GITAM, Department of Engineering Physics

Electrons and Holes

In an intrinsic semiconductor like silicon at temperatures above absolute zero, there will be some electrons which are excited across the band gap into the conduction band and which can produce current. When the electron in pure silicon crosses the gap, it leaves behind an electron vacancy or "hole" in the regular silicon lattice. Under the influence of an external voltage, both the electron and the hole can move across the material. In an n-type semiconductor, the dopant contributes extra electrons, dramatically increasing the conductivity. In a p-type semiconductor, the dopant produces extra vacancies or holes, which likewise increase the conductivity. It is however the behavior of the p-n junction which is the key to the enormous variety of solid-state electronic devices.

Semiconductor Current

Both electrons and holes contribute to current flow in an intrinsic semiconductor. That is, the electrons which have been freed from their lattice positions into the conduction band can move through the material.

In addition, other electrons can hop between lattice positions to fill the vacancies left by the freed electrons. This additional mechanism is called hole conduction because it is as if the holes are migrating across the material in the direction opposite to the free electron movement.

Carrier Concentration of an Intrinsic Semiconductor

The conduction electron population for a semiconductor is calculated by multiplying the density of conduction electron states r(E) times the Fermi function f(E). The number of conduction electrons as a function of energy is then given by

This can be simplified by noting that for the energies of the conduction band, E-E_F>>1, so the 1 in the denominator of the Fermi function becomes insignificant. I.e., the tail of the function which extends into the conduction band is so far out that it can be approximated by the Boltzmann function. Using the fact that

E_F = E_gap/2

The population density can then be written

The total number of electrons in the conduction band, N_cb, can then be obtained by integrating the above function from the bottom of the conduction band upward. For all practical purposes, the upper limit of the integral can be taken to be infinity since by the time we reach the top of the conduction band, the integrand will be essentially zero.