We noted in Section 12.5 (Figure 12.22 ) that in FeO (wüstite), the iron ions can exist in both \(\mathrm{Fe}^{2+}\) and \(\mathrm{Fe}^{3+}\) states. The number of each of these ion types depends on temperature and the ambient oxygen pressure. Furthermore, we also noted that in order to retain electroneutrality, one \(\mathrm{Fe}^{2+}\) vacancy will be created for every two \(\mathrm{Fe}^{3+}\) ions that are formed; consequently, in order to reflect the existence of these vacancies the formula for wüstite is often represented as \(\mathrm{Fe}_{(1-x)} \mathrm{O}\) where \(x\) is some small fraction less than unity. In this nonstoichiometric \(\mathrm{Fe}_{(1-x)} \mathrm{O}\) material, conduction is electronic, and, in fact, it behaves as a \(p\) -type semiconductor. That is, the \(\mathrm{Fe}^{3+}\) ions act as electron acceptors, and it is relatively easy to excite an electron from the valence band into an \(\mathrm{Fe}^{3+}\) acceptor state, with the formation of a hole. Determine the electrical conductivity of a specimen of wüstite that has a hole mobility of \(1.0 \times 10^{-5} \mathrm{m}^{2} / \mathrm{V}\) -s and for which the value of \(x\) is \(0.040 .\) Assume that the acceptor states are saturated (i.e., one hole exists for every \(\left.\mathrm{Fe}^{3+} \text { ion }\right) .\) Wüstite has the sodium chloride crystal structure with a unit cell edge length of \(0.437 \mathrm{nm}\)

Step by step solution

01

Determine the concentration of the \(\mathrm{Fe}^{3+}\) ions

First, we need to calculate the concentration of \(\mathrm{Fe}^{3+}\) ions, which are equal to the concentration of holes in the material. We know that for every two \(\mathrm{Fe}^{3+}\) ions that are formed, one \(\mathrm{Fe}^{2+}\) vacancy will be created. Consequently, we can say that \(2x\) is the fractional concentration of the \(\mathrm{Fe}^{3+}\) ions, where \(x = 0.040\) is given. Fractional concentration of \(\mathrm{Fe}^{3+}\) ions: \(2 \times x = 2 \times 0.040 = 0.080\)

02

Calculate the number of atoms per unit cell

Wüstite has the sodium chloride crystal structure with a unit cell edge length of \(0.437 \mathrm{nm}\). In the sodium chloride structure, there are 4 formula units in each unit cell. Thus, there are 4 \(\mathrm{FeO}\) units per unit cell.

03

Calculate the concentration of holes

Now, we can find the concentration of holes by multiplying the fractional concentration of \(\mathrm{Fe}^{3+}\) ions by the number of atoms per unit cell divided by the volume of the unit cell. Concentration of holes: \(0.080 \times \frac{4}{(0.437 \times 10^{-9})^3 \mathrm{m}^3} = 1.003 \times 10^{28} \mathrm{m}^{-3}\)

04

Calculate the electrical conductivity

Finally, we will calculate the electrical conductivity using the formula: \(\sigma = e \times p \times \mu_{h}\) Where \(\sigma\) is the electrical conductivity, \(e\) is the elementary charge (\(1.6 \times 10^{-19}\) C), \(p\) is the concentration of holes, and \(\mu_{h}\) is the hole mobility. \(\sigma = (1.6 \times 10^{-19})(1.003 \times 10^{28})(1.0 \times 10^{-5})\) \(\sigma = 1.6048 \times 10^{4} \Omega^{-1} \mathrm{m}^{-1}\) The electrical conductivity of the wüstite specimen is approximately \(1.6048 \times 10^{4} \Omega^{-1} \mathrm{m}^{-1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Semiconductor Materials

Semiconductor materials play a crucial role in the realm of modern electronics. They possess electrical conductivity levels that fall between those of conductors and insulators, thereby allowing them to effectively control electrical current.

Semiconductors are commonly made of silicon, but other materials, such as germanium or gallium arsenide, are also used, depending on the application. In semiconductors like silicon, the addition of impurities, known as doping, can alter the electrical properties significantly, thus enhancing their conductivity.

Doping achieves this by introducing free charge carriers in the material; these can be negative charge carriers (electrons) in n-type semiconductors, or positive charge carriers (holes) in p-type semiconductors. The FeO (wüstite) mentioned in the exercise is an example of a p-type semiconductor where the presence of holes results in electrical conductivity. In essence, semiconductor materials act as the backbone of various electronic components like diodes, transistors, and integrated circuits, facilitating the development of a broad range of technological devices.

Nonstoichiometry

Nonstoichiometry refers to the phenomenon wherein the ratio of elements in a compound deviates from what would be expected based on its empirical formula. This deviation occurs due to the presence of vacancies, interstitial atoms, or both, leading to a non-stoichiometric composition.

In the case of wüstite (FeO), for instance, the stoichiometric ratio of iron to oxygen should ideally be 1:1. However, the formation of Fe³⁺ ions and the corresponding iron vacancies, result in a formula represented as Fe_(1-x)O. The parameter 'x' quantifies the level of nonstoichiometry, which has a pronounced effect on the material's electrical properties.

In nonstoichiometric compounds, the defect structures directly influence the behavior of charge carriers, and as such, the electrical conductivity of the material. Students should note that a rich variety of properties in advanced materials stem from nonstoichiometric variations, including not only electrical conductivity but also optical properties and catalytic behavior.

Hole Mobility

Hole mobility is a measure of how quickly holes, which are effectively the absence of an electron in the atomic lattice, can move through a semiconductor material under the influence of an electric field.

It's analogous to the concept of electron mobility but pertains to the movement of these positively charged carriers. High hole mobility is a desired property in p-type semiconductors as it correlates with better electrical conductivity. When an electric field is applied, it exerts a force on the holes, causing them to migrate towards the negative electrode.

The mobility of holes is affected by the temperature, the impurity levels in the material, and the crystal structure. In the exercise given, the hole mobility is provided as part of the information needed to calculate the electrical conductivity of the wüstite specimen. It is a critical factor in determining how swiftly the semiconductor can respond to changes in electrical signals, impacting the performance of electronic devices that utilize such materials.

Crystal Structure

The crystal structure of a solid refers to the ordered arrangement of atoms in a three-dimensional lattice. Each crystal structure type has a definite geometric shape with fixed angles between its faces and characteristic lengths and angles of the unit cells.

Many materials used in semiconductor applications crystallize in the diamond cubic structure, like pure silicon. However, some materials, such as wüstite (FeO), adopt different crystal structures, like the sodium chloride lattice as mentioned in the exercise. The sodium chloride or rock salt structure is characterized by each atom being surrounded by six opposite charged ions, leading to a tightly packed matrix.

The underlying crystal structure profoundly influences various material properties, including electronic band structure, carrier mobility, and overall electrical conductivity. It's essential for students to understand how the intricacies of a crystal lattice affect the charge carrier dynamics, which in turn governs the performance of semiconducting materials in electronic applications.