## Equilibrium Carrier Concentrations

Once we know how to determine the carrier distribution, we can find the carrier concentration by integrating over all energies:

Electron concentration:

Hole concentration:

After a lot of words and math we derive simple equations we can understand and use:

Electron concentration:

Hole concentration:

and finally the n_{o}p_{o} product relationship:

These equations are only valid when the semiconductor is in equilibrium and nondegenerate . Another way to say a semiconductor is nondegenerate is that the Fermi level, E_{F} , is more than 3kT from any of the states for which we are counting electrons. The n_{o}p_{o} product relationship is one of the most useful equations because once you know one of the carrier concentrations (using the equations for n_{o} or p_{o}), the other can be easily calculated.

We typically deal with uniformly doped semiconductors and if they are at room temperature, we also assume total ionization of the dopant atoms. With these assumptions we can use the charge neutrality relationship and the n_{o}p_{o} product relationship from above to derive equations for n_{o} and p_{o} that take into account the doping concentrations:

Charge neutrality relationship: p_{o} – n_{o} + N_{D} – N_{A} = 0

Electron concentration:

Hole concentration:

These equations can be simplified under a number of situations. Below are the most common:

1. When a semiconductor is not doped, N_{A} = 0 and N_{D} = 0, the semiconductor is intrinsic and n_{o} = p_{o} = n_{i}. This also occurs when N_{A} and N_{D} are approximately equal, or n_{i} >> |N_{D} – N_{A}|.

2. The equations for the carrier concentrations for a *p*-type semiconductor, N_{A} >> n_{i} and N_{D} = 0, can be simplified. Since N_{A} >> n_{i} , we can neglect n_{i} in the equation for p_{o} and obtain the carrier concentrations using the following equations:

3. The equations for the carrier concentrations for an *n*-type semiconductor, N_{D} >> n_{i} and N_{A} = 0, can be simplified. Since N_{D} >> n_{i} , we can neglect n_{i} in the equation for no and obtain the carrier concentrations using the following equations:

4. When a *p*-type semiconductor is compensated, doped with both acceptors and donors (N_{A} – N_{D} >> n_{i} and N_{D} is nonzero), the equations may be simplified similarly to Case 2 because we can still neglect n_{i} in the equation for p_{o}. The n_{o}p_{o} product relationship can then be used to solve for the electron concentration:

5. When an *n*-type semiconductor is compensated, doped with both acceptors and donors (N_{D} – N_{A} >> n_{i} and N_{A} is nonzero), the equations may be simplified similarly to Case 3 because we can still neglect n_{i} in the equation for n_{o}. The n_{o}p_{o} product relationship can then be used to solve for the hole concentration:

6. If the doping concentration, or the difference in doping concentrations if the semiconductor is compensated, is comparable to n_{i} , we cannot simplify the equations. The full expression must be used.