The generation of free carriers requires not only the presence of impurities, but also that the impurities give off electrons to the conduction band in which case they are called donors or that they give off holes to the valence band in which case they are called acceptors (since they effectively accept an electron from the filled valence band).
A semiconductor doped with impurities which are ionized (meaning that the impurity atoms either have donated or accepted an electron) will therefore contain free carriers. Shallow impurities are impurities which require little energy - typically around the thermal energy or less - to ionize. Deep impurities require energies larger than the thermal energy to ionize so that only a fraction of the impurities present in the semiconductor contribute to free carriers. Deep impurities which are more than five times the thermal energy away from either band edge are very unlikely to ionize. Such impurities can be effective recombination centers in which electrons and holes fall and annihilate each other. Such deep impurities are also called traps.
Ionized donors provide free electrons in a semiconductor which is then called n-type, while ionized acceptors provide free holes in a semiconductor which we refer to as being a p-type semiconductor.
Shallow impurities readily ionize so that the free carrier density equals the impurity concentration. For donors this implies that the electron density equals the donor concentration, or:
While for accepters the hole density equals the acceptor concentration, or:
If a semiconductor contains both shallow donors and shallow acceptors it is called compensated since equal amounts of donors and acceptors compensate each other, yielding no free carriers. The presence of shallow donors and shallow acceptors in a semiconductor cause the electrons given off by the donor to fall into the acceptor state which ionizes the acceptor without yielding a free electron or hole. The resulting carrier density in compensated material which contains both shallow donors and shallow acceptors is approximately equal to the difference between the donor and acceptor concentration if the donor concentration is larger, yielding n-type material, or:
If the acceptor concentration is larger than the donor concentration, the hole density of the resulting p-type material equals the difference between the acceptor and donor concentration, or:
Solving this quadratic equation yields a solution for the electron density, namely:
The same derivation can be repeated for holes, rather than electrons and yields:
The above expressions yield the free carrier densities for compensated semiconductors assuming that all donors and acceptors are ionized.
From the carrier densities one then obtains the Fermi energies using:
The Fermi energies in n-type and p-type silicon at 300 K as a function of doping density is shown in the figure below:
The analysis still begins by assuming that there is no net charge in the semiconductor. This condition is refered to as charge neutrality. This also means that the total density of positively charged particles (holes and ionized donors) must equals the total density of negatively charged particles (electrons and ionized acceptors) yielding:
The electron and hole densities are then written as a function of the Fermi energy using the expressions for non-degenerate semiconductors, while the ionized impurity densities equal the impurity density multiplied with the probability of occupancy for the acceptors and one minus the probability of occupancy for the donors, yielding:
The figure shows the positive and negative charge densities as well as the electron and hole densities as a function of the Fermi energy for silicon containing 1016 donors and 1014 acceptors. The arrow indicates the intersection of both curves. This intersection provides the Fermi energy and the electron density in the presence of both donors and acceptors.
while at low temperatures the carrier density is dominated by the ionization of the donors. For non-degenerate semiconductors the density at low temperature equals:
The temperature dependence is related to an activation energy by fitting the carrier density versus 1/T on a semi-logarithmic scale to a straight line of the form n(T) = C x exp(-EA/kT), where C is a constant. At high temperatures this activation energy equals half the bandgap energy or EA = Eg/2.
The temperature dependence at low temperatures is somewhat more complex as it depends on whether or not the material is compensated. The figure above was calculated for silicon containing both donors and acceptors. At a 70 K the electron density is below the donor density but still larger than the acceptor density. Under such conditions the activation energy equals half of the ionization energy of the donors or EA = (EC - ED)/2. At lower temperatures where the electron density is lower than the acceptor density, the activation energy equals the ionization energy or EA = (EC - ED). This behavior is explained by the fact that the Fermi energy in compensated material is fixed at the donor energy since the donors levels are always partially empty as electrons are removed from the donor atoms to fill the acceptor energy levels. If the acceptor density is smaller than the electron density, the Fermi energy does change with temperature and the activation energy approaches half of the ionization energy.
Lightly doped semiconductors suffer from freeze-out at relatively high temperature. Higher-doped semiconductor freeze-out at lower temperatures. Highly-doped semiconductors do not contain a single donor energy levels, but rather an impurity band which overlaps with the conduction or valence band. The overlap of the two bands results in free carriers even at zero Kelvin. Degenerately doping a semiconductor therefore eliminates freeze-out effects.