Germanium to which \(10^{24} \mathrm{~m}^{-3}\) As atoms have been added is an extrinsic semiconductor at room temperature, and virtually all the As atoms may be thought of as being ionized (i.e., one charge carrier exists for each As atom). (a) Is this material \(n\)-type or \(p\)-type? (b) Calculate the electrical conductivity of this material, assuming electron and hole mobilities of \(0.1\) and \(0.05 \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}\), respectively.

In the realm of semiconductor technology, an n-type semiconductor plays a pivotal role. This type of material is crafted by introducing pentavalent impurities—elements with five valence electrons—into a pure semiconductor. For instance, when arsenic (As) is added to germanium (Ge), each As atom brings an extra electron which is not required for bonding with the germanium atoms. This fifth electron is relatively free to move, thus significantly improving the material's conductivity.

One might wonder, does this make the semiconductor positively charged? The answer is no. While the extra electrons move freely, the arsenic atoms remain positively charged and fixed in the lattice, ensuring the overall electrical neutrality of the material. This abundance of mobile electrons defines an n-type semiconductor, making electron flow its primary means of electrical conduction.

This concept of electron donor impurities is fundamental for students to understand, as it lays the groundwork for all semiconductor-based electronics, including diodes and transistors, which are the building blocks of modern electronic devices.

When examining materials like those in the exercise problem, electrical conductivity is a measure of a material's ability to allow the flow of charge. This property hinges on the concentration of charge carriers—which can be electrons and holes—and their mobility, reflecting how swiftly these carriers can move through a material under the influence of an electric field.

For instance, in the exercise, the electrical conductivity ( a)) is found by capitalizing on the known quantities, including the charge of an electron, the concentration of electrons, and the electron mobility. It's essential to spotlight that for n-type semiconductors, the contribution of holes to conductivity is negligible due to their drastically lower concentration, which is why the term involving the hole mobility can be omitted in calculations.

Electrical conductivity is not just an abstract concept; it has practical implications. Higher conductivity implies better performance for electronic components, which is why materials with high electron mobility, like the n-type germanium, are sought after in the manufacturing of electronic devices.

Delving into the exercise, the notions of electron and hole mobilities can be quite fascinating. These terms describe the ease with which electrons and holes can navigate through a semiconductor. Electron mobility ( ) measures the velocity that an electron reaches under the action of an electric field, and similarly, hole mobility ( ) does the same for holes.

In n-type semiconductors, electrons are the majority charge carriers, and thus their mobility is key in determining the overall electrical conductivity of the material. On the flip side, holes act as the majority carriers in p-type semiconductors. The mobilities of these carriers are influenced by various factors such as impurity concentrations, lattice imperfections, and temperature. A higher mobility means that charge carriers can move more freely, resulting in better electrical conductivity.

Understanding these mobilities enables students to predict and explain the behavior of semiconductor components under different conditions, a critical skill for those interested in the fields of electronics and materials science.