Charge Flow in Semiconductors
Charge flow in a semiconductor is characterized by the movement of electrons and holes. Considering that the density and availability of electrons and holes in a material is determined by the valence and conduction bands of that material, it follows that for different materials, there will be different densities of electrons and holes. The electron and hole density will determine the current throughput in the semiconductor, which makes it useful to map out the density of holes and electrons in a semiconductor.
Density of States
The density of electrons and holes is related to the density of states function and the Fermi distribution function. States are the formations of electrons and holes that can be formed in a semiconductor. A density of states is the amount of possible formations that can exist in a semiconductor. The Fermi-Dirac probability function is used for determining the the density of quantum states. The following formula determines the most probable formation distribution or state. By varying Ni (number of particles) along energy levels, the most probable state can be found, while gi refers to remaining particle positions in the distribution.
Density of States Calculation using ATLAS
By integration of Fermi-Dirac statistics for the density of states in the conduction and valence bands arises the formulae for electron and hole concentration in a semiconductor:
where Nc and Nv are the effective density of states for the conduction bands and valence bands, which are characteristics of a chosen material. If using a program such as ATLAS, the material selection will contain parameters NC300 and NV300.
Charge Carrier Density
Charge carriers simply refer to electrons and holes, which both contribute to the flow of charge in a semiconductor. The electron distribution in the conduction band is given by the density of quantum states multiplied by the probability (Fermi-Dirac probability function) that a state is occupied by an electron.
Conduction Band Electron Distribution:
The distribution of holes in the valence band is the density of quantum states in the valence band multiplied by the probability that a state is not occupied by an electron:
An intrinsic semiconductor maintains the same concentration of electrons in the conduction band as holes in the valence band. Where n is the electron concentration and p is the hole concentration, the following formulae apply:
The overall intrinsic carrier concentration is:
Eg is the band gap energy, which is equal to the difference of the energy is the conduction band and the energy in the valence band. Eg = Ec – Ev.
Electron and Hole concentrations expressed in terms of the intrinsic carrier concentration, where Ψ is the intrinsic potential and φ is the potential corresponding to the Fermi level (Ef = qφ):
Donor Atoms Effect on Distribution of Electrons and Holes (Extrinsic Semiconductor)
Adding donor or acceptor impurity atoms to a semiconductor will change the distribution of electrons and holes in the material. The Fermi energy will change as dopant atoms are added. If the density of holes is greater than the density of electrons, the semiconductor is a p-type and when the density of electrons is greater than the density of holes, the semiconductor is n-type (see Density of States formulas above).