Estimate the electrical conductivity, at \(135^{\circ} \mathrm{C}\) of silicon that has been doped with \(10^{24} \mathrm{m}\) of aluminum atoms

Semiconductors play a critical role in modern electronics, and their conductivity can be tailored through a process called doping. The incorporation of specific impurities into the semiconductor material determines its conductivity characteristics. P-type semiconductors, one of the two main types of doped semiconductors, result from introducing acceptor impurities, like aluminum, into the silicon crystal lattice.

P-type semiconductors have an abundance of 'holes,' which are essentially the absence of electrons that act as positive charge carriers. When doped with elements such as aluminum, which has one less valence electron than silicon, each dopant atom can accept an electron from the silicon lattice, thereby creating a hole. These holes can move through the lattice and carry charge, which contributes to the material's electrical conductivity. The more the dopant atoms, the higher the number of available holes, resulting in increased conductivity.

Hole mobility is a property that describes how quickly 'holes' can move through a semiconductor when an electric field is applied. It is a key factor in the conductivity of p-type semiconductors, as it affects how easily the holes can flow and, thus, how well the material conducts electricity. Mobility can vary with temperature; as the temperature increases, the mobility tends to decrease due to more frequent collisions of charge carriers with lattice vibrations (phonons).

In the example provided, the mobility of holes in p-type silicon needs to be corrected for the operating temperature, which is higher than room temperature. Using an empirical relationship, the mobility can be calculated for the specific temperature in question. This calculation is vital for predicting the material's behavior under various operating conditions and is a direct factor in determining the overall conductivity of the semiconductor.

Carrier concentration in semiconductors refers to the number of charge carriers available per unit volume to participate in the conduction process. In the case of p-type semiconductors, this would be the hole concentration. The carrier concentration is directly related to the amount of doping; the more dopant atoms introduced into the semiconductor, the higher the carrier concentration.

In the doping process described in the exercise, the introduction of a high concentration of aluminum atoms (\(10^{24} \text{m}^{-3}\)) significantly increases the total number of holes in the silicon lattice. By knowing this concentration and the mobility of these carriers, one can accurately determine the material's electrical conductivity using the formula provided. It is important to note that the electrical conductivity is proportional to the product of the carrier concentration and the mobility of those carriers. The carrier concentration is a vital parameter in designing semiconductor devices with desired electrical characteristics.