Problem 57 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

Germanium shows higher conductivity than silicon at 298 K due to its smaller band gap.

Step 1: Understand Conductivity and Semiconductors

Conductivity in materials is essentially the ability of a material to allow the flow of an electric current. Metals typically have high conductivity due to free electrons, while semiconductors like silicon and germanium have moderate conductivity that can be altered with temperature or doping. At a specific temperature like 298 K, intrinsic semiconductors also conduct to different extents due to differences in band gaps.

Step 2: Consider the Band Gap Energies

The band gap energy is crucial in determining conductivity for semiconductors. Silicon has a band gap of approximately 1.12 eV, while germanium has a band gap of around 0.66 eV. A smaller band gap indicates that less energy is required for electrons to move to the conduction band, making it easier for the material to conduct electricity.

Step 3: Analyze Conductivity at 298 K

At room temperature (298 K), thermally generated electrons and holes contribute to conductivity. Germanium, with its smaller band gap, tends to have higher intrinsic conductivity at this temperature because it requires less thermal energy to excite electrons across the gap compared to silicon.

Step 4: Conclusion

Based on the analysis, germanium will have a higher conductivity than silicon at 298 K because its smaller band gap allows more electrons to be thermally excited to the conduction band.

Key Concepts

Band Gap Energy

When discussing semiconductors, understanding band gap energy is fundamental. The band gap energy is the energy difference between the top of the valence band and the bottom of the conduction band. It determines how easily electrons can move within the material to conduct electricity.
In semiconductors, the band gap is neither too large nor too small. For example, metals have no significant band gap, while insulators have a very large band gap, preventing electron flow.
For silicon, the band gap energy is approximately 1.12 electron volts (eV). Germanium has a smaller band gap of about 0.66 eV.

• A smaller band gap means that less energy is required for electrons to jump from the valence band to the conduction band.
• This jump allows electrons to move and create an electric current.
• Materials with smaller band gaps will have a higher capacity to conduct electricity, especially when thermal energy from heat is available.

Thus, smaller band gap energies are favorable for conductivity.

Intrinsic Conductivity

Intrinsic conductivity refers to the ability of a pure semiconductor to conduct electricity without any doping or impurities. This is a property inherent to the material and is affected by temperature and band gap energy.
As temperature increases, semiconductors exhibit higher conductivity because thermal energy helps electrons cross the band gap.
In intrinsic semiconductors, like silicon and germanium, an increase in thermal energy leads to the production of charge carriers (electrons and holes).

• The generation of electron-hole pairs increases the conductivity since both electrons and holes can carry electric charge.
• The smaller the band gap energy, the easier it is for these electron-hole pairs to form, and therefore, the higher the intrinsic conductivity.
• At room temperature, semiconductors naturally conduct electricity to a limited extent due to these properties.

This explains why germanium, with a smaller band gap, has higher intrinsic conductivity than silicon at room temperature.

Silicon vs Germanium Conductivity

When comparing the conductivity of silicon and germanium at a standard temperature like 298 K, the key lies in their intrinsic properties, especially their band gap energies.
Germanium, with a smaller band gap of 0.66 eV, requires less energy to excite electrons into the conduction band compared to silicon's band gap of 1.12 eV.
This means:

• Germanium will have a higher intrinsic conductivity at this temperature.
• Electrons can be excited more easily in germanium, leading to more charge carriers available for conduction.
• Silicon, while having superior thermal stability, does not inherently conduct as well as germanium at the same temperature due to its larger band gap.

Thus, even though both are semiconductors, germanium will exhibit higher conductivity than silicon under similar conditions, especially at room temperature, owing to its smaller band gap.