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Chapter 18: Problem 23

For each of the following pairs of semiconductors, decide which will have the smaller band gap energy, \(E_{g},\) and then cite the reason for your choice. (a) \(C\) (diamond) and Ge, (b) AlP and InSb, (c) GaAs and ZnSe, (d) ZnSe and CdTe, and (e) \(\mathrm{CdS}\) and \(\mathrm{NaCl}\)

Short Answer

Expert verified
(a) C (diamond) and Ge (b) AlP and InSb (c) GaAs and ZnSe (d) ZnSe and CdTe (e) CdS and NaCl Answer: (a) Ge has a smaller band gap energy due to larger atomic size and greater electron mobility. (b) InSb has a smaller band gap energy because it has a smaller electron effective mass. (c) GaAs has a smaller band gap energy due to its smaller lattice constant and effective mass. (d) CdTe has a smaller band gap energy because of its larger atomic size and heavier elements. (e) CdS is a semiconductor, while NaCl is not, so CdS is the semiconductor in this pair.

Step by step solution

01

Understanding semiconductors and band gap energy

A semiconductor is a type of material that has electrical conductivity between that of a conductor and an insulator. The band gap energy, \(E_{g}\), is the energy required for an electron to jump from the valence band (the outermost electron shell in the semiconductor) to the conduction band, where the electron can move freely and participate in the conduction of electricity. Generally, materials with a smaller band gap energy will exhibit better electrical conductivity and will be better semiconductors.
02

Comparing band gap energies for given pairs of semiconductors

(a) C (diamond) and Ge: - Diamond (C) has a band gap energy of \(5.47 eV\) - Germanium (Ge) has a band gap energy of \(0.66 eV\) So, Ge has a smaller band gap energy. The reason is that Ge is composed of heavier elements and has a larger atomic size, which leads to greater electron mobility. (b) AlP and InSb: - Aluminum Phosphide (AlP) has a band gap energy of \(2.45 eV\) - Indium Antimonide (InSb) has a band gap energy of \(0.17 eV\) Therefore, InSb has a smaller band gap energy. The reason is that InSb has a smaller electron effective mass so that electrons can move easily with a relatively low-energy input. (c) GaAs and ZnSe: - Gallium Arsenide (GaAs) has a band gap energy of \(1.42 eV\) - Zinc Selenide (ZnSe) has a band gap energy of \(2.70 eV\) In this case, GaAs has a smaller band gap energy. The reason is that GaAs has a smaller lattice constant and a smaller effective mass, allowing for better electron mobility. (d) ZnSe and CdTe: - Zinc Selenide (ZnSe) has a band gap energy of \(2.70 eV\) - Cadmium Telluride (CdTe) has a band gap energy of \(1.49 eV\) CdTe has the smaller band gap energy. The reason can be attributed to the larger atomic size and the presence of heavier elements in CdTe, which increase electron mobility. (e) \(\mathrm{CdS}\) and \(\mathrm{NaCl}\): - Cadmium Sulfide (CdS) is a semiconductor with a band gap energy of \(2.42 eV\) - Sodium Chloride (NaCl) is not a semiconductor - it's an ionic solid. In this case, we cannot directly compare their band gap energies because NaCl is not a semiconductor. Thus, CdS is the semiconductor in this pair. In summary, the materials with smaller band gap energies are Germanium (Ge), Indium Antimonide (InSb), Gallium Arsenide (GaAs), Cadmium Telluride (CdTe), and Cadmium Sulfide (CdS). Each material's smaller band gap energy can be attributed to unique characteristics that affect electron mobility and conductivity.

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Key Concepts

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

Semiconductor Materials
Semiconductor materials occupy the middle ground between conductors and insulators in the electrical conductivity spectrum. They are essential in a vast array of electronic devices, from microprocessors to solar cells.

At the heart of semiconductor functionality is the band gap energy, \(E_g\). This energy gap between the valence band and the conduction band dictates how easily electrons can be excited to a state where they can conduct electricity. A smaller band gap means it's easier for electrons to move from the valence band to the conduction band, making the material a better conductor at room temperature.

Semiconductors can be 'doped' with impurities to alter their conductivity. N-type doping adds extra electrons, while P-type doping creates 'holes' or positive charges. The balance between these two types can tailor the electrical properties of a semiconductor for specific applications.
Electron Mobility
Electron mobility is a measure of how quickly an electron can move through a material when subjected to an electric field. It is a critical factor affecting how well a material will perform as a semiconductor.

In materials like Germanium (Ge), electron mobility is high due to the larger atomic size and heavier elements that cause lower effective mass for electrons. This characteristic means that Ge can facilitate electron flow with less disruption, thereby enhancing electrical conductivity.

The step-by-step solution illustrates how Ge's structure leads to greater electron mobility compared to C (diamond), which has a higher band gap energy and therefore lower electron mobility. The solution also shows that features such as lattice constant and effective mass can greatly influence the ability of an electron to move within a solid, which is vital for the functionality of semiconductors in electronic devices.
Electrical Conductivity
Electrical conductivity is the ability of a material to conduct an electric current, and it is intimately related to the concepts of band gap energy and electron mobility. For semiconductors, the level of conductivity is variable and can be manipulated for various uses.

For example, Cadmium Telluride (CdTe) exhibits higher electrical conductivity than Zinc Selenide (ZnSe) because of its smaller band gap energy, as described in the solutions. This allows electrons to be excited more easily and flow with less resistance towards conducting electricity.

The exercise steps show how each material's unique properties, including band gap energy, affect its electrical conductivity. These solutions provide a window into why certain materials are chosen for specific electronic components, illustrating the importance of semiconductor physics in the design of modern technology.

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Most popular questions from this chapter

Some hypothetical metal is known to have an electrical resistivity of \(3.3 \times 10^{-8}(\Omega-\mathrm{m})\) Through a specimen of this metal \(15 \mathrm{mm}\) thick is passed a current of 25 A; when a magnetic field of 0.95 tesla is simultaneously imposed in a direction perpendicular to that of the current, a Hall voltage of \(-2.4 \times 10^{-7} \mathrm{V}\) is measured. Compute (a) the electron mobility for this metal, and (b) the number of free electrons per cubic meter

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

Is it possible for compound semiconductors to exhibit intrinsic behavior? Explain your answer.

Cite the differences in operation and application for junction transistors and MOSFETs.

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 \(0.437 \mathrm{nm}\)
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