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Chapter 13: Problem 53

Which will show the highest conductivity at \(298 \mathrm{K}\) silicon or germanium?

Short Answer

Expert verified
Germanium will show higher conductivity at 298 K due to its smaller band gap.

Step by step solution

01

Understand Conductivity in Semiconductors

Conductivity in semiconductors depends on the movement of charge carriers, which are electrons and holes. The conductivity ( 蟽 ) is also influenced by temperature, material properties, and the energy gap ( E_g ) between the valence and conduction bands. Both silicon and germanium are group IV semiconductors, but they have different band gaps.
02

Compare Band Gaps

Silicon has a band gap of approximately 1.1 eV, while germanium has a smaller band gap of about 0.66 eV. A smaller band gap means that electrons can be excited from the valence band to the conduction band more easily at a given temperature, increasing conductivity.
03

Consider Temperature Effects on Conductivity

At a higher temperature like 298 K, the number of thermally excited electrons increases. Germanium, having a smaller band gap, will have more electrons excited to the conduction band compared to silicon. Hence, at 298 K, germanium will show higher conductivity than silicon due to more significant thermal excitation.

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Key Concepts

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

Silicon and Germanium Comparison
When comparing silicon and germanium, it's essential to note that both are widely used semiconductors. Although they belong to the same group in the periodic table, their properties differ significantly, impacting their electronic applications.
  • Silicon: Silicon is the most abundantly used semiconductor in the electronics industry. It's renowned for its robustness and ability to function efficiently at higher temperatures.
  • Germanium: Germanium, while less commonly used today, was one of the first materials utilized in semiconductor technology. It offers higher conductivity at room temperature compared to silicon, due to its smaller band gap.
Despite these differences, both elements are pivotal in the development of electronic devices, with the choice depending on the specific requirements of the application.
Band Gap
The concept of a band gap is fundamental in understanding semiconductor behavior. The band gap ( E_g ) is the energy difference between the valence band and the conduction band of a semiconductor. This energy difference determines how easily electrons can move to the conduction band to participate in electrical conduction.
  • Silicon: The band gap of silicon is about 1.1 eV. This moderate band gap makes silicon suitable for general electronic applications like solar cells and integrated circuits.
  • Germanium: Germanium has a smaller band gap of approximately 0.66 eV. This smaller gap allows for easier electron transition at lower energies, contributing to its higher conductivity compared to silicon.
Therefore, germanium can be a better choice in contexts where higher conductivity at less energy expenditure is required.
Temperature Effects on Conductivity
Temperature plays a crucial role in the conductivity of semiconductors. At higher temperatures, the kinetic energy of electrons increases, leading to more frequent transitions across the band gap.
  • As temperature rises, more electrons gain the energy needed to jump from the valence band to the conduction band.
  • This increase in charge carriers results in higher conductivity.
At a temperature like 298 K, germanium benefits more from increased thermal energy due to its already smaller band gap. This characteristic allows germanium to surpass silicon in conductivity at this temperature, demonstrating the direct relationship between temperature effects and the energy gap of semiconductors.
Charge Carriers in Semiconductors
In semiconductors, charge carriers, which include electrons and holes, are crucial for conducting electricity. These carriers determine the material's conductivity.
  • Electrons: Negatively charged particles that move from the valence band to the conduction band, enabling electrical current.
  • Holes: Positively charged particles created when an electron vacates its position in the valence band.
In silicon and germanium, these carriers behave differently because of the varied band gaps. With a smaller band gap in germanium, electrons require less energy to transition into the conduction band, increasing the density of charge carriers and thereby boosting conductivity. This mechanism highlights why germanium can be more conductive than silicon under certain conditions.

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Most popular questions from this chapter

Calcium fluoride is the well-known mineral fluorite. Each unit cell contains four \(\mathrm{Ca}^{2+}\) ions and eight \(\mathrm{F}^{-}\) ions. The \(\mathrm{F}^{-}\) ions fill all the tetrahedral holes in a facecentered cubic lattice of \(\mathrm{Ca}^{2+}\) ions. The edge of the \(\mathrm{CaF}_{2}\) unit cell is \(5.46295 \times 10^{-8} \mathrm{cm}\) in length. The density of the solid is \(3.1805 \mathrm{g} / \mathrm{cm}^{3} .\) Use this information to calculate Avogadro's number.

Boron phosphide, BP, is a semiconductor and a hard, abrasion-resistant material. It is made by reacting boron tribromide and phosphorus tribromide in a hydrogen atmosphere at high temperature \(\left(>750^{\circ} \mathrm{C}\right)\) (a) Write a balanced chemical equation for the synthesis of BP. (Hint: Hydrogen is a reducing agent.) (b) Boron phosphide crystallizes in a zinc-blend structure, formed from boron atoms in a face-centered cubic lattice and phosphorus atoms in tetrahedral holes. How many tetrahedral holes are filled with P atoms in each unit cell? (c) The length of a unit cell of BP is 478 pm. What is the density of the solid in \(\mathrm{g} / \mathrm{cm}^{3} ?^{-}\) (d) Calculate the closest distance between a \(\mathrm{B}\) and a P atom in the unit cell. (Assume the B atoms do not touch along the cell edge. The \(\mathrm{B}\) atoms in the faces touch the \(\mathrm{B}\) atoms at the corners of the unit cell.)

Calcium metal crystallizes in a face-centered cubic unit cell. The density of the solid is \(1.54 \mathrm{g} / \mathrm{cm}^{3} .\) What is the radius of a calcium atom?

Vanadium metal has a density of \(6.11 \mathrm{g} / \mathrm{cm}^{3} .\) Assuming the vanadium atomic radius is \(132 \mathrm{pm},\) is the vanadium unit cell primitive cubic, body-centered cubic, or facecentered cubic?

Iron has a body-centered cubic unit cell with a cell dimension of \(286.65 \mathrm{pm} .\) The density of iron is \(7.874 \mathrm{g} / \mathrm{cm}^{3} .\) Use this information to calculate Avogadro's number.
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