The equilibrium carrier concentration can be increased through doping.
3. The total number of carriers in the conduction and valence band is called the equilibrium carrier concentration.
4. The product of minority and majority charge carriers is a constant.

The number of carriers in the conduction and valence band with no externally applied bias is called the equilibrium carrier concentration. For majority carriers, the equilibrium carrier concentration is equal to the intrinsic carrier concentration plus the number of free carriers added by doping the semiconductor. Under most conditions, the doping of the semiconductor is several orders of magnitude greater than the intrinsic carrier concentration, such that the number of majority carriers is approximately equal to the doping.

At equilibrium, the product of the majority and minority carrier concentration is a constant, and this is mathematically expressed by the Law of Mass Action.

$$n_{0} p_{0}=n_{i}^{2}$$

where ni is the intrinsic carrier concentration and n0 and p0 are the electron and hole equilibrium carrier concentrations.

Using the Law of Mass Action above, the majority and minority carrier concentrations are given as:

$$\text { n-type: } n_{0}=N_{D}, p_{0}=\frac{n_{i}^{2}}{N_{D}}$$

$$\text { p-type: } p_{0}=N_{A}, n_{0}=\frac{n_{i}^{2}}{N_{A}}$$

where ND is the concentration of donor atoms and NA is the concentration of acceptor atoms.

The above equations show that the number of minority carriers decreases as the doping level increases. For example, in n-type material, some of the extra electrons added by doping the material will occupy the empty spots (i.e., holes) in the valence band, thus lowering the number of holes.

Equilibrium carrier concentration for low and high doping, showing that as the doping increases, the minority carrier concentration decreases.