Doped Semiconductors
In practical applications it is common to modify the properties of the semiconductor to add impurity atoms or dopants in a controlled manner. Using silicon as an example (with a chemical valency of 4), impurity atoms such as phosphorous or arsenic (with a valency of 5), have the effect of adding extra electrons to the conduction band. Such impurities are called donors. Similarly impurities such as boron and gallium (with a valency of 3) can capture an electron from the valence band, creating holes. These impurities are called acceptors.
Consider the case of a donor atom. If the atom loses its additional electron, it has enough remaining electrons to fit into the lattice of the crystal. However the atom is now positively charged, and consequently a perturbing coulomb potential exists around it. Equation 3-37 can be used to compute the states that the additional electron can occupy within this potential. The problem becomes that of the hydrogen atom, except that the medium typically has a large relative dielectric permittivity (εr≈12 for silicon) and the electron has an effective mass m* that is typically less than the electron mass. The solutions to the corresponding Schrödinger equation are swollen orbits (typically 2.5–5.0 nm in radius) similar to that of a hydrogen atom. The reduced mass and larger dielectric constant cause a reduction in the effective ionization energy of the electron so that it is significantly less than that of a hydrogen atom — donor ionization levels are typically 10s of meV. The energy of the donor states is consequently just below that of the conduction band as shown in Figure 3-5. Similarly, the energy of the acceptor states lies just above the valence band (also in the figure). Figure 3-5 shows an n-type semiconductor. This is a material with significantly more donors than acceptors so that the majority carriers are electrons.
Figure 3-5: Density of states and Fermi function for an n-type semiconductor. The Fermi function is shown at zero temperature and at a finite temperature.
Since donor and acceptor states cannot hold two electrons (unlike states within the band), the carrier statistics are modified slightly. The occupancy of the states in equilibrium is determined from statistical mechanics as:
(3-41)
where gd is the donor degeneracy factor, which is 2 if the conduction band minimum is nondegenerate but which varies when the degeneracy of the donor levels is altered by the band structure having degenerate conduction bands. Similarly, the occupancy of the acceptor states is given by:
(3-42)
where ga is the acceptor degeneracy factor.
In equilibrium the value of the Fermi level is determined self-consistently from the charge neutrality condition, along with the requirement that the number of holes and electrons is determined from the Fermi function:
Here gc(E) is the density of states in the conduction band, gv(E) is the density of states in the valence band, and f(E) is the (equilibrium) Fermi function.