Calculate the room-temperature electrical conductivity of silicon that has been doped with \(5 \times 10^{22} \mathrm{~m}^{-3}\) of boron atoms.

Doping a semiconductor means intentionally adding impurities to an intrinsic (pure) semiconductor material to change its electrical properties. In the creation of an n-type semiconductor, donor atoms such as phosphorus or arsenic are added to the silicon lattice. These atoms have five valence electrons, one more than silicon's four, and this extra electron becomes a free charge carrier, ready to conduct electricity. As a result of doping, the electrical conductivity of the silicon increases due to the higher number of free electrons, which essentially serve as the charge carriers in the n-type material.

One interesting point to note about n-type doping is that, although extra electrons are introduced into the system, only a small fraction may become free charge carriers at any given time. This is largely dependent on the temperature and the energy levels of the dopant atoms. The added electron from each donor atom occupies an energy level very close to the conduction band, which allows it to be excited into the conduction band relatively easily, contributing to the ability of the semiconductor to conduct current.

Charge carrier concentration is the number of charge carriers per unit volume in a material. In semiconductors, these are typically electrons and holes. The concentration of charge carriers plays a crucial role in determining a semiconductor's conductivity. In n-type semiconductors, electrons are the majority carriers, and their concentration is roughly equal to the concentration of donor atoms, assuming each dopant atom releases one electron into the conduction band.

In the example from our exercise, the doping concentration is given as the number of donor atoms per cubic meter, symbolized by 'n'. This directly influences the number of free electrons available to conduct electricity through the material: the higher the doping concentration, the higher the charge carrier concentration, and, hence, the greater the electrical conductivity, up to a certain limit beyond which other effects might come into play, such as electron scattering.

Electron mobility is a measure of how quickly an electron can move through a semiconductor when an electric field is applied. It is influenced by factors such as the temperature, the purity of the semiconductor, and the lattice structure. High electron mobility indicates that electrons can move more easily through the material, which contributes to higher electrical conductivity. The mobility of electrons is often different in intrinsic and doped semiconductors due to the scattering by impurities and imperfections introduced during the doping process.

In n-type semiconductors like the one described in our exercise, dopants increase the number of free electrons but can also lead to increased scattering, potentially reducing mobility. Nevertheless, in many cases, the increase in charge carrier concentration far outweighs the reduction in mobility, leading to a net increase in conductivity. Understanding electron mobility is crucial when designing electronic devices as it directly affects how well a semiconductor can conduct electric current and how quickly it can operate.